Optimal. Leaf size=314 \[ \frac{\sqrt{a+b x} (c+d x)^{5/2} \left (63 a^2 d^2+14 a b c d+3 b^2 c^2\right )}{240 b^3 d^2}+\frac{\sqrt{a+b x} (c+d x)^{3/2} (b c-a d) \left (63 a^2 d^2+14 a b c d+3 b^2 c^2\right )}{192 b^4 d^2}+\frac{\sqrt{a+b x} \sqrt{c+d x} (b c-a d)^2 \left (63 a^2 d^2+14 a b c d+3 b^2 c^2\right )}{128 b^5 d^2}+\frac{(b c-a d)^3 \left (63 a^2 d^2+14 a b c d+3 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{128 b^{11/2} d^{5/2}}-\frac{3 \sqrt{a+b x} (c+d x)^{7/2} (3 a d+b c)}{40 b^2 d^2}+\frac{x \sqrt{a+b x} (c+d x)^{7/2}}{5 b d} \]
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Rubi [A] time = 0.28086, 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} \[ \frac{\sqrt{a+b x} (c+d x)^{5/2} \left (63 a^2 d^2+14 a b c d+3 b^2 c^2\right )}{240 b^3 d^2}+\frac{\sqrt{a+b x} (c+d x)^{3/2} (b c-a d) \left (63 a^2 d^2+14 a b c d+3 b^2 c^2\right )}{192 b^4 d^2}+\frac{\sqrt{a+b x} \sqrt{c+d x} (b c-a d)^2 \left (63 a^2 d^2+14 a b c d+3 b^2 c^2\right )}{128 b^5 d^2}+\frac{(b c-a d)^3 \left (63 a^2 d^2+14 a b c d+3 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{128 b^{11/2} d^{5/2}}-\frac{3 \sqrt{a+b x} (c+d x)^{7/2} (3 a d+b c)}{40 b^2 d^2}+\frac{x \sqrt{a+b x} (c+d x)^{7/2}}{5 b d} \]
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 (c+d x)^{5/2}}{\sqrt{a+b x}} \, dx &=\frac{x \sqrt{a+b x} (c+d x)^{7/2}}{5 b d}+\frac{\int \frac{(c+d x)^{5/2} \left (-a c-\frac{3}{2} (b c+3 a d) x\right )}{\sqrt{a+b x}} \, dx}{5 b d}\\ &=-\frac{3 (b c+3 a d) \sqrt{a+b x} (c+d x)^{7/2}}{40 b^2 d^2}+\frac{x \sqrt{a+b x} (c+d x)^{7/2}}{5 b d}+\frac{\left (3 b^2 c^2+14 a b c d+63 a^2 d^2\right ) \int \frac{(c+d x)^{5/2}}{\sqrt{a+b x}} \, dx}{80 b^2 d^2}\\ &=\frac{\left (3 b^2 c^2+14 a b c d+63 a^2 d^2\right ) \sqrt{a+b x} (c+d x)^{5/2}}{240 b^3 d^2}-\frac{3 (b c+3 a d) \sqrt{a+b x} (c+d x)^{7/2}}{40 b^2 d^2}+\frac{x \sqrt{a+b x} (c+d x)^{7/2}}{5 b d}+\frac{\left ((b c-a d) \left (3 b^2 c^2+14 a b c d+63 a^2 d^2\right )\right ) \int \frac{(c+d x)^{3/2}}{\sqrt{a+b x}} \, dx}{96 b^3 d^2}\\ &=\frac{(b c-a d) \left (3 b^2 c^2+14 a b c d+63 a^2 d^2\right ) \sqrt{a+b x} (c+d x)^{3/2}}{192 b^4 d^2}+\frac{\left (3 b^2 c^2+14 a b c d+63 a^2 d^2\right ) \sqrt{a+b x} (c+d x)^{5/2}}{240 b^3 d^2}-\frac{3 (b c+3 a d) \sqrt{a+b x} (c+d x)^{7/2}}{40 b^2 d^2}+\frac{x \sqrt{a+b x} (c+d x)^{7/2}}{5 b d}+\frac{\left ((b c-a d)^2 \left (3 b^2 c^2+14 a b c d+63 a^2 d^2\right )\right ) \int \frac{\sqrt{c+d x}}{\sqrt{a+b x}} \, dx}{128 b^4 d^2}\\ &=\frac{(b c-a d)^2 \left (3 b^2 c^2+14 a b c d+63 a^2 d^2\right ) \sqrt{a+b x} \sqrt{c+d x}}{128 b^5 d^2}+\frac{(b c-a d) \left (3 b^2 c^2+14 a b c d+63 a^2 d^2\right ) \sqrt{a+b x} (c+d x)^{3/2}}{192 b^4 d^2}+\frac{\left (3 b^2 c^2+14 a b c d+63 a^2 d^2\right ) \sqrt{a+b x} (c+d x)^{5/2}}{240 b^3 d^2}-\frac{3 (b c+3 a d) \sqrt{a+b x} (c+d x)^{7/2}}{40 b^2 d^2}+\frac{x \sqrt{a+b x} (c+d x)^{7/2}}{5 b d}+\frac{\left ((b c-a d)^3 \left (3 b^2 c^2+14 a b c d+63 a^2 d^2\right )\right ) \int \frac{1}{\sqrt{a+b x} \sqrt{c+d x}} \, dx}{256 b^5 d^2}\\ &=\frac{(b c-a d)^2 \left (3 b^2 c^2+14 a b c d+63 a^2 d^2\right ) \sqrt{a+b x} \sqrt{c+d x}}{128 b^5 d^2}+\frac{(b c-a d) \left (3 b^2 c^2+14 a b c d+63 a^2 d^2\right ) \sqrt{a+b x} (c+d x)^{3/2}}{192 b^4 d^2}+\frac{\left (3 b^2 c^2+14 a b c d+63 a^2 d^2\right ) \sqrt{a+b x} (c+d x)^{5/2}}{240 b^3 d^2}-\frac{3 (b c+3 a d) \sqrt{a+b x} (c+d x)^{7/2}}{40 b^2 d^2}+\frac{x \sqrt{a+b x} (c+d x)^{7/2}}{5 b d}+\frac{\left ((b c-a d)^3 \left (3 b^2 c^2+14 a b c d+63 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^6 d^2}\\ &=\frac{(b c-a d)^2 \left (3 b^2 c^2+14 a b c d+63 a^2 d^2\right ) \sqrt{a+b x} \sqrt{c+d x}}{128 b^5 d^2}+\frac{(b c-a d) \left (3 b^2 c^2+14 a b c d+63 a^2 d^2\right ) \sqrt{a+b x} (c+d x)^{3/2}}{192 b^4 d^2}+\frac{\left (3 b^2 c^2+14 a b c d+63 a^2 d^2\right ) \sqrt{a+b x} (c+d x)^{5/2}}{240 b^3 d^2}-\frac{3 (b c+3 a d) \sqrt{a+b x} (c+d x)^{7/2}}{40 b^2 d^2}+\frac{x \sqrt{a+b x} (c+d x)^{7/2}}{5 b d}+\frac{\left ((b c-a d)^3 \left (3 b^2 c^2+14 a b c d+63 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^6 d^2}\\ &=\frac{(b c-a d)^2 \left (3 b^2 c^2+14 a b c d+63 a^2 d^2\right ) \sqrt{a+b x} \sqrt{c+d x}}{128 b^5 d^2}+\frac{(b c-a d) \left (3 b^2 c^2+14 a b c d+63 a^2 d^2\right ) \sqrt{a+b x} (c+d x)^{3/2}}{192 b^4 d^2}+\frac{\left (3 b^2 c^2+14 a b c d+63 a^2 d^2\right ) \sqrt{a+b x} (c+d x)^{5/2}}{240 b^3 d^2}-\frac{3 (b c+3 a d) \sqrt{a+b x} (c+d x)^{7/2}}{40 b^2 d^2}+\frac{x \sqrt{a+b x} (c+d x)^{7/2}}{5 b d}+\frac{(b c-a d)^3 \left (3 b^2 c^2+14 a b c d+63 a^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{128 b^{11/2} d^{5/2}}\\ \end{align*}
Mathematica [A] time = 1.21091, size = 239, normalized size = 0.76 \[ \frac{\sqrt{a+b x} (c+d x)^{7/2} \left (\frac{\left (63 a^2 d^2+14 a b c d+3 b^2 c^2\right ) \left (\sqrt{d} \sqrt{a+b x} \sqrt{\frac{b (c+d x)}{b c-a d}} \left (15 a^2 d^2-10 a b d (4 c+d x)+b^2 \left (33 c^2+26 c d x+8 d^2 x^2\right )\right )+15 (b c-a d)^{5/2} \sinh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b c-a d}}\right )\right )}{48 b^4 d^{3/2} \sqrt{a+b x} (c+d x)^3 \sqrt{\frac{b (c+d x)}{b c-a d}}}-\frac{9 a}{b}-\frac{3 c}{d}+8 x\right )}{40 b d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.022, 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 = 2.64971, size = 1600, normalized size = 5.1 \begin{align*} \left [-\frac{15 \,{\left (3 \, b^{5} c^{5} + 5 \, a b^{4} c^{4} d + 30 \, a^{2} b^{3} c^{3} d^{2} - 150 \, a^{3} b^{2} c^{2} d^{3} + 175 \, a^{4} b c d^{4} - 63 \, 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} - 45 \, b^{5} c^{4} d - 90 \, a b^{4} c^{3} d^{2} + 1564 \, a^{2} b^{3} c^{2} d^{3} - 2310 \, a^{3} b^{2} c d^{4} + 945 \, a^{4} b d^{5} + 144 \,{\left (7 \, b^{5} c d^{4} - 3 \, a b^{4} d^{5}\right )} x^{3} + 8 \,{\left (93 \, b^{5} c^{2} d^{3} - 148 \, a b^{4} c d^{4} + 63 \, a^{2} b^{3} d^{5}\right )} x^{2} + 2 \,{\left (15 \, b^{5} c^{3} d^{2} - 481 \, a b^{4} c^{2} d^{3} + 749 \, a^{2} b^{3} c d^{4} - 315 \, a^{3} b^{2} d^{5}\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{7680 \, b^{6} d^{3}}, -\frac{15 \,{\left (3 \, b^{5} c^{5} + 5 \, a b^{4} c^{4} d + 30 \, a^{2} b^{3} c^{3} d^{2} - 150 \, a^{3} b^{2} c^{2} d^{3} + 175 \, a^{4} b c d^{4} - 63 \, 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} - 45 \, b^{5} c^{4} d - 90 \, a b^{4} c^{3} d^{2} + 1564 \, a^{2} b^{3} c^{2} d^{3} - 2310 \, a^{3} b^{2} c d^{4} + 945 \, a^{4} b d^{5} + 144 \,{\left (7 \, b^{5} c d^{4} - 3 \, a b^{4} d^{5}\right )} x^{3} + 8 \,{\left (93 \, b^{5} c^{2} d^{3} - 148 \, a b^{4} c d^{4} + 63 \, a^{2} b^{3} d^{5}\right )} x^{2} + 2 \,{\left (15 \, b^{5} c^{3} d^{2} - 481 \, a b^{4} c^{2} d^{3} + 749 \, a^{2} b^{3} c d^{4} - 315 \, a^{3} b^{2} d^{5}\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{3840 \, b^{6} d^{3}}\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 [B] time = 1.9273, size = 1195, normalized size = 3.81 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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