Optimal. Leaf size=468 \[ -\frac{2 a^2 \left (c+d x^2\right )^{5/2}}{5 c e (e x)^{5/2}}+\frac{4 \sqrt [4]{c} \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (9 a d (a d+2 b c)+b^2 c^2\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{15 d^{3/4} e^{7/2} \sqrt{c+d x^2}}-\frac{8 \sqrt [4]{c} \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (9 a d (a d+2 b c)+b^2 c^2\right ) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{15 d^{3/4} e^{7/2} \sqrt{c+d x^2}}+\frac{2 (e x)^{3/2} \left (c+d x^2\right )^{3/2} \left (9 a d (a d+2 b c)+b^2 c^2\right )}{9 c^2 e^5}+\frac{4 (e x)^{3/2} \sqrt{c+d x^2} \left (9 a d (a d+2 b c)+b^2 c^2\right )}{15 c e^5}+\frac{8 \sqrt{e x} \sqrt{c+d x^2} \left (9 a d (a d+2 b c)+b^2 c^2\right )}{15 \sqrt{d} e^4 \left (\sqrt{c}+\sqrt{d} x\right )}-\frac{2 a \left (c+d x^2\right )^{5/2} (a d+2 b c)}{c^2 e^3 \sqrt{e x}} \]
[Out]
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Rubi [A] time = 1.07944, antiderivative size = 468, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{2 a^2 \left (c+d x^2\right )^{5/2}}{5 c e (e x)^{5/2}}+\frac{4 \sqrt [4]{c} \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (9 a d (a d+2 b c)+b^2 c^2\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{15 d^{3/4} e^{7/2} \sqrt{c+d x^2}}-\frac{8 \sqrt [4]{c} \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (9 a d (a d+2 b c)+b^2 c^2\right ) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{15 d^{3/4} e^{7/2} \sqrt{c+d x^2}}+\frac{2 (e x)^{3/2} \left (c+d x^2\right )^{3/2} \left (9 a d (a d+2 b c)+b^2 c^2\right )}{9 c^2 e^5}+\frac{4 (e x)^{3/2} \sqrt{c+d x^2} \left (9 a d (a d+2 b c)+b^2 c^2\right )}{15 c e^5}+\frac{8 \sqrt{e x} \sqrt{c+d x^2} \left (9 a d (a d+2 b c)+b^2 c^2\right )}{15 \sqrt{d} e^4 \left (\sqrt{c}+\sqrt{d} x\right )}-\frac{2 a \left (c+d x^2\right )^{5/2} (a d+2 b c)}{c^2 e^3 \sqrt{e x}} \]
Antiderivative was successfully verified.
[In] Int[((a + b*x^2)^2*(c + d*x^2)^(3/2))/(e*x)^(7/2),x]
[Out]
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Rubi in Sympy [A] time = 105.925, size = 444, normalized size = 0.95 \[ - \frac{2 a^{2} \left (c + d x^{2}\right )^{\frac{5}{2}}}{5 c e \left (e x\right )^{\frac{5}{2}}} - \frac{2 a \left (c + d x^{2}\right )^{\frac{5}{2}} \left (a d + 2 b c\right )}{c^{2} e^{3} \sqrt{e x}} - \frac{8 \sqrt [4]{c} \sqrt{\frac{c + d x^{2}}{\left (\sqrt{c} + \sqrt{d} x\right )^{2}}} \left (\sqrt{c} + \sqrt{d} x\right ) \left (9 a d \left (a d + 2 b c\right ) + b^{2} c^{2}\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}} \right )}\middle | \frac{1}{2}\right )}{15 d^{\frac{3}{4}} e^{\frac{7}{2}} \sqrt{c + d x^{2}}} + \frac{4 \sqrt [4]{c} \sqrt{\frac{c + d x^{2}}{\left (\sqrt{c} + \sqrt{d} x\right )^{2}}} \left (\sqrt{c} + \sqrt{d} x\right ) \left (9 a d \left (a d + 2 b c\right ) + b^{2} c^{2}\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}} \right )}\middle | \frac{1}{2}\right )}{15 d^{\frac{3}{4}} e^{\frac{7}{2}} \sqrt{c + d x^{2}}} + \frac{8 \sqrt{e x} \sqrt{c + d x^{2}} \left (9 a d \left (a d + 2 b c\right ) + b^{2} c^{2}\right )}{15 \sqrt{d} e^{4} \left (\sqrt{c} + \sqrt{d} x\right )} + \frac{4 \left (e x\right )^{\frac{3}{2}} \sqrt{c + d x^{2}} \left (9 a d \left (a d + 2 b c\right ) + b^{2} c^{2}\right )}{15 c e^{5}} + \frac{2 \left (e x\right )^{\frac{3}{2}} \left (c + d x^{2}\right )^{\frac{3}{2}} \left (9 a d \left (a d + 2 b c\right ) + b^{2} c^{2}\right )}{9 c^{2} e^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x**2+a)**2*(d*x**2+c)**(3/2)/(e*x)**(7/2),x)
[Out]
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Mathematica [C] time = 1.51636, size = 240, normalized size = 0.51 \[ \frac{x^{7/2} \left (\frac{2 \sqrt{c+d x^2} \left (-9 a^2 \left (c+7 d x^2\right )+18 a b x^2 \left (d x^2-5 c\right )+b^2 x^4 \left (11 c+5 d x^2\right )\right )}{3 x^{5/2}}-\frac{8 x \left (9 a^2 d^2+18 a b c d+b^2 c^2\right ) \left (-\sqrt{x} \left (\frac{c}{x^2}+d\right )+\frac{i c \sqrt{\frac{c}{d x^2}+1} \left (E\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{c}}{\sqrt{d}}}}{\sqrt{x}}\right )\right |-1\right )-F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{c}}{\sqrt{d}}}}{\sqrt{x}}\right )\right |-1\right )\right )}{\left (\frac{i \sqrt{c}}{\sqrt{d}}\right )^{3/2}}\right )}{d \sqrt{c+d x^2}}\right )}{15 (e x)^{7/2}} \]
Antiderivative was successfully verified.
[In] Integrate[((a + b*x^2)^2*(c + d*x^2)^(3/2))/(e*x)^(7/2),x]
[Out]
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Maple [A] time = 0.031, size = 668, normalized size = 1.4 \[{\frac{2}{45\,d{x}^{2}{e}^{3}} \left ( 5\,{b}^{2}{d}^{3}{x}^{8}+108\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticE} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ){x}^{2}{a}^{2}c{d}^{2}+216\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticE} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ){x}^{2}ab{c}^{2}d+12\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticE} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ){x}^{2}{b}^{2}{c}^{3}-54\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticF} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ){x}^{2}{a}^{2}c{d}^{2}-108\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticF} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ){x}^{2}ab{c}^{2}d-6\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticF} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ){x}^{2}{b}^{2}{c}^{3}+18\,{x}^{6}ab{d}^{3}+16\,{x}^{6}{b}^{2}c{d}^{2}-63\,{x}^{4}{a}^{2}{d}^{3}-72\,{x}^{4}abc{d}^{2}+11\,{x}^{4}{b}^{2}{c}^{2}d-72\,{x}^{2}{a}^{2}c{d}^{2}-90\,{x}^{2}ab{c}^{2}d-9\,{a}^{2}{c}^{2}d \right ){\frac{1}{\sqrt{d{x}^{2}+c}}}{\frac{1}{\sqrt{ex}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x^2+a)^2*(d*x^2+c)^(3/2)/(e*x)^(7/2),x)
[Out]
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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")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (b^{2} d x^{6} +{\left (b^{2} c + 2 \, a b d\right )} x^{4} + a^{2} c +{\left (2 \, a b c + a^{2} d\right )} x^{2}\right )} \sqrt{d x^{2} + c}}{\sqrt{e x} e^{3} x^{3}}, x\right ) \]
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")
[Out]
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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*(d*x**2+c)**(3/2)/(e*x)**(7/2),x)
[Out]
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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]