3.131 \(\int \frac{x^3 (A+B x)}{\left (b x+c x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=84 \[ -\frac{2 x^3 (b B-A c)}{3 b c \left (b x+c x^2\right )^{3/2}}+\frac{2 B \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{c^{5/2}}-\frac{2 B x}{c^2 \sqrt{b x+c x^2}} \]

[Out]

(-2*(b*B - A*c)*x^3)/(3*b*c*(b*x + c*x^2)^(3/2)) - (2*B*x)/(c^2*Sqrt[b*x + c*x^2
]) + (2*B*ArcTanh[(Sqrt[c]*x)/Sqrt[b*x + c*x^2]])/c^(5/2)

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Rubi [A]  time = 0.208617, antiderivative size = 84, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ -\frac{2 x^3 (b B-A c)}{3 b c \left (b x+c x^2\right )^{3/2}}+\frac{2 B \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{c^{5/2}}-\frac{2 B x}{c^2 \sqrt{b x+c x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(b*B - A*c)*x^3)/(3*b*c*(b*x + c*x^2)^(3/2)) - (2*B*x)/(c^2*Sqrt[b*x + c*x^2
]) + (2*B*ArcTanh[(Sqrt[c]*x)/Sqrt[b*x + c*x^2]])/c^(5/2)

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Rubi in Sympy [A]  time = 12.7894, size = 76, normalized size = 0.9 \[ - \frac{2 B x}{c^{2} \sqrt{b x + c x^{2}}} + \frac{2 B \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{b x + c x^{2}}} \right )}}{c^{\frac{5}{2}}} + \frac{2 x^{3} \left (A c - B b\right )}{3 b c \left (b x + c x^{2}\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*B*x/(c**2*sqrt(b*x + c*x**2)) + 2*B*atanh(sqrt(c)*x/sqrt(b*x + c*x**2))/c**(5
/2) + 2*x**3*(A*c - B*b)/(3*b*c*(b*x + c*x**2)**(3/2))

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Mathematica [A]  time = 0.12108, size = 94, normalized size = 1.12 \[ \frac{x \left (2 \sqrt{c} x \left (A c^2 x-3 b^2 B-4 b B c x\right )+6 b B \sqrt{x} (b+c x)^{3/2} \log \left (\sqrt{c} \sqrt{b+c x}+c \sqrt{x}\right )\right )}{3 b c^{5/2} (x (b+c x))^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(x*(2*Sqrt[c]*x*(-3*b^2*B - 4*b*B*c*x + A*c^2*x) + 6*b*B*Sqrt[x]*(b + c*x)^(3/2)
*Log[c*Sqrt[x] + Sqrt[c]*Sqrt[b + c*x]]))/(3*b*c^(5/2)*(x*(b + c*x))^(3/2))

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Maple [B]  time = 0.01, size = 206, normalized size = 2.5 \[ -{\frac{A{x}^{2}}{c} \left ( c{x}^{2}+bx \right ) ^{-{\frac{3}{2}}}}-{\frac{Abx}{3\,{c}^{2}} \left ( c{x}^{2}+bx \right ) ^{-{\frac{3}{2}}}}+{\frac{2\,Ax}{3\,bc}{\frac{1}{\sqrt{c{x}^{2}+bx}}}}+{\frac{A}{3\,{c}^{2}}{\frac{1}{\sqrt{c{x}^{2}+bx}}}}-{\frac{{x}^{3}B}{3\,c} \left ( c{x}^{2}+bx \right ) ^{-{\frac{3}{2}}}}+{\frac{Bb{x}^{2}}{2\,{c}^{2}} \left ( c{x}^{2}+bx \right ) ^{-{\frac{3}{2}}}}+{\frac{{b}^{2}Bx}{6\,{c}^{3}} \left ( c{x}^{2}+bx \right ) ^{-{\frac{3}{2}}}}-{\frac{7\,Bx}{3\,{c}^{2}}{\frac{1}{\sqrt{c{x}^{2}+bx}}}}-{\frac{Bb}{6\,{c}^{3}}{\frac{1}{\sqrt{c{x}^{2}+bx}}}}+{B\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-A*x^2/c/(c*x^2+b*x)^(3/2)-1/3*A*b/c^2/(c*x^2+b*x)^(3/2)*x+2/3*A/b/c/(c*x^2+b*x)
^(1/2)*x+1/3*A/c^2/(c*x^2+b*x)^(1/2)-1/3*B*x^3/c/(c*x^2+b*x)^(3/2)+1/2*B*b/c^2*x
^2/(c*x^2+b*x)^(3/2)+1/6*B*b^2/c^3/(c*x^2+b*x)^(3/2)*x-7/3*B*x/c^2/(c*x^2+b*x)^(
1/2)-1/6*B*b/c^3/(c*x^2+b*x)^(1/2)+B/c^(5/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x)^
(1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.315349, size = 1, normalized size = 0.01 \[ \left [\frac{3 \,{\left (B b c x + B b^{2}\right )} \sqrt{c x^{2} + b x} \log \left ({\left (2 \, c x + b\right )} \sqrt{c} + 2 \, \sqrt{c x^{2} + b x} c\right ) - 2 \,{\left (3 \, B b^{2} x +{\left (4 \, B b c - A c^{2}\right )} x^{2}\right )} \sqrt{c}}{3 \,{\left (b c^{3} x + b^{2} c^{2}\right )} \sqrt{c x^{2} + b x} \sqrt{c}}, \frac{2 \,{\left (3 \,{\left (B b c x + B b^{2}\right )} \sqrt{c x^{2} + b x} \arctan \left (\frac{\sqrt{c x^{2} + b x} \sqrt{-c}}{c x}\right ) -{\left (3 \, B b^{2} x +{\left (4 \, B b c - A c^{2}\right )} x^{2}\right )} \sqrt{-c}\right )}}{3 \,{\left (b c^{3} x + b^{2} c^{2}\right )} \sqrt{c x^{2} + b x} \sqrt{-c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/3*(3*(B*b*c*x + B*b^2)*sqrt(c*x^2 + b*x)*log((2*c*x + b)*sqrt(c) + 2*sqrt(c*x
^2 + b*x)*c) - 2*(3*B*b^2*x + (4*B*b*c - A*c^2)*x^2)*sqrt(c))/((b*c^3*x + b^2*c^
2)*sqrt(c*x^2 + b*x)*sqrt(c)), 2/3*(3*(B*b*c*x + B*b^2)*sqrt(c*x^2 + b*x)*arctan
(sqrt(c*x^2 + b*x)*sqrt(-c)/(c*x)) - (3*B*b^2*x + (4*B*b*c - A*c^2)*x^2)*sqrt(-c
))/((b*c^3*x + b^2*c^2)*sqrt(c*x^2 + b*x)*sqrt(-c))]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{3} \left (A + B x\right )}{\left (x \left (b + c x\right )\right )^{\frac{5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3*(B*x+A)/(c*x**2+b*x)**(5/2),x)

[Out]

Integral(x**3*(A + B*x)/(x*(b + c*x))**(5/2), x)

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: TypeError