3.1374 \(\int \frac{(b d+2 c d x)^{5/2}}{\left (a+b x+c x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=219 \[ -\frac{12 d^{5/2} \left (b^2-4 a c\right )^{3/4} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{\sqrt{a+b x+c x^2}}+\frac{12 d^{5/2} \left (b^2-4 a c\right )^{3/4} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{\sqrt{a+b x+c x^2}}-\frac{2 d (b d+2 c d x)^{3/2}}{\sqrt{a+b x+c x^2}} \]

[Out]

(-2*d*(b*d + 2*c*d*x)^(3/2))/Sqrt[a + b*x + c*x^2] + (12*(b^2 - 4*a*c)^(3/4)*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])/Sqrt[a + b*x + c*x^2] - (12*(b^2 - 4
*a*c)^(3/4)*d^(5/2)*Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))]*EllipticF[ArcSi
n[Sqrt[b*d + 2*c*d*x]/((b^2 - 4*a*c)^(1/4)*Sqrt[d])], -1])/Sqrt[a + b*x + c*x^2]

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Rubi [A]  time = 0.708007, antiderivative size = 219, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{12 d^{5/2} \left (b^2-4 a c\right )^{3/4} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{\sqrt{a+b x+c x^2}}+\frac{12 d^{5/2} \left (b^2-4 a c\right )^{3/4} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{\sqrt{a+b x+c x^2}}-\frac{2 d (b d+2 c d x)^{3/2}}{\sqrt{a+b x+c x^2}} \]

Antiderivative was successfully verified.

[In]  Int[(b*d + 2*c*d*x)^(5/2)/(a + b*x + c*x^2)^(3/2),x]

[Out]

(-2*d*(b*d + 2*c*d*x)^(3/2))/Sqrt[a + b*x + c*x^2] + (12*(b^2 - 4*a*c)^(3/4)*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])/Sqrt[a + b*x + c*x^2] - (12*(b^2 - 4
*a*c)^(3/4)*d^(5/2)*Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))]*EllipticF[ArcSi
n[Sqrt[b*d + 2*c*d*x]/((b^2 - 4*a*c)^(1/4)*Sqrt[d])], -1])/Sqrt[a + b*x + c*x^2]

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Rubi in Sympy [A]  time = 126.702, size = 212, normalized size = 0.97 \[ \frac{12 d^{\frac{5}{2}} \sqrt{\frac{c \left (a + b x + c x^{2}\right )}{4 a c - b^{2}}} \left (- 4 a c + b^{2}\right )^{\frac{3}{4}} E\left (\operatorname{asin}{\left (\frac{\sqrt{b d + 2 c d x}}{\sqrt{d} \sqrt [4]{- 4 a c + b^{2}}} \right )}\middle | -1\right )}{\sqrt{a + b x + c x^{2}}} - \frac{12 d^{\frac{5}{2}} \sqrt{\frac{c \left (a + b x + c x^{2}\right )}{4 a c - b^{2}}} \left (- 4 a c + b^{2}\right )^{\frac{3}{4}} F\left (\operatorname{asin}{\left (\frac{\sqrt{b d + 2 c d x}}{\sqrt{d} \sqrt [4]{- 4 a c + b^{2}}} \right )}\middle | -1\right )}{\sqrt{a + b x + c x^{2}}} - \frac{2 d \left (b d + 2 c d x\right )^{\frac{3}{2}}}{\sqrt{a + b x + c x^{2}}} \]

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)**(3/2),x)

[Out]

12*d**(5/2)*sqrt(c*(a + b*x + c*x**2)/(4*a*c - b**2))*(-4*a*c + b**2)**(3/4)*ell
iptic_e(asin(sqrt(b*d + 2*c*d*x)/(sqrt(d)*(-4*a*c + b**2)**(1/4))), -1)/sqrt(a +
b*x + c*x**2) - 12*d**(5/2)*sqrt(c*(a + b*x + c*x**2)/(4*a*c - b**2))*(-4*a*c +
b**2)**(3/4)*elliptic_f(asin(sqrt(b*d + 2*c*d*x)/(sqrt(d)*(-4*a*c + b**2)**(1/4
))), -1)/sqrt(a + b*x + c*x**2) - 2*d*(b*d + 2*c*d*x)**(3/2)/sqrt(a + b*x + c*x*
*2)

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Mathematica [C]  time = 1.6777, size = 229, normalized size = 1.05 \[ \frac{2 i d^3 \left (-6 \left (b^2-4 a c\right ) \sqrt{-\frac{b+2 c x}{\sqrt{b^2-4 a c}}} \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}} F\left (\left .i \sinh ^{-1}\left (\sqrt{-\frac{b+2 c x}{\sqrt{b^2-4 a c}}}\right )\right |-1\right )+6 \left (b^2-4 a c\right ) \sqrt{-\frac{b+2 c x}{\sqrt{b^2-4 a c}}} \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}} E\left (\left .i \sinh ^{-1}\left (\sqrt{-\frac{b+2 c x}{\sqrt{b^2-4 a c}}}\right )\right |-1\right )+i (b+2 c x)^2\right )}{\sqrt{a+x (b+c x)} \sqrt{d (b+2 c x)}} \]

Antiderivative was successfully verified.

[In]  Integrate[(b*d + 2*c*d*x)^(5/2)/(a + b*x + c*x^2)^(3/2),x]

[Out]

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

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Maple [A]  time = 0.03, size = 323, normalized size = 1.5 \[ 2\,{\frac{{d}^{2}\sqrt{c{x}^{2}+bx+a}\sqrt{d \left ( 2\,cx+b \right ) }}{2\,{x}^{3}{c}^{2}+3\,{x}^{2}bc+2\,acx+{b}^{2}x+ab} \left ( 12\,{\it EllipticE} \left ( 1/2\,\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{2},\sqrt{2} \right ) ac\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{-{\frac{2\,cx+b}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{{\frac{-b-2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}-3\,{\it EllipticE} \left ( 1/2\,\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{2},\sqrt{2} \right ){b}^{2}\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{-{\frac{2\,cx+b}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{{\frac{-b-2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}-4\,{c}^{2}{x}^{2}-4\,bxc-{b}^{2} \right ) } \]

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)^(3/2),x)

[Out]

2*(12*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*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)-3*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))*b^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)-4*c^2*x^2-4*b*x*c-b^2)*d^2*(c*x^2+b*x+a)^(1/2)*(d*(2*c*x+b))^(1/2)/(
2*c^2*x^3+3*b*c*x^2+2*a*c*x+b^2*x+a*b)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (2 \, c d x + b d\right )}^{\frac{5}{2}}}{{\left (c x^{2} + b x + a\right )}^{\frac{3}{2}}}\,{d x} \]

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)^(3/2),x, algorithm="maxima")

[Out]

integrate((2*c*d*x + b*d)^(5/2)/(c*x^2 + b*x + a)^(3/2), x)

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

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)^(3/2),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*x^2 + b*
x + a)^(3/2), 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)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (2 \, c d x + b d\right )}^{\frac{5}{2}}}{{\left (c x^{2} + b x + a\right )}^{\frac{3}{2}}}\,{d x} \]

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)^(3/2),x, algorithm="giac")

[Out]

integrate((2*c*d*x + b*d)^(5/2)/(c*x^2 + b*x + a)^(3/2), x)