Optimal. Leaf size=357 \[ \frac{\sqrt [4]{c} \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} \left (9 \sqrt{a} B-5 A \sqrt{c}\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{6 a^{9/4} e^2 \sqrt{e x} \sqrt{a+c x^2}}-\frac{3 B \sqrt [4]{c} \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{a^{7/4} e^2 \sqrt{e x} \sqrt{a+c x^2}}-\frac{5 A \sqrt{a+c x^2}}{3 a^2 e (e x)^{3/2}}-\frac{3 B \sqrt{a+c x^2}}{a^2 e^2 \sqrt{e x}}+\frac{3 B \sqrt{c} x \sqrt{a+c x^2}}{a^2 e^2 \sqrt{e x} \left (\sqrt{a}+\sqrt{c} x\right )}+\frac{A+B x}{a e (e x)^{3/2} \sqrt{a+c x^2}} \]
[Out]
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Rubi [A] time = 0.999568, antiderivative size = 357, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.292 \[ \frac{\sqrt [4]{c} \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} \left (9 \sqrt{a} B-5 A \sqrt{c}\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{6 a^{9/4} e^2 \sqrt{e x} \sqrt{a+c x^2}}-\frac{3 B \sqrt [4]{c} \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{a^{7/4} e^2 \sqrt{e x} \sqrt{a+c x^2}}-\frac{5 A \sqrt{a+c x^2}}{3 a^2 e (e x)^{3/2}}-\frac{3 B \sqrt{a+c x^2}}{a^2 e^2 \sqrt{e x}}+\frac{3 B \sqrt{c} x \sqrt{a+c x^2}}{a^2 e^2 \sqrt{e x} \left (\sqrt{a}+\sqrt{c} x\right )}+\frac{A+B x}{a e (e x)^{3/2} \sqrt{a+c x^2}} \]
Antiderivative was successfully verified.
[In] Int[(A + B*x)/((e*x)^(5/2)*(a + c*x^2)^(3/2)),x]
[Out]
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Rubi in Sympy [A] time = 141.251, size = 333, normalized size = 0.93 \[ - \frac{5 A \sqrt{a + c x^{2}}}{3 a^{2} e \left (e x\right )^{\frac{3}{2}}} + \frac{3 B \sqrt{c} x \sqrt{a + c x^{2}}}{a^{2} e^{2} \sqrt{e x} \left (\sqrt{a} + \sqrt{c} x\right )} - \frac{3 B \sqrt{a + c x^{2}}}{a^{2} e^{2} \sqrt{e x}} - \frac{3 B \sqrt [4]{c} \sqrt{x} \sqrt{\frac{a + c x^{2}}{\left (\sqrt{a} + \sqrt{c} x\right )^{2}}} \left (\sqrt{a} + \sqrt{c} x\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{a^{\frac{7}{4}} e^{2} \sqrt{e x} \sqrt{a + c x^{2}}} + \frac{A + B x}{a e \left (e x\right )^{\frac{3}{2}} \sqrt{a + c x^{2}}} - \frac{\sqrt [4]{c} \sqrt{x} \sqrt{\frac{a + c x^{2}}{\left (\sqrt{a} + \sqrt{c} x\right )^{2}}} \left (\sqrt{a} + \sqrt{c} x\right ) \left (5 A \sqrt{c} - 9 B \sqrt{a}\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{6 a^{\frac{9}{4}} e^{2} \sqrt{e x} \sqrt{a + c x^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)/(e*x)**(5/2)/(c*x**2+a)**(3/2),x)
[Out]
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Mathematica [C] time = 0.936249, size = 219, normalized size = 0.61 \[ \frac{x \left (\sqrt{\frac{i \sqrt{a}}{\sqrt{c}}} \left (-2 a A+3 a B x-5 A c x^2\right )+\sqrt{c} x^{5/2} \sqrt{\frac{a}{c x^2}+1} \left (9 \sqrt{a} B-5 i A \sqrt{c}\right ) F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{a}}{\sqrt{c}}}}{\sqrt{x}}\right )\right |-1\right )-9 \sqrt{a} B \sqrt{c} x^{5/2} \sqrt{\frac{a}{c x^2}+1} E\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{a}}{\sqrt{c}}}}{\sqrt{x}}\right )\right |-1\right )\right )}{3 a^2 \sqrt{\frac{i \sqrt{a}}{\sqrt{c}}} (e x)^{5/2} \sqrt{a+c x^2}} \]
Antiderivative was successfully verified.
[In] Integrate[(A + B*x)/((e*x)^(5/2)*(a + c*x^2)^(3/2)),x]
[Out]
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Maple [A] time = 0.051, size = 307, normalized size = 0.9 \[ -{\frac{1}{6\,{a}^{2}x{e}^{2}} \left ( 5\,A\sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{-{\frac{cx}{\sqrt{-ac}}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}},1/2\,\sqrt{2} \right ) \sqrt{-ac}x+9\,B\sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{-{\frac{cx}{\sqrt{-ac}}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}},1/2\,\sqrt{2} \right ) xa-18\,B\sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{-{\frac{cx}{\sqrt{-ac}}}}{\it EllipticE} \left ( \sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}},1/2\,\sqrt{2} \right ) xa+18\,Bc{x}^{3}+10\,Ac{x}^{2}+12\,aBx+4\,aA \right ){\frac{1}{\sqrt{c{x}^{2}+a}}}{\frac{1}{\sqrt{ex}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)/(e*x)^(5/2)/(c*x^2+a)^(3/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{B x + A}{{\left (c x^{2} + a\right )}^{\frac{3}{2}} \left (e x\right )^{\frac{5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)/((c*x^2 + a)^(3/2)*(e*x)^(5/2)),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{B x + A}{{\left (c e^{2} x^{4} + a e^{2} x^{2}\right )} \sqrt{c x^{2} + a} \sqrt{e x}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)/((c*x^2 + a)^(3/2)*(e*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((B*x+A)/(e*x)**(5/2)/(c*x**2+a)**(3/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{B x + A}{{\left (c x^{2} + a\right )}^{\frac{3}{2}} \left (e x\right )^{\frac{5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)/((c*x^2 + a)^(3/2)*(e*x)^(5/2)),x, algorithm="giac")
[Out]