Optimal. Leaf size=217 \[ \frac{2 d \sqrt{c+d x^2} \left (3 a^2 d^2-14 a b c d+35 b^2 c^2\right )}{105 b^2 \sqrt{a+b x^2} (b c-a d)^4}-\frac{\sqrt{c+d x^2} \left (3 a^2 d^2-14 a b c d+35 b^2 c^2\right )}{105 b^2 \left (a+b x^2\right )^{3/2} (b c-a d)^3}-\frac{a^2 \sqrt{c+d x^2}}{7 b^2 \left (a+b x^2\right )^{7/2} (b c-a d)}+\frac{2 a \sqrt{c+d x^2} (7 b c-4 a d)}{35 b^2 \left (a+b x^2\right )^{5/2} (b c-a d)^2} \]
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Rubi [A] time = 0.268655, antiderivative size = 217, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192, Rules used = {446, 89, 78, 45, 37} \[ \frac{2 d \sqrt{c+d x^2} \left (3 a^2 d^2-14 a b c d+35 b^2 c^2\right )}{105 b^2 \sqrt{a+b x^2} (b c-a d)^4}-\frac{\sqrt{c+d x^2} \left (3 a^2 d^2-14 a b c d+35 b^2 c^2\right )}{105 b^2 \left (a+b x^2\right )^{3/2} (b c-a d)^3}-\frac{a^2 \sqrt{c+d x^2}}{7 b^2 \left (a+b x^2\right )^{7/2} (b c-a d)}+\frac{2 a \sqrt{c+d x^2} (7 b c-4 a d)}{35 b^2 \left (a+b x^2\right )^{5/2} (b c-a d)^2} \]
Antiderivative was successfully verified.
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Rule 446
Rule 89
Rule 78
Rule 45
Rule 37
Rubi steps
\begin{align*} \int \frac{x^5}{\left (a+b x^2\right )^{9/2} \sqrt{c+d x^2}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x^2}{(a+b x)^{9/2} \sqrt{c+d x}} \, dx,x,x^2\right )\\ &=-\frac{a^2 \sqrt{c+d x^2}}{7 b^2 (b c-a d) \left (a+b x^2\right )^{7/2}}+\frac{\operatorname{Subst}\left (\int \frac{-\frac{1}{2} a (7 b c-a d)+\frac{7}{2} b (b c-a d) x}{(a+b x)^{7/2} \sqrt{c+d x}} \, dx,x,x^2\right )}{7 b^2 (b c-a d)}\\ &=-\frac{a^2 \sqrt{c+d x^2}}{7 b^2 (b c-a d) \left (a+b x^2\right )^{7/2}}+\frac{2 a (7 b c-4 a d) \sqrt{c+d x^2}}{35 b^2 (b c-a d)^2 \left (a+b x^2\right )^{5/2}}+\frac{\left (35 b^2 c^2-14 a b c d+3 a^2 d^2\right ) \operatorname{Subst}\left (\int \frac{1}{(a+b x)^{5/2} \sqrt{c+d x}} \, dx,x,x^2\right )}{70 b^2 (b c-a d)^2}\\ &=-\frac{a^2 \sqrt{c+d x^2}}{7 b^2 (b c-a d) \left (a+b x^2\right )^{7/2}}+\frac{2 a (7 b c-4 a d) \sqrt{c+d x^2}}{35 b^2 (b c-a d)^2 \left (a+b x^2\right )^{5/2}}-\frac{\left (35 b^2 c^2-14 a b c d+3 a^2 d^2\right ) \sqrt{c+d x^2}}{105 b^2 (b c-a d)^3 \left (a+b x^2\right )^{3/2}}-\frac{\left (d \left (35 b^2 c^2-14 a b c d+3 a^2 d^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{(a+b x)^{3/2} \sqrt{c+d x}} \, dx,x,x^2\right )}{105 b^2 (b c-a d)^3}\\ &=-\frac{a^2 \sqrt{c+d x^2}}{7 b^2 (b c-a d) \left (a+b x^2\right )^{7/2}}+\frac{2 a (7 b c-4 a d) \sqrt{c+d x^2}}{35 b^2 (b c-a d)^2 \left (a+b x^2\right )^{5/2}}-\frac{\left (35 b^2 c^2-14 a b c d+3 a^2 d^2\right ) \sqrt{c+d x^2}}{105 b^2 (b c-a d)^3 \left (a+b x^2\right )^{3/2}}+\frac{2 d \left (35 b^2 c^2-14 a b c d+3 a^2 d^2\right ) \sqrt{c+d x^2}}{105 b^2 (b c-a d)^4 \sqrt{a+b x^2}}\\ \end{align*}
Mathematica [A] time = 0.0763455, size = 151, normalized size = 0.7 \[ \frac{\sqrt{c+d x^2} \left (a^2 b \left (200 c^2 d x^2-8 c^3-101 c d^2 x^4+6 d^3 x^6\right )+7 a^3 d \left (8 c^2-4 c d x^2+3 d^2 x^4\right )-7 a b^2 c x^2 \left (4 c^2-37 c d x^2+4 d^2 x^4\right )-35 b^3 c^2 x^4 \left (c-2 d x^2\right )\right )}{105 \left (a+b x^2\right )^{7/2} (b c-a d)^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 213, normalized size = 1. \begin{align*}{\frac{6\,{a}^{2}b{d}^{3}{x}^{6}-28\,a{b}^{2}c{d}^{2}{x}^{6}+70\,{b}^{3}{c}^{2}d{x}^{6}+21\,{a}^{3}{d}^{3}{x}^{4}-101\,{a}^{2}bc{d}^{2}{x}^{4}+259\,a{b}^{2}{c}^{2}d{x}^{4}-35\,{b}^{3}{c}^{3}{x}^{4}-28\,{a}^{3}c{d}^{2}{x}^{2}+200\,{a}^{2}b{c}^{2}d{x}^{2}-28\,a{b}^{2}{c}^{3}{x}^{2}+56\,{a}^{3}{c}^{2}d-8\,{a}^{2}b{c}^{3}}{105\,{a}^{4}{d}^{4}-420\,{a}^{3}bc{d}^{3}+630\,{a}^{2}{c}^{2}{d}^{2}{b}^{2}-420\,a{c}^{3}d{b}^{3}+105\,{c}^{4}{b}^{4}}\sqrt{d{x}^{2}+c} \left ( b{x}^{2}+a \right ) ^{-{\frac{7}{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 = 8.54495, size = 914, normalized size = 4.21 \begin{align*} \frac{{\left (2 \,{\left (35 \, b^{3} c^{2} d - 14 \, a b^{2} c d^{2} + 3 \, a^{2} b d^{3}\right )} x^{6} - 8 \, a^{2} b c^{3} + 56 \, a^{3} c^{2} d -{\left (35 \, b^{3} c^{3} - 259 \, a b^{2} c^{2} d + 101 \, a^{2} b c d^{2} - 21 \, a^{3} d^{3}\right )} x^{4} - 4 \,{\left (7 \, a b^{2} c^{3} - 50 \, a^{2} b c^{2} d + 7 \, a^{3} c d^{2}\right )} x^{2}\right )} \sqrt{b x^{2} + a} \sqrt{d x^{2} + c}}{105 \,{\left (a^{4} b^{4} c^{4} - 4 \, a^{5} b^{3} c^{3} d + 6 \, a^{6} b^{2} c^{2} d^{2} - 4 \, a^{7} b c d^{3} + a^{8} d^{4} +{\left (b^{8} c^{4} - 4 \, a b^{7} c^{3} d + 6 \, a^{2} b^{6} c^{2} d^{2} - 4 \, a^{3} b^{5} c d^{3} + a^{4} b^{4} d^{4}\right )} x^{8} + 4 \,{\left (a b^{7} c^{4} - 4 \, a^{2} b^{6} c^{3} d + 6 \, a^{3} b^{5} c^{2} d^{2} - 4 \, a^{4} b^{4} c d^{3} + a^{5} b^{3} d^{4}\right )} x^{6} + 6 \,{\left (a^{2} b^{6} c^{4} - 4 \, a^{3} b^{5} c^{3} d + 6 \, a^{4} b^{4} c^{2} d^{2} - 4 \, a^{5} b^{3} c d^{3} + a^{6} b^{2} d^{4}\right )} x^{4} + 4 \,{\left (a^{3} b^{5} c^{4} - 4 \, a^{4} b^{4} c^{3} d + 6 \, a^{5} b^{3} c^{2} d^{2} - 4 \, a^{6} b^{2} c d^{3} + a^{7} b d^{4}\right )} x^{2}\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 = 1.48882, size = 1407, normalized size = 6.48 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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