Optimal. Leaf size=428 \[ \frac{a^{5/4} e^7 \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} \left (325 \sqrt{a} B-539 A \sqrt{c}\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{140 c^{17/4} \sqrt{e x} \sqrt{a+c x^2}}+\frac{77 a^{5/4} A e^7 \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{10 c^{15/4} \sqrt{e x} \sqrt{a+c x^2}}-\frac{e^3 (e x)^{7/2} (11 A+13 B x)}{6 c^2 \sqrt{a+c x^2}}-\frac{e (e x)^{11/2} (A+B x)}{3 c \left (a+c x^2\right )^{3/2}}-\frac{77 a A e^7 x \sqrt{a+c x^2}}{10 c^{7/2} \sqrt{e x} \left (\sqrt{a}+\sqrt{c} x\right )}+\frac{77 A e^5 (e x)^{3/2} \sqrt{a+c x^2}}{30 c^3}-\frac{65 a B e^6 \sqrt{e x} \sqrt{a+c x^2}}{14 c^4}+\frac{39 B e^4 (e x)^{5/2} \sqrt{a+c x^2}}{14 c^3} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 1.3332, antiderivative size = 428, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.292 \[ \frac{a^{5/4} e^7 \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} \left (325 \sqrt{a} B-539 A \sqrt{c}\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{140 c^{17/4} \sqrt{e x} \sqrt{a+c x^2}}+\frac{77 a^{5/4} A e^7 \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{10 c^{15/4} \sqrt{e x} \sqrt{a+c x^2}}-\frac{e^3 (e x)^{7/2} (11 A+13 B x)}{6 c^2 \sqrt{a+c x^2}}-\frac{e (e x)^{11/2} (A+B x)}{3 c \left (a+c x^2\right )^{3/2}}-\frac{77 a A e^7 x \sqrt{a+c x^2}}{10 c^{7/2} \sqrt{e x} \left (\sqrt{a}+\sqrt{c} x\right )}+\frac{77 A e^5 (e x)^{3/2} \sqrt{a+c x^2}}{30 c^3}-\frac{65 a B e^6 \sqrt{e x} \sqrt{a+c x^2}}{14 c^4}+\frac{39 B e^4 (e x)^{5/2} \sqrt{a+c x^2}}{14 c^3} \]
Antiderivative was successfully verified.
[In] Int[((e*x)^(13/2)*(A + B*x))/(a + c*x^2)^(5/2),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x)**(13/2)*(B*x+A)/(c*x**2+a)**(5/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [C] time = 1.10246, size = 284, normalized size = 0.66 \[ \frac{e^7 \left (-3 a^{3/2} x^{3/2} \sqrt{\frac{a}{c x^2}+1} \left (a+c x^2\right ) \left (539 A \sqrt{c}-325 i \sqrt{a} B\right ) F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{a}}{\sqrt{c}}}}{\sqrt{x}}\right )\right |-1\right )+1617 a^{3/2} A \sqrt{c} x^{3/2} \sqrt{\frac{a}{c x^2}+1} \left (a+c x^2\right ) E\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{a}}{\sqrt{c}}}}{\sqrt{x}}\right )\right |-1\right )-\sqrt{\frac{i \sqrt{a}}{\sqrt{c}}} \left (3 a^3 (539 A+325 B x)+35 a^2 c x^2 (77 A+39 B x)+4 a c^2 x^4 (231 A+65 B x)-12 c^3 x^6 (7 A+5 B x)\right )\right )}{210 c^4 \sqrt{\frac{i \sqrt{a}}{\sqrt{c}}} \sqrt{e x} \left (a+c x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
[In] Integrate[((e*x)^(13/2)*(A + B*x))/(a + c*x^2)^(5/2),x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.06, size = 637, normalized size = 1.5 \[ -{\frac{{e}^{6}}{420\,x{c}^{5}} \left ( 3234\,A\sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{-{\frac{cx}{\sqrt{-ac}}}}{\it EllipticE} \left ( \sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}},1/2\,\sqrt{2} \right ){x}^{2}{a}^{2}{c}^{2}-1617\,A\sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{-{\frac{cx}{\sqrt{-ac}}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}},1/2\,\sqrt{2} \right ){x}^{2}{a}^{2}{c}^{2}-975\,B\sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{-{\frac{cx}{\sqrt{-ac}}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}},1/2\,\sqrt{2} \right ) \sqrt{-ac}{x}^{2}{a}^{2}c-120\,B{c}^{4}{x}^{7}-168\,A{c}^{4}{x}^{6}+3234\,A\sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{-{\frac{cx}{\sqrt{-ac}}}}{\it EllipticE} \left ( \sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}},1/2\,\sqrt{2} \right ){a}^{3}c-1617\,A\sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{-{\frac{cx}{\sqrt{-ac}}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}},1/2\,\sqrt{2} \right ){a}^{3}c-975\,B\sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{-{\frac{cx}{\sqrt{-ac}}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}},1/2\,\sqrt{2} \right ) \sqrt{-ac}{a}^{3}+520\,aB{c}^{3}{x}^{5}-1386\,aA{c}^{3}{x}^{4}+2730\,{a}^{2}B{c}^{2}{x}^{3}-1078\,{a}^{2}A{c}^{2}{x}^{2}+1950\,{a}^{3}Bcx \right ) \sqrt{ex} \left ( c{x}^{2}+a \right ) ^{-{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x)^(13/2)*(B*x+A)/(c*x^2+a)^(5/2),x)
[Out]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (B x + A\right )} \left (e x\right )^{\frac{13}{2}}}{{\left (c x^{2} + a\right )}^{\frac{5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*(e*x)^(13/2)/(c*x^2 + a)^(5/2),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (B e^{6} x^{7} + A e^{6} x^{6}\right )} \sqrt{e x}}{{\left (c^{2} x^{4} + 2 \, a c x^{2} + a^{2}\right )} \sqrt{c x^{2} + a}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*(e*x)^(13/2)/(c*x^2 + a)^(5/2),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
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)**(13/2)*(B*x+A)/(c*x**2+a)**(5/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (B x + A\right )} \left (e x\right )^{\frac{13}{2}}}{{\left (c x^{2} + a\right )}^{\frac{5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*(e*x)^(13/2)/(c*x^2 + a)^(5/2),x, algorithm="giac")
[Out]