Optimal. Leaf size=169 \[ \frac{\sqrt{a+b x} \sqrt{c+d x} \left (15 a^2 d^2-10 b d x (a d+b c)+14 a b c d+15 b^2 c^2\right )}{24 b^3 d^3}-\frac{(a d+b c) \left (5 a^2 d^2-2 a b c d+5 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{8 b^{7/2} d^{7/2}}+\frac{x^2 \sqrt{a+b x} \sqrt{c+d x}}{3 b d} \]
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Rubi [A] time = 0.114809, antiderivative size = 169, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227, Rules used = {100, 147, 63, 217, 206} \[ \frac{\sqrt{a+b x} \sqrt{c+d x} \left (15 a^2 d^2-10 b d x (a d+b c)+14 a b c d+15 b^2 c^2\right )}{24 b^3 d^3}-\frac{(a d+b c) \left (5 a^2 d^2-2 a b c d+5 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{8 b^{7/2} d^{7/2}}+\frac{x^2 \sqrt{a+b x} \sqrt{c+d x}}{3 b d} \]
Antiderivative was successfully verified.
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Rule 100
Rule 147
Rule 63
Rule 217
Rule 206
Rubi steps
\begin{align*} \int \frac{x^3}{\sqrt{a+b x} \sqrt{c+d x}} \, dx &=\frac{x^2 \sqrt{a+b x} \sqrt{c+d x}}{3 b d}+\frac{\int \frac{x \left (-2 a c-\frac{5}{2} (b c+a d) x\right )}{\sqrt{a+b x} \sqrt{c+d x}} \, dx}{3 b d}\\ &=\frac{x^2 \sqrt{a+b x} \sqrt{c+d x}}{3 b d}+\frac{\sqrt{a+b x} \sqrt{c+d x} \left (15 b^2 c^2+14 a b c d+15 a^2 d^2-10 b d (b c+a d) x\right )}{24 b^3 d^3}-\frac{\left ((b c+a d) \left (5 b^2 c^2-2 a b c d+5 a^2 d^2\right )\right ) \int \frac{1}{\sqrt{a+b x} \sqrt{c+d x}} \, dx}{16 b^3 d^3}\\ &=\frac{x^2 \sqrt{a+b x} \sqrt{c+d x}}{3 b d}+\frac{\sqrt{a+b x} \sqrt{c+d x} \left (15 b^2 c^2+14 a b c d+15 a^2 d^2-10 b d (b c+a d) x\right )}{24 b^3 d^3}-\frac{\left ((b c+a d) \left (5 b^2 c^2-2 a b c d+5 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 )}{8 b^4 d^3}\\ &=\frac{x^2 \sqrt{a+b x} \sqrt{c+d x}}{3 b d}+\frac{\sqrt{a+b x} \sqrt{c+d x} \left (15 b^2 c^2+14 a b c d+15 a^2 d^2-10 b d (b c+a d) x\right )}{24 b^3 d^3}-\frac{\left ((b c+a d) \left (5 b^2 c^2-2 a b c d+5 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 )}{8 b^4 d^3}\\ &=\frac{x^2 \sqrt{a+b x} \sqrt{c+d x}}{3 b d}+\frac{\sqrt{a+b x} \sqrt{c+d x} \left (15 b^2 c^2+14 a b c d+15 a^2 d^2-10 b d (b c+a d) x\right )}{24 b^3 d^3}-\frac{(b c+a d) \left (5 b^2 c^2-2 a b c d+5 a^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{8 b^{7/2} d^{7/2}}\\ \end{align*}
Mathematica [A] time = 0.745289, size = 188, normalized size = 1.11 \[ \frac{b \sqrt{d} \sqrt{a+b x} (c+d x) \left (15 a^2 d^2+2 a b d (7 c-5 d x)+b^2 \left (15 c^2-10 c d x+8 d^2 x^2\right )\right )-3 \sqrt{b c-a d} \left (3 a^2 b c d^2+5 a^3 d^3+3 a b^2 c^2 d+5 b^3 c^3\right ) \sqrt{\frac{b (c+d x)}{b c-a d}} \sinh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b c-a d}}\right )}{24 b^4 d^{7/2} \sqrt{c+d x}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.023, size = 395, normalized size = 2.3 \begin{align*} -{\frac{1}{48\,{b}^{3}{d}^{3}} \left ( -16\,{x}^{2}{b}^{2}{d}^{2}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+15\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){a}^{3}{d}^{3}+9\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){a}^{2}bc{d}^{2}+9\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ) a{b}^{2}{c}^{2}d+15\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){b}^{3}{c}^{3}+20\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}xab{d}^{2}+20\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}x{b}^{2}cd-30\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}{a}^{2}{d}^{2}-28\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}abcd-30\,\sqrt{bd}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }{b}^{2}{c}^{2} \right ) \sqrt{bx+a}\sqrt{dx+c}{\frac{1}{\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }}}{\frac{1}{\sqrt{bd}}}} \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 = 2.70651, size = 933, normalized size = 5.52 \begin{align*} \left [\frac{3 \,{\left (5 \, b^{3} c^{3} + 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} + 5 \, a^{3} d^{3}\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 (8 \, b^{3} d^{3} x^{2} + 15 \, b^{3} c^{2} d + 14 \, a b^{2} c d^{2} + 15 \, a^{2} b d^{3} - 10 \,{\left (b^{3} c d^{2} + a b^{2} d^{3}\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{96 \, b^{4} d^{4}}, \frac{3 \,{\left (5 \, b^{3} c^{3} + 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} + 5 \, a^{3} d^{3}\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 (8 \, b^{3} d^{3} x^{2} + 15 \, b^{3} c^{2} d + 14 \, a b^{2} c d^{2} + 15 \, a^{2} b d^{3} - 10 \,{\left (b^{3} c d^{2} + a b^{2} d^{3}\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{48 \, b^{4} d^{4}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{3}}{\sqrt{a + b x} \sqrt{c + d x}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.27297, size = 290, normalized size = 1.72 \begin{align*} \frac{{\left (\sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d} \sqrt{b x + a}{\left (2 \,{\left (b x + a\right )}{\left (\frac{4 \,{\left (b x + a\right )}}{b^{4} d} - \frac{5 \, b^{12} c d^{3} + 13 \, a b^{11} d^{4}}{b^{15} d^{5}}\right )} + \frac{3 \,{\left (5 \, b^{13} c^{2} d^{2} + 8 \, a b^{12} c d^{3} + 11 \, a^{2} b^{11} d^{4}\right )}}{b^{15} d^{5}}\right )} + \frac{3 \,{\left (5 \, b^{3} c^{3} + 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} + 5 \, a^{3} d^{3}\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^{3} d^{3}}\right )} b}{24 \,{\left | b \right |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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