3.2.15 \(\int \frac {(d+e x+f x^2)^2}{(a+b x+c x^2)^{3/2}} \, dx\) [115]

Optimal. Leaf size=309 \[ \frac {2 \left (2 a b^2 c e f-a b^3 f^2+4 a c^2 e (c d-a f)-b c \left (c^2 d^2-3 a^2 f^2+a c \left (e^2+2 d f\right )\right )-\left (2 c^4 d^2+b^4 f^2-2 b^2 c f (b e+2 a f)-2 c^3 \left (b d e+a \left (e^2+2 d f\right )\right )+c^2 \left (6 a b e f+2 a^2 f^2+b^2 \left (e^2+2 d f\right )\right )\right ) x\right )}{c^3 \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}+\frac {f (8 c e-7 b f) \sqrt {a+b x+c x^2}}{4 c^3}+\frac {f^2 x \sqrt {a+b x+c x^2}}{2 c^2}+\frac {\left (15 b^2 f^2-12 c f (2 b e+a f)+8 c^2 \left (e^2+2 d f\right )\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{8 c^{7/2}} \]

[Out]

1/8*(15*b^2*f^2-12*c*f*(a*f+2*b*e)+8*c^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^(7/
2)+2*(2*a*b^2*c*e*f-a*b^3*f^2+4*a*c^2*e*(-a*f+c*d)-b*c*(c^2*d^2-3*a^2*f^2+a*c*(2*d*f+e^2))-(2*c^4*d^2+b^4*f^2-
2*b^2*c*f*(2*a*f+b*e)-2*c^3*(b*d*e+a*(2*d*f+e^2))+c^2*(6*a*b*e*f+2*a^2*f^2+b^2*(2*d*f+e^2)))*x)/c^3/(-4*a*c+b^
2)/(c*x^2+b*x+a)^(1/2)+1/4*f*(-7*b*f+8*c*e)*(c*x^2+b*x+a)^(1/2)/c^3+1/2*f^2*x*(c*x^2+b*x+a)^(1/2)/c^2

________________________________________________________________________________________

Rubi [A]
time = 0.27, antiderivative size = 309, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {1674, 1675, 654, 635, 212} \begin {gather*} \frac {2 \left (-x \left (c^2 \left (2 a^2 f^2+6 a b e f+b^2 \left (2 d f+e^2\right )\right )-2 b^2 c f (2 a f+b e)-2 c^3 \left (a \left (2 d f+e^2\right )+b d e\right )+b^4 f^2+2 c^4 d^2\right )-b c \left (-3 a^2 f^2+a c \left (2 d f+e^2\right )+c^2 d^2\right )-a b^3 f^2+2 a b^2 c e f+4 a c^2 e (c d-a f)\right )}{c^3 \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}+\frac {\tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (-12 c f (a f+2 b e)+15 b^2 f^2+8 c^2 \left (2 d f+e^2\right )\right )}{8 c^{7/2}}+\frac {f \sqrt {a+b x+c x^2} (8 c e-7 b f)}{4 c^3}+\frac {f^2 x \sqrt {a+b x+c x^2}}{2 c^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x + f*x^2)^2/(a + b*x + c*x^2)^(3/2),x]

[Out]

(2*(2*a*b^2*c*e*f - a*b^3*f^2 + 4*a*c^2*e*(c*d - a*f) - b*c*(c^2*d^2 - 3*a^2*f^2 + a*c*(e^2 + 2*d*f)) - (2*c^4
*d^2 + b^4*f^2 - 2*b^2*c*f*(b*e + 2*a*f) - 2*c^3*(b*d*e + a*(e^2 + 2*d*f)) + c^2*(6*a*b*e*f + 2*a^2*f^2 + b^2*
(e^2 + 2*d*f)))*x))/(c^3*(b^2 - 4*a*c)*Sqrt[a + b*x + c*x^2]) + (f*(8*c*e - 7*b*f)*Sqrt[a + b*x + c*x^2])/(4*c
^3) + (f^2*x*Sqrt[a + b*x + c*x^2])/(2*c^2) + ((15*b^2*f^2 - 12*c*f*(2*b*e + a*f) + 8*c^2*(e^2 + 2*d*f))*ArcTa
nh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(8*c^(7/2))

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 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 1674

Int[(Pq_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq, a + b*x + c*
x^2, x], f = Coeff[PolynomialRemainder[Pq, a + b*x + c*x^2, x], x, 0], g = Coeff[PolynomialRemainder[Pq, a + b
*x + c*x^2, x], x, 1]}, Simp[(b*f - 2*a*g + (2*c*f - b*g)*x)*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c)
)), x] + Dist[1/((p + 1)*(b^2 - 4*a*c)), Int[(a + b*x + c*x^2)^(p + 1)*ExpandToSum[(p + 1)*(b^2 - 4*a*c)*Q - (
2*p + 3)*(2*c*f - b*g), x], x], x]] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[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 \frac {\left (d+e x+f x^2\right )^2}{\left (a+b x+c x^2\right )^{3/2}} \, dx &=\frac {2 \left (2 a b^2 c e f-a b^3 f^2+4 a c^2 e (c d-a f)-b c \left (c^2 d^2-3 a^2 f^2+a c \left (e^2+2 d f\right )\right )-\left (2 c^4 d^2+b^4 f^2-2 b^2 c f (b e+2 a f)-2 c^3 \left (b d e+a \left (e^2+2 d f\right )\right )+c^2 \left (6 a b e f+2 a^2 f^2+b^2 \left (e^2+2 d f\right )\right )\right ) x\right )}{c^3 \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}-\frac {2 \int \frac {-\frac {\left (b^2-4 a c\right ) \left (b^2 f^2-c f (2 b e+a f)+c^2 \left (e^2+2 d f\right )\right )}{2 c^3}-\frac {\left (b^2-4 a c\right ) f (2 c e-b f) x}{2 c^2}-\frac {\left (b^2-4 a c\right ) f^2 x^2}{2 c}}{\sqrt {a+b x+c x^2}} \, dx}{b^2-4 a c}\\ &=\frac {2 \left (2 a b^2 c e f-a b^3 f^2+4 a c^2 e (c d-a f)-b c \left (c^2 d^2-3 a^2 f^2+a c \left (e^2+2 d f\right )\right )-\left (2 c^4 d^2+b^4 f^2-2 b^2 c f (b e+2 a f)-2 c^3 \left (b d e+a \left (e^2+2 d f\right )\right )+c^2 \left (6 a b e f+2 a^2 f^2+b^2 \left (e^2+2 d f\right )\right )\right ) x\right )}{c^3 \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}+\frac {f^2 x \sqrt {a+b x+c x^2}}{2 c^2}-\frac {\int \frac {-\frac {\left (b^2-4 a c\right ) \left (2 b^2 f^2-c f (4 b e+3 a f)+2 c^2 \left (e^2+2 d f\right )\right )}{2 c^2}-\frac {\left (b^2-4 a c\right ) f (8 c e-7 b f) x}{4 c}}{\sqrt {a+b x+c x^2}} \, dx}{c \left (b^2-4 a c\right )}\\ &=\frac {2 \left (2 a b^2 c e f-a b^3 f^2+4 a c^2 e (c d-a f)-b c \left (c^2 d^2-3 a^2 f^2+a c \left (e^2+2 d f\right )\right )-\left (2 c^4 d^2+b^4 f^2-2 b^2 c f (b e+2 a f)-2 c^3 \left (b d e+a \left (e^2+2 d f\right )\right )+c^2 \left (6 a b e f+2 a^2 f^2+b^2 \left (e^2+2 d f\right )\right )\right ) x\right )}{c^3 \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}+\frac {f (8 c e-7 b f) \sqrt {a+b x+c x^2}}{4 c^3}+\frac {f^2 x \sqrt {a+b x+c x^2}}{2 c^2}+\frac {\left (15 b^2 f^2-12 c f (2 b e+a f)+8 c^2 \left (e^2+2 d f\right )\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{8 c^3}\\ &=\frac {2 \left (2 a b^2 c e f-a b^3 f^2+4 a c^2 e (c d-a f)-b c \left (c^2 d^2-3 a^2 f^2+a c \left (e^2+2 d f\right )\right )-\left (2 c^4 d^2+b^4 f^2-2 b^2 c f (b e+2 a f)-2 c^3 \left (b d e+a \left (e^2+2 d f\right )\right )+c^2 \left (6 a b e f+2 a^2 f^2+b^2 \left (e^2+2 d f\right )\right )\right ) x\right )}{c^3 \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}+\frac {f (8 c e-7 b f) \sqrt {a+b x+c x^2}}{4 c^3}+\frac {f^2 x \sqrt {a+b x+c x^2}}{2 c^2}+\frac {\left (15 b^2 f^2-12 c f (2 b e+a f)+8 c^2 \left (e^2+2 d f\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 )}{4 c^3}\\ &=\frac {2 \left (2 a b^2 c e f-a b^3 f^2+4 a c^2 e (c d-a f)-b c \left (c^2 d^2-3 a^2 f^2+a c \left (e^2+2 d f\right )\right )-\left (2 c^4 d^2+b^4 f^2-2 b^2 c f (b e+2 a f)-2 c^3 \left (b d e+a \left (e^2+2 d f\right )\right )+c^2 \left (6 a b e f+2 a^2 f^2+b^2 \left (e^2+2 d f\right )\right )\right ) x\right )}{c^3 \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}+\frac {f (8 c e-7 b f) \sqrt {a+b x+c x^2}}{4 c^3}+\frac {f^2 x \sqrt {a+b x+c x^2}}{2 c^2}+\frac {\left (15 b^2 f^2-12 c f (2 b e+a f)+8 c^2 \left (e^2+2 d f\right )\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{8 c^{7/2}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 1.50, size = 291, normalized size = 0.94 \begin {gather*} \frac {-\frac {2 \sqrt {c} \left (15 b^4 f^2 x+b^3 f (15 a f+c x (-24 e+5 f x))+4 b c \left (-13 a^2 f^2+2 c^2 d (d-2 e x)+a c \left (2 e^2+4 d f+20 e f x-5 f^2 x^2\right )\right )-2 b^2 c \left (a f (12 e+31 f x)+c x \left (-4 e^2-8 d f+4 e f x+f^2 x^2\right )\right )+8 c^2 \left (2 c^2 d^2 x+a^2 f (8 e+3 f x)+a c \left (-4 d (e+f x)+x \left (-2 e^2+4 e f x+f^2 x^2\right )\right )\right )\right )}{\left (b^2-4 a c\right ) \sqrt {a+x (b+c x)}}+\left (-15 b^2 f^2+12 c f (2 b e+a f)-8 c^2 \left (e^2+2 d f\right )\right ) \log \left (c^3 \left (b+2 c x-2 \sqrt {c} \sqrt {a+x (b+c x)}\right )\right )}{8 c^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x + f*x^2)^2/(a + b*x + c*x^2)^(3/2),x]

[Out]

((-2*Sqrt[c]*(15*b^4*f^2*x + b^3*f*(15*a*f + c*x*(-24*e + 5*f*x)) + 4*b*c*(-13*a^2*f^2 + 2*c^2*d*(d - 2*e*x) +
a*c*(2*e^2 + 4*d*f + 20*e*f*x - 5*f^2*x^2)) - 2*b^2*c*(a*f*(12*e + 31*f*x) + c*x*(-4*e^2 - 8*d*f + 4*e*f*x +
f^2*x^2)) + 8*c^2*(2*c^2*d^2*x + a^2*f*(8*e + 3*f*x) + a*c*(-4*d*(e + f*x) + x*(-2*e^2 + 4*e*f*x + f^2*x^2))))
)/((b^2 - 4*a*c)*Sqrt[a + x*(b + c*x)]) + (-15*b^2*f^2 + 12*c*f*(2*b*e + a*f) - 8*c^2*(e^2 + 2*d*f))*Log[c^3*(
b + 2*c*x - 2*Sqrt[c]*Sqrt[a + x*(b + c*x)])])/(8*c^(7/2))

________________________________________________________________________________________

Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(748\) vs. \(2(289)=578\).
time = 0.16, size = 749, normalized size = 2.42

method	result	size
default	\(f^{2} \left (\frac {x^{3}}{2 c \sqrt {c \,x^{2}+b x +a}}-\frac {5 b \left (\frac {x^{2}}{c \sqrt {c \,x^{2}+b x +a}}-\frac {3 b \left (-\frac {x}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (-\frac {1}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (2 c x +b \right )}{c \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}\right )}{2 c}+\frac {\ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{c^{\frac {3}{2}}}\right )}{2 c}-\frac {2 a \left (-\frac {1}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (2 c x +b \right )}{c \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}\right )}{c}\right )}{4 c}-\frac {3 a \left (-\frac {x}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (-\frac {1}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (2 c x +b \right )}{c \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}\right )}{2 c}+\frac {\ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{c^{\frac {3}{2}}}\right )}{2 c}\right )+2 e f \left (\frac {x^{2}}{c \sqrt {c \,x^{2}+b x +a}}-\frac {3 b \left (-\frac {x}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (-\frac {1}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (2 c x +b \right )}{c \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}\right )}{2 c}+\frac {\ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{c^{\frac {3}{2}}}\right )}{2 c}-\frac {2 a \left (-\frac {1}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (2 c x +b \right )}{c \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}\right )}{c}\right )+\left (2 d f +e^{2}\right ) \left (-\frac {x}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (-\frac {1}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (2 c x +b \right )}{c \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}\right )}{2 c}+\frac {\ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{c^{\frac {3}{2}}}\right )+2 d e \left (-\frac {1}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (2 c x +b \right )}{c \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}\right )+\frac {2 d^{2} \left (2 c x +b \right )}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}\)	\(749\)
risch	\(\frac {2 b^{2} x d f}{c \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}+\frac {b^{2} x a \,f^{2}}{c^{2} \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}-\frac {4 b x d e}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}+\frac {b^{4} x \,f^{2}}{8 c^{3} \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}+\frac {b^{2} x \,e^{2}}{c \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}+\frac {b^{3} a \,f^{2}}{2 c^{3} \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}-\frac {b^{4} e f}{2 c^{3} \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}+\frac {b^{3} d f}{c^{2} \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}-\frac {2 b^{2} d e}{c \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}-\frac {2 a^{2} f^{2} x}{c \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}-\frac {a^{2} f^{2} b}{c^{2} \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}+\frac {3 x b e f}{c^{2} \sqrt {c \,x^{2}+b x +a}}-\frac {x \,e^{2}}{c \sqrt {c \,x^{2}+b x +a}}+\frac {b^{3} f^{2}}{16 c^{4} \sqrt {c \,x^{2}+b x +a}}+\frac {b \,e^{2}}{2 c^{2} \sqrt {c \,x^{2}+b x +a}}-\frac {2 d e}{c \sqrt {c \,x^{2}+b x +a}}+\frac {2 d^{2} b}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}+\frac {2 \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right ) d f}{c^{\frac {3}{2}}}-\frac {3 \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right ) a \,f^{2}}{2 c^{\frac {5}{2}}}+\frac {15 \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right ) b^{2} f^{2}}{8 c^{\frac {7}{2}}}+\frac {\ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right ) e^{2}}{c^{\frac {3}{2}}}-\frac {2 x d f}{c \sqrt {c \,x^{2}+b x +a}}-\frac {5 b a \,f^{2}}{4 c^{3} \sqrt {c \,x^{2}+b x +a}}-\frac {b^{2} e f}{2 c^{3} \sqrt {c \,x^{2}+b x +a}}+\frac {b d f}{c^{2} \sqrt {c \,x^{2}+b x +a}}+\frac {b^{5} f^{2}}{16 c^{4} \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}+\frac {b^{3} e^{2}}{2 c^{2} \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}+\frac {2 a e f}{c^{2} \sqrt {c \,x^{2}+b x +a}}+\frac {4 c \,d^{2} x}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}+\frac {3 x a \,f^{2}}{2 c^{2} \sqrt {c \,x^{2}+b x +a}}-\frac {15 x \,b^{2} f^{2}}{8 c^{3} \sqrt {c \,x^{2}+b x +a}}-\frac {f \left (-2 c f x +7 b f -8 c e \right ) \sqrt {c \,x^{2}+b x +a}}{4 c^{3}}-\frac {b^{3} x e f}{c^{2} \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}-\frac {3 \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right ) b e f}{c^{\frac {5}{2}}}\)	\(1002\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((f*x^2+e*x+d)^2/(c*x^2+b*x+a)^(3/2),x,method=_RETURNVERBOSE)

[Out]

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

________________________________________________________________________________________

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((f*x^2+e*x+d)^2/(c*x^2+b*x+a)^(3/2),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

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 637 vs. \(2 (289) = 578\).
time = 3.19, size = 1277, normalized size = 4.13 \begin {gather*} \left [\frac {{\left (16 \, {\left (a b^{2} c^{2} - 4 \, a^{2} c^{3}\right )} d f + 3 \, {\left (5 \, a b^{4} - 24 \, a^{2} b^{2} c + 16 \, a^{3} c^{2}\right )} f^{2} + {\left (16 \, {\left (b^{2} c^{3} - 4 \, a c^{4}\right )} d f + 3 \, {\left (5 \, b^{4} c - 24 \, a b^{2} c^{2} + 16 \, a^{2} c^{3}\right )} f^{2}\right )} x^{2} + {\left (16 \, {\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} d f + 3 \, {\left (5 \, b^{5} - 24 \, a b^{3} c + 16 \, a^{2} b c^{2}\right )} f^{2}\right )} x + 8 \, {\left (a b^{2} c^{2} - 4 \, a^{2} c^{3} + {\left (b^{2} c^{3} - 4 \, a c^{4}\right )} x^{2} + {\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} x\right )} e^{2} - 24 \, {\left ({\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} f x^{2} + {\left (b^{4} c - 4 \, a b^{2} c^{2}\right )} f x + {\left (a b^{3} c - 4 \, a^{2} b c^{2}\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 (8 \, b c^{4} d^{2} + 16 \, a b c^{3} d f - 2 \, {\left (b^{2} c^{3} - 4 \, a c^{4}\right )} f^{2} x^{3} + 5 \, {\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} f^{2} x^{2} + {\left (15 \, a b^{3} c - 52 \, a^{2} b c^{2}\right )} f^{2} + {\left (16 \, c^{5} d^{2} + 16 \, {\left (b^{2} c^{3} - 2 \, a c^{4}\right )} d f + {\left (15 \, b^{4} c - 62 \, a b^{2} c^{2} + 24 \, a^{2} c^{3}\right )} f^{2}\right )} x + 8 \, {\left (a b c^{3} + {\left (b^{2} c^{3} - 2 \, a c^{4}\right )} x\right )} e^{2} - 8 \, {\left (4 \, a c^{4} d + {\left (b^{2} c^{3} - 4 \, a c^{4}\right )} f x^{2} + {\left (3 \, a b^{2} c^{2} - 8 \, a^{2} c^{3}\right )} f + {\left (2 \, b c^{4} d + {\left (3 \, b^{3} c^{2} - 10 \, a b c^{3}\right )} f\right )} x\right )} e\right )} \sqrt {c x^{2} + b x + a}}{16 \, {\left (a b^{2} c^{4} - 4 \, a^{2} c^{5} + {\left (b^{2} c^{5} - 4 \, a c^{6}\right )} x^{2} + {\left (b^{3} c^{4} - 4 \, a b c^{5}\right )} x\right )}}, -\frac {{\left (16 \, {\left (a b^{2} c^{2} - 4 \, a^{2} c^{3}\right )} d f + 3 \, {\left (5 \, a b^{4} - 24 \, a^{2} b^{2} c + 16 \, a^{3} c^{2}\right )} f^{2} + {\left (16 \, {\left (b^{2} c^{3} - 4 \, a c^{4}\right )} d f + 3 \, {\left (5 \, b^{4} c - 24 \, a b^{2} c^{2} + 16 \, a^{2} c^{3}\right )} f^{2}\right )} x^{2} + {\left (16 \, {\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} d f + 3 \, {\left (5 \, b^{5} - 24 \, a b^{3} c + 16 \, a^{2} b c^{2}\right )} f^{2}\right )} x + 8 \, {\left (a b^{2} c^{2} - 4 \, a^{2} c^{3} + {\left (b^{2} c^{3} - 4 \, a c^{4}\right )} x^{2} + {\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} x\right )} e^{2} - 24 \, {\left ({\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} f x^{2} + {\left (b^{4} c - 4 \, a b^{2} c^{2}\right )} f x + {\left (a b^{3} c - 4 \, a^{2} b c^{2}\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 (8 \, b c^{4} d^{2} + 16 \, a b c^{3} d f - 2 \, {\left (b^{2} c^{3} - 4 \, a c^{4}\right )} f^{2} x^{3} + 5 \, {\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} f^{2} x^{2} + {\left (15 \, a b^{3} c - 52 \, a^{2} b c^{2}\right )} f^{2} + {\left (16 \, c^{5} d^{2} + 16 \, {\left (b^{2} c^{3} - 2 \, a c^{4}\right )} d f + {\left (15 \, b^{4} c - 62 \, a b^{2} c^{2} + 24 \, a^{2} c^{3}\right )} f^{2}\right )} x + 8 \, {\left (a b c^{3} + {\left (b^{2} c^{3} - 2 \, a c^{4}\right )} x\right )} e^{2} - 8 \, {\left (4 \, a c^{4} d + {\left (b^{2} c^{3} - 4 \, a c^{4}\right )} f x^{2} + {\left (3 \, a b^{2} c^{2} - 8 \, a^{2} c^{3}\right )} f + {\left (2 \, b c^{4} d + {\left (3 \, b^{3} c^{2} - 10 \, a b c^{3}\right )} f\right )} x\right )} e\right )} \sqrt {c x^{2} + b x + a}}{8 \, {\left (a b^{2} c^{4} - 4 \, a^{2} c^{5} + {\left (b^{2} c^{5} - 4 \, a c^{6}\right )} x^{2} + {\left (b^{3} c^{4} - 4 \, a b c^{5}\right )} x\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x^2+e*x+d)^2/(c*x^2+b*x+a)^(3/2),x, algorithm="fricas")

[Out]

[1/16*((16*(a*b^2*c^2 - 4*a^2*c^3)*d*f + 3*(5*a*b^4 - 24*a^2*b^2*c + 16*a^3*c^2)*f^2 + (16*(b^2*c^3 - 4*a*c^4)
*d*f + 3*(5*b^4*c - 24*a*b^2*c^2 + 16*a^2*c^3)*f^2)*x^2 + (16*(b^3*c^2 - 4*a*b*c^3)*d*f + 3*(5*b^5 - 24*a*b^3*
c + 16*a^2*b*c^2)*f^2)*x + 8*(a*b^2*c^2 - 4*a^2*c^3 + (b^2*c^3 - 4*a*c^4)*x^2 + (b^3*c^2 - 4*a*b*c^3)*x)*e^2 -
24*((b^3*c^2 - 4*a*b*c^3)*f*x^2 + (b^4*c - 4*a*b^2*c^2)*f*x + (a*b^3*c - 4*a^2*b*c^2)*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*(8*b*c^4*d^2 + 16*a*b*c^3*d*f
- 2*(b^2*c^3 - 4*a*c^4)*f^2*x^3 + 5*(b^3*c^2 - 4*a*b*c^3)*f^2*x^2 + (15*a*b^3*c - 52*a^2*b*c^2)*f^2 + (16*c^5
*d^2 + 16*(b^2*c^3 - 2*a*c^4)*d*f + (15*b^4*c - 62*a*b^2*c^2 + 24*a^2*c^3)*f^2)*x + 8*(a*b*c^3 + (b^2*c^3 - 2*
a*c^4)*x)*e^2 - 8*(4*a*c^4*d + (b^2*c^3 - 4*a*c^4)*f*x^2 + (3*a*b^2*c^2 - 8*a^2*c^3)*f + (2*b*c^4*d + (3*b^3*c
^2 - 10*a*b*c^3)*f)*x)*e)*sqrt(c*x^2 + b*x + a))/(a*b^2*c^4 - 4*a^2*c^5 + (b^2*c^5 - 4*a*c^6)*x^2 + (b^3*c^4 -
4*a*b*c^5)*x), -1/8*((16*(a*b^2*c^2 - 4*a^2*c^3)*d*f + 3*(5*a*b^4 - 24*a^2*b^2*c + 16*a^3*c^2)*f^2 + (16*(b^2
*c^3 - 4*a*c^4)*d*f + 3*(5*b^4*c - 24*a*b^2*c^2 + 16*a^2*c^3)*f^2)*x^2 + (16*(b^3*c^2 - 4*a*b*c^3)*d*f + 3*(5*
b^5 - 24*a*b^3*c + 16*a^2*b*c^2)*f^2)*x + 8*(a*b^2*c^2 - 4*a^2*c^3 + (b^2*c^3 - 4*a*c^4)*x^2 + (b^3*c^2 - 4*a*
b*c^3)*x)*e^2 - 24*((b^3*c^2 - 4*a*b*c^3)*f*x^2 + (b^4*c - 4*a*b^2*c^2)*f*x + (a*b^3*c - 4*a^2*b*c^2)*f)*e)*sq
rt(-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*(8*b*c^4*d^2 + 16*a*
b*c^3*d*f - 2*(b^2*c^3 - 4*a*c^4)*f^2*x^3 + 5*(b^3*c^2 - 4*a*b*c^3)*f^2*x^2 + (15*a*b^3*c - 52*a^2*b*c^2)*f^2
+ (16*c^5*d^2 + 16*(b^2*c^3 - 2*a*c^4)*d*f + (15*b^4*c - 62*a*b^2*c^2 + 24*a^2*c^3)*f^2)*x + 8*(a*b*c^3 + (b^2
*c^3 - 2*a*c^4)*x)*e^2 - 8*(4*a*c^4*d + (b^2*c^3 - 4*a*c^4)*f*x^2 + (3*a*b^2*c^2 - 8*a^2*c^3)*f + (2*b*c^4*d +
(3*b^3*c^2 - 10*a*b*c^3)*f)*x)*e)*sqrt(c*x^2 + b*x + a))/(a*b^2*c^4 - 4*a^2*c^5 + (b^2*c^5 - 4*a*c^6)*x^2 + (
b^3*c^4 - 4*a*b*c^5)*x)]

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (d + e x + f x^{2}\right )^{2}}{\left (a + b x + c x^{2}\right )^{\frac {3}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x**2+e*x+d)**2/(c*x**2+b*x+a)**(3/2),x)

[Out]

Integral((d + e*x + f*x**2)**2/(a + b*x + c*x**2)**(3/2), x)

________________________________________________________________________________________

Giac [A]
time = 6.40, size = 407, normalized size = 1.32 \begin {gather*} \frac {{\left ({\left (\frac {2 \, {\left (b^{2} c^{2} f^{2} - 4 \, a c^{3} f^{2}\right )} x}{b^{2} c^{3} - 4 \, a c^{4}} - \frac {5 \, b^{3} c f^{2} - 20 \, a b c^{2} f^{2} - 8 \, b^{2} c^{2} f e + 32 \, a c^{3} f e}{b^{2} c^{3} - 4 \, a c^{4}}\right )} x - \frac {16 \, c^{4} d^{2} + 16 \, b^{2} c^{2} d f - 32 \, a c^{3} d f + 15 \, b^{4} f^{2} - 62 \, a b^{2} c f^{2} + 24 \, a^{2} c^{2} f^{2} - 16 \, b c^{3} d e - 24 \, b^{3} c f e + 80 \, a b c^{2} f e + 8 \, b^{2} c^{2} e^{2} - 16 \, a c^{3} e^{2}}{b^{2} c^{3} - 4 \, a c^{4}}\right )} x - \frac {8 \, b c^{3} d^{2} + 16 \, a b c^{2} d f + 15 \, a b^{3} f^{2} - 52 \, a^{2} b c f^{2} - 32 \, a c^{3} d e - 24 \, a b^{2} c f e + 64 \, a^{2} c^{2} f e + 8 \, a b c^{2} e^{2}}{b^{2} c^{3} - 4 \, a c^{4}}}{4 \, \sqrt {c x^{2} + b x + a}} - \frac {{\left (16 \, c^{2} d f + 15 \, b^{2} f^{2} - 12 \, a c f^{2} - 24 \, b c f e + 8 \, c^{2} e^{2}\right )} \log \left ({\left | -2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} - b \right |}\right )}{8 \, c^{\frac {7}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x^2+e*x+d)^2/(c*x^2+b*x+a)^(3/2),x, algorithm="giac")

[Out]

1/4*(((2*(b^2*c^2*f^2 - 4*a*c^3*f^2)*x/(b^2*c^3 - 4*a*c^4) - (5*b^3*c*f^2 - 20*a*b*c^2*f^2 - 8*b^2*c^2*f*e + 3
2*a*c^3*f*e)/(b^2*c^3 - 4*a*c^4))*x - (16*c^4*d^2 + 16*b^2*c^2*d*f - 32*a*c^3*d*f + 15*b^4*f^2 - 62*a*b^2*c*f^
2 + 24*a^2*c^2*f^2 - 16*b*c^3*d*e - 24*b^3*c*f*e + 80*a*b*c^2*f*e + 8*b^2*c^2*e^2 - 16*a*c^3*e^2)/(b^2*c^3 - 4
*a*c^4))*x - (8*b*c^3*d^2 + 16*a*b*c^2*d*f + 15*a*b^3*f^2 - 52*a^2*b*c*f^2 - 32*a*c^3*d*e - 24*a*b^2*c*f*e + 6
4*a^2*c^2*f*e + 8*a*b*c^2*e^2)/(b^2*c^3 - 4*a*c^4))/sqrt(c*x^2 + b*x + a) - 1/8*(16*c^2*d*f + 15*b^2*f^2 - 12*
a*c*f^2 - 24*b*c*f*e + 8*c^2*e^2)*log(abs(-2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*sqrt(c) - b))/c^(7/2)

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (f\,x^2+e\,x+d\right )}^2}{{\left (c\,x^2+b\,x+a\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x + f*x^2)^2/(a + b*x + c*x^2)^(3/2),x)

[Out]

int((d + e*x + f*x^2)^2/(a + b*x + c*x^2)^(3/2), x)

________________________________________________________________________________________