Optimal. Leaf size=331 \[ \frac{4 b^{9/4} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a} \sqrt{c}}\right )|\frac{1}{2}\right )}{15 a^{3/4} c^{11/2} \sqrt{a+b x^2}}-\frac{8 b^{9/4} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a} \sqrt{c}}\right )|\frac{1}{2}\right )}{15 a^{3/4} c^{11/2} \sqrt{a+b x^2}}+\frac{8 b^{5/2} \sqrt{c x} \sqrt{a+b x^2}}{15 a c^6 \left (\sqrt{a}+\sqrt{b} x\right )}-\frac{8 b^2 \sqrt{a+b x^2}}{15 a c^5 \sqrt{c x}}-\frac{4 b \sqrt{a+b x^2}}{15 c^3 (c x)^{5/2}}-\frac{2 \left (a+b x^2\right )^{3/2}}{9 c (c x)^{9/2}} \]
[Out]
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Rubi [A] time = 0.686009, antiderivative size = 331, 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 \[ \frac{4 b^{9/4} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a} \sqrt{c}}\right )|\frac{1}{2}\right )}{15 a^{3/4} c^{11/2} \sqrt{a+b x^2}}-\frac{8 b^{9/4} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a} \sqrt{c}}\right )|\frac{1}{2}\right )}{15 a^{3/4} c^{11/2} \sqrt{a+b x^2}}+\frac{8 b^{5/2} \sqrt{c x} \sqrt{a+b x^2}}{15 a c^6 \left (\sqrt{a}+\sqrt{b} x\right )}-\frac{8 b^2 \sqrt{a+b x^2}}{15 a c^5 \sqrt{c x}}-\frac{4 b \sqrt{a+b x^2}}{15 c^3 (c x)^{5/2}}-\frac{2 \left (a+b x^2\right )^{3/2}}{9 c (c x)^{9/2}} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x^2)^(3/2)/(c*x)^(11/2),x]
[Out]
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Rubi in Sympy [A] time = 70.9209, size = 304, normalized size = 0.92 \[ - \frac{4 b \sqrt{a + b x^{2}}}{15 c^{3} \left (c x\right )^{\frac{5}{2}}} - \frac{2 \left (a + b x^{2}\right )^{\frac{3}{2}}}{9 c \left (c x\right )^{\frac{9}{2}}} + \frac{8 b^{\frac{5}{2}} \sqrt{c x} \sqrt{a + b x^{2}}}{15 a c^{6} \left (\sqrt{a} + \sqrt{b} x\right )} - \frac{8 b^{2} \sqrt{a + b x^{2}}}{15 a c^{5} \sqrt{c x}} - \frac{8 b^{\frac{9}{4}} \sqrt{\frac{a + b x^{2}}{\left (\sqrt{a} + \sqrt{b} x\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a} \sqrt{c}} \right )}\middle | \frac{1}{2}\right )}{15 a^{\frac{3}{4}} c^{\frac{11}{2}} \sqrt{a + b x^{2}}} + \frac{4 b^{\frac{9}{4}} \sqrt{\frac{a + b x^{2}}{\left (\sqrt{a} + \sqrt{b} x\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a} \sqrt{c}} \right )}\middle | \frac{1}{2}\right )}{15 a^{\frac{3}{4}} c^{\frac{11}{2}} \sqrt{a + b x^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x**2+a)**(3/2)/(c*x)**(11/2),x)
[Out]
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Mathematica [C] time = 0.424198, size = 213, normalized size = 0.64 \[ -\frac{2 \sqrt{c x} \left (\sqrt{\frac{i \sqrt{b} x}{\sqrt{a}}} \left (5 a^3+16 a^2 b x^2+23 a b^2 x^4+12 b^3 x^6\right )+12 \sqrt{a} b^{5/2} x^5 \sqrt{\frac{b x^2}{a}+1} F\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{b} x}{\sqrt{a}}}\right )\right |-1\right )-12 \sqrt{a} b^{5/2} x^5 \sqrt{\frac{b x^2}{a}+1} E\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{b} x}{\sqrt{a}}}\right )\right |-1\right )\right )}{45 a c^6 x^5 \sqrt{\frac{i \sqrt{b} x}{\sqrt{a}}} \sqrt{a+b x^2}} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x^2)^(3/2)/(c*x)^(11/2),x]
[Out]
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Maple [A] time = 0.046, size = 234, normalized size = 0.7 \[{\frac{2}{45\,a{x}^{4}{c}^{5}} \left ( 12\,\sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{{\frac{-bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-{\frac{bx}{\sqrt{-ab}}}}{\it EllipticE} \left ( \sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}},1/2\,\sqrt{2} \right ) \sqrt{2}{x}^{4}a{b}^{2}-6\,\sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{{\frac{-bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-{\frac{bx}{\sqrt{-ab}}}}{\it EllipticF} \left ( \sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}},1/2\,\sqrt{2} \right ) \sqrt{2}{x}^{4}a{b}^{2}-12\,{b}^{3}{x}^{6}-23\,a{b}^{2}{x}^{4}-16\,{a}^{2}b{x}^{2}-5\,{a}^{3} \right ){\frac{1}{\sqrt{cx}}}{\frac{1}{\sqrt{b{x}^{2}+a}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x^2+a)^(3/2)/(c*x)^(11/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{2} + a\right )}^{\frac{3}{2}}}{\left (c x\right )^{\frac{11}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^(3/2)/(c*x)^(11/2),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (b x^{2} + a\right )}^{\frac{3}{2}}}{\sqrt{c x} c^{5} x^{5}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^(3/2)/(c*x)^(11/2),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)**(11/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{2} + a\right )}^{\frac{3}{2}}}{\left (c x\right )^{\frac{11}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^(3/2)/(c*x)^(11/2),x, algorithm="giac")
[Out]