3.100 \(\int \sqrt{a+b x+c x^2} (d+e x+f x^2)^2 \, dx\)

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

[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))

________________________________________________________________________________________

Rubi [A]  time = 0.789373, antiderivative size = 436, normalized size of antiderivative = 1., 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 = {1661, 640, 612, 621, 206} \[ \frac{(b+2 c x) \sqrt{a+b x+c x^2} \left (8 c^2 \left (2 a^2 f^2+12 a b e f+5 b^2 \left (2 d f+e^2\right )\right )-56 b^2 c f (a f+b e)-32 c^3 \left (a \left (2 d f+e^2\right )+4 b d e\right )+21 b^4 f^2+128 c^4 d^2\right )}{512 c^5}-\frac{\left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right ) \left (8 c^2 \left (2 a^2 f^2+12 a b e f+5 b^2 \left (2 d f+e^2\right )\right )-56 b^2 c f (a f+b e)-32 c^3 \left (a \left (2 d f+e^2\right )+4 b d e\right )+21 b^4 f^2+128 c^4 d^2\right )}{1024 c^{11/2}}+\frac{x \left (a+b x+c x^2\right )^{3/2} \left (-4 c f (5 a f+14 b e)+21 b^2 f^2+40 c^2 \left (2 d f+e^2\right )\right )}{160 c^3}+\frac{\left (a+b x+c x^2\right )^{3/2} \left (-8 c^2 \left (32 a e f+25 b \left (2 d f+e^2\right )\right )+28 b c f (7 a f+10 b e)-105 b^3 f^2+640 c^3 d e\right )}{960 c^4}+\frac{f x^2 \left (a+b x+c x^2\right )^{3/2} (8 c e-3 b f)}{20 c^2}+\frac{f^2 x^3 \left (a+b x+c x^2\right )^{3/2}}{6 c} \]

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 1661

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]

Rule 640

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 612

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 621

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 206

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

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 ) \operatorname{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*}

Mathematica [A]  time = 0.964738, size = 657, normalized size = 1.51 \[ \frac{-f^2 \left (-15 \left (16 a^2 c^2-56 a b^2 c+21 b^4\right ) \left (2 \sqrt{c} (b+2 c x) \sqrt{a+x (b+c x)}-\left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right )\right )+16 c^{3/2} \left (-196 a b c+120 a c^2 x-126 b^2 c x+105 b^3\right ) (a+x (b+c x))^{3/2}+2304 b c^{7/2} x^2 (a+x (b+c x))^{3/2}\right )-1920 c^4 d^2 \left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right )-40 c^2 \left (2 d f+e^2\right ) \left (80 b c^{3/2} (a+x (b+c x))^{3/2}-3 \left (5 b^2-4 a c\right ) \left (2 \sqrt{c} (b+2 c x) \sqrt{a+x (b+c x)}-\left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right )\right )\right )-1920 b c^3 d e \left (2 \sqrt{c} (b+2 c x) \sqrt{a+x (b+c x)}-\left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right )\right )+8 c e f \left (-16 c^{3/2} \left (32 a c-35 b^2+42 b c x\right ) (a+x (b+c x))^{3/2}-15 b \left (7 b^2-12 a c\right ) \left (2 \sqrt{c} (b+2 c x) \sqrt{a+x (b+c x)}-\left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right )\right )\right )+3840 c^{9/2} d^2 (b+2 c x) \sqrt{a+x (b+c x)}+3840 c^{9/2} x \left (2 d f+e^2\right ) (a+x (b+c x))^{3/2}+10240 c^{9/2} d e (a+x (b+c x))^{3/2}+6144 c^{9/2} e f x^2 (a+x (b+c x))^{3/2}+2560 c^{9/2} f^2 x^3 (a+x (b+c x))^{3/2}}{15360 c^{11/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3840*c^(9/2)*d^2*(b + 2*c*x)*Sqrt[a + x*(b + c*x)] + 10240*c^(9/2)*d*e*(a + x*(b + c*x))^(3/2) + 3840*c^(9/2)
*(e^2 + 2*d*f)*x*(a + x*(b + c*x))^(3/2) + 6144*c^(9/2)*e*f*x^2*(a + x*(b + c*x))^(3/2) + 2560*c^(9/2)*f^2*x^3
*(a + x*(b + c*x))^(3/2) - 1920*c^4*(b^2 - 4*a*c)*d^2*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])] -
1920*b*c^3*d*e*(2*Sqrt[c]*(b + 2*c*x)*Sqrt[a + x*(b + c*x)] - (b^2 - 4*a*c)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sq
rt[a + x*(b + c*x)])]) + 8*c*e*f*(-16*c^(3/2)*(-35*b^2 + 32*a*c + 42*b*c*x)*(a + x*(b + c*x))^(3/2) - 15*b*(7*
b^2 - 12*a*c)*(2*Sqrt[c]*(b + 2*c*x)*Sqrt[a + x*(b + c*x)] - (b^2 - 4*a*c)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt
[a + x*(b + c*x)])])) - 40*c^2*(e^2 + 2*d*f)*(80*b*c^(3/2)*(a + x*(b + c*x))^(3/2) - 3*(5*b^2 - 4*a*c)*(2*Sqrt
[c]*(b + 2*c*x)*Sqrt[a + x*(b + c*x)] - (b^2 - 4*a*c)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])]))
- f^2*(2304*b*c^(7/2)*x^2*(a + x*(b + c*x))^(3/2) + 16*c^(3/2)*(105*b^3 - 196*a*b*c - 126*b^2*c*x + 120*a*c^2
*x)*(a + x*(b + c*x))^(3/2) - 15*(21*b^4 - 56*a*b^2*c + 16*a^2*c^2)*(2*Sqrt[c]*(b + 2*c*x)*Sqrt[a + x*(b + c*x
)] - (b^2 - 4*a*c)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])])))/(15360*c^(11/2))

________________________________________________________________________________________

Maple [B]  time = 0.063, size = 1429, normalized size = 3.3 \begin{align*} \text{result too large to display} \end{align*}

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)

[Out]

3/8*e*f*b/c^2*a*x*(c*x^2+b*x+a)^(1/2)-7/32*e*f*b^3/c^3*x*(c*x^2+b*x+a)^(1/2)-5/16*e*f*b^3/c^(7/2)*ln((1/2*b+c*
x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a+3/16*e*f*b^2/c^3*a*(c*x^2+b*x+a)^(1/2)+3/8*e*f*b/c^(5/2)*a^2*ln((1/2*b+c*x)/
c^(1/2)+(c*x^2+b*x+a)^(1/2))+2/3*d*e*(c*x^2+b*x+a)^(3/2)/c-7/64*f^2*b^3/c^4*(c*x^2+b*x+a)^(3/2)+21/512*f^2*b^5
/c^5*(c*x^2+b*x+a)^(1/2)-21/1024*f^2*b^6/c^(11/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))+1/16*f^2*a^3/c^(
5/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))-5/128*b^4/c^(7/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))
*e^2-1/8*a^2/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*e^2+1/4*d^2/c*(c*x^2+b*x+a)^(1/2)*b+1/2*d^2/c
^(1/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a-1/8*d^2/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2
))*b^2-7/32*f^2*b^2/c^3*a*x*(c*x^2+b*x+a)^(1/2)-7/20*e*f*b/c^2*x*(c*x^2+b*x+a)^(3/2)-1/8*a/c^2*(c*x^2+b*x+a)^(
1/2)*b*d*f+5/16*b^2/c^2*x*(c*x^2+b*x+a)^(1/2)*d*f-1/4*a/c*x*(c*x^2+b*x+a)^(1/2)*d*f-1/2*d*e*b/c^(3/2)*ln((1/2*
b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a+3/8*b^2/c^(5/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a*d*f-1/2*d*
e*b/c*x*(c*x^2+b*x+a)^(1/2)+7/24*e*f*b^2/c^3*(c*x^2+b*x+a)^(3/2)-7/64*e*f*b^4/c^4*(c*x^2+b*x+a)^(1/2)-1/4*a^2/
c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*d*f+1/2*d^2*x*(c*x^2+b*x+a)^(1/2)-5/12*b/c^2*(c*x^2+b*x+a)
^(3/2)*d*f-1/16*a/c^2*(c*x^2+b*x+a)^(1/2)*b*e^2+1/8*d*e*b^3/c^(5/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2)
)+5/32*b^2/c^2*x*(c*x^2+b*x+a)^(1/2)*e^2+5/32*b^3/c^3*(c*x^2+b*x+a)^(1/2)*d*f+3/16*b^2/c^(5/2)*ln((1/2*b+c*x)/
c^(1/2)+(c*x^2+b*x+a)^(1/2))*a*e^2-5/64*b^4/c^(7/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*d*f-1/8*a/c*x*
(c*x^2+b*x+a)^(1/2)*e^2+2/5*e*f*x^2*(c*x^2+b*x+a)^(3/2)/c+1/2*x*(c*x^2+b*x+a)^(3/2)/c*d*f-1/8*f^2*a/c^2*x*(c*x
^2+b*x+a)^(3/2)-1/4*d*e*b^2/c^2*(c*x^2+b*x+a)^(1/2)-3/20*f^2*b/c^2*x^2*(c*x^2+b*x+a)^(3/2)+1/16*f^2*a^2/c^2*x*
(c*x^2+b*x+a)^(1/2)+1/32*f^2*a^2/c^3*(c*x^2+b*x+a)^(1/2)*b+21/160*f^2*b^2/c^3*x*(c*x^2+b*x+a)^(3/2)+21/256*f^2
*b^4/c^4*x*(c*x^2+b*x+a)^(1/2)+35/256*f^2*b^4/c^(9/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a-7/64*f^2*b
^3/c^4*a*(c*x^2+b*x+a)^(1/2)-15/64*f^2*b^2/c^(7/2)*a^2*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))+49/240*f^2*
b/c^3*a*(c*x^2+b*x+a)^(3/2)+7/128*e*f*b^5/c^(9/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))-4/15*e*f*a/c^2*(
c*x^2+b*x+a)^(3/2)+1/4*x*(c*x^2+b*x+a)^(3/2)/c*e^2-5/24*b/c^2*(c*x^2+b*x+a)^(3/2)*e^2+5/64*b^3/c^3*(c*x^2+b*x+
a)^(1/2)*e^2+1/6*f^2*x^3*(c*x^2+b*x+a)^(3/2)/c

________________________________________________________________________________________

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

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

________________________________________________________________________________________

Fricas [A]  time = 2.83322, size = 2866, normalized size = 6.57 \begin{align*} \text{result too large to display} \end{align*}

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 - 128*(b^3*c^3 - 4*a*b*c^4)*d*e + 8*(5*b^4*c^2 - 24*a*b^2*c^3 + 16*
a^2*c^4)*e^2 + (21*b^6 - 140*a*b^4*c + 240*a^2*b^2*c^2 - 64*a^3*c^3)*f^2 + 8*(2*(5*b^4*c^2 - 24*a*b^2*c^3 + 16
*a^2*c^4)*d - (7*b^5*c - 40*a*b^3*c^2 + 48*a^2*b*c^3)*e)*f)*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 + 1920*b*c^5*d^2 + 128*(24*c^6*e*f + b*c^5*f
^2)*x^4 + 16*(120*c^6*e^2 - (9*b^2*c^4 - 20*a*c^5)*f^2 + 24*(10*c^6*d + b*c^5*e)*f)*x^3 - 640*(3*b^2*c^4 - 8*a
*c^5)*d*e + 40*(15*b^3*c^3 - 52*a*b*c^4)*e^2 + (315*b^5*c - 1680*a*b^3*c^2 + 1808*a^2*b*c^3)*f^2 + 8*(640*c^6*
d*e + 40*b*c^5*e^2 + (21*b^3*c^3 - 68*a*b*c^4)*f^2 + 8*(10*b*c^5*d - (7*b^2*c^4 - 16*a*c^5)*e)*f)*x^2 + 8*(10*
(15*b^3*c^3 - 52*a*b*c^4)*d - (105*b^4*c^2 - 460*a*b^2*c^3 + 256*a^2*c^4)*e)*f + 2*(1920*c^6*d^2 + 640*b*c^5*d
*e - 40*(5*b^2*c^4 - 12*a*c^5)*e^2 - (105*b^4*c^2 - 448*a*b^2*c^3 + 240*a^2*c^4)*f^2 - 8*(10*(5*b^2*c^4 - 12*a
*c^5)*d - (35*b^3*c^3 - 116*a*b*c^4)*e)*f)*x)*sqrt(c*x^2 + b*x + a))/c^6, 1/15360*(15*(128*(b^2*c^4 - 4*a*c^5)
*d^2 - 128*(b^3*c^3 - 4*a*b*c^4)*d*e + 8*(5*b^4*c^2 - 24*a*b^2*c^3 + 16*a^2*c^4)*e^2 + (21*b^6 - 140*a*b^4*c +
240*a^2*b^2*c^2 - 64*a^3*c^3)*f^2 + 8*(2*(5*b^4*c^2 - 24*a*b^2*c^3 + 16*a^2*c^4)*d - (7*b^5*c - 40*a*b^3*c^2
+ 48*a^2*b*c^3)*e)*f)*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 + 1920*b*c^5*d^2 + 128*(24*c^6*e*f + b*c^5*f^2)*x^4 + 16*(120*c^6*e^2 - (9*b^2*c^4 - 20*
a*c^5)*f^2 + 24*(10*c^6*d + b*c^5*e)*f)*x^3 - 640*(3*b^2*c^4 - 8*a*c^5)*d*e + 40*(15*b^3*c^3 - 52*a*b*c^4)*e^2
+ (315*b^5*c - 1680*a*b^3*c^2 + 1808*a^2*b*c^3)*f^2 + 8*(640*c^6*d*e + 40*b*c^5*e^2 + (21*b^3*c^3 - 68*a*b*c^
4)*f^2 + 8*(10*b*c^5*d - (7*b^2*c^4 - 16*a*c^5)*e)*f)*x^2 + 8*(10*(15*b^3*c^3 - 52*a*b*c^4)*d - (105*b^4*c^2 -
460*a*b^2*c^3 + 256*a^2*c^4)*e)*f + 2*(1920*c^6*d^2 + 640*b*c^5*d*e - 40*(5*b^2*c^4 - 12*a*c^5)*e^2 - (105*b^
4*c^2 - 448*a*b^2*c^3 + 240*a^2*c^4)*f^2 - 8*(10*(5*b^2*c^4 - 12*a*c^5)*d - (35*b^3*c^3 - 116*a*b*c^4)*e)*f)*x
)*sqrt(c*x^2 + b*x + a))/c^6]

________________________________________________________________________________________

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

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)

________________________________________________________________________________________

Giac [A]  time = 1.30159, size = 861, normalized size = 1.97 \begin{align*} \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{align*}

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)