3.1.100 \(\int \sqrt {a+b x+c x^2} (d+e x+f x^2)^2 \, dx\) [100]

Optimal. Leaf size=436 \[ \frac {\left (128 c^4 d^2+21 b^4 f^2-56 b^2 c f (b e+a f)-32 c^3 \left (4 b d e+a \left (e^2+2 d f\right )\right )+8 c^2 \left (12 a b e f+2 a^2 f^2+5 b^2 \left (e^2+2 d f\right )\right )\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{512 c^5}+\frac {\left (640 c^3 d e-105 b^3 f^2+28 b c f (10 b e+7 a f)-8 c^2 \left (32 a e f+25 b \left (e^2+2 d f\right )\right )\right ) \left (a+b x+c x^2\right )^{3/2}}{960 c^4}+\frac {\left (21 b^2 f^2-4 c f (14 b e+5 a f)+40 c^2 \left (e^2+2 d f\right )\right ) x \left (a+b x+c x^2\right )^{3/2}}{160 c^3}+\frac {f (8 c e-3 b f) x^2 \left (a+b x+c x^2\right )^{3/2}}{20 c^2}+\frac {f^2 x^3 \left (a+b x+c x^2\right )^{3/2}}{6 c}-\frac {\left (b^2-4 a c\right ) \left (128 c^4 d^2+21 b^4 f^2-56 b^2 c f (b e+a f)-32 c^3 \left (4 b d e+a \left (e^2+2 d f\right )\right )+8 c^2 \left (12 a b e f+2 a^2 f^2+5 b^2 \left (e^2+2 d f\right )\right )\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{1024 c^{11/2}} \]

[Out]

1/960*(640*c^3*d*e-105*b^3*f^2+28*b*c*f*(7*a*f+10*b*e)-8*c^2*(32*a*e*f+25*b*(2*d*f+e^2)))*(c*x^2+b*x+a)^(3/2)/
c^4+1/160*(21*b^2*f^2-4*c*f*(5*a*f+14*b*e)+40*c^2*(2*d*f+e^2))*x*(c*x^2+b*x+a)^(3/2)/c^3+1/20*f*(-3*b*f+8*c*e)
*x^2*(c*x^2+b*x+a)^(3/2)/c^2+1/6*f^2*x^3*(c*x^2+b*x+a)^(3/2)/c-1/1024*(-4*a*c+b^2)*(128*c^4*d^2+21*b^4*f^2-56*
b^2*c*f*(a*f+b*e)-32*c^3*(4*b*d*e+a*(2*d*f+e^2))+8*c^2*(12*a*b*e*f+2*a^2*f^2+5*b^2*(2*d*f+e^2)))*arctanh(1/2*(
2*c*x+b)/c^(1/2)/(c*x^2+b*x+a)^(1/2))/c^(11/2)+1/512*(128*c^4*d^2+21*b^4*f^2-56*b^2*c*f*(a*f+b*e)-32*c^3*(4*b*
d*e+a*(2*d*f+e^2))+8*c^2*(12*a*b*e*f+2*a^2*f^2+5*b^2*(2*d*f+e^2)))*(2*c*x+b)*(c*x^2+b*x+a)^(1/2)/c^5

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Rubi [A]
time = 0.50, antiderivative size = 436, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {1675, 654, 626, 635, 212} \begin {gather*} -\frac {\left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (8 c^2 \left (2 a^2 f^2+12 a b e f+5 b^2 \left (2 d f+e^2\right )\right )-56 b^2 c f (a f+b e)-32 c^3 \left (a \left (2 d f+e^2\right )+4 b d e\right )+21 b^4 f^2+128 c^4 d^2\right )}{1024 c^{11/2}}+\frac {(b+2 c x) \sqrt {a+b x+c x^2} \left (8 c^2 \left (2 a^2 f^2+12 a b e f+5 b^2 \left (2 d f+e^2\right )\right )-56 b^2 c f (a f+b e)-32 c^3 \left (a \left (2 d f+e^2\right )+4 b d e\right )+21 b^4 f^2+128 c^4 d^2\right )}{512 c^5}+\frac {\left (a+b x+c x^2\right )^{3/2} \left (-8 c^2 \left (32 a e f+25 b \left (2 d f+e^2\right )\right )+28 b c f (7 a f+10 b e)-105 b^3 f^2+640 c^3 d e\right )}{960 c^4}+\frac {x \left (a+b x+c x^2\right )^{3/2} \left (-4 c f (5 a f+14 b e)+21 b^2 f^2+40 c^2 \left (2 d f+e^2\right )\right )}{160 c^3}+\frac {f x^2 \left (a+b x+c x^2\right )^{3/2} (8 c e-3 b f)}{20 c^2}+\frac {f^2 x^3 \left (a+b x+c x^2\right )^{3/2}}{6 c} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[a + b*x + c*x^2]*(d + e*x + f*x^2)^2,x]

[Out]

((128*c^4*d^2 + 21*b^4*f^2 - 56*b^2*c*f*(b*e + a*f) - 32*c^3*(4*b*d*e + a*(e^2 + 2*d*f)) + 8*c^2*(12*a*b*e*f +
2*a^2*f^2 + 5*b^2*(e^2 + 2*d*f)))*(b + 2*c*x)*Sqrt[a + b*x + c*x^2])/(512*c^5) + ((640*c^3*d*e - 105*b^3*f^2
+ 28*b*c*f*(10*b*e + 7*a*f) - 8*c^2*(32*a*e*f + 25*b*(e^2 + 2*d*f)))*(a + b*x + c*x^2)^(3/2))/(960*c^4) + ((21
*b^2*f^2 - 4*c*f*(14*b*e + 5*a*f) + 40*c^2*(e^2 + 2*d*f))*x*(a + b*x + c*x^2)^(3/2))/(160*c^3) + (f*(8*c*e - 3
*b*f)*x^2*(a + b*x + c*x^2)^(3/2))/(20*c^2) + (f^2*x^3*(a + b*x + c*x^2)^(3/2))/(6*c) - ((b^2 - 4*a*c)*(128*c^
4*d^2 + 21*b^4*f^2 - 56*b^2*c*f*(b*e + a*f) - 32*c^3*(4*b*d*e + a*(e^2 + 2*d*f)) + 8*c^2*(12*a*b*e*f + 2*a^2*f
^2 + 5*b^2*(e^2 + 2*d*f)))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(1024*c^(11/2))

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 626

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(b + 2*c*x)*((a + b*x + c*x^2)^p/(2*c*(2*p + 1
))), x] - Dist[p*((b^2 - 4*a*c)/(2*c*(2*p + 1))), Int[(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x]
&& NeQ[b^2 - 4*a*c, 0] && GtQ[p, 0] && IntegerQ[4*p]

Rule 635

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2, Subst[Int[1/(4*c - x^2), x], x, (b + 2*c*x)
/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 654

Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[e*((a + b*x + c*x^2)^(p +
1)/(2*c*(p + 1))), x] + Dist[(2*c*d - b*e)/(2*c), Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}
, x] && NeQ[2*c*d - b*e, 0] && NeQ[p, -1]

Rule 1675

Int[(Pq_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq, x], e = Coeff[Pq, x, Expo
n[Pq, x]]}, Simp[e*x^(q - 1)*((a + b*x + c*x^2)^(p + 1)/(c*(q + 2*p + 1))), x] + Dist[1/(c*(q + 2*p + 1)), Int
[(a + b*x + c*x^2)^p*ExpandToSum[c*(q + 2*p + 1)*Pq - a*e*(q - 1)*x^(q - 2) - b*e*(q + p)*x^(q - 1) - c*e*(q +
2*p + 1)*x^q, x], x], x]] /; FreeQ[{a, b, c, p}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] &&  !LeQ[p, -1]

Rubi steps

\begin {align*} \int \sqrt {a+b x+c x^2} \left (d+e x+f x^2\right )^2 \, dx &=\frac {f^2 x^3 \left (a+b x+c x^2\right )^{3/2}}{6 c}+\frac {\int \sqrt {a+b x+c x^2} \left (6 c d^2+12 c d e x-3 \left (a f^2-2 c \left (e^2+2 d f\right )\right ) x^2+\frac {3}{2} f (8 c e-3 b f) x^3\right ) \, dx}{6 c}\\ &=\frac {f (8 c e-3 b f) x^2 \left (a+b x+c x^2\right )^{3/2}}{20 c^2}+\frac {f^2 x^3 \left (a+b x+c x^2\right )^{3/2}}{6 c}+\frac {\int \sqrt {a+b x+c x^2} \left (30 c^2 d^2+3 \left (20 c^2 d e-8 a c e f+3 a b f^2\right ) x+\frac {3}{4} \left (21 b^2 f^2-4 c f (14 b e+5 a f)+40 c^2 \left (e^2+2 d f\right )\right ) x^2\right ) \, dx}{30 c^2}\\ &=\frac {\left (21 b^2 f^2-4 c f (14 b e+5 a f)+40 c^2 \left (e^2+2 d f\right )\right ) x \left (a+b x+c x^2\right )^{3/2}}{160 c^3}+\frac {f (8 c e-3 b f) x^2 \left (a+b x+c x^2\right )^{3/2}}{20 c^2}+\frac {f^2 x^3 \left (a+b x+c x^2\right )^{3/2}}{6 c}+\frac {\int \left (\frac {3}{4} \left (160 c^3 d^2-21 a b^2 f^2+4 a c f (14 b e+5 a f)-40 a c^2 \left (e^2+2 d f\right )\right )+\frac {3}{8} \left (640 c^3 d e-105 b^3 f^2+28 b c f (10 b e+7 a f)-8 c^2 \left (32 a e f+25 b \left (e^2+2 d f\right )\right )\right ) x\right ) \sqrt {a+b x+c x^2} \, dx}{120 c^3}\\ &=\frac {\left (640 c^3 d e-105 b^3 f^2+28 b c f (10 b e+7 a f)-8 c^2 \left (32 a e f+25 b \left (e^2+2 d f\right )\right )\right ) \left (a+b x+c x^2\right )^{3/2}}{960 c^4}+\frac {\left (21 b^2 f^2-4 c f (14 b e+5 a f)+40 c^2 \left (e^2+2 d f\right )\right ) x \left (a+b x+c x^2\right )^{3/2}}{160 c^3}+\frac {f (8 c e-3 b f) x^2 \left (a+b x+c x^2\right )^{3/2}}{20 c^2}+\frac {f^2 x^3 \left (a+b x+c x^2\right )^{3/2}}{6 c}+\frac {\left (128 c^4 d^2+21 b^4 f^2-56 b^2 c f (b e+a f)-32 c^3 \left (4 b d e+a \left (e^2+2 d f\right )\right )+8 c^2 \left (12 a b e f+2 a^2 f^2+5 b^2 \left (e^2+2 d f\right )\right )\right ) \int \sqrt {a+b x+c x^2} \, dx}{128 c^4}\\ &=\frac {\left (128 c^4 d^2+21 b^4 f^2-56 b^2 c f (b e+a f)-32 c^3 \left (4 b d e+a \left (e^2+2 d f\right )\right )+8 c^2 \left (12 a b e f+2 a^2 f^2+5 b^2 \left (e^2+2 d f\right )\right )\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{512 c^5}+\frac {\left (640 c^3 d e-105 b^3 f^2+28 b c f (10 b e+7 a f)-8 c^2 \left (32 a e f+25 b \left (e^2+2 d f\right )\right )\right ) \left (a+b x+c x^2\right )^{3/2}}{960 c^4}+\frac {\left (21 b^2 f^2-4 c f (14 b e+5 a f)+40 c^2 \left (e^2+2 d f\right )\right ) x \left (a+b x+c x^2\right )^{3/2}}{160 c^3}+\frac {f (8 c e-3 b f) x^2 \left (a+b x+c x^2\right )^{3/2}}{20 c^2}+\frac {f^2 x^3 \left (a+b x+c x^2\right )^{3/2}}{6 c}-\frac {\left (\left (b^2-4 a c\right ) \left (128 c^4 d^2+21 b^4 f^2-56 b^2 c f (b e+a f)-32 c^3 \left (4 b d e+a \left (e^2+2 d f\right )\right )+8 c^2 \left (12 a b e f+2 a^2 f^2+5 b^2 \left (e^2+2 d f\right )\right )\right )\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{1024 c^5}\\ &=\frac {\left (128 c^4 d^2+21 b^4 f^2-56 b^2 c f (b e+a f)-32 c^3 \left (4 b d e+a \left (e^2+2 d f\right )\right )+8 c^2 \left (12 a b e f+2 a^2 f^2+5 b^2 \left (e^2+2 d f\right )\right )\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{512 c^5}+\frac {\left (640 c^3 d e-105 b^3 f^2+28 b c f (10 b e+7 a f)-8 c^2 \left (32 a e f+25 b \left (e^2+2 d f\right )\right )\right ) \left (a+b x+c x^2\right )^{3/2}}{960 c^4}+\frac {\left (21 b^2 f^2-4 c f (14 b e+5 a f)+40 c^2 \left (e^2+2 d f\right )\right ) x \left (a+b x+c x^2\right )^{3/2}}{160 c^3}+\frac {f (8 c e-3 b f) x^2 \left (a+b x+c x^2\right )^{3/2}}{20 c^2}+\frac {f^2 x^3 \left (a+b x+c x^2\right )^{3/2}}{6 c}-\frac {\left (\left (b^2-4 a c\right ) \left (128 c^4 d^2+21 b^4 f^2-56 b^2 c f (b e+a f)-32 c^3 \left (4 b d e+a \left (e^2+2 d f\right )\right )+8 c^2 \left (12 a b e f+2 a^2 f^2+5 b^2 \left (e^2+2 d f\right )\right )\right )\right ) \text {Subst}\left (\int \frac {1}{4 c-x^2} \, dx,x,\frac {b+2 c x}{\sqrt {a+b x+c x^2}}\right )}{512 c^5}\\ &=\frac {\left (128 c^4 d^2+21 b^4 f^2-56 b^2 c f (b e+a f)-32 c^3 \left (4 b d e+a \left (e^2+2 d f\right )\right )+8 c^2 \left (12 a b e f+2 a^2 f^2+5 b^2 \left (e^2+2 d f\right )\right )\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{512 c^5}+\frac {\left (640 c^3 d e-105 b^3 f^2+28 b c f (10 b e+7 a f)-8 c^2 \left (32 a e f+25 b \left (e^2+2 d f\right )\right )\right ) \left (a+b x+c x^2\right )^{3/2}}{960 c^4}+\frac {\left (21 b^2 f^2-4 c f (14 b e+5 a f)+40 c^2 \left (e^2+2 d f\right )\right ) x \left (a+b x+c x^2\right )^{3/2}}{160 c^3}+\frac {f (8 c e-3 b f) x^2 \left (a+b x+c x^2\right )^{3/2}}{20 c^2}+\frac {f^2 x^3 \left (a+b x+c x^2\right )^{3/2}}{6 c}-\frac {\left (b^2-4 a c\right ) \left (128 c^4 d^2+21 b^4 f^2-56 b^2 c f (b e+a f)-32 c^3 \left (4 b d e+a \left (e^2+2 d f\right )\right )+8 c^2 \left (12 a b e f+2 a^2 f^2+5 b^2 \left (e^2+2 d f\right )\right )\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{1024 c^{11/2}}\\ \end {align*}

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Mathematica [A]
time = 1.98, size = 456, normalized size = 1.05 \begin {gather*} \frac {2 \sqrt {c} \sqrt {a+x (b+c x)} \left (315 b^5 f^2-210 b^4 c f (4 e+f x)-16 b^2 c^2 \left (-2 a f (115 e+28 f x)+c \left (120 d e+25 e^2 x+50 d f x+28 e f x^2+9 f^2 x^3\right )\right )+8 b^3 c \left (-210 a f^2+c \left (75 e^2+70 e f x+3 f \left (50 d+7 f x^2\right )\right )\right )+16 b c^2 \left (113 a^2 f^2-2 a c \left (65 e^2+58 e f x+f \left (130 d+17 f x^2\right )\right )+4 c^2 \left (30 d^2+10 d x (2 e+f x)+x^2 \left (5 e^2+6 e f x+2 f^2 x^2\right )\right )\right )-32 c^3 \left (a^2 f (64 e+15 f x)-2 a c \left (80 d e+15 e^2 x+30 d f x+16 e f x^2+5 f^2 x^3\right )-4 c^2 x \left (30 d^2+10 d x (4 e+3 f x)+x^2 \left (15 e^2+24 e f x+10 f^2 x^2\right )\right )\right )\right )+15 \left (b^2-4 a c\right ) \left (128 c^4 d^2+21 b^4 f^2-56 b^2 c f (b e+a f)-32 c^3 \left (4 b d e+a \left (e^2+2 d f\right )\right )+8 c^2 \left (12 a b e f+2 a^2 f^2+5 b^2 \left (e^2+2 d f\right )\right )\right ) \log \left (b+2 c x-2 \sqrt {c} \sqrt {a+x (b+c x)}\right )}{15360 c^{11/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a + b*x + c*x^2]*(d + e*x + f*x^2)^2,x]

[Out]

(2*Sqrt[c]*Sqrt[a + x*(b + c*x)]*(315*b^5*f^2 - 210*b^4*c*f*(4*e + f*x) - 16*b^2*c^2*(-2*a*f*(115*e + 28*f*x)
+ c*(120*d*e + 25*e^2*x + 50*d*f*x + 28*e*f*x^2 + 9*f^2*x^3)) + 8*b^3*c*(-210*a*f^2 + c*(75*e^2 + 70*e*f*x + 3
*f*(50*d + 7*f*x^2))) + 16*b*c^2*(113*a^2*f^2 - 2*a*c*(65*e^2 + 58*e*f*x + f*(130*d + 17*f*x^2)) + 4*c^2*(30*d
^2 + 10*d*x*(2*e + f*x) + x^2*(5*e^2 + 6*e*f*x + 2*f^2*x^2))) - 32*c^3*(a^2*f*(64*e + 15*f*x) - 2*a*c*(80*d*e
+ 15*e^2*x + 30*d*f*x + 16*e*f*x^2 + 5*f^2*x^3) - 4*c^2*x*(30*d^2 + 10*d*x*(4*e + 3*f*x) + x^2*(15*e^2 + 24*e*
f*x + 10*f^2*x^2)))) + 15*(b^2 - 4*a*c)*(128*c^4*d^2 + 21*b^4*f^2 - 56*b^2*c*f*(b*e + a*f) - 32*c^3*(4*b*d*e +
a*(e^2 + 2*d*f)) + 8*c^2*(12*a*b*e*f + 2*a^2*f^2 + 5*b^2*(e^2 + 2*d*f)))*Log[b + 2*c*x - 2*Sqrt[c]*Sqrt[a + x
*(b + c*x)]])/(15360*c^(11/2))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1185\) vs. \(2(406)=812\).
time = 0.20, size = 1186, normalized size = 2.72

method	result	size
risch	\(\frac {\left (1280 f^{2} c^{5} x^{5}+128 b \,c^{4} f^{2} x^{4}+3072 c^{5} e f \,x^{4}+320 a \,c^{4} f^{2} x^{3}-144 b^{2} c^{3} f^{2} x^{3}+384 b \,c^{4} e f \,x^{3}+3840 c^{5} d f \,x^{3}+1920 c^{5} e^{2} x^{3}-544 a b \,c^{3} f^{2} x^{2}+1024 a \,c^{4} e f \,x^{2}+168 b^{3} c^{2} f^{2} x^{2}-448 b^{2} c^{3} e f \,x^{2}+640 b \,c^{4} d f \,x^{2}+320 b \,c^{4} e^{2} x^{2}+5120 c^{5} d e \,x^{2}-480 a^{2} c^{3} f^{2} x +896 a \,b^{2} c^{2} f^{2} x -1856 a b \,c^{3} e f x +1920 a \,c^{4} d f x +960 a \,c^{4} e^{2} x -210 b^{4} c \,f^{2} x +560 b^{3} c^{2} e f x -800 b^{2} c^{3} d f x -400 b^{2} c^{3} e^{2} x +1280 b \,c^{4} d e x +3840 c^{5} d^{2} x +1808 a^{2} b \,c^{2} f^{2}-2048 a^{2} c^{3} e f -1680 a \,b^{3} c \,f^{2}+3680 a \,b^{2} c^{2} e f -4160 a b \,c^{3} d f -2080 a b \,c^{3} e^{2}+5120 a \,c^{4} d e +315 b^{5} f^{2}-840 b^{4} c e f +1200 b^{3} c^{2} d f +600 b^{3} c^{2} e^{2}-1920 b^{2} c^{3} d e +1920 b \,c^{4} d^{2}\right ) \sqrt {c \,x^{2}+b x +a}}{7680 c^{5}}+\frac {\ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right ) a^{3} f^{2}}{16 c^{\frac {5}{2}}}-\frac {15 \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right ) a^{2} b^{2} f^{2}}{64 c^{\frac {7}{2}}}+\frac {3 \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right ) a^{2} b e f}{8 c^{\frac {5}{2}}}-\frac {\ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right ) a^{2} d f}{4 c^{\frac {3}{2}}}-\frac {\ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right ) a^{2} e^{2}}{8 c^{\frac {3}{2}}}+\frac {35 \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right ) a \,b^{4} f^{2}}{256 c^{\frac {9}{2}}}-\frac {5 \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right ) a \,b^{3} e f}{16 c^{\frac {7}{2}}}+\frac {3 \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right ) a \,b^{2} d f}{8 c^{\frac {5}{2}}}+\frac {3 \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right ) a \,b^{2} e^{2}}{16 c^{\frac {5}{2}}}-\frac {\ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right ) a b d e}{2 c^{\frac {3}{2}}}+\frac {\ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right ) a \,d^{2}}{2 \sqrt {c}}-\frac {21 \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right ) b^{6} f^{2}}{1024 c^{\frac {11}{2}}}+\frac {7 \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right ) b^{5} e f}{128 c^{\frac {9}{2}}}-\frac {5 \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right ) b^{4} d f}{64 c^{\frac {7}{2}}}-\frac {5 \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right ) b^{4} e^{2}}{128 c^{\frac {7}{2}}}+\frac {\ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right ) b^{3} d e}{8 c^{\frac {5}{2}}}-\frac {\ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right ) b^{2} d^{2}}{8 c^{\frac {3}{2}}}\)	\(1046\)
default	\(\text {Expression too large to display}\)	\(1186\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2+b*x+a)^(1/2)*(f*x^2+e*x+d)^2,x,method=_RETURNVERBOSE)

[Out]

f^2*(1/6*x^3*(c*x^2+b*x+a)^(3/2)/c-3/4*b/c*(1/5*x^2*(c*x^2+b*x+a)^(3/2)/c-7/10*b/c*(1/4*x*(c*x^2+b*x+a)^(3/2)/
c-5/8*b/c*(1/3*(c*x^2+b*x+a)^(3/2)/c-1/2*b/c*(1/4*(2*c*x+b)/c*(c*x^2+b*x+a)^(1/2)+1/8*(4*a*c-b^2)/c^(3/2)*ln((
1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))))-1/4*a/c*(1/4*(2*c*x+b)/c*(c*x^2+b*x+a)^(1/2)+1/8*(4*a*c-b^2)/c^(3/2)
*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))))-2/5*a/c*(1/3*(c*x^2+b*x+a)^(3/2)/c-1/2*b/c*(1/4*(2*c*x+b)/c*(c*
x^2+b*x+a)^(1/2)+1/8*(4*a*c-b^2)/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2)))))-1/2*a/c*(1/4*x*(c*x^2+
b*x+a)^(3/2)/c-5/8*b/c*(1/3*(c*x^2+b*x+a)^(3/2)/c-1/2*b/c*(1/4*(2*c*x+b)/c*(c*x^2+b*x+a)^(1/2)+1/8*(4*a*c-b^2)
/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))))-1/4*a/c*(1/4*(2*c*x+b)/c*(c*x^2+b*x+a)^(1/2)+1/8*(4*a*c
-b^2)/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2)))))+2*e*f*(1/5*x^2*(c*x^2+b*x+a)^(3/2)/c-7/10*b/c*(1/
4*x*(c*x^2+b*x+a)^(3/2)/c-5/8*b/c*(1/3*(c*x^2+b*x+a)^(3/2)/c-1/2*b/c*(1/4*(2*c*x+b)/c*(c*x^2+b*x+a)^(1/2)+1/8*
(4*a*c-b^2)/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))))-1/4*a/c*(1/4*(2*c*x+b)/c*(c*x^2+b*x+a)^(1/2)
+1/8*(4*a*c-b^2)/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))))-2/5*a/c*(1/3*(c*x^2+b*x+a)^(3/2)/c-1/2*
b/c*(1/4*(2*c*x+b)/c*(c*x^2+b*x+a)^(1/2)+1/8*(4*a*c-b^2)/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))))
)+(2*d*f+e^2)*(1/4*x*(c*x^2+b*x+a)^(3/2)/c-5/8*b/c*(1/3*(c*x^2+b*x+a)^(3/2)/c-1/2*b/c*(1/4*(2*c*x+b)/c*(c*x^2+
b*x+a)^(1/2)+1/8*(4*a*c-b^2)/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))))-1/4*a/c*(1/4*(2*c*x+b)/c*(c
*x^2+b*x+a)^(1/2)+1/8*(4*a*c-b^2)/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))))+2*d*e*(1/3*(c*x^2+b*x+
a)^(3/2)/c-1/2*b/c*(1/4*(2*c*x+b)/c*(c*x^2+b*x+a)^(1/2)+1/8*(4*a*c-b^2)/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+
b*x+a)^(1/2))))+d^2*(1/4*(2*c*x+b)/c*(c*x^2+b*x+a)^(1/2)+1/8*(4*a*c-b^2)/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2
+b*x+a)^(1/2)))

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)^(1/2)*(f*x^2+e*x+d)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*a*c-b^2>0)', see `assume?` f
or more deta

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Fricas [A]
time = 3.81, size = 1267, normalized size = 2.91 \begin {gather*} \left [\frac {15 \, {\left (128 \, {\left (b^{2} c^{4} - 4 \, a c^{5}\right )} d^{2} + 16 \, {\left (5 \, b^{4} c^{2} - 24 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} d f + {\left (21 \, b^{6} - 140 \, a b^{4} c + 240 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} f^{2} + 8 \, {\left (5 \, b^{4} c^{2} - 24 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} e^{2} - 8 \, {\left (16 \, {\left (b^{3} c^{3} - 4 \, a b c^{4}\right )} d + {\left (7 \, b^{5} c - 40 \, a b^{3} c^{2} + 48 \, a^{2} b c^{3}\right )} f\right )} e\right )} \sqrt {c} \log \left (-8 \, c^{2} x^{2} - 8 \, b c x - b^{2} + 4 \, \sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {c} - 4 \, a c\right ) + 4 \, {\left (1280 \, c^{6} f^{2} x^{5} + 128 \, b c^{5} f^{2} x^{4} + 1920 \, b c^{5} d^{2} + 16 \, {\left (240 \, c^{6} d f - {\left (9 \, b^{2} c^{4} - 20 \, a c^{5}\right )} f^{2}\right )} x^{3} + 80 \, {\left (15 \, b^{3} c^{3} - 52 \, a b c^{4}\right )} d f + {\left (315 \, b^{5} c - 1680 \, a b^{3} c^{2} + 1808 \, a^{2} b c^{3}\right )} f^{2} + 8 \, {\left (80 \, b c^{5} d f + {\left (21 \, b^{3} c^{3} - 68 \, a b c^{4}\right )} f^{2}\right )} x^{2} + 2 \, {\left (1920 \, c^{6} d^{2} - 80 \, {\left (5 \, b^{2} c^{4} - 12 \, a c^{5}\right )} d f - {\left (105 \, b^{4} c^{2} - 448 \, a b^{2} c^{3} + 240 \, a^{2} c^{4}\right )} f^{2}\right )} x + 40 \, {\left (48 \, c^{6} x^{3} + 8 \, b c^{5} x^{2} + 15 \, b^{3} c^{3} - 52 \, a b c^{4} - 2 \, {\left (5 \, b^{2} c^{4} - 12 \, a c^{5}\right )} x\right )} e^{2} + 8 \, {\left (384 \, c^{6} f x^{4} + 48 \, b c^{5} f x^{3} + 8 \, {\left (80 \, c^{6} d - {\left (7 \, b^{2} c^{4} - 16 \, a c^{5}\right )} f\right )} x^{2} - 80 \, {\left (3 \, b^{2} c^{4} - 8 \, a c^{5}\right )} d - {\left (105 \, b^{4} c^{2} - 460 \, a b^{2} c^{3} + 256 \, a^{2} c^{4}\right )} f + 2 \, {\left (80 \, b c^{5} d + {\left (35 \, b^{3} c^{3} - 116 \, a b c^{4}\right )} f\right )} x\right )} e\right )} \sqrt {c x^{2} + b x + a}}{30720 \, c^{6}}, \frac {15 \, {\left (128 \, {\left (b^{2} c^{4} - 4 \, a c^{5}\right )} d^{2} + 16 \, {\left (5 \, b^{4} c^{2} - 24 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} d f + {\left (21 \, b^{6} - 140 \, a b^{4} c + 240 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} f^{2} + 8 \, {\left (5 \, b^{4} c^{2} - 24 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} e^{2} - 8 \, {\left (16 \, {\left (b^{3} c^{3} - 4 \, a b c^{4}\right )} d + {\left (7 \, b^{5} c - 40 \, a b^{3} c^{2} + 48 \, a^{2} b c^{3}\right )} f\right )} e\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{2} + b c x + a c\right )}}\right ) + 2 \, {\left (1280 \, c^{6} f^{2} x^{5} + 128 \, b c^{5} f^{2} x^{4} + 1920 \, b c^{5} d^{2} + 16 \, {\left (240 \, c^{6} d f - {\left (9 \, b^{2} c^{4} - 20 \, a c^{5}\right )} f^{2}\right )} x^{3} + 80 \, {\left (15 \, b^{3} c^{3} - 52 \, a b c^{4}\right )} d f + {\left (315 \, b^{5} c - 1680 \, a b^{3} c^{2} + 1808 \, a^{2} b c^{3}\right )} f^{2} + 8 \, {\left (80 \, b c^{5} d f + {\left (21 \, b^{3} c^{3} - 68 \, a b c^{4}\right )} f^{2}\right )} x^{2} + 2 \, {\left (1920 \, c^{6} d^{2} - 80 \, {\left (5 \, b^{2} c^{4} - 12 \, a c^{5}\right )} d f - {\left (105 \, b^{4} c^{2} - 448 \, a b^{2} c^{3} + 240 \, a^{2} c^{4}\right )} f^{2}\right )} x + 40 \, {\left (48 \, c^{6} x^{3} + 8 \, b c^{5} x^{2} + 15 \, b^{3} c^{3} - 52 \, a b c^{4} - 2 \, {\left (5 \, b^{2} c^{4} - 12 \, a c^{5}\right )} x\right )} e^{2} + 8 \, {\left (384 \, c^{6} f x^{4} + 48 \, b c^{5} f x^{3} + 8 \, {\left (80 \, c^{6} d - {\left (7 \, b^{2} c^{4} - 16 \, a c^{5}\right )} f\right )} x^{2} - 80 \, {\left (3 \, b^{2} c^{4} - 8 \, a c^{5}\right )} d - {\left (105 \, b^{4} c^{2} - 460 \, a b^{2} c^{3} + 256 \, a^{2} c^{4}\right )} f + 2 \, {\left (80 \, b c^{5} d + {\left (35 \, b^{3} c^{3} - 116 \, a b c^{4}\right )} f\right )} x\right )} e\right )} \sqrt {c x^{2} + b x + a}}{15360 \, c^{6}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)^(1/2)*(f*x^2+e*x+d)^2,x, algorithm="fricas")

[Out]

[1/30720*(15*(128*(b^2*c^4 - 4*a*c^5)*d^2 + 16*(5*b^4*c^2 - 24*a*b^2*c^3 + 16*a^2*c^4)*d*f + (21*b^6 - 140*a*b
^4*c + 240*a^2*b^2*c^2 - 64*a^3*c^3)*f^2 + 8*(5*b^4*c^2 - 24*a*b^2*c^3 + 16*a^2*c^4)*e^2 - 8*(16*(b^3*c^3 - 4*
a*b*c^4)*d + (7*b^5*c - 40*a*b^3*c^2 + 48*a^2*b*c^3)*f)*e)*sqrt(c)*log(-8*c^2*x^2 - 8*b*c*x - b^2 + 4*sqrt(c*x
^2 + b*x + a)*(2*c*x + b)*sqrt(c) - 4*a*c) + 4*(1280*c^6*f^2*x^5 + 128*b*c^5*f^2*x^4 + 1920*b*c^5*d^2 + 16*(24
0*c^6*d*f - (9*b^2*c^4 - 20*a*c^5)*f^2)*x^3 + 80*(15*b^3*c^3 - 52*a*b*c^4)*d*f + (315*b^5*c - 1680*a*b^3*c^2 +
1808*a^2*b*c^3)*f^2 + 8*(80*b*c^5*d*f + (21*b^3*c^3 - 68*a*b*c^4)*f^2)*x^2 + 2*(1920*c^6*d^2 - 80*(5*b^2*c^4
- 12*a*c^5)*d*f - (105*b^4*c^2 - 448*a*b^2*c^3 + 240*a^2*c^4)*f^2)*x + 40*(48*c^6*x^3 + 8*b*c^5*x^2 + 15*b^3*c
^3 - 52*a*b*c^4 - 2*(5*b^2*c^4 - 12*a*c^5)*x)*e^2 + 8*(384*c^6*f*x^4 + 48*b*c^5*f*x^3 + 8*(80*c^6*d - (7*b^2*c
^4 - 16*a*c^5)*f)*x^2 - 80*(3*b^2*c^4 - 8*a*c^5)*d - (105*b^4*c^2 - 460*a*b^2*c^3 + 256*a^2*c^4)*f + 2*(80*b*c
^5*d + (35*b^3*c^3 - 116*a*b*c^4)*f)*x)*e)*sqrt(c*x^2 + b*x + a))/c^6, 1/15360*(15*(128*(b^2*c^4 - 4*a*c^5)*d^
2 + 16*(5*b^4*c^2 - 24*a*b^2*c^3 + 16*a^2*c^4)*d*f + (21*b^6 - 140*a*b^4*c + 240*a^2*b^2*c^2 - 64*a^3*c^3)*f^2
+ 8*(5*b^4*c^2 - 24*a*b^2*c^3 + 16*a^2*c^4)*e^2 - 8*(16*(b^3*c^3 - 4*a*b*c^4)*d + (7*b^5*c - 40*a*b^3*c^2 + 4
8*a^2*b*c^3)*f)*e)*sqrt(-c)*arctan(1/2*sqrt(c*x^2 + b*x + a)*(2*c*x + b)*sqrt(-c)/(c^2*x^2 + b*c*x + a*c)) + 2
*(1280*c^6*f^2*x^5 + 128*b*c^5*f^2*x^4 + 1920*b*c^5*d^2 + 16*(240*c^6*d*f - (9*b^2*c^4 - 20*a*c^5)*f^2)*x^3 +
80*(15*b^3*c^3 - 52*a*b*c^4)*d*f + (315*b^5*c - 1680*a*b^3*c^2 + 1808*a^2*b*c^3)*f^2 + 8*(80*b*c^5*d*f + (21*b
^3*c^3 - 68*a*b*c^4)*f^2)*x^2 + 2*(1920*c^6*d^2 - 80*(5*b^2*c^4 - 12*a*c^5)*d*f - (105*b^4*c^2 - 448*a*b^2*c^3
+ 240*a^2*c^4)*f^2)*x + 40*(48*c^6*x^3 + 8*b*c^5*x^2 + 15*b^3*c^3 - 52*a*b*c^4 - 2*(5*b^2*c^4 - 12*a*c^5)*x)*
e^2 + 8*(384*c^6*f*x^4 + 48*b*c^5*f*x^3 + 8*(80*c^6*d - (7*b^2*c^4 - 16*a*c^5)*f)*x^2 - 80*(3*b^2*c^4 - 8*a*c^
5)*d - (105*b^4*c^2 - 460*a*b^2*c^3 + 256*a^2*c^4)*f + 2*(80*b*c^5*d + (35*b^3*c^3 - 116*a*b*c^4)*f)*x)*e)*sqr
t(c*x^2 + b*x + a))/c^6]

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {a + b x + c x^{2}} \left (d + e x + f x^{2}\right )^{2}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2+b*x+a)**(1/2)*(f*x**2+e*x+d)**2,x)

[Out]

Integral(sqrt(a + b*x + c*x**2)*(d + e*x + f*x**2)**2, x)

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Giac [A]
time = 3.19, size = 638, normalized size = 1.46 \begin {gather*} \frac {1}{7680} \, \sqrt {c x^{2} + b x + a} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (10 \, f^{2} x + \frac {b c^{4} f^{2} + 24 \, c^{5} f e}{c^{5}}\right )} x + \frac {240 \, c^{5} d f - 9 \, b^{2} c^{3} f^{2} + 20 \, a c^{4} f^{2} + 24 \, b c^{4} f e + 120 \, c^{5} e^{2}}{c^{5}}\right )} x + \frac {80 \, b c^{4} d f + 21 \, b^{3} c^{2} f^{2} - 68 \, a b c^{3} f^{2} + 640 \, c^{5} d e - 56 \, b^{2} c^{3} f e + 128 \, a c^{4} f e + 40 \, b c^{4} e^{2}}{c^{5}}\right )} x + \frac {1920 \, c^{5} d^{2} - 400 \, b^{2} c^{3} d f + 960 \, a c^{4} d f - 105 \, b^{4} c f^{2} + 448 \, a b^{2} c^{2} f^{2} - 240 \, a^{2} c^{3} f^{2} + 640 \, b c^{4} d e + 280 \, b^{3} c^{2} f e - 928 \, a b c^{3} f e - 200 \, b^{2} c^{3} e^{2} + 480 \, a c^{4} e^{2}}{c^{5}}\right )} x + \frac {1920 \, b c^{4} d^{2} + 1200 \, b^{3} c^{2} d f - 4160 \, a b c^{3} d f + 315 \, b^{5} f^{2} - 1680 \, a b^{3} c f^{2} + 1808 \, a^{2} b c^{2} f^{2} - 1920 \, b^{2} c^{3} d e + 5120 \, a c^{4} d e - 840 \, b^{4} c f e + 3680 \, a b^{2} c^{2} f e - 2048 \, a^{2} c^{3} f e + 600 \, b^{3} c^{2} e^{2} - 2080 \, a b c^{3} e^{2}}{c^{5}}\right )} + \frac {{\left (128 \, b^{2} c^{4} d^{2} - 512 \, a c^{5} d^{2} + 80 \, b^{4} c^{2} d f - 384 \, a b^{2} c^{3} d f + 256 \, a^{2} c^{4} d f + 21 \, b^{6} f^{2} - 140 \, a b^{4} c f^{2} + 240 \, a^{2} b^{2} c^{2} f^{2} - 64 \, a^{3} c^{3} f^{2} - 128 \, b^{3} c^{3} d e + 512 \, a b c^{4} d e - 56 \, b^{5} c f e + 320 \, a b^{3} c^{2} f e - 384 \, a^{2} b c^{3} f e + 40 \, b^{4} c^{2} e^{2} - 192 \, a b^{2} c^{3} e^{2} + 128 \, a^{2} c^{4} e^{2}\right )} \log \left ({\left | -2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} - b \right |}\right )}{1024 \, c^{\frac {11}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)^(1/2)*(f*x^2+e*x+d)^2,x, algorithm="giac")

[Out]

1/7680*sqrt(c*x^2 + b*x + a)*(2*(4*(2*(8*(10*f^2*x + (b*c^4*f^2 + 24*c^5*f*e)/c^5)*x + (240*c^5*d*f - 9*b^2*c^
3*f^2 + 20*a*c^4*f^2 + 24*b*c^4*f*e + 120*c^5*e^2)/c^5)*x + (80*b*c^4*d*f + 21*b^3*c^2*f^2 - 68*a*b*c^3*f^2 +
640*c^5*d*e - 56*b^2*c^3*f*e + 128*a*c^4*f*e + 40*b*c^4*e^2)/c^5)*x + (1920*c^5*d^2 - 400*b^2*c^3*d*f + 960*a*
c^4*d*f - 105*b^4*c*f^2 + 448*a*b^2*c^2*f^2 - 240*a^2*c^3*f^2 + 640*b*c^4*d*e + 280*b^3*c^2*f*e - 928*a*b*c^3*
f*e - 200*b^2*c^3*e^2 + 480*a*c^4*e^2)/c^5)*x + (1920*b*c^4*d^2 + 1200*b^3*c^2*d*f - 4160*a*b*c^3*d*f + 315*b^
5*f^2 - 1680*a*b^3*c*f^2 + 1808*a^2*b*c^2*f^2 - 1920*b^2*c^3*d*e + 5120*a*c^4*d*e - 840*b^4*c*f*e + 3680*a*b^2
*c^2*f*e - 2048*a^2*c^3*f*e + 600*b^3*c^2*e^2 - 2080*a*b*c^3*e^2)/c^5) + 1/1024*(128*b^2*c^4*d^2 - 512*a*c^5*d
^2 + 80*b^4*c^2*d*f - 384*a*b^2*c^3*d*f + 256*a^2*c^4*d*f + 21*b^6*f^2 - 140*a*b^4*c*f^2 + 240*a^2*b^2*c^2*f^2
- 64*a^3*c^3*f^2 - 128*b^3*c^3*d*e + 512*a*b*c^4*d*e - 56*b^5*c*f*e + 320*a*b^3*c^2*f*e - 384*a^2*b*c^3*f*e +
40*b^4*c^2*e^2 - 192*a*b^2*c^3*e^2 + 128*a^2*c^4*e^2)*log(abs(-2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*sqrt(c)
- b))/c^(11/2)

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Mupad [B]
time = 5.31, size = 1299, normalized size = 2.98 \begin {gather*} d^2\,\left (\frac {x}{2}+\frac {b}{4\,c}\right )\,\sqrt {c\,x^2+b\,x+a}+\frac {e^2\,x\,{\left (c\,x^2+b\,x+a\right )}^{3/2}}{4\,c}+\frac {a\,f^2\,\left (\frac {5\,b\,\left (\frac {\ln \left (\frac {b+2\,c\,x}{\sqrt {c}}+2\,\sqrt {c\,x^2+b\,x+a}\right )\,\left (b^3-4\,a\,b\,c\right )}{16\,c^{5/2}}+\frac {\left (-3\,b^2+2\,c\,x\,b+8\,c\,\left (c\,x^2+a\right )\right )\,\sqrt {c\,x^2+b\,x+a}}{24\,c^2}\right )}{8\,c}-\frac {x\,{\left (c\,x^2+b\,x+a\right )}^{3/2}}{4\,c}+\frac {a\,\left (\left (\frac {x}{2}+\frac {b}{4\,c}\right )\,\sqrt {c\,x^2+b\,x+a}+\frac {\ln \left (\frac {\frac {b}{2}+c\,x}{\sqrt {c}}+\sqrt {c\,x^2+b\,x+a}\right )\,\left (a\,c-\frac {b^2}{4}\right )}{2\,c^{3/2}}\right )}{4\,c}\right )}{2\,c}-\frac {3\,b\,f^2\,\left (\frac {7\,b\,\left (\frac {5\,b\,\left (\frac {\ln \left (\frac {b+2\,c\,x}{\sqrt {c}}+2\,\sqrt {c\,x^2+b\,x+a}\right )\,\left (b^3-4\,a\,b\,c\right )}{16\,c^{5/2}}+\frac {\left (-3\,b^2+2\,c\,x\,b+8\,c\,\left (c\,x^2+a\right )\right )\,\sqrt {c\,x^2+b\,x+a}}{24\,c^2}\right )}{8\,c}-\frac {x\,{\left (c\,x^2+b\,x+a\right )}^{3/2}}{4\,c}+\frac {a\,\left (\left (\frac {x}{2}+\frac {b}{4\,c}\right )\,\sqrt {c\,x^2+b\,x+a}+\frac {\ln \left (\frac {\frac {b}{2}+c\,x}{\sqrt {c}}+\sqrt {c\,x^2+b\,x+a}\right )\,\left (a\,c-\frac {b^2}{4}\right )}{2\,c^{3/2}}\right )}{4\,c}\right )}{10\,c}-\frac {2\,a\,\left (\frac {\ln \left (\frac {b+2\,c\,x}{\sqrt {c}}+2\,\sqrt {c\,x^2+b\,x+a}\right )\,\left (b^3-4\,a\,b\,c\right )}{16\,c^{5/2}}+\frac {\left (-3\,b^2+2\,c\,x\,b+8\,c\,\left (c\,x^2+a\right )\right )\,\sqrt {c\,x^2+b\,x+a}}{24\,c^2}\right )}{5\,c}+\frac {x^2\,{\left (c\,x^2+b\,x+a\right )}^{3/2}}{5\,c}\right )}{4\,c}+\frac {f^2\,x^3\,{\left (c\,x^2+b\,x+a\right )}^{3/2}}{6\,c}-\frac {a\,e^2\,\left (\left (\frac {x}{2}+\frac {b}{4\,c}\right )\,\sqrt {c\,x^2+b\,x+a}+\frac {\ln \left (\frac {\frac {b}{2}+c\,x}{\sqrt {c}}+\sqrt {c\,x^2+b\,x+a}\right )\,\left (a\,c-\frac {b^2}{4}\right )}{2\,c^{3/2}}\right )}{4\,c}+\frac {d^2\,\ln \left (\frac {\frac {b}{2}+c\,x}{\sqrt {c}}+\sqrt {c\,x^2+b\,x+a}\right )\,\left (a\,c-\frac {b^2}{4}\right )}{2\,c^{3/2}}-\frac {5\,b\,e^2\,\left (\frac {\ln \left (\frac {b+2\,c\,x}{\sqrt {c}}+2\,\sqrt {c\,x^2+b\,x+a}\right )\,\left (b^3-4\,a\,b\,c\right )}{16\,c^{5/2}}+\frac {\left (-3\,b^2+2\,c\,x\,b+8\,c\,\left (c\,x^2+a\right )\right )\,\sqrt {c\,x^2+b\,x+a}}{24\,c^2}\right )}{8\,c}-\frac {4\,a\,e\,f\,\left (\frac {\ln \left (\frac {b+2\,c\,x}{\sqrt {c}}+2\,\sqrt {c\,x^2+b\,x+a}\right )\,\left (b^3-4\,a\,b\,c\right )}{16\,c^{5/2}}+\frac {\left (-3\,b^2+2\,c\,x\,b+8\,c\,\left (c\,x^2+a\right )\right )\,\sqrt {c\,x^2+b\,x+a}}{24\,c^2}\right )}{5\,c}-\frac {5\,b\,d\,f\,\left (\frac {\ln \left (\frac {b+2\,c\,x}{\sqrt {c}}+2\,\sqrt {c\,x^2+b\,x+a}\right )\,\left (b^3-4\,a\,b\,c\right )}{16\,c^{5/2}}+\frac {\left (-3\,b^2+2\,c\,x\,b+8\,c\,\left (c\,x^2+a\right )\right )\,\sqrt {c\,x^2+b\,x+a}}{24\,c^2}\right )}{4\,c}+\frac {d\,e\,\left (-3\,b^2+2\,c\,x\,b+8\,c\,\left (c\,x^2+a\right )\right )\,\sqrt {c\,x^2+b\,x+a}}{12\,c^2}+\frac {d\,f\,x\,{\left (c\,x^2+b\,x+a\right )}^{3/2}}{2\,c}+\frac {7\,b\,e\,f\,\left (\frac {5\,b\,\left (\frac {\ln \left (\frac {b+2\,c\,x}{\sqrt {c}}+2\,\sqrt {c\,x^2+b\,x+a}\right )\,\left (b^3-4\,a\,b\,c\right )}{16\,c^{5/2}}+\frac {\left (-3\,b^2+2\,c\,x\,b+8\,c\,\left (c\,x^2+a\right )\right )\,\sqrt {c\,x^2+b\,x+a}}{24\,c^2}\right )}{8\,c}-\frac {x\,{\left (c\,x^2+b\,x+a\right )}^{3/2}}{4\,c}+\frac {a\,\left (\left (\frac {x}{2}+\frac {b}{4\,c}\right )\,\sqrt {c\,x^2+b\,x+a}+\frac {\ln \left (\frac {\frac {b}{2}+c\,x}{\sqrt {c}}+\sqrt {c\,x^2+b\,x+a}\right )\,\left (a\,c-\frac {b^2}{4}\right )}{2\,c^{3/2}}\right )}{4\,c}\right )}{5\,c}+\frac {2\,e\,f\,x^2\,{\left (c\,x^2+b\,x+a\right )}^{3/2}}{5\,c}-\frac {a\,d\,f\,\left (\left (\frac {x}{2}+\frac {b}{4\,c}\right )\,\sqrt {c\,x^2+b\,x+a}+\frac {\ln \left (\frac {\frac {b}{2}+c\,x}{\sqrt {c}}+\sqrt {c\,x^2+b\,x+a}\right )\,\left (a\,c-\frac {b^2}{4}\right )}{2\,c^{3/2}}\right )}{2\,c}+\frac {d\,e\,\ln \left (\frac {b+2\,c\,x}{\sqrt {c}}+2\,\sqrt {c\,x^2+b\,x+a}\right )\,\left (b^3-4\,a\,b\,c\right )}{8\,c^{5/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x + c*x^2)^(1/2)*(d + e*x + f*x^2)^2,x)

[Out]

d^2*(x/2 + b/(4*c))*(a + b*x + c*x^2)^(1/2) + (e^2*x*(a + b*x + c*x^2)^(3/2))/(4*c) + (a*f^2*((5*b*((log((b +
2*c*x)/c^(1/2) + 2*(a + b*x + c*x^2)^(1/2))*(b^3 - 4*a*b*c))/(16*c^(5/2)) + ((8*c*(a + c*x^2) - 3*b^2 + 2*b*c*
x)*(a + b*x + c*x^2)^(1/2))/(24*c^2)))/(8*c) - (x*(a + b*x + c*x^2)^(3/2))/(4*c) + (a*((x/2 + b/(4*c))*(a + b*
x + c*x^2)^(1/2) + (log((b/2 + c*x)/c^(1/2) + (a + b*x + c*x^2)^(1/2))*(a*c - b^2/4))/(2*c^(3/2))))/(4*c)))/(2
*c) - (3*b*f^2*((7*b*((5*b*((log((b + 2*c*x)/c^(1/2) + 2*(a + b*x + c*x^2)^(1/2))*(b^3 - 4*a*b*c))/(16*c^(5/2)
) + ((8*c*(a + c*x^2) - 3*b^2 + 2*b*c*x)*(a + b*x + c*x^2)^(1/2))/(24*c^2)))/(8*c) - (x*(a + b*x + c*x^2)^(3/2
))/(4*c) + (a*((x/2 + b/(4*c))*(a + b*x + c*x^2)^(1/2) + (log((b/2 + c*x)/c^(1/2) + (a + b*x + c*x^2)^(1/2))*(
a*c - b^2/4))/(2*c^(3/2))))/(4*c)))/(10*c) - (2*a*((log((b + 2*c*x)/c^(1/2) + 2*(a + b*x + c*x^2)^(1/2))*(b^3
- 4*a*b*c))/(16*c^(5/2)) + ((8*c*(a + c*x^2) - 3*b^2 + 2*b*c*x)*(a + b*x + c*x^2)^(1/2))/(24*c^2)))/(5*c) + (x
^2*(a + b*x + c*x^2)^(3/2))/(5*c)))/(4*c) + (f^2*x^3*(a + b*x + c*x^2)^(3/2))/(6*c) - (a*e^2*((x/2 + b/(4*c))*
(a + b*x + c*x^2)^(1/2) + (log((b/2 + c*x)/c^(1/2) + (a + b*x + c*x^2)^(1/2))*(a*c - b^2/4))/(2*c^(3/2))))/(4*
c) + (d^2*log((b/2 + c*x)/c^(1/2) + (a + b*x + c*x^2)^(1/2))*(a*c - b^2/4))/(2*c^(3/2)) - (5*b*e^2*((log((b +
2*c*x)/c^(1/2) + 2*(a + b*x + c*x^2)^(1/2))*(b^3 - 4*a*b*c))/(16*c^(5/2)) + ((8*c*(a + c*x^2) - 3*b^2 + 2*b*c*
x)*(a + b*x + c*x^2)^(1/2))/(24*c^2)))/(8*c) - (4*a*e*f*((log((b + 2*c*x)/c^(1/2) + 2*(a + b*x + c*x^2)^(1/2))
*(b^3 - 4*a*b*c))/(16*c^(5/2)) + ((8*c*(a + c*x^2) - 3*b^2 + 2*b*c*x)*(a + b*x + c*x^2)^(1/2))/(24*c^2)))/(5*c
) - (5*b*d*f*((log((b + 2*c*x)/c^(1/2) + 2*(a + b*x + c*x^2)^(1/2))*(b^3 - 4*a*b*c))/(16*c^(5/2)) + ((8*c*(a +
c*x^2) - 3*b^2 + 2*b*c*x)*(a + b*x + c*x^2)^(1/2))/(24*c^2)))/(4*c) + (d*e*(8*c*(a + c*x^2) - 3*b^2 + 2*b*c*x
)*(a + b*x + c*x^2)^(1/2))/(12*c^2) + (d*f*x*(a + b*x + c*x^2)^(3/2))/(2*c) + (7*b*e*f*((5*b*((log((b + 2*c*x)
/c^(1/2) + 2*(a + b*x + c*x^2)^(1/2))*(b^3 - 4*a*b*c))/(16*c^(5/2)) + ((8*c*(a + c*x^2) - 3*b^2 + 2*b*c*x)*(a
+ b*x + c*x^2)^(1/2))/(24*c^2)))/(8*c) - (x*(a + b*x + c*x^2)^(3/2))/(4*c) + (a*((x/2 + b/(4*c))*(a + b*x + c*
x^2)^(1/2) + (log((b/2 + c*x)/c^(1/2) + (a + b*x + c*x^2)^(1/2))*(a*c - b^2/4))/(2*c^(3/2))))/(4*c)))/(5*c) +
(2*e*f*x^2*(a + b*x + c*x^2)^(3/2))/(5*c) - (a*d*f*((x/2 + b/(4*c))*(a + b*x + c*x^2)^(1/2) + (log((b/2 + c*x)
/c^(1/2) + (a + b*x + c*x^2)^(1/2))*(a*c - b^2/4))/(2*c^(3/2))))/(2*c) + (d*e*log((b + 2*c*x)/c^(1/2) + 2*(a +
b*x + c*x^2)^(1/2))*(b^3 - 4*a*b*c))/(8*c^(5/2))

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