Optimal. Leaf size=388 \[ \frac{a^{7/4} e^4 \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} \left (49 \sqrt{a} B+25 A \sqrt{c}\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{105 c^{11/4} \sqrt{e x} \sqrt{a+c x^2}}-\frac{14 a^{9/4} B e^4 \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{15 c^{11/4} \sqrt{e x} \sqrt{a+c x^2}}+\frac{14 a^2 B e^4 x \sqrt{a+c x^2}}{15 c^{5/2} \sqrt{e x} \left (\sqrt{a}+\sqrt{c} x\right )}-\frac{10 a A e^3 \sqrt{e x} \sqrt{a+c x^2}}{21 c^2}+\frac{2 A e (e x)^{5/2} \sqrt{a+c x^2}}{7 c}-\frac{14 a B e^2 (e x)^{3/2} \sqrt{a+c x^2}}{45 c^2}+\frac{2 B (e x)^{7/2} \sqrt{a+c x^2}}{9 c} \]
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Rubi [A] time = 1.14557, antiderivative size = 388, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{a^{7/4} e^4 \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} \left (49 \sqrt{a} B+25 A \sqrt{c}\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{105 c^{11/4} \sqrt{e x} \sqrt{a+c x^2}}-\frac{14 a^{9/4} B e^4 \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{15 c^{11/4} \sqrt{e x} \sqrt{a+c x^2}}+\frac{14 a^2 B e^4 x \sqrt{a+c x^2}}{15 c^{5/2} \sqrt{e x} \left (\sqrt{a}+\sqrt{c} x\right )}-\frac{10 a A e^3 \sqrt{e x} \sqrt{a+c x^2}}{21 c^2}+\frac{2 A e (e x)^{5/2} \sqrt{a+c x^2}}{7 c}-\frac{14 a B e^2 (e x)^{3/2} \sqrt{a+c x^2}}{45 c^2}+\frac{2 B (e x)^{7/2} \sqrt{a+c x^2}}{9 c} \]
Antiderivative was successfully verified.
[In] Int[((e*x)^(7/2)*(A + B*x))/Sqrt[a + c*x^2],x]
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Rubi in Sympy [A] time = 153.685, size = 367, normalized size = 0.95 \[ - \frac{10 A a e^{3} \sqrt{e x} \sqrt{a + c x^{2}}}{21 c^{2}} + \frac{2 A e \left (e x\right )^{\frac{5}{2}} \sqrt{a + c x^{2}}}{7 c} - \frac{14 B a^{\frac{9}{4}} e^{4} \sqrt{x} \sqrt{\frac{a + c x^{2}}{\left (\sqrt{a} + \sqrt{c} x\right )^{2}}} \left (\sqrt{a} + \sqrt{c} x\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{15 c^{\frac{11}{4}} \sqrt{e x} \sqrt{a + c x^{2}}} + \frac{14 B a^{2} e^{4} x \sqrt{a + c x^{2}}}{15 c^{\frac{5}{2}} \sqrt{e x} \left (\sqrt{a} + \sqrt{c} x\right )} - \frac{14 B a e^{2} \left (e x\right )^{\frac{3}{2}} \sqrt{a + c x^{2}}}{45 c^{2}} + \frac{2 B \left (e x\right )^{\frac{7}{2}} \sqrt{a + c x^{2}}}{9 c} + \frac{a^{\frac{7}{4}} e^{4} \sqrt{x} \sqrt{\frac{a + c x^{2}}{\left (\sqrt{a} + \sqrt{c} x\right )^{2}}} \left (\sqrt{a} + \sqrt{c} x\right ) \left (25 A \sqrt{c} + 49 B \sqrt{a}\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{105 c^{\frac{11}{4}} \sqrt{e x} \sqrt{a + c x^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x)**(7/2)*(B*x+A)/(c*x**2+a)**(1/2),x)
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Mathematica [C] time = 0.922831, size = 251, normalized size = 0.65 \[ \frac{2 e^4 \left (-147 a^{5/2} B \sqrt{c} x^{3/2} \sqrt{\frac{a}{c x^2}+1} E\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{a}}{\sqrt{c}}}}{\sqrt{x}}\right )\right |-1\right )+\sqrt{\frac{i \sqrt{a}}{\sqrt{c}}} \left (a+c x^2\right ) \left (147 a^2 B-a c x (75 A+49 B x)+5 c^2 x^3 (9 A+7 B x)\right )+3 a^2 \sqrt{c} x^{3/2} \sqrt{\frac{a}{c x^2}+1} \left (49 \sqrt{a} B+25 i A \sqrt{c}\right ) F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{a}}{\sqrt{c}}}}{\sqrt{x}}\right )\right |-1\right )\right )}{315 c^3 \sqrt{\frac{i \sqrt{a}}{\sqrt{c}}} \sqrt{e x} \sqrt{a+c x^2}} \]
Antiderivative was successfully verified.
[In] Integrate[((e*x)^(7/2)*(A + B*x))/Sqrt[a + c*x^2],x]
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Maple [A] time = 0.04, size = 344, normalized size = 0.9 \[{\frac{{e}^{3}}{315\,x{c}^{3}}\sqrt{ex} \left ( 70\,B{c}^{3}{x}^{6}+75\,A\sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{-{\frac{cx}{\sqrt{-ac}}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}},1/2\,\sqrt{2} \right ) \sqrt{-ac}{a}^{2}+90\,A{c}^{3}{x}^{5}-147\,B\sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{-{\frac{cx}{\sqrt{-ac}}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}},1/2\,\sqrt{2} \right ){a}^{3}+294\,B\sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{-{\frac{cx}{\sqrt{-ac}}}}{\it EllipticE} \left ( \sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}},1/2\,\sqrt{2} \right ){a}^{3}-28\,aB{c}^{2}{x}^{4}-60\,aA{c}^{2}{x}^{3}-98\,{a}^{2}Bc{x}^{2}-150\,{a}^{2}Acx \right ){\frac{1}{\sqrt{c{x}^{2}+a}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x)^(7/2)*(B*x+A)/(c*x^2+a)^(1/2),x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (B x + A\right )} \left (e x\right )^{\frac{7}{2}}}{\sqrt{c x^{2} + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*(e*x)^(7/2)/sqrt(c*x^2 + a),x, algorithm="maxima")
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (B e^{3} x^{4} + A e^{3} x^{3}\right )} \sqrt{e x}}{\sqrt{c x^{2} + a}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*(e*x)^(7/2)/sqrt(c*x^2 + a),x, algorithm="fricas")
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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((e*x)**(7/2)*(B*x+A)/(c*x**2+a)**(1/2),x)
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (B x + A\right )} \left (e x\right )^{\frac{7}{2}}}{\sqrt{c x^{2} + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*(e*x)^(7/2)/sqrt(c*x^2 + a),x, algorithm="giac")
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