Optimal. Leaf size=267 \[ \frac{(a+b x)^{3/2} \sqrt{c+d x} \left (3 a^2 d^2-30 a b c d+35 b^2 c^2\right )}{6 d^3 (b c-a d)^2}-\frac{\sqrt{a+b x} \sqrt{c+d x} \left (3 a^2 d^2-30 a b c d+35 b^2 c^2\right )}{4 d^4 (b c-a d)}+\frac{\left (3 a^2 d^2-30 a b c d+35 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{4 \sqrt{b} d^{9/2}}+\frac{2 c^2 (a+b x)^{5/2}}{3 d^2 (c+d x)^{3/2} (b c-a d)}-\frac{4 c (a+b x)^{5/2} (4 b c-3 a d)}{3 d^2 \sqrt{c+d x} (b c-a d)^2} \]
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Rubi [A] time = 0.284155, antiderivative size = 267, 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, Rules used = {89, 78, 50, 63, 217, 206} \[ \frac{(a+b x)^{3/2} \sqrt{c+d x} \left (3 a^2 d^2-30 a b c d+35 b^2 c^2\right )}{6 d^3 (b c-a d)^2}-\frac{\sqrt{a+b x} \sqrt{c+d x} \left (3 a^2 d^2-30 a b c d+35 b^2 c^2\right )}{4 d^4 (b c-a d)}+\frac{\left (3 a^2 d^2-30 a b c d+35 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{4 \sqrt{b} d^{9/2}}+\frac{2 c^2 (a+b x)^{5/2}}{3 d^2 (c+d x)^{3/2} (b c-a d)}-\frac{4 c (a+b x)^{5/2} (4 b c-3 a d)}{3 d^2 \sqrt{c+d x} (b c-a d)^2} \]
Antiderivative was successfully verified.
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Rule 89
Rule 78
Rule 50
Rule 63
Rule 217
Rule 206
Rubi steps
\begin{align*} \int \frac{x^2 (a+b x)^{3/2}}{(c+d x)^{5/2}} \, dx &=\frac{2 c^2 (a+b x)^{5/2}}{3 d^2 (b c-a d) (c+d x)^{3/2}}-\frac{2 \int \frac{(a+b x)^{3/2} \left (\frac{1}{2} c (5 b c-3 a d)-\frac{3}{2} d (b c-a d) x\right )}{(c+d x)^{3/2}} \, dx}{3 d^2 (b c-a d)}\\ &=\frac{2 c^2 (a+b x)^{5/2}}{3 d^2 (b c-a d) (c+d x)^{3/2}}-\frac{4 c (4 b c-3 a d) (a+b x)^{5/2}}{3 d^2 (b c-a d)^2 \sqrt{c+d x}}+\frac{\left (35 b^2 c^2-30 a b c d+3 a^2 d^2\right ) \int \frac{(a+b x)^{3/2}}{\sqrt{c+d x}} \, dx}{3 d^2 (b c-a d)^2}\\ &=\frac{2 c^2 (a+b x)^{5/2}}{3 d^2 (b c-a d) (c+d x)^{3/2}}-\frac{4 c (4 b c-3 a d) (a+b x)^{5/2}}{3 d^2 (b c-a d)^2 \sqrt{c+d x}}+\frac{\left (35 b^2 c^2-30 a b c d+3 a^2 d^2\right ) (a+b x)^{3/2} \sqrt{c+d x}}{6 d^3 (b c-a d)^2}-\frac{\left (35 b^2 c^2-30 a b c d+3 a^2 d^2\right ) \int \frac{\sqrt{a+b x}}{\sqrt{c+d x}} \, dx}{4 d^3 (b c-a d)}\\ &=\frac{2 c^2 (a+b x)^{5/2}}{3 d^2 (b c-a d) (c+d x)^{3/2}}-\frac{4 c (4 b c-3 a d) (a+b x)^{5/2}}{3 d^2 (b c-a d)^2 \sqrt{c+d x}}-\frac{\left (35 b^2 c^2-30 a b c d+3 a^2 d^2\right ) \sqrt{a+b x} \sqrt{c+d x}}{4 d^4 (b c-a d)}+\frac{\left (35 b^2 c^2-30 a b c d+3 a^2 d^2\right ) (a+b x)^{3/2} \sqrt{c+d x}}{6 d^3 (b c-a d)^2}+\frac{\left (35 b^2 c^2-30 a b c d+3 a^2 d^2\right ) \int \frac{1}{\sqrt{a+b x} \sqrt{c+d x}} \, dx}{8 d^4}\\ &=\frac{2 c^2 (a+b x)^{5/2}}{3 d^2 (b c-a d) (c+d x)^{3/2}}-\frac{4 c (4 b c-3 a d) (a+b x)^{5/2}}{3 d^2 (b c-a d)^2 \sqrt{c+d x}}-\frac{\left (35 b^2 c^2-30 a b c d+3 a^2 d^2\right ) \sqrt{a+b x} \sqrt{c+d x}}{4 d^4 (b c-a d)}+\frac{\left (35 b^2 c^2-30 a b c d+3 a^2 d^2\right ) (a+b x)^{3/2} \sqrt{c+d x}}{6 d^3 (b c-a d)^2}+\frac{\left (35 b^2 c^2-30 a b c d+3 a^2 d^2\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{c-\frac{a d}{b}+\frac{d x^2}{b}}} \, dx,x,\sqrt{a+b x}\right )}{4 b d^4}\\ &=\frac{2 c^2 (a+b x)^{5/2}}{3 d^2 (b c-a d) (c+d x)^{3/2}}-\frac{4 c (4 b c-3 a d) (a+b x)^{5/2}}{3 d^2 (b c-a d)^2 \sqrt{c+d x}}-\frac{\left (35 b^2 c^2-30 a b c d+3 a^2 d^2\right ) \sqrt{a+b x} \sqrt{c+d x}}{4 d^4 (b c-a d)}+\frac{\left (35 b^2 c^2-30 a b c d+3 a^2 d^2\right ) (a+b x)^{3/2} \sqrt{c+d x}}{6 d^3 (b c-a d)^2}+\frac{\left (35 b^2 c^2-30 a b c d+3 a^2 d^2\right ) \operatorname{Subst}\left (\int \frac{1}{1-\frac{d x^2}{b}} \, dx,x,\frac{\sqrt{a+b x}}{\sqrt{c+d x}}\right )}{4 b d^4}\\ &=\frac{2 c^2 (a+b x)^{5/2}}{3 d^2 (b c-a d) (c+d x)^{3/2}}-\frac{4 c (4 b c-3 a d) (a+b x)^{5/2}}{3 d^2 (b c-a d)^2 \sqrt{c+d x}}-\frac{\left (35 b^2 c^2-30 a b c d+3 a^2 d^2\right ) \sqrt{a+b x} \sqrt{c+d x}}{4 d^4 (b c-a d)}+\frac{\left (35 b^2 c^2-30 a b c d+3 a^2 d^2\right ) (a+b x)^{3/2} \sqrt{c+d x}}{6 d^3 (b c-a d)^2}+\frac{\left (35 b^2 c^2-30 a b c d+3 a^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{4 \sqrt{b} d^{9/2}}\\ \end{align*}
Mathematica [A] time = 0.736353, size = 220, normalized size = 0.82 \[ \frac{\frac{\sqrt{d} \left (a^2 d \left (55 c^2+78 c d x+15 d^2 x^2\right )+a b \left (-85 c^2 d x-105 c^3+57 c d^2 x^2+21 d^3 x^3\right )+b^2 x \left (-140 c^2 d x-105 c^3-21 c d^2 x^2+6 d^3 x^3\right )\right )}{\sqrt{a+b x}}+\frac{3 (c+d x)^2 \left (3 a^2 d^2-30 a b c d+35 b^2 c^2\right ) \sinh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b c-a d}}\right )}{\sqrt{b c-a d} \sqrt{\frac{b (c+d x)}{b c-a d}}}}{12 d^{9/2} (c+d x)^{3/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.023, size = 676, normalized size = 2.5 \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 = 6.74664, size = 1326, normalized size = 4.97 \begin{align*} \left [\frac{3 \,{\left (35 \, b^{2} c^{4} - 30 \, a b c^{3} d + 3 \, a^{2} c^{2} d^{2} +{\left (35 \, b^{2} c^{2} d^{2} - 30 \, a b c d^{3} + 3 \, a^{2} d^{4}\right )} x^{2} + 2 \,{\left (35 \, b^{2} c^{3} d - 30 \, a b c^{2} d^{2} + 3 \, a^{2} c d^{3}\right )} x\right )} \sqrt{b d} \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 4 \,{\left (2 \, b d x + b c + a d\right )} \sqrt{b d} \sqrt{b x + a} \sqrt{d x + c} + 8 \,{\left (b^{2} c d + a b d^{2}\right )} x\right ) + 4 \,{\left (6 \, b^{2} d^{4} x^{3} - 105 \, b^{2} c^{3} d + 55 \, a b c^{2} d^{2} - 3 \,{\left (7 \, b^{2} c d^{3} - 5 \, a b d^{4}\right )} x^{2} - 2 \,{\left (70 \, b^{2} c^{2} d^{2} - 39 \, a b c d^{3}\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{48 \,{\left (b d^{7} x^{2} + 2 \, b c d^{6} x + b c^{2} d^{5}\right )}}, -\frac{3 \,{\left (35 \, b^{2} c^{4} - 30 \, a b c^{3} d + 3 \, a^{2} c^{2} d^{2} +{\left (35 \, b^{2} c^{2} d^{2} - 30 \, a b c d^{3} + 3 \, a^{2} d^{4}\right )} x^{2} + 2 \,{\left (35 \, b^{2} c^{3} d - 30 \, a b c^{2} d^{2} + 3 \, a^{2} c d^{3}\right )} x\right )} \sqrt{-b d} \arctan \left (\frac{{\left (2 \, b d x + b c + a d\right )} \sqrt{-b d} \sqrt{b x + a} \sqrt{d x + c}}{2 \,{\left (b^{2} d^{2} x^{2} + a b c d +{\left (b^{2} c d + a b d^{2}\right )} x\right )}}\right ) - 2 \,{\left (6 \, b^{2} d^{4} x^{3} - 105 \, b^{2} c^{3} d + 55 \, a b c^{2} d^{2} - 3 \,{\left (7 \, b^{2} c d^{3} - 5 \, a b d^{4}\right )} x^{2} - 2 \,{\left (70 \, b^{2} c^{2} d^{2} - 39 \, a b c d^{3}\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{24 \,{\left (b d^{7} x^{2} + 2 \, b c d^{6} x + b c^{2} d^{5}\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 [A] time = 1.53624, size = 531, normalized size = 1.99 \begin{align*} \frac{{\left ({\left (3 \,{\left (b x + a\right )}{\left (\frac{2 \,{\left (b^{6} c d^{6} - a b^{5} d^{7}\right )}{\left (b x + a\right )}}{b^{4} c d^{7}{\left | b \right |} - a b^{3} d^{8}{\left | b \right |}} - \frac{7 \, b^{7} c^{2} d^{5} - 6 \, a b^{6} c d^{6} - a^{2} b^{5} d^{7}}{b^{4} c d^{7}{\left | b \right |} - a b^{3} d^{8}{\left | b \right |}}\right )} - \frac{4 \,{\left (35 \, b^{8} c^{3} d^{4} - 65 \, a b^{7} c^{2} d^{5} + 33 \, a^{2} b^{6} c d^{6} - 3 \, a^{3} b^{5} d^{7}\right )}}{b^{4} c d^{7}{\left | b \right |} - a b^{3} d^{8}{\left | b \right |}}\right )}{\left (b x + a\right )} - \frac{3 \,{\left (35 \, b^{9} c^{4} d^{3} - 100 \, a b^{8} c^{3} d^{4} + 98 \, a^{2} b^{7} c^{2} d^{5} - 36 \, a^{3} b^{6} c d^{6} + 3 \, a^{4} b^{5} d^{7}\right )}}{b^{4} c d^{7}{\left | b \right |} - a b^{3} d^{8}{\left | b \right |}}\right )} \sqrt{b x + a}}{12 \,{\left (b^{2} c +{\left (b x + a\right )} b d - a b d\right )}^{\frac{3}{2}}} - \frac{{\left (35 \, b^{3} c^{2} - 30 \, a b^{2} c d + 3 \, a^{2} b d^{2}\right )} \log \left ({\left | -\sqrt{b d} \sqrt{b x + a} + \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d} \right |}\right )}{4 \, \sqrt{b d} d^{4}{\left | b \right |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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