3.477 \(\int \frac{(e x)^{7/2} (A+B x)}{\left (a+c x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=339 \[ \frac{e^4 \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} \left (21 \sqrt{a} B+5 A \sqrt{c}\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{12 \sqrt [4]{a} c^{11/4} \sqrt{e x} \sqrt{a+c x^2}}-\frac{e^3 \sqrt{e x} (5 A+7 B x)}{6 c^2 \sqrt{a+c x^2}}-\frac{e (e x)^{5/2} (A+B x)}{3 c \left (a+c x^2\right )^{3/2}}+\frac{7 B e^4 x \sqrt{a+c x^2}}{2 c^{5/2} \sqrt{e x} \left (\sqrt{a}+\sqrt{c} x\right )}-\frac{7 \sqrt [4]{a} B e^4 \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{2 c^{11/4} \sqrt{e x} \sqrt{a+c x^2}} \]

[Out]

-(e*(e*x)^(5/2)*(A + B*x))/(3*c*(a + c*x^2)^(3/2)) - (e^3*Sqrt[e*x]*(5*A + 7*B*x
))/(6*c^2*Sqrt[a + c*x^2]) + (7*B*e^4*x*Sqrt[a + c*x^2])/(2*c^(5/2)*Sqrt[e*x]*(S
qrt[a] + Sqrt[c]*x)) - (7*a^(1/4)*B*e^4*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*c^(11/4)*Sqrt[e*x]*Sqrt[a + c*x^2]) + ((21*Sqrt[a]*B + 5*A*Sqrt[c])*e^4*
Sqrt[x]*(Sqrt[a] + Sqrt[c]*x)*Sqrt[(a + c*x^2)/(Sqrt[a] + Sqrt[c]*x)^2]*Elliptic
F[2*ArcTan[(c^(1/4)*Sqrt[x])/a^(1/4)], 1/2])/(12*a^(1/4)*c^(11/4)*Sqrt[e*x]*Sqrt
[a + c*x^2])

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Rubi [A]  time = 0.795604, antiderivative size = 339, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{e^4 \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} \left (21 \sqrt{a} B+5 A \sqrt{c}\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{12 \sqrt [4]{a} c^{11/4} \sqrt{e x} \sqrt{a+c x^2}}-\frac{e^3 \sqrt{e x} (5 A+7 B x)}{6 c^2 \sqrt{a+c x^2}}-\frac{e (e x)^{5/2} (A+B x)}{3 c \left (a+c x^2\right )^{3/2}}+\frac{7 B e^4 x \sqrt{a+c x^2}}{2 c^{5/2} \sqrt{e x} \left (\sqrt{a}+\sqrt{c} x\right )}-\frac{7 \sqrt [4]{a} B e^4 \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{2 c^{11/4} \sqrt{e x} \sqrt{a+c x^2}} \]

Antiderivative was successfully verified.

[In]  Int[((e*x)^(7/2)*(A + B*x))/(a + c*x^2)^(5/2),x]

[Out]

-(e*(e*x)^(5/2)*(A + B*x))/(3*c*(a + c*x^2)^(3/2)) - (e^3*Sqrt[e*x]*(5*A + 7*B*x
))/(6*c^2*Sqrt[a + c*x^2]) + (7*B*e^4*x*Sqrt[a + c*x^2])/(2*c^(5/2)*Sqrt[e*x]*(S
qrt[a] + Sqrt[c]*x)) - (7*a^(1/4)*B*e^4*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*c^(11/4)*Sqrt[e*x]*Sqrt[a + c*x^2]) + ((21*Sqrt[a]*B + 5*A*Sqrt[c])*e^4*
Sqrt[x]*(Sqrt[a] + Sqrt[c]*x)*Sqrt[(a + c*x^2)/(Sqrt[a] + Sqrt[c]*x)^2]*Elliptic
F[2*ArcTan[(c^(1/4)*Sqrt[x])/a^(1/4)], 1/2])/(12*a^(1/4)*c^(11/4)*Sqrt[e*x]*Sqrt
[a + c*x^2])

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Rubi in Sympy [A]  time = 89.7788, size = 314, normalized size = 0.93 \[ - \frac{7 B \sqrt [4]{a} e^{4} \sqrt{x} \sqrt{\frac{a + c x^{2}}{\left (\sqrt{a} + \sqrt{c} x\right )^{2}}} \left (\sqrt{a} + \sqrt{c} x\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{2 c^{\frac{11}{4}} \sqrt{e x} \sqrt{a + c x^{2}}} + \frac{7 B e^{4} x \sqrt{a + c x^{2}}}{2 c^{\frac{5}{2}} \sqrt{e x} \left (\sqrt{a} + \sqrt{c} x\right )} - \frac{e \left (e x\right )^{\frac{5}{2}} \left (A + B x\right )}{3 c \left (a + c x^{2}\right )^{\frac{3}{2}}} - \frac{e^{3} \sqrt{e x} \left (10 A + 14 B x\right )}{12 c^{2} \sqrt{a + c x^{2}}} + \frac{e^{4} \sqrt{x} \sqrt{\frac{a + c x^{2}}{\left (\sqrt{a} + \sqrt{c} x\right )^{2}}} \left (\sqrt{a} + \sqrt{c} x\right ) \left (5 A \sqrt{c} + 21 B \sqrt{a}\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{12 \sqrt [4]{a} c^{\frac{11}{4}} \sqrt{e x} \sqrt{a + c x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x)**(7/2)*(B*x+A)/(c*x**2+a)**(5/2),x)

[Out]

-7*B*a**(1/4)*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)/(2*c**(11/4)*sqr
t(e*x)*sqrt(a + c*x**2)) + 7*B*e**4*x*sqrt(a + c*x**2)/(2*c**(5/2)*sqrt(e*x)*(sq
rt(a) + sqrt(c)*x)) - e*(e*x)**(5/2)*(A + B*x)/(3*c*(a + c*x**2)**(3/2)) - e**3*
sqrt(e*x)*(10*A + 14*B*x)/(12*c**2*sqrt(a + c*x**2)) + e**4*sqrt(x)*sqrt((a + c*
x**2)/(sqrt(a) + sqrt(c)*x)**2)*(sqrt(a) + sqrt(c)*x)*(5*A*sqrt(c) + 21*B*sqrt(a
))*elliptic_f(2*atan(c**(1/4)*sqrt(x)/a**(1/4)), 1/2)/(12*a**(1/4)*c**(11/4)*sqr
t(e*x)*sqrt(a + c*x**2))

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Mathematica [C]  time = 1.08461, size = 251, normalized size = 0.74 \[ \frac{e^4 \left (\sqrt{\frac{i \sqrt{a}}{\sqrt{c}}} \left (21 a^2 B-5 a c x (A-7 B x)+c^2 x^3 (12 B x-7 A)\right )+\sqrt{c} x^{3/2} \sqrt{\frac{a}{c x^2}+1} \left (a+c x^2\right ) \left (21 \sqrt{a} B+5 i A \sqrt{c}\right ) F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{a}}{\sqrt{c}}}}{\sqrt{x}}\right )\right |-1\right )-21 \sqrt{a} B \sqrt{c} x^{3/2} \sqrt{\frac{a}{c x^2}+1} \left (a+c x^2\right ) E\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{a}}{\sqrt{c}}}}{\sqrt{x}}\right )\right |-1\right )\right )}{6 c^3 \sqrt{\frac{i \sqrt{a}}{\sqrt{c}}} \sqrt{e x} \left (a+c x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[((e*x)^(7/2)*(A + B*x))/(a + c*x^2)^(5/2),x]

[Out]

(e^4*(Sqrt[(I*Sqrt[a])/Sqrt[c]]*(21*a^2*B - 5*a*c*x*(A - 7*B*x) + c^2*x^3*(-7*A
+ 12*B*x)) - 21*Sqrt[a]*B*Sqrt[c]*Sqrt[1 + a/(c*x^2)]*x^(3/2)*(a + c*x^2)*Ellipt
icE[I*ArcSinh[Sqrt[(I*Sqrt[a])/Sqrt[c]]/Sqrt[x]], -1] + (21*Sqrt[a]*B + (5*I)*A*
Sqrt[c])*Sqrt[c]*Sqrt[1 + a/(c*x^2)]*x^(3/2)*(a + c*x^2)*EllipticF[I*ArcSinh[Sqr
t[(I*Sqrt[a])/Sqrt[c]]/Sqrt[x]], -1]))/(6*Sqrt[(I*Sqrt[a])/Sqrt[c]]*c^3*Sqrt[e*x
]*(a + c*x^2)^(3/2))

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Maple [A]  time = 0.051, size = 584, normalized size = 1.7 \[{\frac{{e}^{3}}{12\,x{c}^{3}} \left ( 5\,A\sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{-{\frac{cx}{\sqrt{-ac}}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}},1/2\,\sqrt{2} \right ) \sqrt{-ac}{x}^{2}c+42\,B\sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{-{\frac{cx}{\sqrt{-ac}}}}{\it EllipticE} \left ( \sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}},1/2\,\sqrt{2} \right ){x}^{2}ac-21\,B\sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{-{\frac{cx}{\sqrt{-ac}}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}},1/2\,\sqrt{2} \right ){x}^{2}ac+5\,A{\it EllipticF} \left ( \sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}},1/2\,\sqrt{2} \right ) \sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{-{\frac{cx}{\sqrt{-ac}}}}\sqrt{-ac}a+42\,B{\it EllipticE} \left ( \sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}},1/2\,\sqrt{2} \right ) \sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{-{\frac{cx}{\sqrt{-ac}}}}{a}^{2}-21\,B{\it EllipticF} \left ( \sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}},1/2\,\sqrt{2} \right ) \sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{-{\frac{cx}{\sqrt{-ac}}}}{a}^{2}-18\,B{c}^{2}{x}^{4}-14\,A{c}^{2}{x}^{3}-14\,aBc{x}^{2}-10\,aAcx \right ) \sqrt{ex} \left ( c{x}^{2}+a \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x)^(7/2)*(B*x+A)/(c*x^2+a)^(5/2),x)

[Out]

1/12*(5*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+42*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-21
*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+5*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+42*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^2-21*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^2-18*B*c^2*x^4-14*A*c^2*x^3-14*a*B*c*x^2-1
0*a*A*c*x)*e^3/x*(e*x)^(1/2)/c^3/(c*x^2+a)^(3/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (B x + A\right )} \left (e x\right )^{\frac{7}{2}}}{{\left (c x^{2} + a\right )}^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(e*x)^(7/2)/(c*x^2 + a)^(5/2),x, algorithm="maxima")

[Out]

integrate((B*x + A)*(e*x)^(7/2)/(c*x^2 + a)^(5/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(e*x)^(7/2)/(c*x^2 + a)^(5/2),x, algorithm="fricas")

[Out]

integral((B*e^3*x^4 + A*e^3*x^3)*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)**(7/2)*(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. \[ \int \frac{{\left (B x + A\right )} \left (e x\right )^{\frac{7}{2}}}{{\left (c x^{2} + a\right )}^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(e*x)^(7/2)/(c*x^2 + a)^(5/2),x, algorithm="giac")

[Out]

integrate((B*x + A)*(e*x)^(7/2)/(c*x^2 + a)^(5/2), x)