Optimal. Leaf size=152 \[ \frac{4 b^{7/4} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a} \sqrt{c}}\right )|\frac{1}{2}\right )}{7 \sqrt [4]{a} c^{9/2} \sqrt{a+b x^2}}-\frac{4 b \sqrt{a+b x^2}}{7 c^3 (c x)^{3/2}}-\frac{2 \left (a+b x^2\right )^{3/2}}{7 c (c x)^{7/2}} \]
[Out]
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Rubi [A] time = 0.256562, antiderivative size = 152, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158 \[ \frac{4 b^{7/4} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a} \sqrt{c}}\right )|\frac{1}{2}\right )}{7 \sqrt [4]{a} c^{9/2} \sqrt{a+b x^2}}-\frac{4 b \sqrt{a+b x^2}}{7 c^3 (c x)^{3/2}}-\frac{2 \left (a+b x^2\right )^{3/2}}{7 c (c x)^{7/2}} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x^2)^(3/2)/(c*x)^(9/2),x]
[Out]
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Rubi in Sympy [A] time = 23.6911, size = 139, normalized size = 0.91 \[ - \frac{4 b \sqrt{a + b x^{2}}}{7 c^{3} \left (c x\right )^{\frac{3}{2}}} - \frac{2 \left (a + b x^{2}\right )^{\frac{3}{2}}}{7 c \left (c x\right )^{\frac{7}{2}}} + \frac{4 b^{\frac{7}{4}} \sqrt{\frac{a + b x^{2}}{\left (\sqrt{a} + \sqrt{b} x\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a} \sqrt{c}} \right )}\middle | \frac{1}{2}\right )}{7 \sqrt [4]{a} c^{\frac{9}{2}} \sqrt{a + b x^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x**2+a)**(3/2)/(c*x)**(9/2),x)
[Out]
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Mathematica [C] time = 0.242044, size = 121, normalized size = 0.8 \[ \frac{x^{9/2} \left (-\frac{2 \left (a+b x^2\right ) \left (a+3 b x^2\right )}{x^{7/2}}+\frac{8 i b^2 x \sqrt{\frac{a}{b x^2}+1} F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{a}}{\sqrt{b}}}}{\sqrt{x}}\right )\right |-1\right )}{\sqrt{\frac{i \sqrt{a}}{\sqrt{b}}}}\right )}{7 (c x)^{9/2} \sqrt{a+b x^2}} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x^2)^(3/2)/(c*x)^(9/2),x]
[Out]
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Maple [A] time = 0.04, size = 135, normalized size = 0.9 \[{\frac{2}{7\,{x}^{3}{c}^{4}} \left ( 2\,\sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{{\frac{-bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-{\frac{bx}{\sqrt{-ab}}}}{\it EllipticF} \left ( \sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}},1/2\,\sqrt{2} \right ) \sqrt{-ab}\sqrt{2}{x}^{3}b-3\,{b}^{2}{x}^{4}-4\,ab{x}^{2}-{a}^{2} \right ){\frac{1}{\sqrt{cx}}}{\frac{1}{\sqrt{b{x}^{2}+a}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x^2+a)^(3/2)/(c*x)^(9/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{2} + a\right )}^{\frac{3}{2}}}{\left (c x\right )^{\frac{9}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^(3/2)/(c*x)^(9/2),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (b x^{2} + a\right )}^{\frac{3}{2}}}{\sqrt{c x} c^{4} x^{4}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^(3/2)/(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)**(3/2)/(c*x)**(9/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{2} + a\right )}^{\frac{3}{2}}}{\left (c x\right )^{\frac{9}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^(3/2)/(c*x)^(9/2),x, algorithm="giac")
[Out]