3.466 \(\int \frac{(e x)^{7/2} (A+B x)}{\left (a+c x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=360 \[ -\frac{a^{3/4} e^4 \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} \left (63 \sqrt{a} B+25 A \sqrt{c}\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{30 c^{11/4} \sqrt{e x} \sqrt{a+c x^2}}+\frac{21 a^{5/4} B e^4 \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{5 c^{11/4} \sqrt{e x} \sqrt{a+c x^2}}-\frac{e (e x)^{5/2} (A+B x)}{c \sqrt{a+c x^2}}+\frac{5 A e^3 \sqrt{e x} \sqrt{a+c x^2}}{3 c^2}-\frac{21 a B e^4 x \sqrt{a+c x^2}}{5 c^{5/2} \sqrt{e x} \left (\sqrt{a}+\sqrt{c} x\right )}+\frac{7 B e^2 (e x)^{3/2} \sqrt{a+c x^2}}{5 c^2} \]

[Out]

-((e*(e*x)^(5/2)*(A + B*x))/(c*Sqrt[a + c*x^2])) + (5*A*e^3*Sqrt[e*x]*Sqrt[a + c
*x^2])/(3*c^2) + (7*B*e^2*(e*x)^(3/2)*Sqrt[a + c*x^2])/(5*c^2) - (21*a*B*e^4*x*S
qrt[a + c*x^2])/(5*c^(5/2)*Sqrt[e*x]*(Sqrt[a] + Sqrt[c]*x)) + (21*a^(5/4)*B*e^4*
Sqrt[x]*(Sqrt[a] + Sqrt[c]*x)*Sqrt[(a + c*x^2)/(Sqrt[a] + Sqrt[c]*x)^2]*Elliptic
E[2*ArcTan[(c^(1/4)*Sqrt[x])/a^(1/4)], 1/2])/(5*c^(11/4)*Sqrt[e*x]*Sqrt[a + c*x^
2]) - (a^(3/4)*(63*Sqrt[a]*B + 25*A*Sqrt[c])*e^4*Sqrt[x]*(Sqrt[a] + Sqrt[c]*x)*S
qrt[(a + c*x^2)/(Sqrt[a] + Sqrt[c]*x)^2]*EllipticF[2*ArcTan[(c^(1/4)*Sqrt[x])/a^
(1/4)], 1/2])/(30*c^(11/4)*Sqrt[e*x]*Sqrt[a + c*x^2])

_______________________________________________________________________________________

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]

-((e*(e*x)^(5/2)*(A + B*x))/(c*Sqrt[a + c*x^2])) + (5*A*e^3*Sqrt[e*x]*Sqrt[a + c
*x^2])/(3*c^2) + (7*B*e^2*(e*x)^(3/2)*Sqrt[a + c*x^2])/(5*c^2) - (21*a*B*e^4*x*S
qrt[a + c*x^2])/(5*c^(5/2)*Sqrt[e*x]*(Sqrt[a] + Sqrt[c]*x)) + (21*a^(5/4)*B*e^4*
Sqrt[x]*(Sqrt[a] + Sqrt[c]*x)*Sqrt[(a + c*x^2)/(Sqrt[a] + Sqrt[c]*x)^2]*Elliptic
E[2*ArcTan[(c^(1/4)*Sqrt[x])/a^(1/4)], 1/2])/(5*c^(11/4)*Sqrt[e*x]*Sqrt[a + c*x^
2]) - (a^(3/4)*(63*Sqrt[a]*B + 25*A*Sqrt[c])*e^4*Sqrt[x]*(Sqrt[a] + Sqrt[c]*x)*S
qrt[(a + c*x^2)/(Sqrt[a] + Sqrt[c]*x)^2]*EllipticF[2*ArcTan[(c^(1/4)*Sqrt[x])/a^
(1/4)], 1/2])/(30*c^(11/4)*Sqrt[e*x]*Sqrt[a + c*x^2])

_______________________________________________________________________________________

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)

[Out]

5*A*e**3*sqrt(e*x)*sqrt(a + c*x**2)/(3*c**2) + 21*B*a**(5/4)*e**4*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)/(5*c**(11/4)*sqrt(e*x)*sqrt(a + c*x**2)) - 21*B*a
*e**4*x*sqrt(a + c*x**2)/(5*c**(5/2)*sqrt(e*x)*(sqrt(a) + sqrt(c)*x)) + 7*B*e**2
*(e*x)**(3/2)*sqrt(a + c*x**2)/(5*c**2) - a**(3/4)*e**4*sqrt(x)*sqrt((a + c*x**2
)/(sqrt(a) + sqrt(c)*x)**2)*(sqrt(a) + sqrt(c)*x)*(25*A*sqrt(c) + 63*B*sqrt(a))*
elliptic_f(2*atan(c**(1/4)*sqrt(x)/a**(1/4)), 1/2)/(30*c**(11/4)*sqrt(e*x)*sqrt(
a + c*x**2)) - e*(e*x)**(5/2)*(A + B*x)/(c*sqrt(a + c*x**2))

_______________________________________________________________________________________

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]

[Out]

-(e^4*(Sqrt[(I*Sqrt[a])/Sqrt[c]]*(63*a^2*B - 2*c^2*x^3*(5*A + 3*B*x) + a*c*x*(-2
5*A + 42*B*x)) - 63*a^(3/2)*B*Sqrt[c]*Sqrt[1 + a/(c*x^2)]*x^(3/2)*EllipticE[I*Ar
cSinh[Sqrt[(I*Sqrt[a])/Sqrt[c]]/Sqrt[x]], -1] + a*(63*Sqrt[a]*B + (25*I)*A*Sqrt[
c])*Sqrt[c]*Sqrt[1 + a/(c*x^2)]*x^(3/2)*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[a])/Sqr
t[c]]/Sqrt[x]], -1]))/(15*Sqrt[(I*Sqrt[a])/Sqrt[c]]*c^3*Sqrt[e*x]*Sqrt[a + c*x^2
])

_______________________________________________________________________________________

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]

-1/30*e^3/x*(e*x)^(1/2)*(25*A*EllipticF(((c*x+(-a*c)^(1/2))/(-a*c)^(1/2))^(1/2),
1/2*2^(1/2))*((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)*(-a*c)^(1/2)*a+126*B*EllipticE((
(c*x+(-a*c)^(1/2))/(-a*c)^(1/2))^(1/2),1/2*2^(1/2))*((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)*a^2-63*B*EllipticF(((c*x+(-a*c)^(1/2))/(-a*c)^(1/2))^(1/2),1/2*2^(1/2))*(
(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)*a^2-12*B*c^2*x^4-20*A*c^2*x^3-42*a*B*c*x^2-50*
a*A*c*x)/(c*x^2+a)^(1/2)/c^3

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (B x + A\right )} \left (e x\right )^{\frac{7}{2}}}{{\left (c x^{2} + a\right )}^{\frac{3}{2}}}\,{d x} \]

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")

[Out]

integrate((B*x + A)*(e*x)^(7/2)/(c*x^2 + a)^(3/2), x)

_______________________________________________________________________________________

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]

integral((B*e^3*x^4 + A*e^3*x^3)*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)**(7/2)*(B*x+A)/(c*x**2+a)**(3/2),x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (B x + A\right )} \left (e x\right )^{\frac{7}{2}}}{{\left (c x^{2} + a\right )}^{\frac{3}{2}}}\,{d x} \]

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")

[Out]

integrate((B*x + A)*(e*x)^(7/2)/(c*x^2 + a)^(3/2), x)