Optimal. Leaf size=217 \[ -\frac{2 \sqrt{c+d x^2} \left (5 a^2 d^2-22 a b c d+77 b^2 c^2\right )}{231 c^2 x^{3/2}}+\frac{2 d^{3/4} \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (5 a^2 d^2-22 a b c d+77 b^2 c^2\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )|\frac{1}{2}\right )}{231 c^{9/4} \sqrt{c+d x^2}}-\frac{2 a^2 \left (c+d x^2\right )^{3/2}}{11 c x^{11/2}}-\frac{2 a \left (c+d x^2\right )^{3/2} (22 b c-5 a d)}{77 c^2 x^{7/2}} \]
[Out]
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Rubi [A] time = 0.455355, antiderivative size = 213, normalized size of antiderivative = 0.98, number of steps used = 5, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192 \[ \frac{2 d^{3/4} \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (5 a^2 d^2-22 a b c d+77 b^2 c^2\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )|\frac{1}{2}\right )}{231 c^{9/4} \sqrt{c+d x^2}}-\frac{2 a^2 \left (c+d x^2\right )^{3/2}}{11 c x^{11/2}}-\frac{2 \sqrt{c+d x^2} \left (77 b^2-\frac{a d (22 b c-5 a d)}{c^2}\right )}{231 x^{3/2}}-\frac{2 a \left (c+d x^2\right )^{3/2} (22 b c-5 a d)}{77 c^2 x^{7/2}} \]
Antiderivative was successfully verified.
[In] Int[((a + b*x^2)^2*Sqrt[c + d*x^2])/x^(13/2),x]
[Out]
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Rubi in Sympy [A] time = 37.8611, size = 204, normalized size = 0.94 \[ - \frac{2 a^{2} \left (c + d x^{2}\right )^{\frac{3}{2}}}{11 c x^{\frac{11}{2}}} + \frac{2 a \left (c + d x^{2}\right )^{\frac{3}{2}} \left (5 a d - 22 b c\right )}{77 c^{2} x^{\frac{7}{2}}} - \frac{2 \sqrt{c + d x^{2}} \left (a d \left (5 a d - 22 b c\right ) + 77 b^{2} c^{2}\right )}{231 c^{2} x^{\frac{3}{2}}} + \frac{2 d^{\frac{3}{4}} \sqrt{\frac{c + d x^{2}}{\left (\sqrt{c} + \sqrt{d} x\right )^{2}}} \left (\sqrt{c} + \sqrt{d} x\right ) \left (a d \left (5 a d - 22 b c\right ) + 77 b^{2} c^{2}\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}} \right )}\middle | \frac{1}{2}\right )}{231 c^{\frac{9}{4}} \sqrt{c + d x^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x**2+a)**2*(d*x**2+c)**(1/2)/x**(13/2),x)
[Out]
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Mathematica [C] time = 0.244865, size = 187, normalized size = 0.86 \[ -\frac{2 \sqrt{c+d x^2} \left (a^2 \left (21 c^2+6 c d x^2-10 d^2 x^4\right )+22 a b c x^2 \left (3 c+2 d x^2\right )+77 b^2 c^2 x^4\right )}{231 c^2 x^{11/2}}+\frac{4 i d x \sqrt{\frac{c}{d x^2}+1} \left (5 a^2 d^2-22 a b c d+77 b^2 c^2\right ) F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{c}}{\sqrt{d}}}}{\sqrt{x}}\right )\right |-1\right )}{231 c^2 \sqrt{\frac{i \sqrt{c}}{\sqrt{d}}} \sqrt{c+d x^2}} \]
Antiderivative was successfully verified.
[In] Integrate[((a + b*x^2)^2*Sqrt[c + d*x^2])/x^(13/2),x]
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Maple [A] time = 0.049, size = 403, normalized size = 1.9 \[{\frac{2}{231\,{c}^{2}} \left ( 5\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticF} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ) \sqrt{-cd}{x}^{5}{a}^{2}{d}^{2}-22\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticF} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ) \sqrt{-cd}{x}^{5}abcd+77\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticF} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ) \sqrt{-cd}{x}^{5}{b}^{2}{c}^{2}+10\,{x}^{6}{a}^{2}{d}^{3}-44\,{x}^{6}abc{d}^{2}-77\,{x}^{6}{b}^{2}{c}^{2}d+4\,{x}^{4}{a}^{2}c{d}^{2}-110\,{x}^{4}ab{c}^{2}d-77\,{x}^{4}{b}^{2}{c}^{3}-27\,{x}^{2}{a}^{2}{c}^{2}d-66\,{x}^{2}ab{c}^{3}-21\,{a}^{2}{c}^{3} \right ){\frac{1}{\sqrt{d{x}^{2}+c}}}{x}^{-{\frac{11}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x^2+a)^2*(d*x^2+c)^(1/2)/x^(13/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{2} + a\right )}^{2} \sqrt{d x^{2} + c}}{x^{\frac{13}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^2*sqrt(d*x^2 + c)/x^(13/2),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )} \sqrt{d x^{2} + c}}{x^{\frac{13}{2}}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^2*sqrt(d*x^2 + c)/x^(13/2),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)**(1/2)/x**(13/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{2} + a\right )}^{2} \sqrt{d x^{2} + c}}{x^{\frac{13}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^2*sqrt(d*x^2 + c)/x^(13/2),x, algorithm="giac")
[Out]