Optimal. Leaf size=220 \[ \frac{(c+d x)^{5/2} (7 b c-3 a d)}{4 a^2 c x (a+b x)^{3/2}}+\frac{5 (c+d x)^{3/2} (7 b c-3 a d) (b c-a d)}{12 a^3 c (a+b x)^{3/2}}+\frac{5 \sqrt{c+d x} (7 b c-3 a d) (b c-a d)}{4 a^4 \sqrt{a+b x}}-\frac{5 \sqrt{c} (7 b c-3 a d) (b c-a d) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{4 a^{9/2}}-\frac{(c+d x)^{7/2}}{2 a c x^2 (a+b x)^{3/2}} \]
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Rubi [A] time = 0.107583, 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, Rules used = {96, 94, 93, 208} \[ \frac{(c+d x)^{5/2} (7 b c-3 a d)}{4 a^2 c x (a+b x)^{3/2}}+\frac{5 (c+d x)^{3/2} (7 b c-3 a d) (b c-a d)}{12 a^3 c (a+b x)^{3/2}}+\frac{5 \sqrt{c+d x} (7 b c-3 a d) (b c-a d)}{4 a^4 \sqrt{a+b x}}-\frac{5 \sqrt{c} (7 b c-3 a d) (b c-a d) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{4 a^{9/2}}-\frac{(c+d x)^{7/2}}{2 a c x^2 (a+b x)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 96
Rule 94
Rule 93
Rule 208
Rubi steps
\begin{align*} \int \frac{(c+d x)^{5/2}}{x^3 (a+b x)^{5/2}} \, dx &=-\frac{(c+d x)^{7/2}}{2 a c x^2 (a+b x)^{3/2}}-\frac{\left (\frac{7 b c}{2}-\frac{3 a d}{2}\right ) \int \frac{(c+d x)^{5/2}}{x^2 (a+b x)^{5/2}} \, dx}{2 a c}\\ &=\frac{(7 b c-3 a d) (c+d x)^{5/2}}{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}}+\frac{(5 (7 b c-3 a d) (b c-a d)) \int \frac{(c+d x)^{3/2}}{x (a+b x)^{5/2}} \, dx}{8 a^2 c}\\ &=\frac{5 (7 b c-3 a d) (b c-a d) (c+d x)^{3/2}}{12 a^3 c (a+b x)^{3/2}}+\frac{(7 b c-3 a d) (c+d x)^{5/2}}{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}}+\frac{(5 (7 b c-3 a d) (b c-a d)) \int \frac{\sqrt{c+d x}}{x (a+b x)^{3/2}} \, dx}{8 a^3}\\ &=\frac{5 (7 b c-3 a d) (b c-a d) \sqrt{c+d x}}{4 a^4 \sqrt{a+b x}}+\frac{5 (7 b c-3 a d) (b c-a d) (c+d x)^{3/2}}{12 a^3 c (a+b x)^{3/2}}+\frac{(7 b c-3 a d) (c+d x)^{5/2}}{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}}+\frac{(5 c (7 b c-3 a d) (b c-a d)) \int \frac{1}{x \sqrt{a+b x} \sqrt{c+d x}} \, dx}{8 a^4}\\ &=\frac{5 (7 b c-3 a d) (b c-a d) \sqrt{c+d x}}{4 a^4 \sqrt{a+b x}}+\frac{5 (7 b c-3 a d) (b c-a d) (c+d x)^{3/2}}{12 a^3 c (a+b x)^{3/2}}+\frac{(7 b c-3 a d) (c+d x)^{5/2}}{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}}+\frac{(5 c (7 b c-3 a d) (b c-a d)) \operatorname{Subst}\left (\int \frac{1}{-a+c x^2} \, dx,x,\frac{\sqrt{a+b x}}{\sqrt{c+d x}}\right )}{4 a^4}\\ &=\frac{5 (7 b c-3 a d) (b c-a d) \sqrt{c+d x}}{4 a^4 \sqrt{a+b x}}+\frac{5 (7 b c-3 a d) (b c-a d) (c+d x)^{3/2}}{12 a^3 c (a+b x)^{3/2}}+\frac{(7 b c-3 a d) (c+d x)^{5/2}}{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}}-\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}}\\ \end{align*}
Mathematica [A] time = 0.213662, size = 159, normalized size = 0.72 \[ \frac{\frac{1}{2} x (7 b c-3 a d) \left (3 a^{5/2} (c+d x)^{5/2}+5 x (b c-a d) \left (\sqrt{a} \sqrt{c+d x} (4 a c+a d x+3 b c x)-3 c^{3/2} (a+b x)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )\right )\right )-3 a^{7/2} (c+d x)^{7/2}}{6 a^{9/2} c x^2 (a+b x)^{3/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.027, size = 758, normalized size = 3.5 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 18.4938, size = 1422, normalized size = 6.46 \begin{align*} \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{{\left (2 \, a c +{\left (b c + a d\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c} \sqrt{-\frac{c}{a}}}{2 \,{\left (b c d x^{2} + a c^{2} +{\left (b c^{2} + a c d\right )} x\right )}}\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 ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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