Optimal. Leaf size=245 \[ \frac{e^{3/2} \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (21 a^2 d^2-70 a b c d+45 b^2 c^2\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{42 \sqrt [4]{c} d^{13/4} \sqrt{c+d x^2}}-\frac{e \sqrt{e x} \sqrt{c+d x^2} \left (21 a^2 d^2-70 a b c d+45 b^2 c^2\right )}{21 c d^3}+\frac{(e x)^{5/2} (b c-a d)^2}{c d^2 e \sqrt{c+d x^2}}+\frac{2 b^2 (e x)^{5/2} \sqrt{c+d x^2}}{7 d^2 e} \]
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Rubi [A] time = 0.52173, antiderivative size = 245, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179 \[ \frac{e^{3/2} \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (21 a^2 d^2-70 a b c d+45 b^2 c^2\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{42 \sqrt [4]{c} d^{13/4} \sqrt{c+d x^2}}-\frac{e \sqrt{e x} \sqrt{c+d x^2} \left (21 a^2 d^2-70 a b c d+45 b^2 c^2\right )}{21 c d^3}+\frac{(e x)^{5/2} (b c-a d)^2}{c d^2 e \sqrt{c+d x^2}}+\frac{2 b^2 (e x)^{5/2} \sqrt{c+d x^2}}{7 d^2 e} \]
Antiderivative was successfully verified.
[In] Int[((e*x)^(3/2)*(a + b*x^2)^2)/(c + d*x^2)^(3/2),x]
[Out]
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Rubi in Sympy [A] time = 60.1852, size = 228, normalized size = 0.93 \[ \frac{2 b^{2} \left (e x\right )^{\frac{5}{2}} \sqrt{c + d x^{2}}}{7 d^{2} e} + \frac{\left (e x\right )^{\frac{5}{2}} \left (a d - b c\right )^{2}}{c d^{2} e \sqrt{c + d x^{2}}} - \frac{e \sqrt{e x} \sqrt{c + d x^{2}} \left (21 a^{2} d^{2} - 70 a b c d + 45 b^{2} c^{2}\right )}{21 c d^{3}} + \frac{e^{\frac{3}{2}} \sqrt{\frac{c + d x^{2}}{\left (\sqrt{c} + \sqrt{d} x\right )^{2}}} \left (\sqrt{c} + \sqrt{d} x\right ) \left (21 a^{2} d^{2} - 70 a b c d + 45 b^{2} c^{2}\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}} \right )}\middle | \frac{1}{2}\right )}{42 \sqrt [4]{c} d^{\frac{13}{4}} \sqrt{c + d x^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x)**(3/2)*(b*x**2+a)**2/(d*x**2+c)**(3/2),x)
[Out]
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Mathematica [C] time = 0.328986, size = 191, normalized size = 0.78 \[ \frac{e \sqrt{e x} \left (i \sqrt{x} \sqrt{\frac{c}{d x^2}+1} \left (21 a^2 d^2-70 a b c d+45 b^2 c^2\right ) F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{c}}{\sqrt{d}}}}{\sqrt{x}}\right )\right |-1\right )+\sqrt{\frac{i \sqrt{c}}{\sqrt{d}}} \left (-21 a^2 d^2+14 a b d \left (5 c+2 d x^2\right )-3 b^2 \left (15 c^2+6 c d x^2-2 d^2 x^4\right )\right )\right )}{21 d^3 \sqrt{\frac{i \sqrt{c}}{\sqrt{d}}} \sqrt{c+d x^2}} \]
Antiderivative was successfully verified.
[In] Integrate[((e*x)^(3/2)*(a + b*x^2)^2)/(c + d*x^2)^(3/2),x]
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Maple [A] time = 0.032, size = 363, normalized size = 1.5 \[{\frac{e}{42\,x{d}^{4}}\sqrt{ex} \left ( 21\,\sqrt{-cd}\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticF} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ){a}^{2}{d}^{2}-70\,\sqrt{-cd}\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticF} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ) abcd+45\,\sqrt{-cd}\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticF} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ){b}^{2}{c}^{2}+12\,{x}^{5}{b}^{2}{d}^{3}+56\,{x}^{3}ab{d}^{3}-36\,{x}^{3}{b}^{2}c{d}^{2}-42\,x{a}^{2}{d}^{3}+140\,xabc{d}^{2}-90\,x{b}^{2}{c}^{2}d \right ){\frac{1}{\sqrt{d{x}^{2}+c}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x)^(3/2)*(b*x^2+a)^2/(d*x^2+c)^(3/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{2} + a\right )}^{2} \left (e x\right )^{\frac{3}{2}}}{{\left (d x^{2} + c\right )}^{\frac{3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^2*(e*x)^(3/2)/(d*x^2 + c)^(3/2),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (b^{2} e x^{5} + 2 \, a b e x^{3} + a^{2} e x\right )} \sqrt{e x}}{{\left (d x^{2} + c\right )}^{\frac{3}{2}}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^2*(e*x)^(3/2)/(d*x^2 + c)^(3/2),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((e*x)**(3/2)*(b*x**2+a)**2/(d*x**2+c)**(3/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{2} + a\right )}^{2} \left (e x\right )^{\frac{3}{2}}}{{\left (d x^{2} + c\right )}^{\frac{3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^2*(e*x)^(3/2)/(d*x^2 + c)^(3/2),x, algorithm="giac")
[Out]