Optimal. Leaf size=360 \[ -\frac{a^{3/4} e^4 \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} \left (63 \sqrt{a} B+25 A \sqrt{c}\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{30 c^{11/4} \sqrt{e x} \sqrt{a+c x^2}}+\frac{21 a^{5/4} B e^4 \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{5 c^{11/4} \sqrt{e x} \sqrt{a+c x^2}}-\frac{e (e x)^{5/2} (A+B x)}{c \sqrt{a+c x^2}}+\frac{5 A e^3 \sqrt{e x} \sqrt{a+c x^2}}{3 c^2}-\frac{21 a B e^4 x \sqrt{a+c x^2}}{5 c^{5/2} \sqrt{e x} \left (\sqrt{a}+\sqrt{c} x\right )}+\frac{7 B e^2 (e x)^{3/2} \sqrt{a+c x^2}}{5 c^2} \]
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Rubi [A] time = 0.982463, antiderivative size = 360, 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 \[ -\frac{a^{3/4} e^4 \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} \left (63 \sqrt{a} B+25 A \sqrt{c}\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{30 c^{11/4} \sqrt{e x} \sqrt{a+c x^2}}+\frac{21 a^{5/4} B e^4 \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{5 c^{11/4} \sqrt{e x} \sqrt{a+c x^2}}-\frac{e (e x)^{5/2} (A+B x)}{c \sqrt{a+c x^2}}+\frac{5 A e^3 \sqrt{e x} \sqrt{a+c x^2}}{3 c^2}-\frac{21 a B e^4 x \sqrt{a+c x^2}}{5 c^{5/2} \sqrt{e x} \left (\sqrt{a}+\sqrt{c} x\right )}+\frac{7 B e^2 (e x)^{3/2} \sqrt{a+c x^2}}{5 c^2} \]
Antiderivative was successfully verified.
[In] Int[((e*x)^(7/2)*(A + B*x))/(a + c*x^2)^(3/2),x]
[Out]
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Rubi in Sympy [A] time = 128.175, size = 338, normalized size = 0.94 \[ \frac{5 A e^{3} \sqrt{e x} \sqrt{a + c x^{2}}}{3 c^{2}} + \frac{21 B a^{\frac{5}{4}} e^{4} \sqrt{x} \sqrt{\frac{a + c x^{2}}{\left (\sqrt{a} + \sqrt{c} x\right )^{2}}} \left (\sqrt{a} + \sqrt{c} x\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{5 c^{\frac{11}{4}} \sqrt{e x} \sqrt{a + c x^{2}}} - \frac{21 B a e^{4} x \sqrt{a + c x^{2}}}{5 c^{\frac{5}{2}} \sqrt{e x} \left (\sqrt{a} + \sqrt{c} x\right )} + \frac{7 B e^{2} \left (e x\right )^{\frac{3}{2}} \sqrt{a + c x^{2}}}{5 c^{2}} - \frac{a^{\frac{3}{4}} e^{4} \sqrt{x} \sqrt{\frac{a + c x^{2}}{\left (\sqrt{a} + \sqrt{c} x\right )^{2}}} \left (\sqrt{a} + \sqrt{c} x\right ) \left (25 A \sqrt{c} + 63 B \sqrt{a}\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{30 c^{\frac{11}{4}} \sqrt{e x} \sqrt{a + c x^{2}}} - \frac{e \left (e x\right )^{\frac{5}{2}} \left (A + B x\right )}{c \sqrt{a + c x^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x)**(7/2)*(B*x+A)/(c*x**2+a)**(3/2),x)
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Mathematica [C] time = 1.14957, size = 240, normalized size = 0.67 \[ -\frac{e^4 \left (-63 a^{3/2} B \sqrt{c} x^{3/2} \sqrt{\frac{a}{c x^2}+1} E\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{a}}{\sqrt{c}}}}{\sqrt{x}}\right )\right |-1\right )+\sqrt{\frac{i \sqrt{a}}{\sqrt{c}}} \left (63 a^2 B+a c x (42 B x-25 A)-2 c^2 x^3 (5 A+3 B x)\right )+a \sqrt{c} x^{3/2} \sqrt{\frac{a}{c x^2}+1} \left (63 \sqrt{a} B+25 i A \sqrt{c}\right ) F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{a}}{\sqrt{c}}}}{\sqrt{x}}\right )\right |-1\right )\right )}{15 c^3 \sqrt{\frac{i \sqrt{a}}{\sqrt{c}}} \sqrt{e x} \sqrt{a+c x^2}} \]
Antiderivative was successfully verified.
[In] Integrate[((e*x)^(7/2)*(A + B*x))/(a + c*x^2)^(3/2),x]
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Maple [A] time = 0.065, size = 318, normalized size = 0.9 \[ -{\frac{{e}^{3}}{30\,x{c}^{3}}\sqrt{ex} \left ( 25\,A{\it EllipticF} \left ( \sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}},1/2\,\sqrt{2} \right ) \sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{-{\frac{cx}{\sqrt{-ac}}}}\sqrt{-ac}a+126\,B{\it EllipticE} \left ( \sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}},1/2\,\sqrt{2} \right ) \sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{-{\frac{cx}{\sqrt{-ac}}}}{a}^{2}-63\,B{\it EllipticF} \left ( \sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}},1/2\,\sqrt{2} \right ) \sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{-{\frac{cx}{\sqrt{-ac}}}}{a}^{2}-12\,B{c}^{2}{x}^{4}-20\,A{c}^{2}{x}^{3}-42\,aBc{x}^{2}-50\,aAcx \right ){\frac{1}{\sqrt{c{x}^{2}+a}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x)^(7/2)*(B*x+A)/(c*x^2+a)^(3/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (B x + A\right )} \left (e x\right )^{\frac{7}{2}}}{{\left (c x^{2} + a\right )}^{\frac{3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*(e*x)^(7/2)/(c*x^2 + a)^(3/2),x, algorithm="maxima")
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (B e^{3} x^{4} + A e^{3} x^{3}\right )} \sqrt{e x}}{{\left (c x^{2} + a\right )}^{\frac{3}{2}}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*(e*x)^(7/2)/(c*x^2 + a)^(3/2),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((e*x)**(7/2)*(B*x+A)/(c*x**2+a)**(3/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (B x + A\right )} \left (e x\right )^{\frac{7}{2}}}{{\left (c x^{2} + a\right )}^{\frac{3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*(e*x)^(7/2)/(c*x^2 + a)^(3/2),x, algorithm="giac")
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