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

Optimal. Leaf size=341 \[ -\frac{e^2 \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} \left (3 \sqrt{a} B-A \sqrt{c}\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{12 a^{5/4} c^{7/4} \sqrt{e x} \sqrt{a+c x^2}}+\frac{B e^2 \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{2 a^{3/4} c^{7/4} \sqrt{e x} \sqrt{a+c x^2}}+\frac{e \sqrt{e x} (A+3 B x)}{6 a c \sqrt{a+c x^2}}-\frac{e \sqrt{e x} (A+B x)}{3 c \left (a+c x^2\right )^{3/2}}-\frac{B e^2 x \sqrt{a+c x^2}}{2 a c^{3/2} \sqrt{e x} \left (\sqrt{a}+\sqrt{c} x\right )} \]

[Out]

-(e*Sqrt[e*x]*(A + B*x))/(3*c*(a + c*x^2)^(3/2)) + (e*Sqrt[e*x]*(A + 3*B*x))/(6*
a*c*Sqrt[a + c*x^2]) - (B*e^2*x*Sqrt[a + c*x^2])/(2*a*c^(3/2)*Sqrt[e*x]*(Sqrt[a]
+ Sqrt[c]*x)) + (B*e^2*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])/(2*a^(3/4)*
c^(7/4)*Sqrt[e*x]*Sqrt[a + c*x^2]) - ((3*Sqrt[a]*B - A*Sqrt[c])*e^2*Sqrt[x]*(Sqr
t[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])/(12*a^(5/4)*c^(7/4)*Sqrt[e*x]*Sqrt[a + c*x^2])

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Rubi [A]  time = 0.81944, antiderivative size = 341, 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 \[ -\frac{e^2 \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} \left (3 \sqrt{a} B-A \sqrt{c}\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{12 a^{5/4} c^{7/4} \sqrt{e x} \sqrt{a+c x^2}}+\frac{B e^2 \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{2 a^{3/4} c^{7/4} \sqrt{e x} \sqrt{a+c x^2}}+\frac{e \sqrt{e x} (A+3 B x)}{6 a c \sqrt{a+c x^2}}-\frac{e \sqrt{e x} (A+B x)}{3 c \left (a+c x^2\right )^{3/2}}-\frac{B e^2 x \sqrt{a+c x^2}}{2 a c^{3/2} \sqrt{e x} \left (\sqrt{a}+\sqrt{c} x\right )} \]

Antiderivative was successfully verified.

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

[Out]

-(e*Sqrt[e*x]*(A + B*x))/(3*c*(a + c*x^2)^(3/2)) + (e*Sqrt[e*x]*(A + 3*B*x))/(6*
a*c*Sqrt[a + c*x^2]) - (B*e^2*x*Sqrt[a + c*x^2])/(2*a*c^(3/2)*Sqrt[e*x]*(Sqrt[a]
+ Sqrt[c]*x)) + (B*e^2*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])/(2*a^(3/4)*
c^(7/4)*Sqrt[e*x]*Sqrt[a + c*x^2]) - ((3*Sqrt[a]*B - A*Sqrt[c])*e^2*Sqrt[x]*(Sqr
t[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])/(12*a^(5/4)*c^(7/4)*Sqrt[e*x]*Sqrt[a + c*x^2])

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Rubi in Sympy [A]  time = 105.416, size = 308, normalized size = 0.9 \[ - \frac{B e^{2} x \sqrt{a + c x^{2}}}{2 a c^{\frac{3}{2}} \sqrt{e x} \left (\sqrt{a} + \sqrt{c} x\right )} + \frac{B e^{2} \sqrt{x} \sqrt{\frac{a + c x^{2}}{\left (\sqrt{a} + \sqrt{c} x\right )^{2}}} \left (\sqrt{a} + \sqrt{c} x\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{2 a^{\frac{3}{4}} c^{\frac{7}{4}} \sqrt{e x} \sqrt{a + c x^{2}}} - \frac{e \sqrt{e x} \left (A + B x\right )}{3 c \left (a + c x^{2}\right )^{\frac{3}{2}}} + \frac{e \sqrt{e x} \left (A + 3 B x\right )}{6 a c \sqrt{a + c x^{2}}} + \frac{e^{2} \sqrt{x} \sqrt{\frac{a + c x^{2}}{\left (\sqrt{a} + \sqrt{c} x\right )^{2}}} \left (\sqrt{a} + \sqrt{c} x\right ) \left (A \sqrt{c} - 3 B \sqrt{a}\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{12 a^{\frac{5}{4}} c^{\frac{7}{4}} \sqrt{e x} \sqrt{a + c x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-B*e**2*x*sqrt(a + c*x**2)/(2*a*c**(3/2)*sqrt(e*x)*(sqrt(a) + sqrt(c)*x)) + B*e*
*2*sqrt(x)*sqrt((a + c*x**2)/(sqrt(a) + sqrt(c)*x)**2)*(sqrt(a) + sqrt(c)*x)*ell
iptic_e(2*atan(c**(1/4)*sqrt(x)/a**(1/4)), 1/2)/(2*a**(3/4)*c**(7/4)*sqrt(e*x)*s
qrt(a + c*x**2)) - e*sqrt(e*x)*(A + B*x)/(3*c*(a + c*x**2)**(3/2)) + e*sqrt(e*x)
*(A + 3*B*x)/(6*a*c*sqrt(a + c*x**2)) + e**2*sqrt(x)*sqrt((a + c*x**2)/(sqrt(a)
+ sqrt(c)*x)**2)*(sqrt(a) + sqrt(c)*x)*(A*sqrt(c) - 3*B*sqrt(a))*elliptic_f(2*at
an(c**(1/4)*sqrt(x)/a**(1/4)), 1/2)/(12*a**(5/4)*c**(7/4)*sqrt(e*x)*sqrt(a + c*x
**2))

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Mathematica [C]  time = 0.985692, size = 249, normalized size = 0.73 \[ \frac{e^2 \left (\sqrt{\frac{i \sqrt{a}}{\sqrt{c}}} \left (-3 a^2 B-a c x (A+5 B x)+A c^2 x^3\right )+i \sqrt{c} x^{3/2} \sqrt{\frac{a}{c x^2}+1} \left (a+c x^2\right ) \left (A \sqrt{c}+3 i \sqrt{a} B\right ) F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{a}}{\sqrt{c}}}}{\sqrt{x}}\right )\right |-1\right )+3 \sqrt{a} B \sqrt{c} x^{3/2} \sqrt{\frac{a}{c x^2}+1} \left (a+c x^2\right ) E\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{a}}{\sqrt{c}}}}{\sqrt{x}}\right )\right |-1\right )\right )}{6 a c^2 \sqrt{\frac{i \sqrt{a}}{\sqrt{c}}} \sqrt{e x} \left (a+c x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(e^2*(Sqrt[(I*Sqrt[a])/Sqrt[c]]*(-3*a^2*B + A*c^2*x^3 - a*c*x*(A + 5*B*x)) + 3*S
qrt[a]*B*Sqrt[c]*Sqrt[1 + a/(c*x^2)]*x^(3/2)*(a + c*x^2)*EllipticE[I*ArcSinh[Sqr
t[(I*Sqrt[a])/Sqrt[c]]/Sqrt[x]], -1] + I*((3*I)*Sqrt[a]*B + A*Sqrt[c])*Sqrt[c]*S
qrt[1 + a/(c*x^2)]*x^(3/2)*(a + c*x^2)*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[a])/Sqrt
[c]]/Sqrt[x]], -1]))/(6*a*Sqrt[(I*Sqrt[a])/Sqrt[c]]*c^2*Sqrt[e*x]*(a + c*x^2)^(3
/2))

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Maple [A]  time = 0.05, size = 583, normalized size = 1.7 \[{\frac{e}{12\,{c}^{2}xa} \left ( A\sqrt{{1 \left ( cx+\sqrt{-ac} \right ){\frac{1}{\sqrt{-ac}}}}}\sqrt{2}\sqrt{{1 \left ( -cx+\sqrt{-ac} \right ){\frac{1}{\sqrt{-ac}}}}}\sqrt{-{cx{\frac{1}{\sqrt{-ac}}}}}{\it EllipticF} \left ( \sqrt{{1 \left ( cx+\sqrt{-ac} \right ){\frac{1}{\sqrt{-ac}}}}},{\frac{\sqrt{2}}{2}} \right ) \sqrt{-ac}{x}^{2}c-6\,B\sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{-{\frac{cx}{\sqrt{-ac}}}}{\it EllipticE} \left ( \sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}},1/2\,\sqrt{2} \right ){x}^{2}ac+3\,B\sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{-{\frac{cx}{\sqrt{-ac}}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}},1/2\,\sqrt{2} \right ){x}^{2}ac+A{\it EllipticF} \left ( \sqrt{{1 \left ( cx+\sqrt{-ac} \right ){\frac{1}{\sqrt{-ac}}}}},{\frac{\sqrt{2}}{2}} \right ) \sqrt{{1 \left ( cx+\sqrt{-ac} \right ){\frac{1}{\sqrt{-ac}}}}}\sqrt{2}\sqrt{{1 \left ( -cx+\sqrt{-ac} \right ){\frac{1}{\sqrt{-ac}}}}}\sqrt{-{cx{\frac{1}{\sqrt{-ac}}}}}\sqrt{-ac}a-6\,B{\it EllipticE} \left ( \sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}},1/2\,\sqrt{2} \right ) \sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{-{\frac{cx}{\sqrt{-ac}}}}{a}^{2}+3\,B{\it EllipticF} \left ( \sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}},1/2\,\sqrt{2} \right ) \sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{-{\frac{cx}{\sqrt{-ac}}}}{a}^{2}+6\,B{c}^{2}{x}^{4}+2\,A{c}^{2}{x}^{3}+2\,aBc{x}^{2}-2\,aAcx \right ) \sqrt{ex} \left ( c{x}^{2}+a \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/12*(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)^(1/2)*x^2*c-6*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)*EllipticE(((c*x+(-a*c)^(1/2))/(-a*c)^(1/2))^(1/2),1/2*2^(1/2))*x^2*a*c+3*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))*x^2*a*c+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-6*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+3*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+6*B*c^2*x^4+2*A*c^2*x^3+2*a*B*c*x^2-2*a*A*c*x)*e
/x*(e*x)^(1/2)/c^2/a/(c*x^2+a)^(3/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((B*e*x^2 + A*e*x)*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)**(3/2)*(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. \[ \int \frac{{\left (B x + A\right )} \left (e x\right )^{\frac{3}{2}}}{{\left (c x^{2} + a\right )}^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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