Optimal. Leaf size=329 \[ -\frac{4 a^{13/4} c^{5/2} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a} \sqrt{c}}\right )|\frac{1}{2}\right )}{65 b^{7/4} \sqrt{a+b x^2}}+\frac{8 a^{13/4} c^{5/2} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a} \sqrt{c}}\right )|\frac{1}{2}\right )}{65 b^{7/4} \sqrt{a+b x^2}}-\frac{8 a^3 c^2 \sqrt{c x} \sqrt{a+b x^2}}{65 b^{3/2} \left (\sqrt{a}+\sqrt{b} x\right )}+\frac{8 a^2 c (c x)^{3/2} \sqrt{a+b x^2}}{195 b}+\frac{2 (c x)^{7/2} \left (a+b x^2\right )^{3/2}}{13 c}+\frac{4 a (c x)^{7/2} \sqrt{a+b x^2}}{39 c} \]
[Out]
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Rubi [A] time = 0.670314, antiderivative size = 329, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.316 \[ -\frac{4 a^{13/4} c^{5/2} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a} \sqrt{c}}\right )|\frac{1}{2}\right )}{65 b^{7/4} \sqrt{a+b x^2}}+\frac{8 a^{13/4} c^{5/2} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a} \sqrt{c}}\right )|\frac{1}{2}\right )}{65 b^{7/4} \sqrt{a+b x^2}}-\frac{8 a^3 c^2 \sqrt{c x} \sqrt{a+b x^2}}{65 b^{3/2} \left (\sqrt{a}+\sqrt{b} x\right )}+\frac{8 a^2 c (c x)^{3/2} \sqrt{a+b x^2}}{195 b}+\frac{2 (c x)^{7/2} \left (a+b x^2\right )^{3/2}}{13 c}+\frac{4 a (c x)^{7/2} \sqrt{a+b x^2}}{39 c} \]
Antiderivative was successfully verified.
[In] Int[(c*x)^(5/2)*(a + b*x^2)^(3/2),x]
[Out]
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Rubi in Sympy [A] time = 68.9929, size = 303, normalized size = 0.92 \[ \frac{8 a^{\frac{13}{4}} c^{\frac{5}{2}} \sqrt{\frac{a + b x^{2}}{\left (\sqrt{a} + \sqrt{b} x\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a} \sqrt{c}} \right )}\middle | \frac{1}{2}\right )}{65 b^{\frac{7}{4}} \sqrt{a + b x^{2}}} - \frac{4 a^{\frac{13}{4}} c^{\frac{5}{2}} \sqrt{\frac{a + b x^{2}}{\left (\sqrt{a} + \sqrt{b} x\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a} \sqrt{c}} \right )}\middle | \frac{1}{2}\right )}{65 b^{\frac{7}{4}} \sqrt{a + b x^{2}}} - \frac{8 a^{3} c^{2} \sqrt{c x} \sqrt{a + b x^{2}}}{65 b^{\frac{3}{2}} \left (\sqrt{a} + \sqrt{b} x\right )} + \frac{8 a^{2} c \left (c x\right )^{\frac{3}{2}} \sqrt{a + b x^{2}}}{195 b} + \frac{4 a \left (c x\right )^{\frac{7}{2}} \sqrt{a + b x^{2}}}{39 c} + \frac{2 \left (c x\right )^{\frac{7}{2}} \left (a + b x^{2}\right )^{\frac{3}{2}}}{13 c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c*x)**(5/2)*(b*x**2+a)**(3/2),x)
[Out]
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Mathematica [C] time = 0.310453, size = 202, normalized size = 0.61 \[ \frac{2 c^2 \sqrt{c x} \left (12 a^{7/2} \sqrt{\frac{b x^2}{a}+1} F\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{b} x}{\sqrt{a}}}\right )\right |-1\right )-12 a^{7/2} \sqrt{\frac{b x^2}{a}+1} E\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{b} x}{\sqrt{a}}}\right )\right |-1\right )+\sqrt{b} x \sqrt{\frac{i \sqrt{b} x}{\sqrt{a}}} \left (4 a^3+29 a^2 b x^2+40 a b^2 x^4+15 b^3 x^6\right )\right )}{195 b^{3/2} \sqrt{\frac{i \sqrt{b} x}{\sqrt{a}}} \sqrt{a+b x^2}} \]
Antiderivative was successfully verified.
[In] Integrate[(c*x)^(5/2)*(a + b*x^2)^(3/2),x]
[Out]
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Maple [A] time = 0.033, size = 232, normalized size = 0.7 \[ -{\frac{2\,{c}^{2}}{195\,{b}^{2}x}\sqrt{cx} \left ( -15\,{x}^{8}{b}^{4}-40\,{x}^{6}a{b}^{3}+12\,\sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{{\frac{-bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-{\frac{bx}{\sqrt{-ab}}}}{\it EllipticE} \left ( \sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}},1/2\,\sqrt{2} \right ) \sqrt{2}{a}^{4}-6\,\sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{{\frac{-bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-{\frac{bx}{\sqrt{-ab}}}}{\it EllipticF} \left ( \sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}},1/2\,\sqrt{2} \right ) \sqrt{2}{a}^{4}-29\,{x}^{4}{a}^{2}{b}^{2}-4\,{x}^{2}{a}^{3}b \right ){\frac{1}{\sqrt{b{x}^{2}+a}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c*x)^(5/2)*(b*x^2+a)^(3/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (b x^{2} + a\right )}^{\frac{3}{2}} \left (c x\right )^{\frac{5}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^(3/2)*(c*x)^(5/2),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (b c^{2} x^{4} + a c^{2} x^{2}\right )} \sqrt{b x^{2} + a} \sqrt{c x}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^(3/2)*(c*x)^(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((c*x)**(5/2)*(b*x**2+a)**(3/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (b x^{2} + a\right )}^{\frac{3}{2}} \left (c x\right )^{\frac{5}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^(3/2)*(c*x)^(5/2),x, algorithm="giac")
[Out]