Optimal. Leaf size=138 \[ \frac{a \sqrt{c+d x^2}}{5 b \left (a+b x^2\right )^{5/2} (b c-a d)}+\frac{2 d \sqrt{c+d x^2} (5 b c-a d)}{15 b \sqrt{a+b x^2} (b c-a d)^3}-\frac{\sqrt{c+d x^2} (5 b c-a d)}{15 b \left (a+b x^2\right )^{3/2} (b c-a d)^2} \]
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Rubi [A] time = 0.100783, antiderivative size = 138, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {446, 78, 45, 37} \[ \frac{a \sqrt{c+d x^2}}{5 b \left (a+b x^2\right )^{5/2} (b c-a d)}+\frac{2 d \sqrt{c+d x^2} (5 b c-a d)}{15 b \sqrt{a+b x^2} (b c-a d)^3}-\frac{\sqrt{c+d x^2} (5 b c-a d)}{15 b \left (a+b x^2\right )^{3/2} (b c-a d)^2} \]
Antiderivative was successfully verified.
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Rule 446
Rule 78
Rule 45
Rule 37
Rubi steps
\begin{align*} \int \frac{x^3}{\left (a+b x^2\right )^{7/2} \sqrt{c+d x^2}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x}{(a+b x)^{7/2} \sqrt{c+d x}} \, dx,x,x^2\right )\\ &=\frac{a \sqrt{c+d x^2}}{5 b (b c-a d) \left (a+b x^2\right )^{5/2}}+\frac{(5 b c-a d) \operatorname{Subst}\left (\int \frac{1}{(a+b x)^{5/2} \sqrt{c+d x}} \, dx,x,x^2\right )}{10 b (b c-a d)}\\ &=\frac{a \sqrt{c+d x^2}}{5 b (b c-a d) \left (a+b x^2\right )^{5/2}}-\frac{(5 b c-a d) \sqrt{c+d x^2}}{15 b (b c-a d)^2 \left (a+b x^2\right )^{3/2}}-\frac{(d (5 b c-a d)) \operatorname{Subst}\left (\int \frac{1}{(a+b x)^{3/2} \sqrt{c+d x}} \, dx,x,x^2\right )}{15 b (b c-a d)^2}\\ &=\frac{a \sqrt{c+d x^2}}{5 b (b c-a d) \left (a+b x^2\right )^{5/2}}-\frac{(5 b c-a d) \sqrt{c+d x^2}}{15 b (b c-a d)^2 \left (a+b x^2\right )^{3/2}}+\frac{2 d (5 b c-a d) \sqrt{c+d x^2}}{15 b (b c-a d)^3 \sqrt{a+b x^2}}\\ \end{align*}
Mathematica [A] time = 0.0353247, size = 91, normalized size = 0.66 \[ \frac{\sqrt{c+d x^2} \left (-5 a^2 d \left (d x^2-2 c\right )-2 a b \left (c^2-13 c d x^2+d^2 x^4\right )-5 b^2 c x^2 \left (c-2 d x^2\right )\right )}{15 \left (a+b x^2\right )^{5/2} (b c-a d)^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 125, normalized size = 0.9 \begin{align*} -{\frac{-2\,ab{d}^{2}{x}^{4}+10\,{b}^{2}cd{x}^{4}-5\,{a}^{2}{d}^{2}{x}^{2}+26\,abcd{x}^{2}-5\,{b}^{2}{c}^{2}{x}^{2}+10\,{a}^{2}cd-2\,ab{c}^{2}}{15\,{a}^{3}{d}^{3}-45\,{a}^{2}c{d}^{2}b+45\,a{c}^{2}d{b}^{2}-15\,{c}^{3}{b}^{3}}\sqrt{d{x}^{2}+c} \left ( b{x}^{2}+a \right ) ^{-{\frac{5}{2}}}} \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 [B] time = 4.32376, size = 537, normalized size = 3.89 \begin{align*} \frac{{\left (2 \,{\left (5 \, b^{2} c d - a b d^{2}\right )} x^{4} - 2 \, a b c^{2} + 10 \, a^{2} c d -{\left (5 \, b^{2} c^{2} - 26 \, a b c d + 5 \, a^{2} d^{2}\right )} x^{2}\right )} \sqrt{b x^{2} + a} \sqrt{d x^{2} + c}}{15 \,{\left (a^{3} b^{3} c^{3} - 3 \, a^{4} b^{2} c^{2} d + 3 \, a^{5} b c d^{2} - a^{6} d^{3} +{\left (b^{6} c^{3} - 3 \, a b^{5} c^{2} d + 3 \, a^{2} b^{4} c d^{2} - a^{3} b^{3} d^{3}\right )} x^{6} + 3 \,{\left (a b^{5} c^{3} - 3 \, a^{2} b^{4} c^{2} d + 3 \, a^{3} b^{3} c d^{2} - a^{4} b^{2} d^{3}\right )} x^{4} + 3 \,{\left (a^{2} b^{4} c^{3} - 3 \, a^{3} b^{3} c^{2} d + 3 \, a^{4} b^{2} c d^{2} - a^{5} b d^{3}\right )} x^{2}\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}}{\left (a + b x^{2}\right )^{\frac{7}{2}} \sqrt{c + d x^{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.31265, size = 637, normalized size = 4.62 \begin{align*} \frac{4 \,{\left (5 \, \sqrt{b d} b^{8} c^{3} d - 11 \, \sqrt{b d} a b^{7} c^{2} d^{2} + 7 \, \sqrt{b d} a^{2} b^{6} c d^{3} - \sqrt{b d} a^{3} b^{5} d^{4} - 25 \, \sqrt{b d}{\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 )}^{2} b^{6} c^{2} d + 30 \, \sqrt{b d}{\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 )}^{2} a b^{5} c d^{2} - 5 \, \sqrt{b d}{\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 )}^{2} a^{2} b^{4} d^{3} + 35 \, \sqrt{b d}{\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 )}^{4} b^{4} c d + 5 \, \sqrt{b d}{\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 )}^{4} a b^{3} d^{2} - 15 \, \sqrt{b d}{\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 )}^{6} b^{2} d\right )}}{15 \,{\left (b^{2} c - a b d -{\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 )}^{2}\right )}^{5} b{\left | b \right |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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