Optimal. Leaf size=146 \[ \frac{5 a c^3 \sqrt{c x}}{2 b^2 \sqrt [4]{a+b x^2}}-\frac{5 a c^{7/2} \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt{c} \sqrt [4]{a+b x^2}}\right )}{4 b^{9/4}}-\frac{5 a c^{7/2} \tanh ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt{c} \sqrt [4]{a+b x^2}}\right )}{4 b^{9/4}}+\frac{c (c x)^{5/2}}{2 b \sqrt [4]{a+b x^2}} \]
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Rubi [A] time = 0.0763353, antiderivative size = 146, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.368, Rules used = {285, 288, 329, 240, 212, 208, 205} \[ \frac{5 a c^3 \sqrt{c x}}{2 b^2 \sqrt [4]{a+b x^2}}-\frac{5 a c^{7/2} \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt{c} \sqrt [4]{a+b x^2}}\right )}{4 b^{9/4}}-\frac{5 a c^{7/2} \tanh ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt{c} \sqrt [4]{a+b x^2}}\right )}{4 b^{9/4}}+\frac{c (c x)^{5/2}}{2 b \sqrt [4]{a+b x^2}} \]
Antiderivative was successfully verified.
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Rule 285
Rule 288
Rule 329
Rule 240
Rule 212
Rule 208
Rule 205
Rubi steps
\begin{align*} \int \frac{(c x)^{7/2}}{\left (a+b x^2\right )^{5/4}} \, dx &=\frac{c (c x)^{5/2}}{2 b \sqrt [4]{a+b x^2}}-\frac{\left (5 a c^2\right ) \int \frac{(c x)^{3/2}}{\left (a+b x^2\right )^{5/4}} \, dx}{4 b}\\ &=\frac{5 a c^3 \sqrt{c x}}{2 b^2 \sqrt [4]{a+b x^2}}+\frac{c (c x)^{5/2}}{2 b \sqrt [4]{a+b x^2}}-\frac{\left (5 a c^4\right ) \int \frac{1}{\sqrt{c x} \sqrt [4]{a+b x^2}} \, dx}{4 b^2}\\ &=\frac{5 a c^3 \sqrt{c x}}{2 b^2 \sqrt [4]{a+b x^2}}+\frac{c (c x)^{5/2}}{2 b \sqrt [4]{a+b x^2}}-\frac{\left (5 a c^3\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt [4]{a+\frac{b x^4}{c^2}}} \, dx,x,\sqrt{c x}\right )}{2 b^2}\\ &=\frac{5 a c^3 \sqrt{c x}}{2 b^2 \sqrt [4]{a+b x^2}}+\frac{c (c x)^{5/2}}{2 b \sqrt [4]{a+b x^2}}-\frac{\left (5 a c^3\right ) \operatorname{Subst}\left (\int \frac{1}{1-\frac{b x^4}{c^2}} \, dx,x,\frac{\sqrt{c x}}{\sqrt [4]{a+b x^2}}\right )}{2 b^2}\\ &=\frac{5 a c^3 \sqrt{c x}}{2 b^2 \sqrt [4]{a+b x^2}}+\frac{c (c x)^{5/2}}{2 b \sqrt [4]{a+b x^2}}-\frac{\left (5 a c^4\right ) \operatorname{Subst}\left (\int \frac{1}{c-\sqrt{b} x^2} \, dx,x,\frac{\sqrt{c x}}{\sqrt [4]{a+b x^2}}\right )}{4 b^2}-\frac{\left (5 a c^4\right ) \operatorname{Subst}\left (\int \frac{1}{c+\sqrt{b} x^2} \, dx,x,\frac{\sqrt{c x}}{\sqrt [4]{a+b x^2}}\right )}{4 b^2}\\ &=\frac{5 a c^3 \sqrt{c x}}{2 b^2 \sqrt [4]{a+b x^2}}+\frac{c (c x)^{5/2}}{2 b \sqrt [4]{a+b x^2}}-\frac{5 a c^{7/2} \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt{c} \sqrt [4]{a+b x^2}}\right )}{4 b^{9/4}}-\frac{5 a c^{7/2} \tanh ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt{c} \sqrt [4]{a+b x^2}}\right )}{4 b^{9/4}}\\ \end{align*}
Mathematica [C] time = 0.0365638, size = 63, normalized size = 0.43 \[ \frac{c (c x)^{5/2} \left (1-\sqrt [4]{\frac{b x^2}{a}+1} \, _2F_1\left (\frac{5}{4},\frac{5}{4};\frac{9}{4};-\frac{b x^2}{a}\right )\right )}{2 b \sqrt [4]{a+b x^2}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.06, size = 0, normalized size = 0. \begin{align*} \int{ \left ( cx \right ) ^{{\frac{7}{2}}} \left ( b{x}^{2}+a \right ) ^{-{\frac{5}{4}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (c x\right )^{\frac{7}{2}}}{{\left (b x^{2} + a\right )}^{\frac{5}{4}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.29662, size = 841, normalized size = 5.76 \begin{align*} \frac{4 \,{\left (b c^{3} x^{2} + 5 \, a c^{3}\right )}{\left (b x^{2} + a\right )}^{\frac{3}{4}} \sqrt{c x} + 20 \, \left (\frac{a^{4} c^{14}}{b^{9}}\right )^{\frac{1}{4}}{\left (b^{3} x^{2} + a b^{2}\right )} \arctan \left (-\frac{\left (\frac{a^{4} c^{14}}{b^{9}}\right )^{\frac{3}{4}}{\left (b x^{2} + a\right )}^{\frac{3}{4}} \sqrt{c x} a b^{7} c^{3} -{\left (b^{8} x^{2} + a b^{7}\right )} \left (\frac{a^{4} c^{14}}{b^{9}}\right )^{\frac{3}{4}} \sqrt{\frac{\sqrt{b x^{2} + a} a^{2} c^{7} x + \sqrt{\frac{a^{4} c^{14}}{b^{9}}}{\left (b^{5} x^{2} + a b^{4}\right )}}{b x^{2} + a}}}{a^{4} b c^{14} x^{2} + a^{5} c^{14}}\right ) - 5 \, \left (\frac{a^{4} c^{14}}{b^{9}}\right )^{\frac{1}{4}}{\left (b^{3} x^{2} + a b^{2}\right )} \log \left (\frac{5 \,{\left ({\left (b x^{2} + a\right )}^{\frac{3}{4}} \sqrt{c x} a c^{3} + \left (\frac{a^{4} c^{14}}{b^{9}}\right )^{\frac{1}{4}}{\left (b^{3} x^{2} + a b^{2}\right )}\right )}}{b x^{2} + a}\right ) + 5 \, \left (\frac{a^{4} c^{14}}{b^{9}}\right )^{\frac{1}{4}}{\left (b^{3} x^{2} + a b^{2}\right )} \log \left (\frac{5 \,{\left ({\left (b x^{2} + a\right )}^{\frac{3}{4}} \sqrt{c x} a c^{3} - \left (\frac{a^{4} c^{14}}{b^{9}}\right )^{\frac{1}{4}}{\left (b^{3} x^{2} + a b^{2}\right )}\right )}}{b x^{2} + a}\right )}{8 \,{\left (b^{3} x^{2} + a b^{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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (c x\right )^{\frac{7}{2}}}{{\left (b x^{2} + a\right )}^{\frac{5}{4}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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