Optimal. Leaf size=441 \[ \frac{2 d^{5/4} \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (7 a^2 d^2-26 a b c d+39 b^2 c^2\right ) \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right ),\frac{1}{2}\right )}{195 c^{11/4} \sqrt{c+d x^2}}-\frac{4 d \sqrt{c+d x^2} \left (7 a^2 d^2-26 a b c d+39 b^2 c^2\right )}{195 c^3 \sqrt{x}}+\frac{4 d^{3/2} \sqrt{x} \sqrt{c+d x^2} \left (7 a^2 d^2-26 a b c d+39 b^2 c^2\right )}{195 c^3 \left (\sqrt{c}+\sqrt{d} x\right )}-\frac{2 \sqrt{c+d x^2} \left (7 a^2 d^2-26 a b c d+39 b^2 c^2\right )}{195 c^2 x^{5/2}}-\frac{4 d^{5/4} \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (7 a^2 d^2-26 a b c d+39 b^2 c^2\right ) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )|\frac{1}{2}\right )}{195 c^{11/4} \sqrt{c+d x^2}}-\frac{2 a^2 \left (c+d x^2\right )^{3/2}}{13 c x^{13/2}}-\frac{2 a \left (c+d x^2\right )^{3/2} (26 b c-7 a d)}{117 c^2 x^{9/2}} \]
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Rubi [A] time = 0.392979, antiderivative size = 437, normalized size of antiderivative = 0.99, number of steps used = 8, number of rules used = 8, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308, Rules used = {462, 453, 277, 325, 329, 305, 220, 1196} \[ -\frac{4 d \sqrt{c+d x^2} \left (7 a^2 d^2-26 a b c d+39 b^2 c^2\right )}{195 c^3 \sqrt{x}}+\frac{4 d^{3/2} \sqrt{x} \sqrt{c+d x^2} \left (7 a^2 d^2-26 a b c d+39 b^2 c^2\right )}{195 c^3 \left (\sqrt{c}+\sqrt{d} x\right )}+\frac{2 d^{5/4} \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (7 a^2 d^2-26 a b c d+39 b^2 c^2\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )|\frac{1}{2}\right )}{195 c^{11/4} \sqrt{c+d x^2}}-\frac{4 d^{5/4} \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (7 a^2 d^2-26 a b c d+39 b^2 c^2\right ) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )|\frac{1}{2}\right )}{195 c^{11/4} \sqrt{c+d x^2}}-\frac{2 a^2 \left (c+d x^2\right )^{3/2}}{13 c x^{13/2}}-\frac{2 \sqrt{c+d x^2} \left (39 b^2-\frac{a d (26 b c-7 a d)}{c^2}\right )}{195 x^{5/2}}-\frac{2 a \left (c+d x^2\right )^{3/2} (26 b c-7 a d)}{117 c^2 x^{9/2}} \]
Antiderivative was successfully verified.
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Rule 462
Rule 453
Rule 277
Rule 325
Rule 329
Rule 305
Rule 220
Rule 1196
Rubi steps
\begin{align*} \int \frac{\left (a+b x^2\right )^2 \sqrt{c+d x^2}}{x^{15/2}} \, dx &=-\frac{2 a^2 \left (c+d x^2\right )^{3/2}}{13 c x^{13/2}}+\frac{2 \int \frac{\left (\frac{1}{2} a (26 b c-7 a d)+\frac{13}{2} b^2 c x^2\right ) \sqrt{c+d x^2}}{x^{11/2}} \, dx}{13 c}\\ &=-\frac{2 a^2 \left (c+d x^2\right )^{3/2}}{13 c x^{13/2}}-\frac{2 a (26 b c-7 a d) \left (c+d x^2\right )^{3/2}}{117 c^2 x^{9/2}}-\frac{1}{39} \left (-39 b^2+\frac{a d (26 b c-7 a d)}{c^2}\right ) \int \frac{\sqrt{c+d x^2}}{x^{7/2}} \, dx\\ &=-\frac{2 \left (39 b^2-\frac{a d (26 b c-7 a d)}{c^2}\right ) \sqrt{c+d x^2}}{195 x^{5/2}}-\frac{2 a^2 \left (c+d x^2\right )^{3/2}}{13 c x^{13/2}}-\frac{2 a (26 b c-7 a d) \left (c+d x^2\right )^{3/2}}{117 c^2 x^{9/2}}+\frac{1}{195} \left (2 d \left (39 b^2-\frac{a d (26 b c-7 a d)}{c^2}\right )\right ) \int \frac{1}{x^{3/2} \sqrt{c+d x^2}} \, dx\\ &=-\frac{2 \left (39 b^2-\frac{a d (26 b c-7 a d)}{c^2}\right ) \sqrt{c+d x^2}}{195 x^{5/2}}-\frac{4 d \left (39 b^2 c^2-26 a b c d+7 a^2 d^2\right ) \sqrt{c+d x^2}}{195 c^3 \sqrt{x}}-\frac{2 a^2 \left (c+d x^2\right )^{3/2}}{13 c x^{13/2}}-\frac{2 a (26 b c-7 a d) \left (c+d x^2\right )^{3/2}}{117 c^2 x^{9/2}}+\frac{\left (2 d^2 \left (39 b^2 c^2-26 a b c d+7 a^2 d^2\right )\right ) \int \frac{\sqrt{x}}{\sqrt{c+d x^2}} \, dx}{195 c^3}\\ &=-\frac{2 \left (39 b^2-\frac{a d (26 b c-7 a d)}{c^2}\right ) \sqrt{c+d x^2}}{195 x^{5/2}}-\frac{4 d \left (39 b^2 c^2-26 a b c d+7 a^2 d^2\right ) \sqrt{c+d x^2}}{195 c^3 \sqrt{x}}-\frac{2 a^2 \left (c+d x^2\right )^{3/2}}{13 c x^{13/2}}-\frac{2 a (26 b c-7 a d) \left (c+d x^2\right )^{3/2}}{117 c^2 x^{9/2}}+\frac{\left (4 d^2 \left (39 b^2 c^2-26 a b c d+7 a^2 d^2\right )\right ) \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{c+d x^4}} \, dx,x,\sqrt{x}\right )}{195 c^3}\\ &=-\frac{2 \left (39 b^2-\frac{a d (26 b c-7 a d)}{c^2}\right ) \sqrt{c+d x^2}}{195 x^{5/2}}-\frac{4 d \left (39 b^2 c^2-26 a b c d+7 a^2 d^2\right ) \sqrt{c+d x^2}}{195 c^3 \sqrt{x}}-\frac{2 a^2 \left (c+d x^2\right )^{3/2}}{13 c x^{13/2}}-\frac{2 a (26 b c-7 a d) \left (c+d x^2\right )^{3/2}}{117 c^2 x^{9/2}}+\frac{\left (4 d^{3/2} \left (39 b^2 c^2-26 a b c d+7 a^2 d^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{c+d x^4}} \, dx,x,\sqrt{x}\right )}{195 c^{5/2}}-\frac{\left (4 d^{3/2} \left (39 b^2 c^2-26 a b c d+7 a^2 d^2\right )\right ) \operatorname{Subst}\left (\int \frac{1-\frac{\sqrt{d} x^2}{\sqrt{c}}}{\sqrt{c+d x^4}} \, dx,x,\sqrt{x}\right )}{195 c^{5/2}}\\ &=-\frac{2 \left (39 b^2-\frac{a d (26 b c-7 a d)}{c^2}\right ) \sqrt{c+d x^2}}{195 x^{5/2}}-\frac{4 d \left (39 b^2 c^2-26 a b c d+7 a^2 d^2\right ) \sqrt{c+d x^2}}{195 c^3 \sqrt{x}}+\frac{4 d^{3/2} \left (39 b^2 c^2-26 a b c d+7 a^2 d^2\right ) \sqrt{x} \sqrt{c+d x^2}}{195 c^3 \left (\sqrt{c}+\sqrt{d} x\right )}-\frac{2 a^2 \left (c+d x^2\right )^{3/2}}{13 c x^{13/2}}-\frac{2 a (26 b c-7 a d) \left (c+d x^2\right )^{3/2}}{117 c^2 x^{9/2}}-\frac{4 d^{5/4} \left (39 b^2 c^2-26 a b c d+7 a^2 d^2\right ) \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )|\frac{1}{2}\right )}{195 c^{11/4} \sqrt{c+d x^2}}+\frac{2 d^{5/4} \left (39 b^2 c^2-26 a b c d+7 a^2 d^2\right ) \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )|\frac{1}{2}\right )}{195 c^{11/4} \sqrt{c+d x^2}}\\ \end{align*}
Mathematica [C] time = 0.173301, size = 182, normalized size = 0.41 \[ \frac{4 d^2 x^8 \sqrt{\frac{d x^2}{c}+1} \left (7 a^2 d^2-26 a b c d+39 b^2 c^2\right ) \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{7}{4};-\frac{d x^2}{c}\right )-2 \left (c+d x^2\right ) \left (a^2 \left (10 c^2 d x^2+45 c^3-14 c d^2 x^4+42 d^3 x^6\right )+26 a b c x^2 \left (5 c^2+2 c d x^2-6 d^2 x^4\right )+117 b^2 c^2 x^4 \left (c+2 d x^2\right )\right )}{585 c^3 x^{13/2} \sqrt{c+d x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.034, size = 706, normalized size = 1.6 \begin{align*}{\frac{2}{585\,{c}^{3}} \left ( 42\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticE} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ){x}^{6}{a}^{2}c{d}^{3}-156\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticE} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ){x}^{6}ab{c}^{2}{d}^{2}+234\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticE} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ){x}^{6}{b}^{2}{c}^{3}d-21\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticF} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ){x}^{6}{a}^{2}c{d}^{3}+78\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticF} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ){x}^{6}ab{c}^{2}{d}^{2}-117\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticF} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ){x}^{6}{b}^{2}{c}^{3}d-42\,{x}^{8}{a}^{2}{d}^{4}+156\,{x}^{8}abc{d}^{3}-234\,{x}^{8}{b}^{2}{c}^{2}{d}^{2}-28\,{x}^{6}{a}^{2}c{d}^{3}+104\,{x}^{6}ab{c}^{2}{d}^{2}-351\,{x}^{6}{b}^{2}{c}^{3}d+4\,{x}^{4}{a}^{2}{c}^{2}{d}^{2}-182\,{x}^{4}ab{c}^{3}d-117\,{x}^{4}{b}^{2}{c}^{4}-55\,{x}^{2}{a}^{2}{c}^{3}d-130\,{x}^{2}ab{c}^{4}-45\,{a}^{2}{c}^{4} \right ){\frac{1}{\sqrt{d{x}^{2}+c}}}{x}^{-{\frac{13}{2}}}} \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 (b x^{2} + a\right )}^{2} \sqrt{d x^{2} + c}}{x^{\frac{15}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )} \sqrt{d x^{2} + c}}{x^{\frac{15}{2}}}, x\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 (b x^{2} + a\right )}^{2} \sqrt{d x^{2} + c}}{x^{\frac{15}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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