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

Optimal. Leaf size=326 \[ -\frac{\sqrt [4]{a} e^3 \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} \left (5 \sqrt{a} B-9 A \sqrt{c}\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{6 c^{9/4} \sqrt{e x} \sqrt{a+c x^2}}-\frac{e (e x)^{3/2} (A+B x)}{c \sqrt{a+c x^2}}+\frac{3 A e^3 x \sqrt{a+c x^2}}{c^{3/2} \sqrt{e x} \left (\sqrt{a}+\sqrt{c} x\right )}-\frac{3 \sqrt [4]{a} A e^3 \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{c^{7/4} \sqrt{e x} \sqrt{a+c x^2}}+\frac{5 B e^2 \sqrt{e x} \sqrt{a+c x^2}}{3 c^2} \]

[Out]

-((e*(e*x)^(3/2)*(A + B*x))/(c*Sqrt[a + c*x^2])) + (5*B*e^2*Sqrt[e*x]*Sqrt[a + c
*x^2])/(3*c^2) + (3*A*e^3*x*Sqrt[a + c*x^2])/(c^(3/2)*Sqrt[e*x]*(Sqrt[a] + Sqrt[
c]*x)) - (3*a^(1/4)*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])/(c^(7/4)*
Sqrt[e*x]*Sqrt[a + c*x^2]) - (a^(1/4)*(5*Sqrt[a]*B - 9*A*Sqrt[c])*e^3*Sqrt[x]*(S
qrt[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])/(6*c^(9/4)*Sqrt[e*x]*Sqrt[a + c*x^2])

_______________________________________________________________________________________

Rubi [A]  time = 0.813675, antiderivative size = 326, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.292 \[ -\frac{\sqrt [4]{a} e^3 \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} \left (5 \sqrt{a} B-9 A \sqrt{c}\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{6 c^{9/4} \sqrt{e x} \sqrt{a+c x^2}}-\frac{e (e x)^{3/2} (A+B x)}{c \sqrt{a+c x^2}}+\frac{3 A e^3 x \sqrt{a+c x^2}}{c^{3/2} \sqrt{e x} \left (\sqrt{a}+\sqrt{c} x\right )}-\frac{3 \sqrt [4]{a} A e^3 \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{c^{7/4} \sqrt{e x} \sqrt{a+c x^2}}+\frac{5 B e^2 \sqrt{e x} \sqrt{a+c x^2}}{3 c^2} \]

Antiderivative was successfully verified.

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

[Out]

-((e*(e*x)^(3/2)*(A + B*x))/(c*Sqrt[a + c*x^2])) + (5*B*e^2*Sqrt[e*x]*Sqrt[a + c
*x^2])/(3*c^2) + (3*A*e^3*x*Sqrt[a + c*x^2])/(c^(3/2)*Sqrt[e*x]*(Sqrt[a] + Sqrt[
c]*x)) - (3*a^(1/4)*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])/(c^(7/4)*
Sqrt[e*x]*Sqrt[a + c*x^2]) - (a^(1/4)*(5*Sqrt[a]*B - 9*A*Sqrt[c])*e^3*Sqrt[x]*(S
qrt[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])/(6*c^(9/4)*Sqrt[e*x]*Sqrt[a + c*x^2])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 96.0799, size = 304, normalized size = 0.93 \[ - \frac{3 A \sqrt [4]{a} e^{3} \sqrt{x} \sqrt{\frac{a + c x^{2}}{\left (\sqrt{a} + \sqrt{c} x\right )^{2}}} \left (\sqrt{a} + \sqrt{c} x\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{c^{\frac{7}{4}} \sqrt{e x} \sqrt{a + c x^{2}}} + \frac{3 A e^{3} x \sqrt{a + c x^{2}}}{c^{\frac{3}{2}} \sqrt{e x} \left (\sqrt{a} + \sqrt{c} x\right )} + \frac{5 B e^{2} \sqrt{e x} \sqrt{a + c x^{2}}}{3 c^{2}} + \frac{\sqrt [4]{a} e^{3} \sqrt{x} \sqrt{\frac{a + c x^{2}}{\left (\sqrt{a} + \sqrt{c} x\right )^{2}}} \left (\sqrt{a} + \sqrt{c} x\right ) \left (9 A \sqrt{c} - 5 B \sqrt{a}\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{6 c^{\frac{9}{4}} \sqrt{e x} \sqrt{a + c x^{2}}} - \frac{e \left (e x\right )^{\frac{3}{2}} \left (A + B x\right )}{c \sqrt{a + c x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3*A*a**(1/4)*e**3*sqrt(x)*sqrt((a + c*x**2)/(sqrt(a) + sqrt(c)*x)**2)*(sqrt(a)
+ sqrt(c)*x)*elliptic_e(2*atan(c**(1/4)*sqrt(x)/a**(1/4)), 1/2)/(c**(7/4)*sqrt(e
*x)*sqrt(a + c*x**2)) + 3*A*e**3*x*sqrt(a + c*x**2)/(c**(3/2)*sqrt(e*x)*(sqrt(a)
+ sqrt(c)*x)) + 5*B*e**2*sqrt(e*x)*sqrt(a + c*x**2)/(3*c**2) + a**(1/4)*e**3*sq
rt(x)*sqrt((a + c*x**2)/(sqrt(a) + sqrt(c)*x)**2)*(sqrt(a) + sqrt(c)*x)*(9*A*sqr
t(c) - 5*B*sqrt(a))*elliptic_f(2*atan(c**(1/4)*sqrt(x)/a**(1/4)), 1/2)/(6*c**(9/
4)*sqrt(e*x)*sqrt(a + c*x**2)) - e*(e*x)**(3/2)*(A + B*x)/(c*sqrt(a + c*x**2))

_______________________________________________________________________________________

Mathematica [C]  time = 0.837476, size = 228, normalized size = 0.7 \[ \frac{e^3 \left (\sqrt{a} x^{3/2} \sqrt{\frac{a}{c x^2}+1} \left (9 A \sqrt{c}-5 i \sqrt{a} B\right ) F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{a}}{\sqrt{c}}}}{\sqrt{x}}\right )\right |-1\right )+\sqrt{\frac{i \sqrt{a}}{\sqrt{c}}} \left (9 a A+5 a B x+6 A c x^2+2 B c x^3\right )-9 \sqrt{a} A \sqrt{c} x^{3/2} \sqrt{\frac{a}{c x^2}+1} E\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{a}}{\sqrt{c}}}}{\sqrt{x}}\right )\right |-1\right )\right )}{3 c^2 \sqrt{\frac{i \sqrt{a}}{\sqrt{c}}} \sqrt{e x} \sqrt{a+c x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(e^3*(Sqrt[(I*Sqrt[a])/Sqrt[c]]*(9*a*A + 5*a*B*x + 6*A*c*x^2 + 2*B*c*x^3) - 9*Sq
rt[a]*A*Sqrt[c]*Sqrt[1 + a/(c*x^2)]*x^(3/2)*EllipticE[I*ArcSinh[Sqrt[(I*Sqrt[a])
/Sqrt[c]]/Sqrt[x]], -1] + Sqrt[a]*((-5*I)*Sqrt[a]*B + 9*A*Sqrt[c])*Sqrt[1 + a/(c
*x^2)]*x^(3/2)*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[a])/Sqrt[c]]/Sqrt[x]], -1]))/(3*
Sqrt[(I*Sqrt[a])/Sqrt[c]]*c^2*Sqrt[e*x]*Sqrt[a + c*x^2])

_______________________________________________________________________________________

Maple [A]  time = 0.048, size = 308, normalized size = 0.9 \[ -{\frac{{e}^{2}}{6\,x{c}^{3}}\sqrt{ex} \left ( 9\,A\sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{-{\frac{cx}{\sqrt{-ac}}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}},1/2\,\sqrt{2} \right ) ac-18\,A\sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{-{\frac{cx}{\sqrt{-ac}}}}{\it EllipticE} \left ( \sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}},1/2\,\sqrt{2} \right ) ac+5\,B\sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{-{\frac{cx}{\sqrt{-ac}}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}},1/2\,\sqrt{2} \right ) \sqrt{-ac}a-4\,B{c}^{2}{x}^{3}+6\,A{c}^{2}{x}^{2}-10\,aBcx \right ){\frac{1}{\sqrt{c{x}^{2}+a}}}} \]

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]

-1/6/x*e^2*(e*x)^(1/2)*(9*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*c-18*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*c+5*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*B*c^2*x^3+6*A*c^2*x^2-10*a*B*c*x)/(c*x^2+a)^
(1/2)/c^3

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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