Optimal. Leaf size=141 \[ -\frac{3 \sqrt{a+b x^2} \sqrt{c+d x^2} (a d+b c)}{8 b^2 d^2}-\frac{\left (4 a b c d-3 (a d+b c)^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x^2}}{\sqrt{b} \sqrt{c+d x^2}}\right )}{8 b^{5/2} d^{5/2}}+\frac{x^2 \sqrt{a+b x^2} \sqrt{c+d x^2}}{4 b d} \]
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Rubi [A] time = 0.158631, antiderivative size = 141, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {446, 90, 80, 63, 217, 206} \[ -\frac{3 \sqrt{a+b x^2} \sqrt{c+d x^2} (a d+b c)}{8 b^2 d^2}-\frac{\left (4 a b c d-3 (a d+b c)^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x^2}}{\sqrt{b} \sqrt{c+d x^2}}\right )}{8 b^{5/2} d^{5/2}}+\frac{x^2 \sqrt{a+b x^2} \sqrt{c+d x^2}}{4 b d} \]
Antiderivative was successfully verified.
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Rule 446
Rule 90
Rule 80
Rule 63
Rule 217
Rule 206
Rubi steps
\begin{align*} \int \frac{x^5}{\sqrt{a+b x^2} \sqrt{c+d x^2}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{a+b x} \sqrt{c+d x}} \, dx,x,x^2\right )\\ &=\frac{x^2 \sqrt{a+b x^2} \sqrt{c+d x^2}}{4 b d}+\frac{\operatorname{Subst}\left (\int \frac{-a c-\frac{3}{2} (b c+a d) x}{\sqrt{a+b x} \sqrt{c+d x}} \, dx,x,x^2\right )}{4 b d}\\ &=-\frac{3 (b c+a d) \sqrt{a+b x^2} \sqrt{c+d x^2}}{8 b^2 d^2}+\frac{x^2 \sqrt{a+b x^2} \sqrt{c+d x^2}}{4 b d}-\frac{\left (4 a b c d-3 (b c+a d)^2\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x} \sqrt{c+d x}} \, dx,x,x^2\right )}{16 b^2 d^2}\\ &=-\frac{3 (b c+a d) \sqrt{a+b x^2} \sqrt{c+d x^2}}{8 b^2 d^2}+\frac{x^2 \sqrt{a+b x^2} \sqrt{c+d x^2}}{4 b d}-\frac{\left (4 a b c d-3 (b c+a 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^2}\right )}{8 b^3 d^2}\\ &=-\frac{3 (b c+a d) \sqrt{a+b x^2} \sqrt{c+d x^2}}{8 b^2 d^2}+\frac{x^2 \sqrt{a+b x^2} \sqrt{c+d x^2}}{4 b d}-\frac{\left (4 a b c d-3 (b c+a d)^2\right ) \operatorname{Subst}\left (\int \frac{1}{1-\frac{d x^2}{b}} \, dx,x,\frac{\sqrt{a+b x^2}}{\sqrt{c+d x^2}}\right )}{8 b^3 d^2}\\ &=-\frac{3 (b c+a d) \sqrt{a+b x^2} \sqrt{c+d x^2}}{8 b^2 d^2}+\frac{x^2 \sqrt{a+b x^2} \sqrt{c+d x^2}}{4 b d}-\frac{\left (4 a b c d-3 (b c+a d)^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x^2}}{\sqrt{b} \sqrt{c+d x^2}}\right )}{8 b^{5/2} d^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.252941, size = 154, normalized size = 1.09 \[ \frac{\sqrt{b c-a d} \left (3 a^2 d^2+2 a b c d+3 b^2 c^2\right ) \sqrt{\frac{b \left (c+d x^2\right )}{b c-a d}} \sinh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x^2}}{\sqrt{b c-a d}}\right )+b \sqrt{d} \sqrt{a+b x^2} \left (c+d x^2\right ) \left (-3 a d-3 b c+2 b d x^2\right )}{8 b^3 d^{5/2} \sqrt{c+d x^2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.029, size = 340, normalized size = 2.4 \begin{align*}{\frac{1}{16\,{b}^{2}{d}^{2}} \left ( 4\,\sqrt{bd}\sqrt{bd{x}^{4}+ad{x}^{2}+bc{x}^{2}+ac}{x}^{2}bd+3\,{d}^{2}\ln \left ( 1/2\,{\frac{2\,d{x}^{2}b+2\,\sqrt{bd{x}^{4}+ad{x}^{2}+bc{x}^{2}+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){a}^{2}+2\,\ln \left ( 1/2\,{\frac{2\,d{x}^{2}b+2\,\sqrt{bd{x}^{4}+ad{x}^{2}+bc{x}^{2}+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ) cabd+3\,{b}^{2}\ln \left ( 1/2\,{\frac{2\,d{x}^{2}b+2\,\sqrt{bd{x}^{4}+ad{x}^{2}+bc{x}^{2}+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){c}^{2}-6\,\sqrt{bd}\sqrt{bd{x}^{4}+ad{x}^{2}+bc{x}^{2}+ac}ad-6\,\sqrt{bd}\sqrt{bd{x}^{4}+ad{x}^{2}+bc{x}^{2}+ac}bc \right ) \sqrt{b{x}^{2}+a}\sqrt{d{x}^{2}+c}{\frac{1}{\sqrt{bd{x}^{4}+ad{x}^{2}+bc{x}^{2}+ac}}}{\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 = 2.05447, size = 761, normalized size = 5.4 \begin{align*} \left [\frac{{\left (3 \, b^{2} c^{2} + 2 \, a b c d + 3 \, a^{2} d^{2}\right )} \sqrt{b d} \log \left (8 \, b^{2} d^{2} x^{4} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 8 \,{\left (b^{2} c d + a b d^{2}\right )} x^{2} + 4 \,{\left (2 \, b d x^{2} + b c + a d\right )} \sqrt{b x^{2} + a} \sqrt{d x^{2} + c} \sqrt{b d}\right ) + 4 \,{\left (2 \, b^{2} d^{2} x^{2} - 3 \, b^{2} c d - 3 \, a b d^{2}\right )} \sqrt{b x^{2} + a} \sqrt{d x^{2} + c}}{32 \, b^{3} d^{3}}, -\frac{{\left (3 \, b^{2} c^{2} + 2 \, a b c d + 3 \, a^{2} d^{2}\right )} \sqrt{-b d} \arctan \left (\frac{{\left (2 \, b d x^{2} + b c + a d\right )} \sqrt{b x^{2} + a} \sqrt{d x^{2} + c} \sqrt{-b d}}{2 \,{\left (b^{2} d^{2} x^{4} + a b c d +{\left (b^{2} c d + a b d^{2}\right )} x^{2}\right )}}\right ) - 2 \,{\left (2 \, b^{2} d^{2} x^{2} - 3 \, b^{2} c d - 3 \, a b d^{2}\right )} \sqrt{b x^{2} + a} \sqrt{d x^{2} + c}}{16 \, b^{3} d^{3}}\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^{5}}{\sqrt{a + b x^{2}} \sqrt{c + d x^{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.23209, size = 212, normalized size = 1.5 \begin{align*} \frac{\sqrt{b^{2} c +{\left (b x^{2} + a\right )} b d - a b d} \sqrt{b x^{2} + a}{\left (\frac{2 \,{\left (b x^{2} + a\right )}}{b d} - \frac{3 \, b^{2} c d + 5 \, a b d^{2}}{b^{2} d^{3}}\right )} - \frac{{\left (3 \, b^{2} c^{2} + 2 \, a b c d + 3 \, a^{2} d^{2}\right )} \log \left ({\left | -\sqrt{b x^{2} + a} \sqrt{b d} + \sqrt{b^{2} c +{\left (b x^{2} + a\right )} b d - a b d} \right |}\right )}{\sqrt{b d} d^{2}}}{8 \, b{\left | b \right |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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