Optimal. Leaf size=271 \[ -\frac{\sqrt{a+b x} (c+d x)^{3/2} \left (-105 a^2 d^2-4 b d x (b c-21 a d)+14 a b c d+3 b^2 c^2\right )}{32 b^4 d^2}+\frac{3 (b c-a d) \left (-105 a^3 d^3+35 a^2 b c d^2+5 a b^2 c^2 d+b^3 c^3\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{64 b^{11/2} d^{5/2}}+\frac{3 \sqrt{a+b x} \sqrt{c+d x} \left (-105 a^3 d^3+35 a^2 b c d^2+5 a b^2 c^2 d+b^3 c^3\right )}{64 b^5 d^2}+\frac{9 x^2 \sqrt{a+b x} (c+d x)^{3/2}}{4 b^2}-\frac{2 x^3 (c+d x)^{3/2}}{b \sqrt{a+b x}} \]
[Out]
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Rubi [A] time = 0.586291, antiderivative size = 271, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273 \[ -\frac{\sqrt{a+b x} (c+d x)^{3/2} \left (-105 a^2 d^2-4 b d x (b c-21 a d)+14 a b c d+3 b^2 c^2\right )}{32 b^4 d^2}+\frac{3 (b c-a d) \left (-105 a^3 d^3+35 a^2 b c d^2+5 a b^2 c^2 d+b^3 c^3\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{64 b^{11/2} d^{5/2}}+\frac{3 \sqrt{a+b x} \sqrt{c+d x} \left (-105 a^3 d^3+35 a^2 b c d^2+5 a b^2 c^2 d+b^3 c^3\right )}{64 b^5 d^2}+\frac{9 x^2 \sqrt{a+b x} (c+d x)^{3/2}}{4 b^2}-\frac{2 x^3 (c+d x)^{3/2}}{b \sqrt{a+b x}} \]
Antiderivative was successfully verified.
[In] Int[(x^3*(c + d*x)^(3/2))/(a + b*x)^(3/2),x]
[Out]
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Rubi in Sympy [A] time = 52.4902, size = 277, normalized size = 1.02 \[ - \frac{2 x^{3} \left (c + d x\right )^{\frac{3}{2}}}{b \sqrt{a + b x}} + \frac{9 x^{2} \sqrt{a + b x} \left (c + d x\right )^{\frac{3}{2}}}{4 b^{2}} + \frac{\sqrt{a + b x} \left (c + d x\right )^{\frac{3}{2}} \left (\frac{315 a^{2} d^{2}}{8} - \frac{21 a b c d}{4} - \frac{9 b^{2} c^{2}}{8} - \frac{3 b d x \left (21 a d - b c\right )}{2}\right )}{12 b^{4} d^{2}} - \frac{3 \sqrt{a + b x} \sqrt{c + d x} \left (105 a^{3} d^{3} - 35 a^{2} b c d^{2} - 5 a b^{2} c^{2} d - b^{3} c^{3}\right )}{64 b^{5} d^{2}} + \frac{3 \left (a d - b c\right ) \left (105 a^{3} d^{3} - 35 a^{2} b c d^{2} - 5 a b^{2} c^{2} d - b^{3} c^{3}\right ) \operatorname{atanh}{\left (\frac{\sqrt{d} \sqrt{a + b x}}{\sqrt{b} \sqrt{c + d x}} \right )}}{64 b^{\frac{11}{2}} d^{\frac{5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**3*(d*x+c)**(3/2)/(b*x+a)**(3/2),x)
[Out]
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Mathematica [A] time = 0.264414, size = 253, normalized size = 0.93 \[ \frac{3 (b c-a d) \left (-105 a^3 d^3+35 a^2 b c d^2+5 a b^2 c^2 d+b^3 c^3\right ) \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right )}{128 b^{11/2} d^{5/2}}-\frac{\sqrt{c+d x} \left (315 a^4 d^3+105 a^3 b d^2 (d x-3 c)+a^2 b^2 d \left (13 c^2-119 c d x-42 d^2 x^2\right )+a b^3 \left (3 c^3+11 c^2 d x+44 c d^2 x^2+24 d^3 x^3\right )-b^4 x \left (-3 c^3+2 c^2 d x+24 c d^2 x^2+16 d^3 x^3\right )\right )}{64 b^5 d^2 \sqrt{a+b x}} \]
Antiderivative was successfully verified.
[In] Integrate[(x^3*(c + d*x)^(3/2))/(a + b*x)^(3/2),x]
[Out]
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Maple [B] time = 0.047, size = 961, normalized size = 3.6 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^3*(d*x+c)^(3/2)/(b*x+a)^(3/2),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((d*x + c)^(3/2)*x^3/(b*x + a)^(3/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.981152, size = 1, normalized size = 0. \[ \left [\frac{4 \,{\left (16 \, b^{4} d^{3} x^{4} - 3 \, a b^{3} c^{3} - 13 \, a^{2} b^{2} c^{2} d + 315 \, a^{3} b c d^{2} - 315 \, a^{4} d^{3} + 24 \,{\left (b^{4} c d^{2} - a b^{3} d^{3}\right )} x^{3} + 2 \,{\left (b^{4} c^{2} d - 22 \, a b^{3} c d^{2} + 21 \, a^{2} b^{2} d^{3}\right )} x^{2} -{\left (3 \, b^{4} c^{3} + 11 \, a b^{3} c^{2} d - 119 \, a^{2} b^{2} c d^{2} + 105 \, a^{3} b d^{3}\right )} x\right )} \sqrt{b d} \sqrt{b x + a} \sqrt{d x + c} + 3 \,{\left (a b^{4} c^{4} + 4 \, a^{2} b^{3} c^{3} d + 30 \, a^{3} b^{2} c^{2} d^{2} - 140 \, a^{4} b c d^{3} + 105 \, a^{5} d^{4} +{\left (b^{5} c^{4} + 4 \, a b^{4} c^{3} d + 30 \, a^{2} b^{3} c^{2} d^{2} - 140 \, a^{3} b^{2} c d^{3} + 105 \, a^{4} b d^{4}\right )} x\right )} \log \left (4 \,{\left (2 \, b^{2} d^{2} x + b^{2} c d + a b d^{2}\right )} \sqrt{b x + a} \sqrt{d x + c} +{\left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 8 \,{\left (b^{2} c d + a b d^{2}\right )} x\right )} \sqrt{b d}\right )}{256 \,{\left (b^{6} d^{2} x + a b^{5} d^{2}\right )} \sqrt{b d}}, \frac{2 \,{\left (16 \, b^{4} d^{3} x^{4} - 3 \, a b^{3} c^{3} - 13 \, a^{2} b^{2} c^{2} d + 315 \, a^{3} b c d^{2} - 315 \, a^{4} d^{3} + 24 \,{\left (b^{4} c d^{2} - a b^{3} d^{3}\right )} x^{3} + 2 \,{\left (b^{4} c^{2} d - 22 \, a b^{3} c d^{2} + 21 \, a^{2} b^{2} d^{3}\right )} x^{2} -{\left (3 \, b^{4} c^{3} + 11 \, a b^{3} c^{2} d - 119 \, a^{2} b^{2} c d^{2} + 105 \, a^{3} b d^{3}\right )} x\right )} \sqrt{-b d} \sqrt{b x + a} \sqrt{d x + c} + 3 \,{\left (a b^{4} c^{4} + 4 \, a^{2} b^{3} c^{3} d + 30 \, a^{3} b^{2} c^{2} d^{2} - 140 \, a^{4} b c d^{3} + 105 \, a^{5} d^{4} +{\left (b^{5} c^{4} + 4 \, a b^{4} c^{3} d + 30 \, a^{2} b^{3} c^{2} d^{2} - 140 \, a^{3} b^{2} c d^{3} + 105 \, a^{4} b d^{4}\right )} x\right )} \arctan \left (\frac{{\left (2 \, b d x + b c + a d\right )} \sqrt{-b d}}{2 \, \sqrt{b x + a} \sqrt{d x + c} b d}\right )}{128 \,{\left (b^{6} d^{2} x + a b^{5} d^{2}\right )} \sqrt{-b d}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x + c)^(3/2)*x^3/(b*x + a)^(3/2),x, algorithm="fricas")
[Out]
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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(x**3*(d*x+c)**(3/2)/(b*x+a)**(3/2),x)
[Out]
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GIAC/XCAS [A] time = 0.618315, size = 4, normalized size = 0.01 \[ \mathit{sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x + c)^(3/2)*x^3/(b*x + a)^(3/2),x, algorithm="giac")
[Out]