Optimal. Leaf size=213 \[ -\frac{2 a^2 \left (c+d x^2\right )^{3/2}}{7 c x^{7/2}}+\frac{2 \sqrt{x} \sqrt{c+d x^2} \left (a d (14 b c-a d)+7 b^2 c^2\right )}{21 c^2}+\frac{2 \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (a d (14 b c-a d)+7 b^2 c^2\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )|\frac{1}{2}\right )}{21 c^{5/4} \sqrt [4]{d} \sqrt{c+d x^2}}-\frac{2 a \left (c+d x^2\right )^{3/2} (14 b c-a d)}{21 c^2 x^{3/2}} \]
[Out]
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Rubi [A] time = 0.432012, antiderivative size = 210, normalized size of antiderivative = 0.99, number of steps used = 5, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192 \[ -\frac{2 a^2 \left (c+d x^2\right )^{3/2}}{7 c x^{7/2}}+\frac{2}{21} \sqrt{x} \sqrt{c+d x^2} \left (\frac{a d (14 b c-a d)}{c^2}+7 b^2\right )+\frac{2 \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (a d (14 b c-a d)+7 b^2 c^2\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )|\frac{1}{2}\right )}{21 c^{5/4} \sqrt [4]{d} \sqrt{c+d x^2}}-\frac{2 a \left (c+d x^2\right )^{3/2} (14 b c-a d)}{21 c^2 x^{3/2}} \]
Antiderivative was successfully verified.
[In] Int[((a + b*x^2)^2*Sqrt[c + d*x^2])/x^(9/2),x]
[Out]
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Rubi in Sympy [A] time = 37.7216, size = 199, normalized size = 0.93 \[ - \frac{2 a^{2} \left (c + d x^{2}\right )^{\frac{3}{2}}}{7 c x^{\frac{7}{2}}} + \frac{2 a \left (c + d x^{2}\right )^{\frac{3}{2}} \left (a d - 14 b c\right )}{21 c^{2} x^{\frac{3}{2}}} + \frac{2 \sqrt{x} \sqrt{c + d x^{2}} \left (- a d \left (a d - 14 b c\right ) + 7 b^{2} c^{2}\right )}{21 c^{2}} + \frac{2 \sqrt{\frac{c + d x^{2}}{\left (\sqrt{c} + \sqrt{d} x\right )^{2}}} \left (\sqrt{c} + \sqrt{d} x\right ) \left (- a d \left (a d - 14 b c\right ) + 7 b^{2} c^{2}\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}} \right )}\middle | \frac{1}{2}\right )}{21 c^{\frac{5}{4}} \sqrt [4]{d} \sqrt{c + d x^{2}}} \]
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**(9/2),x)
[Out]
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Mathematica [C] time = 0.32464, size = 160, normalized size = 0.75 \[ \frac{2 \left (\left (c+d x^2\right ) \left (-a^2 \left (3 c+2 d x^2\right )-14 a b c x^2+7 b^2 c x^4\right )+\frac{2 i x^{9/2} \sqrt{\frac{c}{d x^2}+1} \left (-a^2 d^2+14 a b c d+7 b^2 c^2\right ) F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{c}}{\sqrt{d}}}}{\sqrt{x}}\right )\right |-1\right )}{\sqrt{\frac{i \sqrt{c}}{\sqrt{d}}}}\right )}{21 c x^{7/2} \sqrt{c+d x^2}} \]
Antiderivative was successfully verified.
[In] Integrate[((a + b*x^2)^2*Sqrt[c + d*x^2])/x^(9/2),x]
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Maple [A] time = 0.079, size = 385, normalized size = 1.8 \[ -{\frac{2}{21\,cd} \left ( \sqrt{{1 \left ( dx+\sqrt{-cd} \right ){\frac{1}{\sqrt{-cd}}}}}\sqrt{2}\sqrt{{1 \left ( -dx+\sqrt{-cd} \right ){\frac{1}{\sqrt{-cd}}}}}\sqrt{-{dx{\frac{1}{\sqrt{-cd}}}}}{\it EllipticF} \left ( \sqrt{{1 \left ( dx+\sqrt{-cd} \right ){\frac{1}{\sqrt{-cd}}}}},{\frac{\sqrt{2}}{2}} \right ) \sqrt{-cd}{x}^{3}{a}^{2}{d}^{2}-14\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticF} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ) \sqrt{-cd}{x}^{3}abcd-7\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticF} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ) \sqrt{-cd}{x}^{3}{b}^{2}{c}^{2}-7\,{x}^{6}{b}^{2}c{d}^{2}+2\,{x}^{4}{a}^{2}{d}^{3}+14\,{x}^{4}abc{d}^{2}-7\,{x}^{4}{b}^{2}{c}^{2}d+5\,{x}^{2}{a}^{2}c{d}^{2}+14\,{x}^{2}ab{c}^{2}d+3\,{a}^{2}{c}^{2}d \right ){\frac{1}{\sqrt{d{x}^{2}+c}}}{x}^{-{\frac{7}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x^2+a)^2*(d*x^2+c)^(1/2)/x^(9/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{2} + a\right )}^{2} \sqrt{d x^{2} + c}}{x^{\frac{9}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^2*sqrt(d*x^2 + c)/x^(9/2),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )} \sqrt{d x^{2} + c}}{x^{\frac{9}{2}}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^2*sqrt(d*x^2 + c)/x^(9/2),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**(9/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{2} + a\right )}^{2} \sqrt{d x^{2} + c}}{x^{\frac{9}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^2*sqrt(d*x^2 + c)/x^(9/2),x, algorithm="giac")
[Out]