Optimal. Leaf size=438 \[ \frac{\sqrt [4]{d} \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (7 a^2 d^2-18 a b c d+15 b^2 c^2\right ) \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right ),\frac{1}{2}\right )}{15 c^{11/4} e^{11/2} \sqrt{c+d x^2}}-\frac{2 \sqrt{c+d x^2} \left (7 a^2 d^2-18 a b c d+15 b^2 c^2\right )}{15 c^3 e^5 \sqrt{e x}}+\frac{2 \sqrt{d} \sqrt{e x} \sqrt{c+d x^2} \left (7 a^2 d^2-18 a b c d+15 b^2 c^2\right )}{15 c^3 e^6 \left (\sqrt{c}+\sqrt{d} x\right )}-\frac{2 \sqrt [4]{d} \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (7 a^2 d^2-18 a b c d+15 b^2 c^2\right ) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{15 c^{11/4} e^{11/2} \sqrt{c+d x^2}}-\frac{2 a^2 \sqrt{c+d x^2}}{9 c e (e x)^{9/2}}-\frac{2 a \sqrt{c+d x^2} (18 b c-7 a d)}{45 c^2 e^3 (e x)^{5/2}} \]
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Rubi [A] time = 0.423363, antiderivative size = 438, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {462, 453, 325, 329, 305, 220, 1196} \[ -\frac{2 \sqrt{c+d x^2} \left (7 a^2 d^2-18 a b c d+15 b^2 c^2\right )}{15 c^3 e^5 \sqrt{e x}}+\frac{2 \sqrt{d} \sqrt{e x} \sqrt{c+d x^2} \left (7 a^2 d^2-18 a b c d+15 b^2 c^2\right )}{15 c^3 e^6 \left (\sqrt{c}+\sqrt{d} x\right )}+\frac{\sqrt [4]{d} \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (7 a^2 d^2-18 a b c d+15 b^2 c^2\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{15 c^{11/4} e^{11/2} \sqrt{c+d x^2}}-\frac{2 \sqrt [4]{d} \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (7 a^2 d^2-18 a b c d+15 b^2 c^2\right ) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{15 c^{11/4} e^{11/2} \sqrt{c+d x^2}}-\frac{2 a^2 \sqrt{c+d x^2}}{9 c e (e x)^{9/2}}-\frac{2 a \sqrt{c+d x^2} (18 b c-7 a d)}{45 c^2 e^3 (e x)^{5/2}} \]
Antiderivative was successfully verified.
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Rule 462
Rule 453
Rule 325
Rule 329
Rule 305
Rule 220
Rule 1196
Rubi steps
\begin{align*} \int \frac{\left (a+b x^2\right )^2}{(e x)^{11/2} \sqrt{c+d x^2}} \, dx &=-\frac{2 a^2 \sqrt{c+d x^2}}{9 c e (e x)^{9/2}}+\frac{2 \int \frac{\frac{1}{2} a (18 b c-7 a d)+\frac{9}{2} b^2 c x^2}{(e x)^{7/2} \sqrt{c+d x^2}} \, dx}{9 c e^2}\\ &=-\frac{2 a^2 \sqrt{c+d x^2}}{9 c e (e x)^{9/2}}-\frac{2 a (18 b c-7 a d) \sqrt{c+d x^2}}{45 c^2 e^3 (e x)^{5/2}}-\frac{\left (4 \left (-\frac{45}{4} b^2 c^2+\frac{3}{4} a d (18 b c-7 a d)\right )\right ) \int \frac{1}{(e x)^{3/2} \sqrt{c+d x^2}} \, dx}{45 c^2 e^4}\\ &=-\frac{2 a^2 \sqrt{c+d x^2}}{9 c e (e x)^{9/2}}-\frac{2 a (18 b c-7 a d) \sqrt{c+d x^2}}{45 c^2 e^3 (e x)^{5/2}}-\frac{2 \left (15 b^2 c^2-a d (18 b c-7 a d)\right ) \sqrt{c+d x^2}}{15 c^3 e^5 \sqrt{e x}}-\frac{\left (4 d \left (-\frac{45}{4} b^2 c^2+\frac{3}{4} a d (18 b c-7 a d)\right )\right ) \int \frac{\sqrt{e x}}{\sqrt{c+d x^2}} \, dx}{45 c^3 e^6}\\ &=-\frac{2 a^2 \sqrt{c+d x^2}}{9 c e (e x)^{9/2}}-\frac{2 a (18 b c-7 a d) \sqrt{c+d x^2}}{45 c^2 e^3 (e x)^{5/2}}-\frac{2 \left (15 b^2 c^2-a d (18 b c-7 a d)\right ) \sqrt{c+d x^2}}{15 c^3 e^5 \sqrt{e x}}-\frac{\left (8 d \left (-\frac{45}{4} b^2 c^2+\frac{3}{4} a d (18 b c-7 a d)\right )\right ) \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{c+\frac{d x^4}{e^2}}} \, dx,x,\sqrt{e x}\right )}{45 c^3 e^7}\\ &=-\frac{2 a^2 \sqrt{c+d x^2}}{9 c e (e x)^{9/2}}-\frac{2 a (18 b c-7 a d) \sqrt{c+d x^2}}{45 c^2 e^3 (e x)^{5/2}}-\frac{2 \left (15 b^2 c^2-a d (18 b c-7 a d)\right ) \sqrt{c+d x^2}}{15 c^3 e^5 \sqrt{e x}}-\frac{\left (8 \sqrt{d} \left (-\frac{45}{4} b^2 c^2+\frac{3}{4} a d (18 b c-7 a d)\right )\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{c+\frac{d x^4}{e^2}}} \, dx,x,\sqrt{e x}\right )}{45 c^{5/2} e^6}+\frac{\left (8 \sqrt{d} \left (-\frac{45}{4} b^2 c^2+\frac{3}{4} a d (18 b c-7 a d)\right )\right ) \operatorname{Subst}\left (\int \frac{1-\frac{\sqrt{d} x^2}{\sqrt{c} e}}{\sqrt{c+\frac{d x^4}{e^2}}} \, dx,x,\sqrt{e x}\right )}{45 c^{5/2} e^6}\\ &=-\frac{2 a^2 \sqrt{c+d x^2}}{9 c e (e x)^{9/2}}-\frac{2 a (18 b c-7 a d) \sqrt{c+d x^2}}{45 c^2 e^3 (e x)^{5/2}}-\frac{2 \left (15 b^2 c^2-a d (18 b c-7 a d)\right ) \sqrt{c+d x^2}}{15 c^3 e^5 \sqrt{e x}}+\frac{2 \sqrt{d} \left (15 b^2 c^2-a d (18 b c-7 a d)\right ) \sqrt{e x} \sqrt{c+d x^2}}{15 c^3 e^6 \left (\sqrt{c}+\sqrt{d} x\right )}-\frac{2 \sqrt [4]{d} \left (15 b^2 c^2-a d (18 b c-7 a d)\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{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{15 c^{11/4} e^{11/2} \sqrt{c+d x^2}}+\frac{\sqrt [4]{d} \left (15 b^2 c^2-a d (18 b c-7 a d)\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{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{15 c^{11/4} e^{11/2} \sqrt{c+d x^2}}\\ \end{align*}
Mathematica [C] time = 0.182344, size = 155, normalized size = 0.35 \[ \frac{2 \sqrt{e x} \left (d x^6 \sqrt{\frac{d x^2}{c}+1} \left (7 a^2 d^2-18 a b c d+15 b^2 c^2\right ) \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{7}{4};-\frac{d x^2}{c}\right )-\left (c+d x^2\right ) \left (a^2 \left (5 c^2-7 c d x^2+21 d^2 x^4\right )+18 a b c x^2 \left (c-3 d x^2\right )+45 b^2 c^2 x^4\right )\right )}{45 c^3 e^6 x^5 \sqrt{c+d x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.034, size = 667, normalized size = 1.5 \begin{align*}{\frac{1}{45\,{x}^{4}{e}^{5}{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}^{4}{a}^{2}c{d}^{2}-108\,\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}^{4}ab{c}^{2}d+90\,\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}^{4}{b}^{2}{c}^{3}-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}^{4}{a}^{2}c{d}^{2}+54\,\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}^{4}ab{c}^{2}d-45\,\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}^{4}{b}^{2}{c}^{3}-42\,{x}^{6}{a}^{2}{d}^{3}+108\,{x}^{6}abc{d}^{2}-90\,{x}^{6}{b}^{2}{c}^{2}d-28\,{x}^{4}{a}^{2}c{d}^{2}+72\,{x}^{4}ab{c}^{2}d-90\,{x}^{4}{b}^{2}{c}^{3}+4\,{x}^{2}{a}^{2}{c}^{2}d-36\,{x}^{2}ab{c}^{3}-10\,{a}^{2}{c}^{3} \right ){\frac{1}{\sqrt{d{x}^{2}+c}}}{\frac{1}{\sqrt{ex}}}} \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} \left (e x\right )^{\frac{11}{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} \sqrt{e x}}{d e^{6} x^{8} + c e^{6} x^{6}}, 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} \left (e x\right )^{\frac{11}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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