Optimal. Leaf size=241 \[ -\frac{\sqrt{c+d x} (35 b c-3 a d) (b c-a d)}{24 a^3 c x \sqrt{a+b x}}+\frac{7 \sqrt{c+d x} (b c-a d)}{12 a^2 x^2 \sqrt{a+b x}}+\frac{(b c-a d) \left (-a^2 d^2-10 a b c d+35 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{8 a^{9/2} c^{3/2}}-\frac{b \sqrt{c+d x} \left (3 a^2 d^2-100 a b c d+105 b^2 c^2\right )}{24 a^4 c \sqrt{a+b x}}-\frac{c \sqrt{c+d x}}{3 a x^3 \sqrt{a+b x}} \]
[Out]
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Rubi [A] time = 0.802934, antiderivative size = 241, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273 \[ -\frac{\sqrt{c+d x} (35 b c-3 a d) (b c-a d)}{24 a^3 c x \sqrt{a+b x}}+\frac{7 \sqrt{c+d x} (b c-a d)}{12 a^2 x^2 \sqrt{a+b x}}+\frac{(b c-a d) \left (-a^2 d^2-10 a b c d+35 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{8 a^{9/2} c^{3/2}}-\frac{b \sqrt{c+d x} \left (3 a^2 d^2-100 a b c d+105 b^2 c^2\right )}{24 a^4 c \sqrt{a+b x}}-\frac{c \sqrt{c+d x}}{3 a x^3 \sqrt{a+b x}} \]
Antiderivative was successfully verified.
[In] Int[(c + d*x)^(3/2)/(x^4*(a + b*x)^(3/2)),x]
[Out]
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Rubi in Sympy [A] time = 116.928, size = 223, normalized size = 0.93 \[ - \frac{c \sqrt{c + d x}}{3 a x^{3} \sqrt{a + b x}} - \frac{7 \sqrt{c + d x} \left (a d - b c\right )}{12 a^{2} x^{2} \sqrt{a + b x}} - \frac{\sqrt{c + d x} \left (a d - b c\right ) \left (3 a d - 35 b c\right )}{24 a^{3} c x \sqrt{a + b x}} - \frac{b \sqrt{c + d x} \left (3 a^{2} d^{2} - 100 a b c d + 105 b^{2} c^{2}\right )}{24 a^{4} c \sqrt{a + b x}} + \frac{\left (a d - b c\right ) \left (a^{2} d^{2} + 10 a b c d - 35 b^{2} c^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{a + b x}}{\sqrt{a} \sqrt{c + d x}} \right )}}{8 a^{\frac{9}{2}} c^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((d*x+c)**(3/2)/x**4/(b*x+a)**(3/2),x)
[Out]
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Mathematica [A] time = 0.297871, size = 236, normalized size = 0.98 \[ \frac{-3 \log (x) (a d-b c) \left (a^2 d^2+10 a b c d-35 b^2 c^2\right )+3 (a d-b c) \left (a^2 d^2+10 a b c d-35 b^2 c^2\right ) \log \left (2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x}+2 a c+a d x+b c x\right )-\frac{2 \sqrt{a} \sqrt{c} \sqrt{c+d x} \left (a^3 \left (8 c^2+14 c d x+3 d^2 x^2\right )+a^2 b x \left (-14 c^2-38 c d x+3 d^2 x^2\right )+5 a b^2 c x^2 (7 c-20 d x)+105 b^3 c^2 x^3\right )}{x^3 \sqrt{a+b x}}}{48 a^{9/2} c^{3/2}} \]
Antiderivative was successfully verified.
[In] Integrate[(c + d*x)^(3/2)/(x^4*(a + b*x)^(3/2)),x]
[Out]
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Maple [B] time = 0.045, size = 707, normalized size = 2.9 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((d*x+c)^(3/2)/x^4/(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)/((b*x + a)^(3/2)*x^4),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.712647, size = 1, normalized size = 0. \[ \left [-\frac{4 \,{\left (8 \, a^{3} c^{2} +{\left (105 \, b^{3} c^{2} - 100 \, a b^{2} c d + 3 \, a^{2} b d^{2}\right )} x^{3} +{\left (35 \, a b^{2} c^{2} - 38 \, a^{2} b c d + 3 \, a^{3} d^{2}\right )} x^{2} - 14 \,{\left (a^{2} b c^{2} - a^{3} c d\right )} x\right )} \sqrt{a c} \sqrt{b x + a} \sqrt{d x + c} - 3 \,{\left ({\left (35 \, b^{4} c^{3} - 45 \, a b^{3} c^{2} d + 9 \, a^{2} b^{2} c d^{2} + a^{3} b d^{3}\right )} x^{4} +{\left (35 \, a b^{3} c^{3} - 45 \, a^{2} b^{2} c^{2} d + 9 \, a^{3} b c d^{2} + a^{4} d^{3}\right )} x^{3}\right )} \log \left (\frac{4 \,{\left (2 \, a^{2} c^{2} +{\left (a b c^{2} + a^{2} c d\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c} +{\left (8 \, a^{2} c^{2} +{\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} + 8 \,{\left (a b c^{2} + a^{2} c d\right )} x\right )} \sqrt{a c}}{x^{2}}\right )}{96 \,{\left (a^{4} b c x^{4} + a^{5} c x^{3}\right )} \sqrt{a c}}, -\frac{2 \,{\left (8 \, a^{3} c^{2} +{\left (105 \, b^{3} c^{2} - 100 \, a b^{2} c d + 3 \, a^{2} b d^{2}\right )} x^{3} +{\left (35 \, a b^{2} c^{2} - 38 \, a^{2} b c d + 3 \, a^{3} d^{2}\right )} x^{2} - 14 \,{\left (a^{2} b c^{2} - a^{3} c d\right )} x\right )} \sqrt{-a c} \sqrt{b x + a} \sqrt{d x + c} - 3 \,{\left ({\left (35 \, b^{4} c^{3} - 45 \, a b^{3} c^{2} d + 9 \, a^{2} b^{2} c d^{2} + a^{3} b d^{3}\right )} x^{4} +{\left (35 \, a b^{3} c^{3} - 45 \, a^{2} b^{2} c^{2} d + 9 \, a^{3} b c d^{2} + a^{4} d^{3}\right )} x^{3}\right )} \arctan \left (\frac{{\left (2 \, a c +{\left (b c + a d\right )} x\right )} \sqrt{-a c}}{2 \, \sqrt{b x + a} \sqrt{d x + c} a c}\right )}{48 \,{\left (a^{4} b c x^{4} + a^{5} c x^{3}\right )} \sqrt{-a c}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x + c)^(3/2)/((b*x + a)^(3/2)*x^4),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)**(3/2)/x**4/(b*x+a)**(3/2),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)^(3/2)/((b*x + a)^(3/2)*x^4),x, algorithm="giac")
[Out]