Optimal. Leaf size=153 \[ -\frac{5 a^{3/4} c^{7/2} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a} \sqrt{c}}\right )|\frac{1}{2}\right )}{6 b^{9/4} \sqrt{a+b x^2}}+\frac{5 c^3 \sqrt{c x} \sqrt{a+b x^2}}{3 b^2}-\frac{c (c x)^{5/2}}{b \sqrt{a+b x^2}} \]
[Out]
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Rubi [A] time = 0.251851, antiderivative size = 153, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21 \[ -\frac{5 a^{3/4} c^{7/2} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a} \sqrt{c}}\right )|\frac{1}{2}\right )}{6 b^{9/4} \sqrt{a+b x^2}}+\frac{5 c^3 \sqrt{c x} \sqrt{a+b x^2}}{3 b^2}-\frac{c (c x)^{5/2}}{b \sqrt{a+b x^2}} \]
Antiderivative was successfully verified.
[In] Int[(c*x)^(7/2)/(a + b*x^2)^(3/2),x]
[Out]
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Rubi in Sympy [A] time = 24.5488, size = 139, normalized size = 0.91 \[ - \frac{5 a^{\frac{3}{4}} c^{\frac{7}{2}} \sqrt{\frac{a + b x^{2}}{\left (\sqrt{a} + \sqrt{b} x\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a} \sqrt{c}} \right )}\middle | \frac{1}{2}\right )}{6 b^{\frac{9}{4}} \sqrt{a + b x^{2}}} - \frac{c \left (c x\right )^{\frac{5}{2}}}{b \sqrt{a + b x^{2}}} + \frac{5 c^{3} \sqrt{c x} \sqrt{a + b x^{2}}}{3 b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c*x)**(7/2)/(b*x**2+a)**(3/2),x)
[Out]
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Mathematica [C] time = 0.127496, size = 131, normalized size = 0.86 \[ \frac{c^3 \sqrt{c x} \left (\sqrt{\frac{i \sqrt{a}}{\sqrt{b}}} \left (5 a+2 b x^2\right )-5 i a \sqrt{x} \sqrt{\frac{a}{b x^2}+1} F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{a}}{\sqrt{b}}}}{\sqrt{x}}\right )\right |-1\right )\right )}{3 b^2 \sqrt{\frac{i \sqrt{a}}{\sqrt{b}}} \sqrt{a+b x^2}} \]
Antiderivative was successfully verified.
[In] Integrate[(c*x)^(7/2)/(a + b*x^2)^(3/2),x]
[Out]
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Maple [A] time = 0.048, size = 128, normalized size = 0.8 \[ -{\frac{{c}^{3}}{6\,{b}^{3}x}\sqrt{cx} \left ( 5\,\sqrt{-ab}\sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{{\frac{-bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-{\frac{bx}{\sqrt{-ab}}}}{\it EllipticF} \left ( \sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}},1/2\,\sqrt{2} \right ) \sqrt{2}a-4\,{b}^{2}{x}^{3}-10\,abx \right ){\frac{1}{\sqrt{b{x}^{2}+a}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c*x)^(7/2)/(b*x^2+a)^(3/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (c x\right )^{\frac{7}{2}}}{{\left (b x^{2} + a\right )}^{\frac{3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x)^(7/2)/(b*x^2 + a)^(3/2),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{\sqrt{c x} c^{3} x^{3}}{{\left (b x^{2} + a\right )}^{\frac{3}{2}}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x)^(7/2)/(b*x^2 + a)^(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((c*x)**(7/2)/(b*x**2+a)**(3/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (c x\right )^{\frac{7}{2}}}{{\left (b x^{2} + a\right )}^{\frac{3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x)^(7/2)/(b*x^2 + a)^(3/2),x, algorithm="giac")
[Out]