Optimal. Leaf size=220 \[ -\frac{2 (d+e x)^2 \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{3 b^2 c \left (b x+c x^2\right )^{3/2}}+\frac{2 \left (b c d^2 \left (-4 b c (2 A e+B d)+8 A c^2 d+b^2 B e\right )+x \left (2 b^2 c^2 d e (4 A e+3 B d)-8 b c^3 d^2 (3 A e+B d)+16 A c^4 d^3-3 b^4 B e^3+2 b^3 B c d e^2\right )\right )}{3 b^4 c^2 \sqrt{b x+c x^2}}+\frac{2 B e^3 \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{c^{5/2}} \]
[Out]
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Rubi [A] time = 0.538937, antiderivative size = 220, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ -\frac{2 (d+e x)^2 \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{3 b^2 c \left (b x+c x^2\right )^{3/2}}+\frac{2 \left (b c d^2 \left (-4 b c (2 A e+B d)+8 A c^2 d+b^2 B e\right )+x \left (2 b^2 c^2 d e (4 A e+3 B d)-8 b c^3 d^2 (3 A e+B d)+16 A c^4 d^3-3 b^4 B e^3+2 b^3 B c d e^2\right )\right )}{3 b^4 c^2 \sqrt{b x+c x^2}}+\frac{2 B e^3 \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{c^{5/2}} \]
Antiderivative was successfully verified.
[In] Int[((A + B*x)*(d + e*x)^3)/(b*x + c*x^2)^(5/2),x]
[Out]
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Rubi in Sympy [A] time = 61.1182, size = 246, normalized size = 1.12 \[ \frac{2 B e^{3} \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{b x + c x^{2}}} \right )}}{c^{\frac{5}{2}}} - \frac{2 \left (d + e x\right )^{2} \left (A b c d + x \left (2 A c^{2} d + B b^{2} e - b c \left (A e + B d\right )\right )\right )}{3 b^{2} c \left (b x + c x^{2}\right )^{\frac{3}{2}}} + \frac{4 \left (\frac{b c d^{2} \left (- 8 A b c e + 8 A c^{2} d + B b^{2} e - 4 B b c d\right )}{2} + x \left (4 A b^{2} c^{2} d e^{2} - 12 A b c^{3} d^{2} e + 8 A c^{4} d^{3} - \frac{3 B b^{4} e^{3}}{2} + B b^{3} c d e^{2} + 3 B b^{2} c^{2} d^{2} e - 4 B b c^{3} d^{3}\right )\right )}{3 b^{4} c^{2} \sqrt{b x + c x^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)*(e*x+d)**3/(c*x**2+b*x)**(5/2),x)
[Out]
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Mathematica [A] time = 0.970266, size = 199, normalized size = 0.9 \[ \frac{x^{5/2} \left (\frac{2 B e^3 (b+c x)^{5/2} \log \left (\sqrt{c} \sqrt{b+c x}+c \sqrt{x}\right )}{c^{5/2}}-\frac{2 (b+c x) \left (x^2 (b+c x) (c d-b e)^2 \left (b c (5 B d-A e)-8 A c^2 d+4 b^2 B e\right )+c^2 d^2 x (b+c x)^2 (9 A b e-8 A c d+3 b B d)+b x^2 (b B-A c) (c d-b e)^3+A b c^2 d^3 (b+c x)^2\right )}{3 b^4 c^2 x^{3/2}}\right )}{(x (b+c x))^{5/2}} \]
Antiderivative was successfully verified.
[In] Integrate[((A + B*x)*(d + e*x)^3)/(b*x + c*x^2)^(5/2),x]
[Out]
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Maple [B] time = 0.014, size = 680, normalized size = 3.1 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)*(e*x+d)^3/(c*x^2+b*x)^(5/2),x)
[Out]
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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)*(e*x + d)^3/(c*x^2 + b*x)^(5/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.308858, size = 1, normalized size = 0. \[ \left [\frac{3 \,{\left (B b^{4} c e^{3} x^{2} + B b^{5} e^{3} x\right )} \sqrt{c x^{2} + b x} \log \left ({\left (2 \, c x + b\right )} \sqrt{c} + 2 \, \sqrt{c x^{2} + b x} c\right ) - 2 \,{\left (A b^{3} c^{2} d^{3} +{\left (8 \,{\left (B b c^{4} - 2 \, A c^{5}\right )} d^{3} - 6 \,{\left (B b^{2} c^{3} - 4 \, A b c^{4}\right )} d^{2} e - 3 \,{\left (B b^{3} c^{2} + 2 \, A b^{2} c^{3}\right )} d e^{2} +{\left (4 \, B b^{4} c - A b^{3} c^{2}\right )} e^{3}\right )} x^{3} - 3 \,{\left (3 \, A b^{3} c^{2} d e^{2} - B b^{5} e^{3} - 4 \,{\left (B b^{2} c^{3} - 2 \, A b c^{4}\right )} d^{3} + 3 \,{\left (B b^{3} c^{2} - 4 \, A b^{2} c^{3}\right )} d^{2} e\right )} x^{2} + 3 \,{\left (3 \, A b^{3} c^{2} d^{2} e +{\left (B b^{3} c^{2} - 2 \, A b^{2} c^{3}\right )} d^{3}\right )} x\right )} \sqrt{c}}{3 \,{\left (b^{4} c^{3} x^{2} + b^{5} c^{2} x\right )} \sqrt{c x^{2} + b x} \sqrt{c}}, \frac{2 \,{\left (3 \,{\left (B b^{4} c e^{3} x^{2} + B b^{5} e^{3} x\right )} \sqrt{c x^{2} + b x} \arctan \left (\frac{\sqrt{c x^{2} + b x} \sqrt{-c}}{c x}\right ) -{\left (A b^{3} c^{2} d^{3} +{\left (8 \,{\left (B b c^{4} - 2 \, A c^{5}\right )} d^{3} - 6 \,{\left (B b^{2} c^{3} - 4 \, A b c^{4}\right )} d^{2} e - 3 \,{\left (B b^{3} c^{2} + 2 \, A b^{2} c^{3}\right )} d e^{2} +{\left (4 \, B b^{4} c - A b^{3} c^{2}\right )} e^{3}\right )} x^{3} - 3 \,{\left (3 \, A b^{3} c^{2} d e^{2} - B b^{5} e^{3} - 4 \,{\left (B b^{2} c^{3} - 2 \, A b c^{4}\right )} d^{3} + 3 \,{\left (B b^{3} c^{2} - 4 \, A b^{2} c^{3}\right )} d^{2} e\right )} x^{2} + 3 \,{\left (3 \, A b^{3} c^{2} d^{2} e +{\left (B b^{3} c^{2} - 2 \, A b^{2} c^{3}\right )} d^{3}\right )} x\right )} \sqrt{-c}\right )}}{3 \,{\left (b^{4} c^{3} x^{2} + b^{5} c^{2} x\right )} \sqrt{c x^{2} + b x} \sqrt{-c}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*(e*x + d)^3/(c*x^2 + b*x)^(5/2),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (A + B x\right ) \left (d + e x\right )^{3}}{\left (x \left (b + c x\right )\right )^{\frac{5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x+A)*(e*x+d)**3/(c*x**2+b*x)**(5/2),x)
[Out]
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GIAC/XCAS [A] time = 0.309001, size = 390, normalized size = 1.77 \[ -\frac{B e^{3}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} \sqrt{c} - b \right |}\right )}{c^{\frac{5}{2}}} - \frac{2 \,{\left (\frac{A d^{3}}{b} +{\left (x{\left (\frac{{\left (8 \, B b c^{4} d^{3} - 16 \, A c^{5} d^{3} - 6 \, B b^{2} c^{3} d^{2} e + 24 \, A b c^{4} d^{2} e - 3 \, B b^{3} c^{2} d e^{2} - 6 \, A b^{2} c^{3} d e^{2} + 4 \, B b^{4} c e^{3} - A b^{3} c^{2} e^{3}\right )} x}{b^{4} c^{2}} + \frac{3 \,{\left (4 \, B b^{2} c^{3} d^{3} - 8 \, A b c^{4} d^{3} - 3 \, B b^{3} c^{2} d^{2} e + 12 \, A b^{2} c^{3} d^{2} e - 3 \, A b^{3} c^{2} d e^{2} + B b^{5} e^{3}\right )}}{b^{4} c^{2}}\right )} + \frac{3 \,{\left (B b^{3} c^{2} d^{3} - 2 \, A b^{2} c^{3} d^{3} + 3 \, A b^{3} c^{2} d^{2} e\right )}}{b^{4} c^{2}}\right )} x\right )}}{3 \,{\left (c x^{2} + b x\right )}^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*(e*x + d)^3/(c*x^2 + b*x)^(5/2),x, algorithm="giac")
[Out]