3.1381 \(\int \frac{(b d+2 c d x)^{3/2}}{\left (a+b x+c x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=173 \[ -\frac{8 c d^{3/2} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{3 \left (b^2-4 a c\right )^{3/4} \sqrt{a+b x+c x^2}}-\frac{4 c d \sqrt{b d+2 c d x}}{3 \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}}-\frac{2 d \sqrt{b d+2 c d x}}{3 \left (a+b x+c x^2\right )^{3/2}} \]

[Out]

(-2*d*Sqrt[b*d + 2*c*d*x])/(3*(a + b*x + c*x^2)^(3/2)) - (4*c*d*Sqrt[b*d + 2*c*d
*x])/(3*(b^2 - 4*a*c)*Sqrt[a + b*x + c*x^2]) - (8*c*d^(3/2)*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])/(3*(b^2 - 4*a*c)^(3/4)*Sqrt[a + b*x + c*x^2])

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Rubi [A]  time = 0.418829, antiderivative size = 173, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179 \[ -\frac{8 c d^{3/2} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{3 \left (b^2-4 a c\right )^{3/4} \sqrt{a+b x+c x^2}}-\frac{4 c d \sqrt{b d+2 c d x}}{3 \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}}-\frac{2 d \sqrt{b d+2 c d x}}{3 \left (a+b x+c x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*d*Sqrt[b*d + 2*c*d*x])/(3*(a + b*x + c*x^2)^(3/2)) - (4*c*d*Sqrt[b*d + 2*c*d
*x])/(3*(b^2 - 4*a*c)*Sqrt[a + b*x + c*x^2]) - (8*c*d^(3/2)*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])/(3*(b^2 - 4*a*c)^(3/4)*Sqrt[a + b*x + c*x^2])

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Rubi in Sympy [A]  time = 91.8256, size = 170, normalized size = 0.98 \[ - \frac{8 c d^{\frac{3}{2}} \sqrt{\frac{c \left (a + b x + c x^{2}\right )}{4 a c - b^{2}}} F\left (\operatorname{asin}{\left (\frac{\sqrt{b d + 2 c d x}}{\sqrt{d} \sqrt [4]{- 4 a c + b^{2}}} \right )}\middle | -1\right )}{3 \left (- 4 a c + b^{2}\right )^{\frac{3}{4}} \sqrt{a + b x + c x^{2}}} - \frac{4 c d \sqrt{b d + 2 c d x}}{3 \left (- 4 a c + b^{2}\right ) \sqrt{a + b x + c x^{2}}} - \frac{2 d \sqrt{b d + 2 c d x}}{3 \left (a + b x + c x^{2}\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-8*c*d**(3/2)*sqrt(c*(a + b*x + c*x**2)/(4*a*c - b**2))*elliptic_f(asin(sqrt(b*d
+ 2*c*d*x)/(sqrt(d)*(-4*a*c + b**2)**(1/4))), -1)/(3*(-4*a*c + b**2)**(3/4)*sqr
t(a + b*x + c*x**2)) - 4*c*d*sqrt(b*d + 2*c*d*x)/(3*(-4*a*c + b**2)*sqrt(a + b*x
+ c*x**2)) - 2*d*sqrt(b*d + 2*c*d*x)/(3*(a + b*x + c*x**2)**(3/2))

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Mathematica [C]  time = 0.758028, size = 168, normalized size = 0.97 \[ -\frac{2 d \sqrt{d (b+2 c x)} \left (\sqrt{b^2-4 a c} \left (2 c \left (c x^2-a\right )+b^2+2 b c x\right )-4 i \sqrt{-\sqrt{b^2-4 a c}} (b+2 c x)^{5/2} \left (\frac{c (a+x (b+c x))}{(b+2 c x)^2}\right )^{3/2} F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{-\sqrt{b^2-4 a c}}}{\sqrt{b+2 c x}}\right )\right |-1\right )\right )}{3 \left (b^2-4 a c\right )^{3/2} (a+x (b+c x))^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.069, size = 487, normalized size = 2.8 \[{\frac{2\,d}{ \left ( 12\,ac-3\,{b}^{2} \right ) \left ( 2\,cx+b \right ) } \left ( 2\,{\it EllipticF} \left ( 1/2\,\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{2},\sqrt{2} \right ){x}^{2}{c}^{2}\sqrt{{\frac{-b-2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{-4\,ac+{b}^{2}}\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{-{\frac{2\,cx+b}{\sqrt{-4\,ac+{b}^{2}}}}}+2\,{\it EllipticF} \left ( 1/2\,\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{2},\sqrt{2} \right ) xbc\sqrt{{\frac{-b-2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{-4\,ac+{b}^{2}}\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{-{\frac{2\,cx+b}{\sqrt{-4\,ac+{b}^{2}}}}}+2\,\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{-{\frac{2\,cx+b}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{{\frac{-b-2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}{\it EllipticF} \left ( 1/2\,\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{2},\sqrt{2} \right ) \sqrt{-4\,ac+{b}^{2}}ac+4\,{c}^{3}{x}^{3}+6\,b{c}^{2}{x}^{2}-4\,a{c}^{2}x+4\,x{b}^{2}c-2\,abc+{b}^{3} \right ) \sqrt{d \left ( 2\,cx+b \right ) } \left ( c{x}^{2}+bx+a \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*(2*EllipticF(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*c^2*((-b-2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*(
-4*a*c+b^2)^(1/2)*((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)+2*EllipticF(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*c*((-b-2*c*x+(-4*a*c+b^2)^(1/2))/(-
4*a*c+b^2)^(1/2))^(1/2)*(-4*a*c+b^2)^(1/2)*((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)+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)*EllipticF(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))*(-4*a*c+b^2)^(1/2)*a*c+
4*c^3*x^3+6*b*c^2*x^2-4*a*c^2*x+4*x*b^2*c-2*a*b*c+b^3)*d*(d*(2*c*x+b))^(1/2)/(4*
a*c-b^2)/(2*c*x+b)/(c*x^2+b*x+a)^(3/2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*d*x + b*d)^(3/2)/(c*x^2 + b*x + a)^(5/2),x, algorithm="maxima")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*d*x + b*d)^(3/2)/(c*x^2 + b*x + a)^(5/2),x, algorithm="fricas")

[Out]

integral((2*c*d*x + b*d)^(3/2)/((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)**(3/2)/(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. \[ \int \frac{{\left (2 \, c d x + b d\right )}^{\frac{3}{2}}}{{\left (c x^{2} + b x + a\right )}^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*d*x + b*d)^(3/2)/(c*x^2 + b*x + a)^(5/2),x, algorithm="giac")

[Out]

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