Optimal. Leaf size=314 \[ \frac{(a+b x)^{5/2} \sqrt{c+d x} \left (3 a^2 d^2+14 a b c d+63 b^2 c^2\right )}{240 b^2 d^3}-\frac{(a+b x)^{3/2} \sqrt{c+d x} (b c-a d) \left (3 a^2 d^2+14 a b c d+63 b^2 c^2\right )}{192 b^2 d^4}+\frac{\sqrt{a+b x} \sqrt{c+d x} (b c-a d)^2 \left (3 a^2 d^2+14 a b c d+63 b^2 c^2\right )}{128 b^2 d^5}-\frac{(b c-a d)^3 \left (3 a^2 d^2+14 a b c d+63 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{128 b^{5/2} d^{11/2}}-\frac{3 (a+b x)^{7/2} \sqrt{c+d x} (a d+3 b c)}{40 b^2 d^2}+\frac{x (a+b x)^{7/2} \sqrt{c+d x}}{5 b d} \]
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Rubi [A] time = 0.285383, antiderivative size = 314, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {90, 80, 50, 63, 217, 206} \[ \frac{(a+b x)^{5/2} \sqrt{c+d x} \left (3 a^2 d^2+14 a b c d+63 b^2 c^2\right )}{240 b^2 d^3}-\frac{(a+b x)^{3/2} \sqrt{c+d x} (b c-a d) \left (3 a^2 d^2+14 a b c d+63 b^2 c^2\right )}{192 b^2 d^4}+\frac{\sqrt{a+b x} \sqrt{c+d x} (b c-a d)^2 \left (3 a^2 d^2+14 a b c d+63 b^2 c^2\right )}{128 b^2 d^5}-\frac{(b c-a d)^3 \left (3 a^2 d^2+14 a b c d+63 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{128 b^{5/2} d^{11/2}}-\frac{3 (a+b x)^{7/2} \sqrt{c+d x} (a d+3 b c)}{40 b^2 d^2}+\frac{x (a+b x)^{7/2} \sqrt{c+d x}}{5 b d} \]
Antiderivative was successfully verified.
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Rule 90
Rule 80
Rule 50
Rule 63
Rule 217
Rule 206
Rubi steps
\begin{align*} \int \frac{x^2 (a+b x)^{5/2}}{\sqrt{c+d x}} \, dx &=\frac{x (a+b x)^{7/2} \sqrt{c+d x}}{5 b d}+\frac{\int \frac{(a+b x)^{5/2} \left (-a c-\frac{3}{2} (3 b c+a d) x\right )}{\sqrt{c+d x}} \, dx}{5 b d}\\ &=-\frac{3 (3 b c+a d) (a+b x)^{7/2} \sqrt{c+d x}}{40 b^2 d^2}+\frac{x (a+b x)^{7/2} \sqrt{c+d x}}{5 b d}+\frac{\left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right ) \int \frac{(a+b x)^{5/2}}{\sqrt{c+d x}} \, dx}{80 b^2 d^2}\\ &=\frac{\left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right ) (a+b x)^{5/2} \sqrt{c+d x}}{240 b^2 d^3}-\frac{3 (3 b c+a d) (a+b x)^{7/2} \sqrt{c+d x}}{40 b^2 d^2}+\frac{x (a+b x)^{7/2} \sqrt{c+d x}}{5 b d}-\frac{\left ((b c-a d) \left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right )\right ) \int \frac{(a+b x)^{3/2}}{\sqrt{c+d x}} \, dx}{96 b^2 d^3}\\ &=-\frac{(b c-a d) \left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right ) (a+b x)^{3/2} \sqrt{c+d x}}{192 b^2 d^4}+\frac{\left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right ) (a+b x)^{5/2} \sqrt{c+d x}}{240 b^2 d^3}-\frac{3 (3 b c+a d) (a+b x)^{7/2} \sqrt{c+d x}}{40 b^2 d^2}+\frac{x (a+b x)^{7/2} \sqrt{c+d x}}{5 b d}+\frac{\left ((b c-a d)^2 \left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right )\right ) \int \frac{\sqrt{a+b x}}{\sqrt{c+d x}} \, dx}{128 b^2 d^4}\\ &=\frac{(b c-a d)^2 \left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right ) \sqrt{a+b x} \sqrt{c+d x}}{128 b^2 d^5}-\frac{(b c-a d) \left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right ) (a+b x)^{3/2} \sqrt{c+d x}}{192 b^2 d^4}+\frac{\left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right ) (a+b x)^{5/2} \sqrt{c+d x}}{240 b^2 d^3}-\frac{3 (3 b c+a d) (a+b x)^{7/2} \sqrt{c+d x}}{40 b^2 d^2}+\frac{x (a+b x)^{7/2} \sqrt{c+d x}}{5 b d}-\frac{\left ((b c-a d)^3 \left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right )\right ) \int \frac{1}{\sqrt{a+b x} \sqrt{c+d x}} \, dx}{256 b^2 d^5}\\ &=\frac{(b c-a d)^2 \left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right ) \sqrt{a+b x} \sqrt{c+d x}}{128 b^2 d^5}-\frac{(b c-a d) \left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right ) (a+b x)^{3/2} \sqrt{c+d x}}{192 b^2 d^4}+\frac{\left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right ) (a+b x)^{5/2} \sqrt{c+d x}}{240 b^2 d^3}-\frac{3 (3 b c+a d) (a+b x)^{7/2} \sqrt{c+d x}}{40 b^2 d^2}+\frac{x (a+b x)^{7/2} \sqrt{c+d x}}{5 b d}-\frac{\left ((b c-a d)^3 \left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right )\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 )}{128 b^3 d^5}\\ &=\frac{(b c-a d)^2 \left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right ) \sqrt{a+b x} \sqrt{c+d x}}{128 b^2 d^5}-\frac{(b c-a d) \left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right ) (a+b x)^{3/2} \sqrt{c+d x}}{192 b^2 d^4}+\frac{\left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right ) (a+b x)^{5/2} \sqrt{c+d x}}{240 b^2 d^3}-\frac{3 (3 b c+a d) (a+b x)^{7/2} \sqrt{c+d x}}{40 b^2 d^2}+\frac{x (a+b x)^{7/2} \sqrt{c+d x}}{5 b d}-\frac{\left ((b c-a d)^3 \left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right )\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 )}{128 b^3 d^5}\\ &=\frac{(b c-a d)^2 \left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right ) \sqrt{a+b x} \sqrt{c+d x}}{128 b^2 d^5}-\frac{(b c-a d) \left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right ) (a+b x)^{3/2} \sqrt{c+d x}}{192 b^2 d^4}+\frac{\left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right ) (a+b x)^{5/2} \sqrt{c+d x}}{240 b^2 d^3}-\frac{3 (3 b c+a d) (a+b x)^{7/2} \sqrt{c+d x}}{40 b^2 d^2}+\frac{x (a+b x)^{7/2} \sqrt{c+d x}}{5 b d}-\frac{(b c-a d)^3 \left (63 b^2 c^2+14 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 )}{128 b^{5/2} d^{11/2}}\\ \end{align*}
Mathematica [A] time = 1.07725, size = 249, normalized size = 0.79 \[ \frac{\sqrt{c+d x} \left (\frac{5 (b c-a d)^3 \left (3 a^2 d^2+14 a b c d+63 b^2 c^2\right ) \left (\frac{16 d^3 (a+b x)^3}{15 (b c-a d)^3}-\frac{4 d^2 (a+b x)^2}{3 (b c-a d)^2}+\frac{2 d (a+b x)}{b c-a d}-\frac{2 \sqrt{d} \sqrt{a+b x} \sinh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b c-a d}}\right )}{\sqrt{b c-a d} \sqrt{\frac{b (c+d x)}{b c-a d}}}\right )}{4 b d^5}-\frac{24 (a+b x)^4 (a d+3 b c)}{b d}+64 x (a+b x)^4\right )}{320 b d \sqrt{a+b x}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.02, size = 788, 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 = 2.7204, size = 1598, normalized size = 5.09 \begin{align*} \left [-\frac{15 \,{\left (63 \, b^{5} c^{5} - 175 \, a b^{4} c^{4} d + 150 \, a^{2} b^{3} c^{3} d^{2} - 30 \, a^{3} b^{2} c^{2} d^{3} - 5 \, a^{4} b c d^{4} - 3 \, a^{5} d^{5}\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 (384 \, b^{5} d^{5} x^{4} + 945 \, b^{5} c^{4} d - 2310 \, a b^{4} c^{3} d^{2} + 1564 \, a^{2} b^{3} c^{2} d^{3} - 90 \, a^{3} b^{2} c d^{4} - 45 \, a^{4} b d^{5} - 144 \,{\left (3 \, b^{5} c d^{4} - 7 \, a b^{4} d^{5}\right )} x^{3} + 8 \,{\left (63 \, b^{5} c^{2} d^{3} - 148 \, a b^{4} c d^{4} + 93 \, a^{2} b^{3} d^{5}\right )} x^{2} - 2 \,{\left (315 \, b^{5} c^{3} d^{2} - 749 \, a b^{4} c^{2} d^{3} + 481 \, a^{2} b^{3} c d^{4} - 15 \, a^{3} b^{2} d^{5}\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{7680 \, b^{3} d^{6}}, \frac{15 \,{\left (63 \, b^{5} c^{5} - 175 \, a b^{4} c^{4} d + 150 \, a^{2} b^{3} c^{3} d^{2} - 30 \, a^{3} b^{2} c^{2} d^{3} - 5 \, a^{4} b c d^{4} - 3 \, a^{5} d^{5}\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 (384 \, b^{5} d^{5} x^{4} + 945 \, b^{5} c^{4} d - 2310 \, a b^{4} c^{3} d^{2} + 1564 \, a^{2} b^{3} c^{2} d^{3} - 90 \, a^{3} b^{2} c d^{4} - 45 \, a^{4} b d^{5} - 144 \,{\left (3 \, b^{5} c d^{4} - 7 \, a b^{4} d^{5}\right )} x^{3} + 8 \,{\left (63 \, b^{5} c^{2} d^{3} - 148 \, a b^{4} c d^{4} + 93 \, a^{2} b^{3} d^{5}\right )} x^{2} - 2 \,{\left (315 \, b^{5} c^{3} d^{2} - 749 \, a b^{4} c^{2} d^{3} + 481 \, a^{2} b^{3} c d^{4} - 15 \, a^{3} b^{2} d^{5}\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{3840 \, b^{3} d^{6}}\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 = 1.31021, size = 516, normalized size = 1.64 \begin{align*} \frac{{\left (\sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}{\left (2 \,{\left (4 \,{\left (b x + a\right )}{\left (6 \,{\left (b x + a\right )}{\left (\frac{8 \,{\left (b x + a\right )}}{b^{3} d} - \frac{9 \, b^{7} c d^{7} + 11 \, a b^{6} d^{8}}{b^{9} d^{9}}\right )} + \frac{63 \, b^{8} c^{2} d^{6} + 14 \, a b^{7} c d^{7} + 3 \, a^{2} b^{6} d^{8}}{b^{9} d^{9}}\right )} - \frac{5 \,{\left (63 \, b^{9} c^{3} d^{5} - 49 \, a b^{8} c^{2} d^{6} - 11 \, a^{2} b^{7} c d^{7} - 3 \, a^{3} b^{6} d^{8}\right )}}{b^{9} d^{9}}\right )}{\left (b x + a\right )} + \frac{15 \,{\left (63 \, b^{10} c^{4} d^{4} - 112 \, a b^{9} c^{3} d^{5} + 38 \, a^{2} b^{8} c^{2} d^{6} + 8 \, a^{3} b^{7} c d^{7} + 3 \, a^{4} b^{6} d^{8}\right )}}{b^{9} d^{9}}\right )} \sqrt{b x + a} + \frac{15 \,{\left (63 \, b^{5} c^{5} - 175 \, a b^{4} c^{4} d + 150 \, a^{2} b^{3} c^{3} d^{2} - 30 \, a^{3} b^{2} c^{2} d^{3} - 5 \, a^{4} b c d^{4} - 3 \, a^{5} d^{5}\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 )}{\sqrt{b d} b^{2} d^{5}}\right )} b}{1920 \,{\left | b \right |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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