Optimal. Leaf size=442 \[ -\frac{e^2 \sqrt{e x} \sqrt{c+d x^2} \left (5 a^2 d^2-70 a b c d+77 b^2 c^2\right )}{10 c d^{7/2} \left (\sqrt{c}+\sqrt{d} x\right )}+\frac{e (e x)^{3/2} \left (5 a^2 d^2-70 a b c d+77 b^2 c^2\right )}{30 c d^3 \sqrt{c+d x^2}}-\frac{e^{5/2} \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (5 a^2 d^2-70 a b c d+77 b^2 c^2\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{20 c^{3/4} d^{15/4} \sqrt{c+d x^2}}+\frac{e^{5/2} \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (5 a^2 d^2-70 a b c d+77 b^2 c^2\right ) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{10 c^{3/4} d^{15/4} \sqrt{c+d x^2}}+\frac{(e x)^{7/2} (b c-a d)^2}{3 c d^2 e \left (c+d x^2\right )^{3/2}}+\frac{2 b^2 (e x)^{7/2}}{5 d^2 e \sqrt{c+d x^2}} \]
[Out]
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Rubi [A] time = 0.923458, antiderivative size = 442, 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 \[ -\frac{e^2 \sqrt{e x} \sqrt{c+d x^2} \left (5 a^2 d^2-70 a b c d+77 b^2 c^2\right )}{10 c d^{7/2} \left (\sqrt{c}+\sqrt{d} x\right )}+\frac{e (e x)^{3/2} \left (5 a^2 d^2-70 a b c d+77 b^2 c^2\right )}{30 c d^3 \sqrt{c+d x^2}}-\frac{e^{5/2} \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (5 a^2 d^2-70 a b c d+77 b^2 c^2\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{20 c^{3/4} d^{15/4} \sqrt{c+d x^2}}+\frac{e^{5/2} \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (5 a^2 d^2-70 a b c d+77 b^2 c^2\right ) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{10 c^{3/4} d^{15/4} \sqrt{c+d x^2}}+\frac{(e x)^{7/2} (b c-a d)^2}{3 c d^2 e \left (c+d x^2\right )^{3/2}}+\frac{2 b^2 (e x)^{7/2}}{5 d^2 e \sqrt{c+d x^2}} \]
Antiderivative was successfully verified.
[In] Int[((e*x)^(5/2)*(a + b*x^2)^2)/(c + d*x^2)^(5/2),x]
[Out]
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Rubi in Sympy [A] time = 106.014, size = 413, normalized size = 0.93 \[ \frac{2 b^{2} \left (e x\right )^{\frac{7}{2}}}{5 d^{2} e \sqrt{c + d x^{2}}} + \frac{\left (e x\right )^{\frac{7}{2}} \left (a d - b c\right )^{2}}{3 c d^{2} e \left (c + d x^{2}\right )^{\frac{3}{2}}} + \frac{e \left (e x\right )^{\frac{3}{2}} \left (5 a^{2} d^{2} - 70 a b c d + 77 b^{2} c^{2}\right )}{30 c d^{3} \sqrt{c + d x^{2}}} - \frac{e^{2} \sqrt{e x} \sqrt{c + d x^{2}} \left (5 a^{2} d^{2} - 70 a b c d + 77 b^{2} c^{2}\right )}{10 c d^{\frac{7}{2}} \left (\sqrt{c} + \sqrt{d} x\right )} + \frac{e^{\frac{5}{2}} \sqrt{\frac{c + d x^{2}}{\left (\sqrt{c} + \sqrt{d} x\right )^{2}}} \left (\sqrt{c} + \sqrt{d} x\right ) \left (5 a^{2} d^{2} - 70 a b c d + 77 b^{2} c^{2}\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}} \right )}\middle | \frac{1}{2}\right )}{10 c^{\frac{3}{4}} d^{\frac{15}{4}} \sqrt{c + d x^{2}}} - \frac{e^{\frac{5}{2}} \sqrt{\frac{c + d x^{2}}{\left (\sqrt{c} + \sqrt{d} x\right )^{2}}} \left (\sqrt{c} + \sqrt{d} x\right ) \left (5 a^{2} d^{2} - 70 a b c d + 77 b^{2} c^{2}\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}} \right )}\middle | \frac{1}{2}\right )}{20 c^{\frac{3}{4}} d^{\frac{15}{4}} \sqrt{c + d x^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x)**(5/2)*(b*x**2+a)**2/(d*x**2+c)**(5/2),x)
[Out]
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Mathematica [C] time = 1.37026, size = 298, normalized size = 0.67 \[ \frac{(e x)^{5/2} \left (-d x^2 \left (-5 a^2 d^2 \left (c+3 d x^2\right )+10 a b c d \left (7 c+9 d x^2\right )+b^2 (-c) \left (77 c^2+99 c d x^2+12 d^2 x^4\right )\right )-\frac{3 \left (c+d x^2\right ) \left (5 a^2 d^2-70 a b c d+77 b^2 c^2\right ) \left (\sqrt{\frac{i \sqrt{c}}{\sqrt{d}}} \left (c+d x^2\right )+\sqrt{c} \sqrt{d} x^{3/2} \sqrt{\frac{c}{d x^2}+1} F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{c}}{\sqrt{d}}}}{\sqrt{x}}\right )\right |-1\right )-\sqrt{c} \sqrt{d} x^{3/2} \sqrt{\frac{c}{d x^2}+1} E\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{c}}{\sqrt{d}}}}{\sqrt{x}}\right )\right |-1\right )\right )}{\sqrt{\frac{i \sqrt{c}}{\sqrt{d}}}}\right )}{30 c d^4 x^3 \left (c+d x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
[In] Integrate[((e*x)^(5/2)*(a + b*x^2)^2)/(c + d*x^2)^(5/2),x]
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Maple [B] time = 0.059, size = 1191, normalized size = 2.7 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x)^(5/2)*(b*x^2+a)^2/(d*x^2+c)^(5/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{2} + a\right )}^{2} \left (e x\right )^{\frac{5}{2}}}{{\left (d x^{2} + c\right )}^{\frac{5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^2*(e*x)^(5/2)/(d*x^2 + c)^(5/2),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (b^{2} e^{2} x^{6} + 2 \, a b e^{2} x^{4} + a^{2} e^{2} x^{2}\right )} \sqrt{e x}}{{\left (d^{2} x^{4} + 2 \, c d x^{2} + c^{2}\right )} \sqrt{d x^{2} + c}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^2*(e*x)^(5/2)/(d*x^2 + c)^(5/2),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)**(5/2)*(b*x**2+a)**2/(d*x**2+c)**(5/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{2} + a\right )}^{2} \left (e x\right )^{\frac{5}{2}}}{{\left (d x^{2} + c\right )}^{\frac{5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^2*(e*x)^(5/2)/(d*x^2 + c)^(5/2),x, algorithm="giac")
[Out]