Optimal. Leaf size=434 \[ -\frac{2 a^2}{5 c e (e x)^{5/2} \sqrt{c+d x^2}}-\frac{\left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (5 b^2 c^2-3 a d (10 b c-7 a d)\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{10 c^{11/4} d^{3/4} e^{7/2} \sqrt{c+d x^2}}+\frac{\left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (5 b^2 c^2-3 a d (10 b c-7 a d)\right ) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{5 c^{11/4} d^{3/4} e^{7/2} \sqrt{c+d x^2}}+\frac{(e x)^{3/2} \left (5 b^2 c^2-3 a d (10 b c-7 a d)\right )}{5 c^3 e^5 \sqrt{c+d x^2}}-\frac{\sqrt{e x} \sqrt{c+d x^2} \left (5 b^2 c^2-3 a d (10 b c-7 a d)\right )}{5 c^3 \sqrt{d} e^4 \left (\sqrt{c}+\sqrt{d} x\right )}-\frac{2 a (10 b c-7 a d)}{5 c^2 e^3 \sqrt{e x} \sqrt{c+d x^2}} \]
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Rubi [A] time = 1.01458, antiderivative size = 434, 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 \[ -\frac{2 a^2}{5 c e (e x)^{5/2} \sqrt{c+d x^2}}-\frac{\left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (5 b^2 c^2-3 a d (10 b c-7 a d)\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{10 c^{11/4} d^{3/4} e^{7/2} \sqrt{c+d x^2}}+\frac{\left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (5 b^2 c^2-3 a d (10 b c-7 a d)\right ) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{5 c^{11/4} d^{3/4} e^{7/2} \sqrt{c+d x^2}}+\frac{(e x)^{3/2} \left (5 b^2 c^2-3 a d (10 b c-7 a d)\right )}{5 c^3 e^5 \sqrt{c+d x^2}}-\frac{\sqrt{e x} \sqrt{c+d x^2} \left (5 b^2 c^2-3 a d (10 b c-7 a d)\right )}{5 c^3 \sqrt{d} e^4 \left (\sqrt{c}+\sqrt{d} x\right )}-\frac{2 a (10 b c-7 a d)}{5 c^2 e^3 \sqrt{e x} \sqrt{c+d x^2}} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x^2)^2/((e*x)^(7/2)*(c + d*x^2)^(3/2)),x]
[Out]
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Rubi in Sympy [A] time = 96.8345, size = 411, normalized size = 0.95 \[ - \frac{2 a^{2}}{5 c e \left (e x\right )^{\frac{5}{2}} \sqrt{c + d x^{2}}} + \frac{2 a \left (7 a d - 10 b c\right )}{5 c^{2} e^{3} \sqrt{e x} \sqrt{c + d x^{2}}} + \frac{\left (e x\right )^{\frac{3}{2}} \left (3 a d \left (7 a d - 10 b c\right ) + 5 b^{2} c^{2}\right )}{5 c^{3} e^{5} \sqrt{c + d x^{2}}} - \frac{\sqrt{e x} \sqrt{c + d x^{2}} \left (3 a d \left (7 a d - 10 b c\right ) + 5 b^{2} c^{2}\right )}{5 c^{3} \sqrt{d} e^{4} \left (\sqrt{c} + \sqrt{d} x\right )} + \frac{\sqrt{\frac{c + d x^{2}}{\left (\sqrt{c} + \sqrt{d} x\right )^{2}}} \left (\sqrt{c} + \sqrt{d} x\right ) \left (3 a d \left (7 a d - 10 b c\right ) + 5 b^{2} c^{2}\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}} \right )}\middle | \frac{1}{2}\right )}{5 c^{\frac{11}{4}} d^{\frac{3}{4}} e^{\frac{7}{2}} \sqrt{c + d x^{2}}} - \frac{\sqrt{\frac{c + d x^{2}}{\left (\sqrt{c} + \sqrt{d} x\right )^{2}}} \left (\sqrt{c} + \sqrt{d} x\right ) \left (3 a d \left (7 a d - 10 b c\right ) + 5 b^{2} c^{2}\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}} \right )}\middle | \frac{1}{2}\right )}{10 c^{\frac{11}{4}} d^{\frac{3}{4}} e^{\frac{7}{2}} \sqrt{c + d x^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x**2+a)**2/(e*x)**(7/2)/(d*x**2+c)**(3/2),x)
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Mathematica [C] time = 0.574978, size = 277, normalized size = 0.64 \[ \frac{i \left (\sqrt{d} \sqrt{\frac{i \sqrt{d} x}{\sqrt{c}}} \left (a^2 \left (-2 c^2+14 c d x^2+21 d^2 x^4\right )-10 a b c x^2 \left (2 c+3 d x^2\right )+5 b^2 c^2 x^4\right )+\sqrt{c} x^3 \sqrt{\frac{d x^2}{c}+1} \left (21 a^2 d^2-30 a b c d+5 b^2 c^2\right ) F\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{d} x}{\sqrt{c}}}\right )\right |-1\right )-\sqrt{c} x^3 \sqrt{\frac{d x^2}{c}+1} \left (21 a^2 d^2-30 a b c d+5 b^2 c^2\right ) E\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{d} x}{\sqrt{c}}}\right )\right |-1\right )\right )}{5 c^{7/2} e^2 (e x)^{3/2} \left (\frac{i \sqrt{d} x}{\sqrt{c}}\right )^{3/2} \sqrt{c+d x^2}} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x^2)^2/((e*x)^(7/2)*(c + d*x^2)^(3/2)),x]
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Maple [A] time = 0.036, size = 638, normalized size = 1.5 \[ -{\frac{1}{10\,d{x}^{2}{e}^{3}{c}^{3}} \left ( 42\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticE} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ){x}^{2}{a}^{2}c{d}^{2}-60\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticE} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ){x}^{2}ab{c}^{2}d+10\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticE} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ){x}^{2}{b}^{2}{c}^{3}-21\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticF} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ){x}^{2}{a}^{2}c{d}^{2}+30\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticF} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ){x}^{2}ab{c}^{2}d-5\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticF} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ){x}^{2}{b}^{2}{c}^{3}-42\,{x}^{4}{a}^{2}{d}^{3}+60\,{x}^{4}abc{d}^{2}-10\,{x}^{4}{b}^{2}{c}^{2}d-28\,{x}^{2}{a}^{2}c{d}^{2}+40\,{x}^{2}ab{c}^{2}d+4\,{a}^{2}{c}^{2}d \right ){\frac{1}{\sqrt{d{x}^{2}+c}}}{\frac{1}{\sqrt{ex}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x^2+a)^2/(e*x)^(7/2)/(d*x^2+c)^(3/2),x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{2} + a\right )}^{2}}{{\left (d x^{2} + c\right )}^{\frac{3}{2}} \left (e x\right )^{\frac{7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^2/((d*x^2 + c)^(3/2)*(e*x)^(7/2)),x, algorithm="maxima")
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}{{\left (d e^{3} x^{5} + c e^{3} x^{3}\right )} \sqrt{d x^{2} + c} \sqrt{e x}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^2/((d*x^2 + c)^(3/2)*(e*x)^(7/2)),x, algorithm="fricas")
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x**2+a)**2/(e*x)**(7/2)/(d*x**2+c)**(3/2),x)
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{2} + a\right )}^{2}}{{\left (d x^{2} + c\right )}^{\frac{3}{2}} \left (e x\right )^{\frac{7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^2/((d*x^2 + c)^(3/2)*(e*x)^(7/2)),x, algorithm="giac")
[Out]