3.474 \(\int \frac{(e x)^{13/2} (A+B x)}{\left (a+c x^2\right )^{5/2}} \, dx\)

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]

-(e*(e*x)^(11/2)*(A + B*x))/(3*c*(a + c*x^2)^(3/2)) - (e^3*(e*x)^(7/2)*(11*A + 1
3*B*x))/(6*c^2*Sqrt[a + c*x^2]) - (65*a*B*e^6*Sqrt[e*x]*Sqrt[a + c*x^2])/(14*c^4
) + (77*A*e^5*(e*x)^(3/2)*Sqrt[a + c*x^2])/(30*c^3) + (39*B*e^4*(e*x)^(5/2)*Sqrt
[a + c*x^2])/(14*c^3) - (77*a*A*e^7*x*Sqrt[a + c*x^2])/(10*c^(7/2)*Sqrt[e*x]*(Sq
rt[a] + Sqrt[c]*x)) + (77*a^(5/4)*A*e^7*Sqrt[x]*(Sqrt[a] + Sqrt[c]*x)*Sqrt[(a +
c*x^2)/(Sqrt[a] + Sqrt[c]*x)^2]*EllipticE[2*ArcTan[(c^(1/4)*Sqrt[x])/a^(1/4)], 1
/2])/(10*c^(15/4)*Sqrt[e*x]*Sqrt[a + c*x^2]) + (a^(5/4)*(325*Sqrt[a]*B - 539*A*S
qrt[c])*e^7*Sqrt[x]*(Sqrt[a] + Sqrt[c]*x)*Sqrt[(a + c*x^2)/(Sqrt[a] + Sqrt[c]*x)
^2]*EllipticF[2*ArcTan[(c^(1/4)*Sqrt[x])/a^(1/4)], 1/2])/(140*c^(17/4)*Sqrt[e*x]
*Sqrt[a + c*x^2])

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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]

-(e*(e*x)^(11/2)*(A + B*x))/(3*c*(a + c*x^2)^(3/2)) - (e^3*(e*x)^(7/2)*(11*A + 1
3*B*x))/(6*c^2*Sqrt[a + c*x^2]) - (65*a*B*e^6*Sqrt[e*x]*Sqrt[a + c*x^2])/(14*c^4
) + (77*A*e^5*(e*x)^(3/2)*Sqrt[a + c*x^2])/(30*c^3) + (39*B*e^4*(e*x)^(5/2)*Sqrt
[a + c*x^2])/(14*c^3) - (77*a*A*e^7*x*Sqrt[a + c*x^2])/(10*c^(7/2)*Sqrt[e*x]*(Sq
rt[a] + Sqrt[c]*x)) + (77*a^(5/4)*A*e^7*Sqrt[x]*(Sqrt[a] + Sqrt[c]*x)*Sqrt[(a +
c*x^2)/(Sqrt[a] + Sqrt[c]*x)^2]*EllipticE[2*ArcTan[(c^(1/4)*Sqrt[x])/a^(1/4)], 1
/2])/(10*c^(15/4)*Sqrt[e*x]*Sqrt[a + c*x^2]) + (a^(5/4)*(325*Sqrt[a]*B - 539*A*S
qrt[c])*e^7*Sqrt[x]*(Sqrt[a] + Sqrt[c]*x)*Sqrt[(a + c*x^2)/(Sqrt[a] + Sqrt[c]*x)
^2]*EllipticF[2*ArcTan[(c^(1/4)*Sqrt[x])/a^(1/4)], 1/2])/(140*c^(17/4)*Sqrt[e*x]
*Sqrt[a + c*x^2])

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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]

Timed out

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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]

(e^7*(-(Sqrt[(I*Sqrt[a])/Sqrt[c]]*(-12*c^3*x^6*(7*A + 5*B*x) + 35*a^2*c*x^2*(77*
A + 39*B*x) + 4*a*c^2*x^4*(231*A + 65*B*x) + 3*a^3*(539*A + 325*B*x))) + 1617*a^
(3/2)*A*Sqrt[c]*Sqrt[1 + a/(c*x^2)]*x^(3/2)*(a + c*x^2)*EllipticE[I*ArcSinh[Sqrt
[(I*Sqrt[a])/Sqrt[c]]/Sqrt[x]], -1] - 3*a^(3/2)*((-325*I)*Sqrt[a]*B + 539*A*Sqrt
[c])*Sqrt[1 + a/(c*x^2)]*x^(3/2)*(a + c*x^2)*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[a]
)/Sqrt[c]]/Sqrt[x]], -1]))/(210*Sqrt[(I*Sqrt[a])/Sqrt[c]]*c^4*Sqrt[e*x]*(a + c*x
^2)^(3/2))

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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]

-1/420*(3234*A*((c*x+(-a*c)^(1/2))/(-a*c)^(1/2))^(1/2)*2^(1/2)*((-c*x+(-a*c)^(1/
2))/(-a*c)^(1/2))^(1/2)*(-x*c/(-a*c)^(1/2))^(1/2)*EllipticE(((c*x+(-a*c)^(1/2))/
(-a*c)^(1/2))^(1/2),1/2*2^(1/2))*x^2*a^2*c^2-1617*A*((c*x+(-a*c)^(1/2))/(-a*c)^(
1/2))^(1/2)*2^(1/2)*((-c*x+(-a*c)^(1/2))/(-a*c)^(1/2))^(1/2)*(-x*c/(-a*c)^(1/2))
^(1/2)*EllipticF(((c*x+(-a*c)^(1/2))/(-a*c)^(1/2))^(1/2),1/2*2^(1/2))*x^2*a^2*c^
2-975*B*((c*x+(-a*c)^(1/2))/(-a*c)^(1/2))^(1/2)*2^(1/2)*((-c*x+(-a*c)^(1/2))/(-a
*c)^(1/2))^(1/2)*(-x*c/(-a*c)^(1/2))^(1/2)*EllipticF(((c*x+(-a*c)^(1/2))/(-a*c)^
(1/2))^(1/2),1/2*2^(1/2))*(-a*c)^(1/2)*x^2*a^2*c-120*B*c^4*x^7-168*A*c^4*x^6+323
4*A*((c*x+(-a*c)^(1/2))/(-a*c)^(1/2))^(1/2)*2^(1/2)*((-c*x+(-a*c)^(1/2))/(-a*c)^
(1/2))^(1/2)*(-x*c/(-a*c)^(1/2))^(1/2)*EllipticE(((c*x+(-a*c)^(1/2))/(-a*c)^(1/2
))^(1/2),1/2*2^(1/2))*a^3*c-1617*A*((c*x+(-a*c)^(1/2))/(-a*c)^(1/2))^(1/2)*2^(1/
2)*((-c*x+(-a*c)^(1/2))/(-a*c)^(1/2))^(1/2)*(-x*c/(-a*c)^(1/2))^(1/2)*EllipticF(
((c*x+(-a*c)^(1/2))/(-a*c)^(1/2))^(1/2),1/2*2^(1/2))*a^3*c-975*B*((c*x+(-a*c)^(1
/2))/(-a*c)^(1/2))^(1/2)*2^(1/2)*((-c*x+(-a*c)^(1/2))/(-a*c)^(1/2))^(1/2)*(-x*c/
(-a*c)^(1/2))^(1/2)*EllipticF(((c*x+(-a*c)^(1/2))/(-a*c)^(1/2))^(1/2),1/2*2^(1/2
))*(-a*c)^(1/2)*a^3+520*a*B*c^3*x^5-1386*a*A*c^3*x^4+2730*a^2*B*c^2*x^3-1078*a^2
*A*c^2*x^2+1950*a^3*B*c*x)*e^6/x*(e*x)^(1/2)/c^5/(c*x^2+a)^(3/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (B x + A\right )} \left (e x\right )^{\frac{13}{2}}}{{\left (c x^{2} + a\right )}^{\frac{5}{2}}}\,{d x} \]

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]

integrate((B*x + A)*(e*x)^(13/2)/(c*x^2 + a)^(5/2), x)

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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]

integral((B*e^6*x^7 + A*e^6*x^6)*sqrt(e*x)/((c^2*x^4 + 2*a*c*x^2 + a^2)*sqrt(c*x
^2 + a)), x)

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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)**(13/2)*(B*x+A)/(c*x**2+a)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (B x + A\right )} \left (e x\right )^{\frac{13}{2}}}{{\left (c x^{2} + a\right )}^{\frac{5}{2}}}\,{d x} \]

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]

integrate((B*x + A)*(e*x)^(13/2)/(c*x^2 + a)^(5/2), x)