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

Optimal. Leaf size=369 \[ \frac{8 a^{11/4} \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} \left (77 \sqrt{a} B+195 A \sqrt{c}\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{3003 c^{3/4} \sqrt{e x} \sqrt{a+c x^2}}-\frac{16 a^{13/4} B \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{39 c^{3/4} \sqrt{e x} \sqrt{a+c x^2}}+\frac{16 a^3 B x \sqrt{a+c x^2}}{39 \sqrt{c} \sqrt{e x} \left (\sqrt{a}+\sqrt{c} x\right )}+\frac{8 a^2 \sqrt{e x} \sqrt{a+c x^2} (195 A+77 B x)}{3003 e}+\frac{20 a \sqrt{e x} \left (a+c x^2\right )^{3/2} (117 A+77 B x)}{9009 e}+\frac{2 \sqrt{e x} \left (a+c x^2\right )^{5/2} (13 A+11 B x)}{143 e} \]

[Out]

(8*a^2*Sqrt[e*x]*(195*A + 77*B*x)*Sqrt[a + c*x^2])/(3003*e) + (16*a^3*B*x*Sqrt[a
+ c*x^2])/(39*Sqrt[c]*Sqrt[e*x]*(Sqrt[a] + Sqrt[c]*x)) + (20*a*Sqrt[e*x]*(117*A
+ 77*B*x)*(a + c*x^2)^(3/2))/(9009*e) + (2*Sqrt[e*x]*(13*A + 11*B*x)*(a + c*x^2
)^(5/2))/(143*e) - (16*a^(13/4)*B*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])/(
39*c^(3/4)*Sqrt[e*x]*Sqrt[a + c*x^2]) + (8*a^(11/4)*(77*Sqrt[a]*B + 195*A*Sqrt[c
])*Sqrt[x]*(Sqrt[a] + Sqrt[c]*x)*Sqrt[(a + c*x^2)/(Sqrt[a] + Sqrt[c]*x)^2]*Ellip
ticF[2*ArcTan[(c^(1/4)*Sqrt[x])/a^(1/4)], 1/2])/(3003*c^(3/4)*Sqrt[e*x]*Sqrt[a +
c*x^2])

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Rubi [A]  time = 0.992372, antiderivative size = 369, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{8 a^{11/4} \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} \left (77 \sqrt{a} B+195 A \sqrt{c}\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{3003 c^{3/4} \sqrt{e x} \sqrt{a+c x^2}}-\frac{16 a^{13/4} B \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{39 c^{3/4} \sqrt{e x} \sqrt{a+c x^2}}+\frac{16 a^3 B x \sqrt{a+c x^2}}{39 \sqrt{c} \sqrt{e x} \left (\sqrt{a}+\sqrt{c} x\right )}+\frac{8 a^2 \sqrt{e x} \sqrt{a+c x^2} (195 A+77 B x)}{3003 e}+\frac{20 a \sqrt{e x} \left (a+c x^2\right )^{3/2} (117 A+77 B x)}{9009 e}+\frac{2 \sqrt{e x} \left (a+c x^2\right )^{5/2} (13 A+11 B x)}{143 e} \]

Antiderivative was successfully verified.

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

[Out]

(8*a^2*Sqrt[e*x]*(195*A + 77*B*x)*Sqrt[a + c*x^2])/(3003*e) + (16*a^3*B*x*Sqrt[a
+ c*x^2])/(39*Sqrt[c]*Sqrt[e*x]*(Sqrt[a] + Sqrt[c]*x)) + (20*a*Sqrt[e*x]*(117*A
+ 77*B*x)*(a + c*x^2)^(3/2))/(9009*e) + (2*Sqrt[e*x]*(13*A + 11*B*x)*(a + c*x^2
)^(5/2))/(143*e) - (16*a^(13/4)*B*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])/(
39*c^(3/4)*Sqrt[e*x]*Sqrt[a + c*x^2]) + (8*a^(11/4)*(77*Sqrt[a]*B + 195*A*Sqrt[c
])*Sqrt[x]*(Sqrt[a] + Sqrt[c]*x)*Sqrt[(a + c*x^2)/(Sqrt[a] + Sqrt[c]*x)^2]*Ellip
ticF[2*ArcTan[(c^(1/4)*Sqrt[x])/a^(1/4)], 1/2])/(3003*c^(3/4)*Sqrt[e*x]*Sqrt[a +
c*x^2])

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Rubi in Sympy [A]  time = 139.965, size = 355, normalized size = 0.96 \[ - \frac{16 B a^{\frac{13}{4}} \sqrt{x} \sqrt{\frac{a + c x^{2}}{\left (\sqrt{a} + \sqrt{c} x\right )^{2}}} \left (\sqrt{a} + \sqrt{c} x\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{39 c^{\frac{3}{4}} \sqrt{e x} \sqrt{a + c x^{2}}} + \frac{16 B a^{3} x \sqrt{a + c x^{2}}}{39 \sqrt{c} \sqrt{e x} \left (\sqrt{a} + \sqrt{c} x\right )} + \frac{8 a^{\frac{11}{4}} \sqrt{x} \sqrt{\frac{a + c x^{2}}{\left (\sqrt{a} + \sqrt{c} x\right )^{2}}} \left (\sqrt{a} + \sqrt{c} x\right ) \left (195 A \sqrt{c} + 77 B \sqrt{a}\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{3003 c^{\frac{3}{4}} \sqrt{e x} \sqrt{a + c x^{2}}} + \frac{64 a^{2} \sqrt{e x} \left (\frac{585 A}{8} + \frac{231 B x}{8}\right ) \sqrt{a + c x^{2}}}{9009 e} + \frac{80 a \sqrt{e x} \left (\frac{117 A}{4} + \frac{77 B x}{4}\right ) \left (a + c x^{2}\right )^{\frac{3}{2}}}{9009 e} + \frac{4 \sqrt{e x} \left (\frac{13 A}{2} + \frac{11 B x}{2}\right ) \left (a + c x^{2}\right )^{\frac{5}{2}}}{143 e} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-16*B*a**(13/4)*sqrt(x)*sqrt((a + c*x**2)/(sqrt(a) + sqrt(c)*x)**2)*(sqrt(a) + s
qrt(c)*x)*elliptic_e(2*atan(c**(1/4)*sqrt(x)/a**(1/4)), 1/2)/(39*c**(3/4)*sqrt(e
*x)*sqrt(a + c*x**2)) + 16*B*a**3*x*sqrt(a + c*x**2)/(39*sqrt(c)*sqrt(e*x)*(sqrt
(a) + sqrt(c)*x)) + 8*a**(11/4)*sqrt(x)*sqrt((a + c*x**2)/(sqrt(a) + sqrt(c)*x)*
*2)*(sqrt(a) + sqrt(c)*x)*(195*A*sqrt(c) + 77*B*sqrt(a))*elliptic_f(2*atan(c**(1
/4)*sqrt(x)/a**(1/4)), 1/2)/(3003*c**(3/4)*sqrt(e*x)*sqrt(a + c*x**2)) + 64*a**2
*sqrt(e*x)*(585*A/8 + 231*B*x/8)*sqrt(a + c*x**2)/(9009*e) + 80*a*sqrt(e*x)*(117
*A/4 + 77*B*x/4)*(a + c*x**2)**(3/2)/(9009*e) + 4*sqrt(e*x)*(13*A/2 + 11*B*x/2)*
(a + c*x**2)**(5/2)/(143*e)

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Mathematica [C]  time = 0.923122, size = 267, normalized size = 0.72 \[ \frac{-3696 a^{7/2} B \sqrt{c} x^{3/2} \sqrt{\frac{a}{c x^2}+1} E\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{a}}{\sqrt{c}}}}{\sqrt{x}}\right )\right |-1\right )+48 a^3 \sqrt{c} x^{3/2} \sqrt{\frac{a}{c x^2}+1} \left (77 \sqrt{a} B+195 i A \sqrt{c}\right ) F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{a}}{\sqrt{c}}}}{\sqrt{x}}\right )\right |-1\right )+2 \sqrt{\frac{i \sqrt{a}}{\sqrt{c}}} \left (a+c x^2\right ) \left (1848 a^3 B+a^2 c x (4329 A+2387 B x)+4 a c^2 x^3 (702 A+539 B x)+63 c^3 x^5 (13 A+11 B x)\right )}{9009 c \sqrt{\frac{i \sqrt{a}}{\sqrt{c}}} \sqrt{e x} \sqrt{a+c x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[(I*Sqrt[a])/Sqrt[c]]*(a + c*x^2)*(1848*a^3*B + 63*c^3*x^5*(13*A + 11*B*x
) + 4*a*c^2*x^3*(702*A + 539*B*x) + a^2*c*x*(4329*A + 2387*B*x)) - 3696*a^(7/2)*
B*Sqrt[c]*Sqrt[1 + a/(c*x^2)]*x^(3/2)*EllipticE[I*ArcSinh[Sqrt[(I*Sqrt[a])/Sqrt[
c]]/Sqrt[x]], -1] + 48*a^3*(77*Sqrt[a]*B + (195*I)*A*Sqrt[c])*Sqrt[c]*Sqrt[1 + a
/(c*x^2)]*x^(3/2)*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[a])/Sqrt[c]]/Sqrt[x]], -1])/(
9009*Sqrt[(I*Sqrt[a])/Sqrt[c]]*c*Sqrt[e*x]*Sqrt[a + c*x^2])

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Maple [A]  time = 0.023, size = 362, normalized size = 1. \[{\frac{2}{9009\,c} \left ( 693\,B{c}^{4}{x}^{8}+819\,A{c}^{4}{x}^{7}+2849\,aB{c}^{3}{x}^{6}+2340\,A{\it EllipticF} \left ( \sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}},1/2\,\sqrt{2} \right ) \sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{{\frac{-cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{-{\frac{cx}{\sqrt{-ac}}}}\sqrt{2}\sqrt{-ac}{a}^{3}+3627\,aA{c}^{3}{x}^{5}+1848\,B{\it EllipticE} \left ( \sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}},1/2\,\sqrt{2} \right ) \sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{{\frac{-cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{-{\frac{cx}{\sqrt{-ac}}}}\sqrt{2}{a}^{4}-924\,B{\it EllipticF} \left ( \sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}},1/2\,\sqrt{2} \right ) \sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{{\frac{-cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{-{\frac{cx}{\sqrt{-ac}}}}\sqrt{2}{a}^{4}+4543\,{a}^{2}B{c}^{2}{x}^{4}+7137\,{a}^{2}A{c}^{2}{x}^{3}+2387\,{a}^{3}Bc{x}^{2}+4329\,{a}^{3}Acx \right ){\frac{1}{\sqrt{c{x}^{2}+a}}}{\frac{1}{\sqrt{ex}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/9009/(c*x^2+a)^(1/2)/c*(693*B*c^4*x^8+819*A*c^4*x^7+2849*a*B*c^3*x^6+2340*A*El
lipticF(((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)*((-c*x+(-a*c)^(1/2))/(-a*c)^(1/2))^(1/2)*(-x*c/(-a*c)^(1/2)
)^(1/2)*2^(1/2)*(-a*c)^(1/2)*a^3+3627*a*A*c^3*x^5+1848*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)*
((-c*x+(-a*c)^(1/2))/(-a*c)^(1/2))^(1/2)*(-x*c/(-a*c)^(1/2))^(1/2)*2^(1/2)*a^4-9
24*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)*((-c*x+(-a*c)^(1/2))/(-a*c)^(1/2))^(1/2)*(-x*c/(-a*c
)^(1/2))^(1/2)*2^(1/2)*a^4+4543*a^2*B*c^2*x^4+7137*a^2*A*c^2*x^3+2387*a^3*B*c*x^
2+4329*a^3*A*c*x)/(e*x)^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (B c^{2} x^{5} + A c^{2} x^{4} + 2 \, B a c x^{3} + 2 \, A a c x^{2} + B a^{2} x + A a^{2}\right )} \sqrt{c x^{2} + a}}{\sqrt{e x}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((B*c^2*x^5 + A*c^2*x^4 + 2*B*a*c*x^3 + 2*A*a*c*x^2 + B*a^2*x + A*a^2)*s
qrt(c*x^2 + a)/sqrt(e*x), x)

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Sympy [A]  time = 122.729, size = 301, normalized size = 0.82 \[ \frac{A a^{\frac{5}{2}} \sqrt{x} \Gamma \left (\frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{1}{4} \\ \frac{5}{4} \end{matrix}\middle |{\frac{c x^{2} e^{i \pi }}{a}} \right )}}{2 \sqrt{e} \Gamma \left (\frac{5}{4}\right )} + \frac{A a^{\frac{3}{2}} c x^{\frac{5}{2}} \Gamma \left (\frac{5}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{5}{4} \\ \frac{9}{4} \end{matrix}\middle |{\frac{c x^{2} e^{i \pi }}{a}} \right )}}{\sqrt{e} \Gamma \left (\frac{9}{4}\right )} + \frac{A \sqrt{a} c^{2} x^{\frac{9}{2}} \Gamma \left (\frac{9}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{9}{4} \\ \frac{13}{4} \end{matrix}\middle |{\frac{c x^{2} e^{i \pi }}{a}} \right )}}{2 \sqrt{e} \Gamma \left (\frac{13}{4}\right )} + \frac{B a^{\frac{5}{2}} x^{\frac{3}{2}} \Gamma \left (\frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{3}{4} \\ \frac{7}{4} \end{matrix}\middle |{\frac{c x^{2} e^{i \pi }}{a}} \right )}}{2 \sqrt{e} \Gamma \left (\frac{7}{4}\right )} + \frac{B a^{\frac{3}{2}} c x^{\frac{7}{2}} \Gamma \left (\frac{7}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{7}{4} \\ \frac{11}{4} \end{matrix}\middle |{\frac{c x^{2} e^{i \pi }}{a}} \right )}}{\sqrt{e} \Gamma \left (\frac{11}{4}\right )} + \frac{B \sqrt{a} c^{2} x^{\frac{11}{2}} \Gamma \left (\frac{11}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{11}{4} \\ \frac{15}{4} \end{matrix}\middle |{\frac{c x^{2} e^{i \pi }}{a}} \right )}}{2 \sqrt{e} \Gamma \left (\frac{15}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

A*a**(5/2)*sqrt(x)*gamma(1/4)*hyper((-1/2, 1/4), (5/4,), c*x**2*exp_polar(I*pi)/
a)/(2*sqrt(e)*gamma(5/4)) + A*a**(3/2)*c*x**(5/2)*gamma(5/4)*hyper((-1/2, 5/4),
(9/4,), c*x**2*exp_polar(I*pi)/a)/(sqrt(e)*gamma(9/4)) + A*sqrt(a)*c**2*x**(9/2)
*gamma(9/4)*hyper((-1/2, 9/4), (13/4,), c*x**2*exp_polar(I*pi)/a)/(2*sqrt(e)*gam
ma(13/4)) + B*a**(5/2)*x**(3/2)*gamma(3/4)*hyper((-1/2, 3/4), (7/4,), c*x**2*exp
_polar(I*pi)/a)/(2*sqrt(e)*gamma(7/4)) + B*a**(3/2)*c*x**(7/2)*gamma(7/4)*hyper(
(-1/2, 7/4), (11/4,), c*x**2*exp_polar(I*pi)/a)/(sqrt(e)*gamma(11/4)) + B*sqrt(a
)*c**2*x**(11/2)*gamma(11/4)*hyper((-1/2, 11/4), (15/4,), c*x**2*exp_polar(I*pi)
/a)/(2*sqrt(e)*gamma(15/4))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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