Optimal. Leaf size=368 \[ \frac{2 c^{5/4} \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} \left (21 \sqrt{a} B-5 A \sqrt{c}\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{105 a^{5/4} e^4 \sqrt{e x} \sqrt{a+c x^2}}-\frac{4 B c^{5/4} \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{5 a^{3/4} e^4 \sqrt{e x} \sqrt{a+c x^2}}-\frac{2 \sqrt{a+c x^2} (5 A+7 B x)}{35 e (e x)^{7/2}}-\frac{4 A c \sqrt{a+c x^2}}{21 a e^3 (e x)^{3/2}}+\frac{4 B c^{3/2} x \sqrt{a+c x^2}}{5 a e^4 \sqrt{e x} \left (\sqrt{a}+\sqrt{c} x\right )}-\frac{4 B c \sqrt{a+c x^2}}{5 a e^4 \sqrt{e x}} \]
[Out]
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Rubi [A] time = 0.994203, antiderivative size = 368, 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 \[ \frac{2 c^{5/4} \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} \left (21 \sqrt{a} B-5 A \sqrt{c}\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{105 a^{5/4} e^4 \sqrt{e x} \sqrt{a+c x^2}}-\frac{4 B c^{5/4} \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{5 a^{3/4} e^4 \sqrt{e x} \sqrt{a+c x^2}}-\frac{2 \sqrt{a+c x^2} (5 A+7 B x)}{35 e (e x)^{7/2}}-\frac{4 A c \sqrt{a+c x^2}}{21 a e^3 (e x)^{3/2}}+\frac{4 B c^{3/2} x \sqrt{a+c x^2}}{5 a e^4 \sqrt{e x} \left (\sqrt{a}+\sqrt{c} x\right )}-\frac{4 B c \sqrt{a+c x^2}}{5 a e^4 \sqrt{e x}} \]
Antiderivative was successfully verified.
[In] Int[((A + B*x)*Sqrt[a + c*x^2])/(e*x)^(9/2),x]
[Out]
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Rubi in Sympy [A] time = 137.94, size = 348, normalized size = 0.95 \[ - \frac{4 A c \sqrt{a + c x^{2}}}{21 a e^{3} \left (e x\right )^{\frac{3}{2}}} + \frac{4 B c^{\frac{3}{2}} x \sqrt{a + c x^{2}}}{5 a e^{4} \sqrt{e x} \left (\sqrt{a} + \sqrt{c} x\right )} - \frac{4 B c \sqrt{a + c x^{2}}}{5 a e^{4} \sqrt{e x}} - \frac{4 B c^{\frac{5}{4}} \sqrt{x} \sqrt{\frac{a + c x^{2}}{\left (\sqrt{a} + \sqrt{c} x\right )^{2}}} \left (\sqrt{a} + \sqrt{c} x\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{5 a^{\frac{3}{4}} e^{4} \sqrt{e x} \sqrt{a + c x^{2}}} - \frac{4 \left (\frac{5 A}{2} + \frac{7 B x}{2}\right ) \sqrt{a + c x^{2}}}{35 e \left (e x\right )^{\frac{7}{2}}} - \frac{2 c^{\frac{5}{4}} \sqrt{x} \sqrt{\frac{a + c x^{2}}{\left (\sqrt{a} + \sqrt{c} x\right )^{2}}} \left (\sqrt{a} + \sqrt{c} x\right ) \left (5 A \sqrt{c} - 21 B \sqrt{a}\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{105 a^{\frac{5}{4}} e^{4} \sqrt{e x} \sqrt{a + c x^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)*(c*x**2+a)**(1/2)/(e*x)**(9/2),x)
[Out]
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Mathematica [C] time = 1.15194, size = 236, normalized size = 0.64 \[ -\frac{2 \sqrt{e x} \left (2 i c^{3/2} x^{9/2} \sqrt{\frac{a}{c x^2}+1} \left (5 A \sqrt{c}+21 i \sqrt{a} B\right ) F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{a}}{\sqrt{c}}}}{\sqrt{x}}\right )\right |-1\right )+\sqrt{\frac{i \sqrt{a}}{\sqrt{c}}} \left (a+c x^2\right ) \left (3 a (5 A+7 B x)+10 A c x^2\right )+42 \sqrt{a} B c^{3/2} x^{9/2} \sqrt{\frac{a}{c x^2}+1} E\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{a}}{\sqrt{c}}}}{\sqrt{x}}\right )\right |-1\right )\right )}{105 a e^5 x^4 \sqrt{\frac{i \sqrt{a}}{\sqrt{c}}} \sqrt{a+c x^2}} \]
Antiderivative was successfully verified.
[In] Integrate[((A + B*x)*Sqrt[a + c*x^2])/(e*x)^(9/2),x]
[Out]
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Maple [A] time = 0.048, size = 340, normalized size = 0.9 \[ -{\frac{2}{105\,{x}^{3}{e}^{4}a} \left ( 5\,A\sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{-{\frac{cx}{\sqrt{-ac}}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}},1/2\,\sqrt{2} \right ) \sqrt{-ac}{x}^{3}c+21\,B\sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{-{\frac{cx}{\sqrt{-ac}}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}},1/2\,\sqrt{2} \right ){x}^{3}ac-42\,B\sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{-{\frac{cx}{\sqrt{-ac}}}}{\it EllipticE} \left ( \sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}},1/2\,\sqrt{2} \right ){x}^{3}ac+42\,B{c}^{2}{x}^{5}+10\,A{c}^{2}{x}^{4}+63\,aBc{x}^{3}+25\,aAc{x}^{2}+21\,{a}^{2}Bx+15\,A{a}^{2} \right ){\frac{1}{\sqrt{c{x}^{2}+a}}}{\frac{1}{\sqrt{ex}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)*(c*x^2+a)^(1/2)/(e*x)^(9/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{c x^{2} + a}{\left (B x + A\right )}}{\left (e x\right )^{\frac{9}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(c*x^2 + a)*(B*x + A)/(e*x)^(9/2),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{\sqrt{c x^{2} + a}{\left (B x + A\right )}}{\sqrt{e x} e^{4} x^{4}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(c*x^2 + a)*(B*x + A)/(e*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+A)*(c*x**2+a)**(1/2)/(e*x)**(9/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{c x^{2} + a}{\left (B x + A\right )}}{\left (e x\right )^{\frac{9}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(c*x^2 + a)*(B*x + A)/(e*x)^(9/2),x, algorithm="giac")
[Out]