Optimal. Leaf size=220 \[ -\frac{5 \sqrt{c} (7 b c-3 a d) (b c-a d) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{4 a^{9/2}}+\frac{5 \sqrt{c+d x} (7 b c-3 a d) (b c-a d)}{4 a^4 \sqrt{a+b x}}+\frac{5 (c+d x)^{3/2} (7 b c-3 a d) (b c-a d)}{12 a^3 c (a+b x)^{3/2}}+\frac{(c+d x)^{5/2} (7 b c-3 a d)}{4 a^2 c x (a+b x)^{3/2}}-\frac{(c+d x)^{7/2}}{2 a c x^2 (a+b x)^{3/2}} \]
[Out]
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Rubi [A] time = 0.40489, antiderivative size = 220, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ -\frac{5 \sqrt{c} (7 b c-3 a d) (b c-a d) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{4 a^{9/2}}+\frac{5 \sqrt{c+d x} (7 b c-3 a d) (b c-a d)}{4 a^4 \sqrt{a+b x}}+\frac{5 (c+d x)^{3/2} (7 b c-3 a d) (b c-a d)}{12 a^3 c (a+b x)^{3/2}}+\frac{(c+d x)^{5/2} (7 b c-3 a d)}{4 a^2 c x (a+b x)^{3/2}}-\frac{(c+d x)^{7/2}}{2 a c x^2 (a+b x)^{3/2}} \]
Antiderivative was successfully verified.
[In] Int[(c + d*x)^(5/2)/(x^3*(a + b*x)^(5/2)),x]
[Out]
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Rubi in Sympy [A] time = 34.4425, size = 207, normalized size = 0.94 \[ - \frac{2 b \left (c + d x\right )^{\frac{7}{2}}}{3 a x^{2} \left (a + b x\right )^{\frac{3}{2}} \left (a d - b c\right )} - \frac{\left (c + d x\right )^{\frac{5}{2}} \left (3 a d - 7 b c\right )}{6 a^{2} x^{2} \sqrt{a + b x} \left (a d - b c\right )} + \frac{5 \left (c + d x\right )^{\frac{3}{2}} \left (3 a d - 7 b c\right )}{6 a^{3} x \sqrt{a + b x}} - \frac{5 c \sqrt{a + b x} \sqrt{c + d x} \left (3 a d - 7 b c\right )}{4 a^{4} x} - \frac{5 \sqrt{c} \left (a d - b c\right ) \left (3 a d - 7 b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{a + b x}}{\sqrt{a} \sqrt{c + d x}} \right )}}{4 a^{\frac{9}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((d*x+c)**(5/2)/x**3/(b*x+a)**(5/2),x)
[Out]
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Mathematica [A] time = 0.423981, size = 211, normalized size = 0.96 \[ \frac{\frac{2 \sqrt{a} \sqrt{c+d x} \left (-3 a^3 \left (2 c^2+9 c d x-8 d^2 x^2\right )+a^2 b x \left (21 c^2-158 c d x+16 d^2 x^2\right )+5 a b^2 c x^2 (28 c-23 d x)+105 b^3 c^2 x^3\right )}{x^2 (a+b x)^{3/2}}+15 \sqrt{c} \log (x) (7 b c-3 a d) (b c-a d)-15 \sqrt{c} (7 b c-3 a d) (b c-a d) \log \left (2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x}+2 a c+a d x+b c x\right )}{24 a^{9/2}} \]
Antiderivative was successfully verified.
[In] Integrate[(c + d*x)^(5/2)/(x^3*(a + b*x)^(5/2)),x]
[Out]
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Maple [B] time = 0.045, size = 758, normalized size = 3.5 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((d*x+c)^(5/2)/x^3/(b*x+a)^(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((d*x + c)^(5/2)/((b*x + a)^(5/2)*x^3),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.886089, size = 1, normalized size = 0. \[ \left [\frac{15 \,{\left ({\left (7 \, b^{4} c^{2} - 10 \, a b^{3} c d + 3 \, a^{2} b^{2} d^{2}\right )} x^{4} + 2 \,{\left (7 \, a b^{3} c^{2} - 10 \, a^{2} b^{2} c d + 3 \, a^{3} b d^{2}\right )} x^{3} +{\left (7 \, a^{2} b^{2} c^{2} - 10 \, a^{3} b c d + 3 \, a^{4} d^{2}\right )} x^{2}\right )} \sqrt{\frac{c}{a}} \log \left (\frac{8 \, a^{2} c^{2} +{\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} - 4 \,{\left (2 \, a^{2} c +{\left (a b c + a^{2} d\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c} \sqrt{\frac{c}{a}} + 8 \,{\left (a b c^{2} + a^{2} c d\right )} x}{x^{2}}\right ) - 4 \,{\left (6 \, a^{3} c^{2} -{\left (105 \, b^{3} c^{2} - 115 \, a b^{2} c d + 16 \, a^{2} b d^{2}\right )} x^{3} - 2 \,{\left (70 \, a b^{2} c^{2} - 79 \, a^{2} b c d + 12 \, a^{3} d^{2}\right )} x^{2} - 3 \,{\left (7 \, a^{2} b c^{2} - 9 \, a^{3} c d\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{48 \,{\left (a^{4} b^{2} x^{4} + 2 \, a^{5} b x^{3} + a^{6} x^{2}\right )}}, -\frac{15 \,{\left ({\left (7 \, b^{4} c^{2} - 10 \, a b^{3} c d + 3 \, a^{2} b^{2} d^{2}\right )} x^{4} + 2 \,{\left (7 \, a b^{3} c^{2} - 10 \, a^{2} b^{2} c d + 3 \, a^{3} b d^{2}\right )} x^{3} +{\left (7 \, a^{2} b^{2} c^{2} - 10 \, a^{3} b c d + 3 \, a^{4} d^{2}\right )} x^{2}\right )} \sqrt{-\frac{c}{a}} \arctan \left (\frac{2 \, a c +{\left (b c + a d\right )} x}{2 \, \sqrt{b x + a} \sqrt{d x + c} a \sqrt{-\frac{c}{a}}}\right ) + 2 \,{\left (6 \, a^{3} c^{2} -{\left (105 \, b^{3} c^{2} - 115 \, a b^{2} c d + 16 \, a^{2} b d^{2}\right )} x^{3} - 2 \,{\left (70 \, a b^{2} c^{2} - 79 \, a^{2} b c d + 12 \, a^{3} d^{2}\right )} x^{2} - 3 \,{\left (7 \, a^{2} b c^{2} - 9 \, a^{3} c d\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{24 \,{\left (a^{4} b^{2} x^{4} + 2 \, a^{5} b x^{3} + a^{6} x^{2}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x + c)^(5/2)/((b*x + a)^(5/2)*x^3),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((d*x+c)**(5/2)/x**3/(b*x+a)**(5/2),x)
[Out]
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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((d*x + c)^(5/2)/((b*x + a)^(5/2)*x^3),x, algorithm="giac")
[Out]