3.442 \(\int (e x)^{5/2} (A+B x) \left (a+c x^2\right )^{3/2} \, dx\)

Optimal. Leaf size=438 \[ \frac{4 a^{13/4} e^3 \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} \left (65 \sqrt{a} B-231 A \sqrt{c}\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{15015 c^{9/4} \sqrt{e x} \sqrt{a+c x^2}}+\frac{8 a^{13/4} A e^3 \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{65 c^{7/4} \sqrt{e x} \sqrt{a+c x^2}}-\frac{8 a^3 A e^3 x \sqrt{a+c x^2}}{65 c^{3/2} \sqrt{e x} \left (\sqrt{a}+\sqrt{c} x\right )}+\frac{4 a^2 e^2 \sqrt{e x} \sqrt{a+c x^2} (65 a B-231 A c x)}{15015 c^2}+\frac{2 a e^2 \sqrt{e x} \left (a+c x^2\right )^{3/2} (13 a B-77 A c x)}{3003 c^2}+\frac{2 A e (e x)^{3/2} \left (a+c x^2\right )^{5/2}}{13 c}-\frac{2 a B e^2 \sqrt{e x} \left (a+c x^2\right )^{5/2}}{33 c^2}+\frac{2 B (e x)^{5/2} \left (a+c x^2\right )^{5/2}}{15 c} \]

[Out]

(-8*a^3*A*e^3*x*Sqrt[a + c*x^2])/(65*c^(3/2)*Sqrt[e*x]*(Sqrt[a] + Sqrt[c]*x)) +
(4*a^2*e^2*Sqrt[e*x]*(65*a*B - 231*A*c*x)*Sqrt[a + c*x^2])/(15015*c^2) + (2*a*e^
2*Sqrt[e*x]*(13*a*B - 77*A*c*x)*(a + c*x^2)^(3/2))/(3003*c^2) - (2*a*B*e^2*Sqrt[
e*x]*(a + c*x^2)^(5/2))/(33*c^2) + (2*A*e*(e*x)^(3/2)*(a + c*x^2)^(5/2))/(13*c)
+ (2*B*(e*x)^(5/2)*(a + c*x^2)^(5/2))/(15*c) + (8*a^(13/4)*A*e^3*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])/(65*c^(7/4)*Sqrt[e*x]*Sqrt[a + c*x^2]) + (4*a^(13/
4)*(65*Sqrt[a]*B - 231*A*Sqrt[c])*e^3*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
])/(15015*c^(9/4)*Sqrt[e*x]*Sqrt[a + c*x^2])

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Rubi [A]  time = 1.35722, antiderivative size = 438, 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 \[ \frac{4 a^{13/4} e^3 \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} \left (65 \sqrt{a} B-231 A \sqrt{c}\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{15015 c^{9/4} \sqrt{e x} \sqrt{a+c x^2}}+\frac{8 a^{13/4} A e^3 \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{65 c^{7/4} \sqrt{e x} \sqrt{a+c x^2}}-\frac{8 a^3 A e^3 x \sqrt{a+c x^2}}{65 c^{3/2} \sqrt{e x} \left (\sqrt{a}+\sqrt{c} x\right )}+\frac{4 a^2 e^2 \sqrt{e x} \sqrt{a+c x^2} (65 a B-231 A c x)}{15015 c^2}+\frac{2 a e^2 \sqrt{e x} \left (a+c x^2\right )^{3/2} (13 a B-77 A c x)}{3003 c^2}+\frac{2 A e (e x)^{3/2} \left (a+c x^2\right )^{5/2}}{13 c}-\frac{2 a B e^2 \sqrt{e x} \left (a+c x^2\right )^{5/2}}{33 c^2}+\frac{2 B (e x)^{5/2} \left (a+c x^2\right )^{5/2}}{15 c} \]

Antiderivative was successfully verified.

[In]  Int[(e*x)^(5/2)*(A + B*x)*(a + c*x^2)^(3/2),x]

[Out]

(-8*a^3*A*e^3*x*Sqrt[a + c*x^2])/(65*c^(3/2)*Sqrt[e*x]*(Sqrt[a] + Sqrt[c]*x)) +
(4*a^2*e^2*Sqrt[e*x]*(65*a*B - 231*A*c*x)*Sqrt[a + c*x^2])/(15015*c^2) + (2*a*e^
2*Sqrt[e*x]*(13*a*B - 77*A*c*x)*(a + c*x^2)^(3/2))/(3003*c^2) - (2*a*B*e^2*Sqrt[
e*x]*(a + c*x^2)^(5/2))/(33*c^2) + (2*A*e*(e*x)^(3/2)*(a + c*x^2)^(5/2))/(13*c)
+ (2*B*(e*x)^(5/2)*(a + c*x^2)^(5/2))/(15*c) + (8*a^(13/4)*A*e^3*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])/(65*c^(7/4)*Sqrt[e*x]*Sqrt[a + c*x^2]) + (4*a^(13/
4)*(65*Sqrt[a]*B - 231*A*Sqrt[c])*e^3*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
])/(15015*c^(9/4)*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)**(5/2)*(B*x+A)*(c*x**2+a)**(3/2),x)

[Out]

Timed out

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Mathematica [C]  time = 1.43962, size = 276, normalized size = 0.63 \[ -\frac{2 e^3 \left (4 a^{7/2} x^{3/2} \sqrt{\frac{a}{c x^2}+1} \left (231 A \sqrt{c}-65 i \sqrt{a} B\right ) F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{a}}{\sqrt{c}}}}{\sqrt{x}}\right )\right |-1\right )-924 a^{7/2} A \sqrt{c} x^{3/2} \sqrt{\frac{a}{c x^2}+1} E\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{a}}{\sqrt{c}}}}{\sqrt{x}}\right )\right |-1\right )+\sqrt{\frac{i \sqrt{a}}{\sqrt{c}}} \left (a+c x^2\right ) \left (4 a^3 (231 A+65 B x)-4 a^2 c x^2 (77 A+39 B x)-7 a c^2 x^4 (275 A+221 B x)-77 c^3 x^6 (15 A+13 B x)\right )\right )}{15015 c^2 \sqrt{\frac{i \sqrt{a}}{\sqrt{c}}} \sqrt{e x} \sqrt{a+c x^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(e*x)^(5/2)*(A + B*x)*(a + c*x^2)^(3/2),x]

[Out]

(-2*e^3*(Sqrt[(I*Sqrt[a])/Sqrt[c]]*(a + c*x^2)*(-77*c^3*x^6*(15*A + 13*B*x) - 4*
a^2*c*x^2*(77*A + 39*B*x) + 4*a^3*(231*A + 65*B*x) - 7*a*c^2*x^4*(275*A + 221*B*
x)) - 924*a^(7/2)*A*Sqrt[c]*Sqrt[1 + a/(c*x^2)]*x^(3/2)*EllipticE[I*ArcSinh[Sqrt
[(I*Sqrt[a])/Sqrt[c]]/Sqrt[x]], -1] + 4*a^(7/2)*((-65*I)*Sqrt[a]*B + 231*A*Sqrt[
c])*Sqrt[1 + a/(c*x^2)]*x^(3/2)*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[a])/Sqrt[c]]/Sq
rt[x]], -1]))/(15015*Sqrt[(I*Sqrt[a])/Sqrt[c]]*c^2*Sqrt[e*x]*Sqrt[a + c*x^2])

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Maple [A]  time = 0.039, size = 384, normalized size = 0.9 \[ -{\frac{2\,{e}^{2}}{15015\,x{c}^{3}}\sqrt{ex} \left ( -1001\,B{x}^{9}{c}^{5}-1155\,A{x}^{8}{c}^{5}-2548\,B{x}^{7}a{c}^{4}-3080\,A{x}^{6}a{c}^{4}+924\,A\sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{-{\frac{cx}{\sqrt{-ac}}}}{\it EllipticE} \left ( \sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}},1/2\,\sqrt{2} \right ){a}^{4}c-462\,A\sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{-{\frac{cx}{\sqrt{-ac}}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}},1/2\,\sqrt{2} \right ){a}^{4}c-130\,B\sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{-{\frac{cx}{\sqrt{-ac}}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}},1/2\,\sqrt{2} \right ) \sqrt{-ac}{a}^{4}-1703\,B{x}^{5}{a}^{2}{c}^{3}-2233\,A{x}^{4}{a}^{2}{c}^{3}+104\,B{x}^{3}{a}^{3}{c}^{2}-308\,A{x}^{2}{a}^{3}{c}^{2}+260\,Bx{a}^{4}c \right ){\frac{1}{\sqrt{c{x}^{2}+a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x)^(5/2)*(B*x+A)*(c*x^2+a)^(3/2),x)

[Out]

-2/15015/x*e^2*(e*x)^(1/2)/(c*x^2+a)^(1/2)*(-1001*B*x^9*c^5-1155*A*x^8*c^5-2548*
B*x^7*a*c^4-3080*A*x^6*a*c^4+924*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^4*c-462*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^4*c-130*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^4-1703*B*x^5*a^2*c^3-2233*A*x^4*
a^2*c^3+104*B*x^3*a^3*c^2-308*A*x^2*a^3*c^2+260*B*x*a^4*c)/c^3

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (c x^{2} + a\right )}^{\frac{3}{2}}{\left (B x + A\right )} \left (e x\right )^{\frac{5}{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^(3/2)*(B*x + A)*(e*x)^(5/2),x, algorithm="maxima")

[Out]

integrate((c*x^2 + a)^(3/2)*(B*x + A)*(e*x)^(5/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (B c e^{2} x^{5} + A c e^{2} x^{4} + B a e^{2} x^{3} + A a e^{2} x^{2}\right )} \sqrt{c x^{2} + a} \sqrt{e x}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^(3/2)*(B*x + A)*(e*x)^(5/2),x, algorithm="fricas")

[Out]

integral((B*c*e^2*x^5 + A*c*e^2*x^4 + B*a*e^2*x^3 + A*a*e^2*x^2)*sqrt(c*x^2 + a)
*sqrt(e*x), 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)**(5/2)*(B*x+A)*(c*x**2+a)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (c x^{2} + a\right )}^{\frac{3}{2}}{\left (B x + A\right )} \left (e x\right )^{\frac{5}{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^(3/2)*(B*x + A)*(e*x)^(5/2),x, algorithm="giac")

[Out]

integrate((c*x^2 + a)^(3/2)*(B*x + A)*(e*x)^(5/2), x)