3.434 \(\int (e x)^{5/2} (A+B x) \sqrt{a+c x^2} \, dx\)

Optimal. Leaf size=397 \[ \frac{2 a^{9/4} e^3 \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} \left (25 \sqrt{a} B-77 A \sqrt{c}\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{1155 c^{9/4} \sqrt{e x} \sqrt{a+c x^2}}+\frac{4 a^{9/4} A e^3 \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{15 c^{7/4} \sqrt{e x} \sqrt{a+c x^2}}-\frac{4 a^2 A e^3 x \sqrt{a+c x^2}}{15 c^{3/2} \sqrt{e x} \left (\sqrt{a}+\sqrt{c} x\right )}+\frac{2 a e^2 \sqrt{e x} \sqrt{a+c x^2} (25 a B-77 A c x)}{1155 c^2}+\frac{2 A e (e x)^{3/2} \left (a+c x^2\right )^{3/2}}{9 c}-\frac{10 a B e^2 \sqrt{e x} \left (a+c x^2\right )^{3/2}}{77 c^2}+\frac{2 B (e x)^{5/2} \left (a+c x^2\right )^{3/2}}{11 c} \]

[Out]

(-4*a^2*A*e^3*x*Sqrt[a + c*x^2])/(15*c^(3/2)*Sqrt[e*x]*(Sqrt[a] + Sqrt[c]*x)) +
(2*a*e^2*Sqrt[e*x]*(25*a*B - 77*A*c*x)*Sqrt[a + c*x^2])/(1155*c^2) - (10*a*B*e^2
*Sqrt[e*x]*(a + c*x^2)^(3/2))/(77*c^2) + (2*A*e*(e*x)^(3/2)*(a + c*x^2)^(3/2))/(
9*c) + (2*B*(e*x)^(5/2)*(a + c*x^2)^(3/2))/(11*c) + (4*a^(9/4)*A*e^3*Sqrt[x]*(Sq
rt[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])/(15*c^(7/4)*Sqrt[e*x]*Sqrt[a + c*x^2]) + (2*a^
(9/4)*(25*Sqrt[a]*B - 77*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])/(1155*c^(9/4)*Sqrt[e*x]*Sqrt[a + c*x^2])

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Rubi [A]  time = 1.13484, antiderivative size = 397, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.292 \[ \frac{2 a^{9/4} e^3 \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} \left (25 \sqrt{a} B-77 A \sqrt{c}\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{1155 c^{9/4} \sqrt{e x} \sqrt{a+c x^2}}+\frac{4 a^{9/4} A e^3 \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{15 c^{7/4} \sqrt{e x} \sqrt{a+c x^2}}-\frac{4 a^2 A e^3 x \sqrt{a+c x^2}}{15 c^{3/2} \sqrt{e x} \left (\sqrt{a}+\sqrt{c} x\right )}+\frac{2 a e^2 \sqrt{e x} \sqrt{a+c x^2} (25 a B-77 A c x)}{1155 c^2}+\frac{2 A e (e x)^{3/2} \left (a+c x^2\right )^{3/2}}{9 c}-\frac{10 a B e^2 \sqrt{e x} \left (a+c x^2\right )^{3/2}}{77 c^2}+\frac{2 B (e x)^{5/2} \left (a+c x^2\right )^{3/2}}{11 c} \]

Antiderivative was successfully verified.

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

[Out]

(-4*a^2*A*e^3*x*Sqrt[a + c*x^2])/(15*c^(3/2)*Sqrt[e*x]*(Sqrt[a] + Sqrt[c]*x)) +
(2*a*e^2*Sqrt[e*x]*(25*a*B - 77*A*c*x)*Sqrt[a + c*x^2])/(1155*c^2) - (10*a*B*e^2
*Sqrt[e*x]*(a + c*x^2)^(3/2))/(77*c^2) + (2*A*e*(e*x)^(3/2)*(a + c*x^2)^(3/2))/(
9*c) + (2*B*(e*x)^(5/2)*(a + c*x^2)^(3/2))/(11*c) + (4*a^(9/4)*A*e^3*Sqrt[x]*(Sq
rt[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])/(15*c^(7/4)*Sqrt[e*x]*Sqrt[a + c*x^2]) + (2*a^
(9/4)*(25*Sqrt[a]*B - 77*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])/(1155*c^(9/4)*Sqrt[e*x]*Sqrt[a + c*x^2])

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Rubi in Sympy [A]  time = 156.522, size = 382, normalized size = 0.96 \[ \frac{4 A a^{\frac{9}{4}} e^{3} \sqrt{x} \sqrt{\frac{a + c x^{2}}{\left (\sqrt{a} + \sqrt{c} x\right )^{2}}} \left (\sqrt{a} + \sqrt{c} x\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{15 c^{\frac{7}{4}} \sqrt{e x} \sqrt{a + c x^{2}}} - \frac{4 A a^{2} e^{3} x \sqrt{a + c x^{2}}}{15 c^{\frac{3}{2}} \sqrt{e x} \left (\sqrt{a} + \sqrt{c} x\right )} + \frac{2 A e \left (e x\right )^{\frac{3}{2}} \left (a + c x^{2}\right )^{\frac{3}{2}}}{9 c} - \frac{10 B a e^{2} \sqrt{e x} \left (a + c x^{2}\right )^{\frac{3}{2}}}{77 c^{2}} + \frac{2 B \left (e x\right )^{\frac{5}{2}} \left (a + c x^{2}\right )^{\frac{3}{2}}}{11 c} - \frac{2 a^{\frac{9}{4}} e^{3} \sqrt{x} \sqrt{\frac{a + c x^{2}}{\left (\sqrt{a} + \sqrt{c} x\right )^{2}}} \left (\sqrt{a} + \sqrt{c} x\right ) \left (77 A \sqrt{c} - 25 B \sqrt{a}\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{1155 c^{\frac{9}{4}} \sqrt{e x} \sqrt{a + c x^{2}}} + \frac{32 a e^{2} \sqrt{e x} \sqrt{a + c x^{2}} \left (- \frac{693 A c x}{16} + \frac{225 B a}{16}\right )}{10395 c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x)**(5/2)*(B*x+A)*(c*x**2+a)**(1/2),x)

[Out]

4*A*a**(9/4)*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)/(15*c**(7/4)*sqrt
(e*x)*sqrt(a + c*x**2)) - 4*A*a**2*e**3*x*sqrt(a + c*x**2)/(15*c**(3/2)*sqrt(e*x
)*(sqrt(a) + sqrt(c)*x)) + 2*A*e*(e*x)**(3/2)*(a + c*x**2)**(3/2)/(9*c) - 10*B*a
*e**2*sqrt(e*x)*(a + c*x**2)**(3/2)/(77*c**2) + 2*B*(e*x)**(5/2)*(a + c*x**2)**(
3/2)/(11*c) - 2*a**(9/4)*e**3*sqrt(x)*sqrt((a + c*x**2)/(sqrt(a) + sqrt(c)*x)**2
)*(sqrt(a) + sqrt(c)*x)*(77*A*sqrt(c) - 25*B*sqrt(a))*elliptic_f(2*atan(c**(1/4)
*sqrt(x)/a**(1/4)), 1/2)/(1155*c**(9/4)*sqrt(e*x)*sqrt(a + c*x**2)) + 32*a*e**2*
sqrt(e*x)*sqrt(a + c*x**2)*(-693*A*c*x/16 + 225*B*a/16)/(10395*c**2)

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Mathematica [C]  time = 1.38773, size = 257, normalized size = 0.65 \[ -\frac{2 e^3 \left (6 a^{5/2} x^{3/2} \sqrt{\frac{a}{c x^2}+1} \left (77 A \sqrt{c}-25 i \sqrt{a} B\right ) F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{a}}{\sqrt{c}}}}{\sqrt{x}}\right )\right |-1\right )-462 a^{5/2} A \sqrt{c} x^{3/2} \sqrt{\frac{a}{c x^2}+1} E\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{a}}{\sqrt{c}}}}{\sqrt{x}}\right )\right |-1\right )+\sqrt{\frac{i \sqrt{a}}{\sqrt{c}}} \left (a+c x^2\right ) \left (6 a^2 (77 A+25 B x)-2 a c x^2 (77 A+45 B x)-35 c^2 x^4 (11 A+9 B x)\right )\right )}{3465 c^2 \sqrt{\frac{i \sqrt{a}}{\sqrt{c}}} \sqrt{e x} \sqrt{a+c x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*e^3*(Sqrt[(I*Sqrt[a])/Sqrt[c]]*(a + c*x^2)*(-35*c^2*x^4*(11*A + 9*B*x) + 6*a
^2*(77*A + 25*B*x) - 2*a*c*x^2*(77*A + 45*B*x)) - 462*a^(5/2)*A*Sqrt[c]*Sqrt[1 +
a/(c*x^2)]*x^(3/2)*EllipticE[I*ArcSinh[Sqrt[(I*Sqrt[a])/Sqrt[c]]/Sqrt[x]], -1]
+ 6*a^(5/2)*((-25*I)*Sqrt[a]*B + 77*A*Sqrt[c])*Sqrt[1 + a/(c*x^2)]*x^(3/2)*Ellip
ticF[I*ArcSinh[Sqrt[(I*Sqrt[a])/Sqrt[c]]/Sqrt[x]], -1]))/(3465*Sqrt[(I*Sqrt[a])/
Sqrt[c]]*c^2*Sqrt[e*x]*Sqrt[a + c*x^2])

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Maple [A]  time = 0.038, size = 360, normalized size = 0.9 \[ -{\frac{2\,{e}^{2}}{3465\,x{c}^{3}}\sqrt{ex} \left ( -315\,B{c}^{4}{x}^{7}-385\,A{c}^{4}{x}^{6}+462\,A\sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{-{\frac{cx}{\sqrt{-ac}}}}{\it EllipticE} \left ( \sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}},1/2\,\sqrt{2} \right ){a}^{3}c-231\,A\sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{-{\frac{cx}{\sqrt{-ac}}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}},1/2\,\sqrt{2} \right ){a}^{3}c-75\,B\sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{-{\frac{cx}{\sqrt{-ac}}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}},1/2\,\sqrt{2} \right ) \sqrt{-ac}{a}^{3}-405\,aB{c}^{3}{x}^{5}-539\,aA{c}^{3}{x}^{4}+60\,{a}^{2}B{c}^{2}{x}^{3}-154\,{a}^{2}A{c}^{2}{x}^{2}+150\,{a}^{3}Bcx \right ){\frac{1}{\sqrt{c{x}^{2}+a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x)^(5/2)*(B*x+A)*(c*x^2+a)^(1/2),x)

[Out]

-2/3465/x*e^2*(e*x)^(1/2)/(c*x^2+a)^(1/2)*(-315*B*c^4*x^7-385*A*c^4*x^6+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)*EllipticE(((c*x+(-a*c)^(1/2))/(-a*c)^(1/2))^(1/
2),1/2*2^(1/2))*a^3*c-231*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^3*c-75*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^3-405*a*B*c^3*x^5-539*a*A*c^3*x^4+60*a^2*B*c^2*x^3-154*a^2*A*c^2*x^2+1
50*a^3*B*c*x)/c^3

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((B*e^2*x^3 + 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)**(1/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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