3.1386 \(\int \frac{(b d+2 c d x)^{5/2}}{\left (a+b x+c x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=264 \[ -\frac{8 c d^{5/2} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{\sqrt [4]{b^2-4 a c} \sqrt{a+b x+c x^2}}+\frac{8 c d^{5/2} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{\sqrt [4]{b^2-4 a c} \sqrt{a+b x+c x^2}}-\frac{4 c d (b d+2 c d x)^{3/2}}{\left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}}-\frac{2 d (b d+2 c d x)^{3/2}}{3 \left (a+b x+c x^2\right )^{3/2}} \]

[Out]

(-2*d*(b*d + 2*c*d*x)^(3/2))/(3*(a + b*x + c*x^2)^(3/2)) - (4*c*d*(b*d + 2*c*d*x
)^(3/2))/((b^2 - 4*a*c)*Sqrt[a + b*x + c*x^2]) + (8*c*d^(5/2)*Sqrt[-((c*(a + b*x
+ c*x^2))/(b^2 - 4*a*c))]*EllipticE[ArcSin[Sqrt[b*d + 2*c*d*x]/((b^2 - 4*a*c)^(
1/4)*Sqrt[d])], -1])/((b^2 - 4*a*c)^(1/4)*Sqrt[a + b*x + c*x^2]) - (8*c*d^(5/2)*
Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))]*EllipticF[ArcSin[Sqrt[b*d + 2*c*d*x
]/((b^2 - 4*a*c)^(1/4)*Sqrt[d])], -1])/((b^2 - 4*a*c)^(1/4)*Sqrt[a + b*x + c*x^2
])

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Rubi [A]  time = 0.81645, antiderivative size = 264, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286 \[ -\frac{8 c d^{5/2} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{\sqrt [4]{b^2-4 a c} \sqrt{a+b x+c x^2}}+\frac{8 c d^{5/2} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{\sqrt [4]{b^2-4 a c} \sqrt{a+b x+c x^2}}-\frac{4 c d (b d+2 c d x)^{3/2}}{\left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}}-\frac{2 d (b d+2 c d x)^{3/2}}{3 \left (a+b x+c x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[(b*d + 2*c*d*x)^(5/2)/(a + b*x + c*x^2)^(5/2),x]

[Out]

(-2*d*(b*d + 2*c*d*x)^(3/2))/(3*(a + b*x + c*x^2)^(3/2)) - (4*c*d*(b*d + 2*c*d*x
)^(3/2))/((b^2 - 4*a*c)*Sqrt[a + b*x + c*x^2]) + (8*c*d^(5/2)*Sqrt[-((c*(a + b*x
+ c*x^2))/(b^2 - 4*a*c))]*EllipticE[ArcSin[Sqrt[b*d + 2*c*d*x]/((b^2 - 4*a*c)^(
1/4)*Sqrt[d])], -1])/((b^2 - 4*a*c)^(1/4)*Sqrt[a + b*x + c*x^2]) - (8*c*d^(5/2)*
Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))]*EllipticF[ArcSin[Sqrt[b*d + 2*c*d*x
]/((b^2 - 4*a*c)^(1/4)*Sqrt[d])], -1])/((b^2 - 4*a*c)^(1/4)*Sqrt[a + b*x + c*x^2
])

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Rubi in Sympy [A]  time = 142.685, size = 258, normalized size = 0.98 \[ \frac{8 c d^{\frac{5}{2}} \sqrt{\frac{c \left (a + b x + c x^{2}\right )}{4 a c - b^{2}}} E\left (\operatorname{asin}{\left (\frac{\sqrt{b d + 2 c d x}}{\sqrt{d} \sqrt [4]{- 4 a c + b^{2}}} \right )}\middle | -1\right )}{\sqrt [4]{- 4 a c + b^{2}} \sqrt{a + b x + c x^{2}}} - \frac{8 c d^{\frac{5}{2}} \sqrt{\frac{c \left (a + b x + c x^{2}\right )}{4 a c - b^{2}}} F\left (\operatorname{asin}{\left (\frac{\sqrt{b d + 2 c d x}}{\sqrt{d} \sqrt [4]{- 4 a c + b^{2}}} \right )}\middle | -1\right )}{\sqrt [4]{- 4 a c + b^{2}} \sqrt{a + b x + c x^{2}}} - \frac{4 c d \left (b d + 2 c d x\right )^{\frac{3}{2}}}{\left (- 4 a c + b^{2}\right ) \sqrt{a + b x + c x^{2}}} - \frac{2 d \left (b d + 2 c d x\right )^{\frac{3}{2}}}{3 \left (a + b x + c x^{2}\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((2*c*d*x+b*d)**(5/2)/(c*x**2+b*x+a)**(5/2),x)

[Out]

8*c*d**(5/2)*sqrt(c*(a + b*x + c*x**2)/(4*a*c - b**2))*elliptic_e(asin(sqrt(b*d
+ 2*c*d*x)/(sqrt(d)*(-4*a*c + b**2)**(1/4))), -1)/((-4*a*c + b**2)**(1/4)*sqrt(a
+ b*x + c*x**2)) - 8*c*d**(5/2)*sqrt(c*(a + b*x + c*x**2)/(4*a*c - b**2))*ellip
tic_f(asin(sqrt(b*d + 2*c*d*x)/(sqrt(d)*(-4*a*c + b**2)**(1/4))), -1)/((-4*a*c +
b**2)**(1/4)*sqrt(a + b*x + c*x**2)) - 4*c*d*(b*d + 2*c*d*x)**(3/2)/((-4*a*c +
b**2)*sqrt(a + b*x + c*x**2)) - 2*d*(b*d + 2*c*d*x)**(3/2)/(3*(a + b*x + c*x**2)
**(3/2))

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Mathematica [C]  time = 2.1194, size = 218, normalized size = 0.83 \[ \frac{(d (b+2 c x))^{5/2} \left (-\frac{2 (a+x (b+c x)) \left (2 c \left (a+3 c x^2\right )+b^2+6 b c x\right )}{3 \left (b^2-4 a c\right ) (b+2 c x)}+\frac{8 i c \sqrt{-\frac{b+2 c x}{\sqrt{b^2-4 a c}}} (a+x (b+c x))^2 \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}} \left (E\left (\left .i \sinh ^{-1}\left (\sqrt{-\frac{b+2 c x}{\sqrt{b^2-4 a c}}}\right )\right |-1\right )-F\left (\left .i \sinh ^{-1}\left (\sqrt{-\frac{b+2 c x}{\sqrt{b^2-4 a c}}}\right )\right |-1\right )\right )}{(b+2 c x)^3}\right )}{(a+x (b+c x))^{5/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(b*d + 2*c*d*x)^(5/2)/(a + b*x + c*x^2)^(5/2),x]

[Out]

((d*(b + 2*c*x))^(5/2)*((-2*(a + x*(b + c*x))*(b^2 + 6*b*c*x + 2*c*(a + 3*c*x^2)
))/(3*(b^2 - 4*a*c)*(b + 2*c*x)) + ((8*I)*c*Sqrt[-((b + 2*c*x)/Sqrt[b^2 - 4*a*c]
)]*(a + x*(b + c*x))^2*Sqrt[(c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)]*(EllipticE[I*A
rcSinh[Sqrt[-((b + 2*c*x)/Sqrt[b^2 - 4*a*c])]], -1] - EllipticF[I*ArcSinh[Sqrt[-
((b + 2*c*x)/Sqrt[b^2 - 4*a*c])]], -1]))/(b + 2*c*x)^3))/(a + x*(b + c*x))^(5/2)

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Maple [B]  time = 0.038, size = 871, normalized size = 3.3 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2*c*d*x+b*d)^(5/2)/(c*x^2+b*x+a)^(5/2),x)

[Out]

-2/3*(d*(2*c*x+b))^(1/2)*(24*EllipticE(1/2*((b+2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c
+b^2)^(1/2))^(1/2)*2^(1/2),2^(1/2))*x^2*a*c^3*((b+2*c*x+(-4*a*c+b^2)^(1/2))/(-4*
a*c+b^2)^(1/2))^(1/2)*(-(2*c*x+b)/(-4*a*c+b^2)^(1/2))^(1/2)*((-b-2*c*x+(-4*a*c+b
^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)-6*EllipticE(1/2*((b+2*c*x+(-4*a*c+b^2)^(1/2
))/(-4*a*c+b^2)^(1/2))^(1/2)*2^(1/2),2^(1/2))*x^2*b^2*c^2*((b+2*c*x+(-4*a*c+b^2)
^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*(-(2*c*x+b)/(-4*a*c+b^2)^(1/2))^(1/2)*((-b-2*c
*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)+24*EllipticE(1/2*((b+2*c*x+(-4*
a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*2^(1/2),2^(1/2))*x*a*b*c^2*((b+2*c*x+(
-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*(-(2*c*x+b)/(-4*a*c+b^2)^(1/2))^(1/
2)*((-b-2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)-6*EllipticE(1/2*((b+
2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*2^(1/2),2^(1/2))*x*b^3*c*((b
+2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*(-(2*c*x+b)/(-4*a*c+b^2)^(1
/2))^(1/2)*((-b-2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)+24*((b+2*c*x
+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*(-(2*c*x+b)/(-4*a*c+b^2)^(1/2))^(
1/2)*((-b-2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*EllipticE(1/2*((b+
2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*2^(1/2),2^(1/2))*a^2*c^2-6*(
(b+2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*(-(2*c*x+b)/(-4*a*c+b^2)^
(1/2))^(1/2)*((-b-2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*EllipticE(
1/2*((b+2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*2^(1/2),2^(1/2))*a*b
^2*c-24*c^4*x^4-48*b*c^3*x^3-8*x^2*a*c^3-34*x^2*b^2*c^2-8*x*a*b*c^2-10*b^3*c*x-2
*a*c*b^2-b^4)*d^2/(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (2 \, c d x + b d\right )}^{\frac{5}{2}}}{{\left (c x^{2} + b x + a\right )}^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*d*x + b*d)^(5/2)/(c*x^2 + b*x + a)^(5/2),x, algorithm="maxima")

[Out]

integrate((2*c*d*x + b*d)^(5/2)/(c*x^2 + b*x + a)^(5/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (4 \, c^{2} d^{2} x^{2} + 4 \, b c d^{2} x + b^{2} d^{2}\right )} \sqrt{2 \, c d x + b d}}{{\left (c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x +{\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}\right )} \sqrt{c x^{2} + b x + a}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*d*x + b*d)^(5/2)/(c*x^2 + b*x + a)^(5/2),x, algorithm="fricas")

[Out]

integral((4*c^2*d^2*x^2 + 4*b*c*d^2*x + b^2*d^2)*sqrt(2*c*d*x + b*d)/((c^2*x^4 +
2*b*c*x^3 + 2*a*b*x + (b^2 + 2*a*c)*x^2 + a^2)*sqrt(c*x^2 + b*x + a)), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*d*x+b*d)**(5/2)/(c*x**2+b*x+a)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (2 \, c d x + b d\right )}^{\frac{5}{2}}}{{\left (c x^{2} + b x + a\right )}^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*d*x + b*d)^(5/2)/(c*x^2 + b*x + a)^(5/2),x, algorithm="giac")

[Out]

integrate((2*c*d*x + b*d)^(5/2)/(c*x^2 + b*x + a)^(5/2), x)