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

Optimal. Leaf size=341 \[ \frac{4 a^{5/4} \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} \left (5 \sqrt{a} B+21 A \sqrt{c}\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{35 \sqrt [4]{c} e \sqrt{e x} \sqrt{a+c x^2}}-\frac{24 a^{5/4} A \sqrt [4]{c} \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{5 e \sqrt{e x} \sqrt{a+c x^2}}+\frac{4 \sqrt{e x} \sqrt{a+c x^2} (5 a B+21 A c x)}{35 e^2}-\frac{2 \left (a+c x^2\right )^{3/2} (7 A-B x)}{7 e \sqrt{e x}}+\frac{24 a A \sqrt{c} x \sqrt{a+c x^2}}{5 e \sqrt{e x} \left (\sqrt{a}+\sqrt{c} x\right )} \]

[Out]

(24*a*A*Sqrt[c]*x*Sqrt[a + c*x^2])/(5*e*Sqrt[e*x]*(Sqrt[a] + Sqrt[c]*x)) + (4*Sq
rt[e*x]*(5*a*B + 21*A*c*x)*Sqrt[a + c*x^2])/(35*e^2) - (2*(7*A - B*x)*(a + c*x^2
)^(3/2))/(7*e*Sqrt[e*x]) - (24*a^(5/4)*A*c^(1/4)*Sqrt[x]*(Sqrt[a] + Sqrt[c]*x)*S
qrt[(a + c*x^2)/(Sqrt[a] + Sqrt[c]*x)^2]*EllipticE[2*ArcTan[(c^(1/4)*Sqrt[x])/a^
(1/4)], 1/2])/(5*e*Sqrt[e*x]*Sqrt[a + c*x^2]) + (4*a^(5/4)*(5*Sqrt[a]*B + 21*A*S
qrt[c])*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])/(35*c^(1/4)*e*Sqrt[e*x]*Sqr
t[a + c*x^2])

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Rubi [A]  time = 0.80738, 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 \[ \frac{4 a^{5/4} \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} \left (5 \sqrt{a} B+21 A \sqrt{c}\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{35 \sqrt [4]{c} e \sqrt{e x} \sqrt{a+c x^2}}-\frac{24 a^{5/4} A \sqrt [4]{c} \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{5 e \sqrt{e x} \sqrt{a+c x^2}}+\frac{4 \sqrt{e x} \sqrt{a+c x^2} (5 a B+21 A c x)}{35 e^2}-\frac{2 \left (a+c x^2\right )^{3/2} (7 A-B x)}{7 e \sqrt{e x}}+\frac{24 a A \sqrt{c} x \sqrt{a+c x^2}}{5 e \sqrt{e x} \left (\sqrt{a}+\sqrt{c} x\right )} \]

Antiderivative was successfully verified.

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

[Out]

(24*a*A*Sqrt[c]*x*Sqrt[a + c*x^2])/(5*e*Sqrt[e*x]*(Sqrt[a] + Sqrt[c]*x)) + (4*Sq
rt[e*x]*(5*a*B + 21*A*c*x)*Sqrt[a + c*x^2])/(35*e^2) - (2*(7*A - B*x)*(a + c*x^2
)^(3/2))/(7*e*Sqrt[e*x]) - (24*a^(5/4)*A*c^(1/4)*Sqrt[x]*(Sqrt[a] + Sqrt[c]*x)*S
qrt[(a + c*x^2)/(Sqrt[a] + Sqrt[c]*x)^2]*EllipticE[2*ArcTan[(c^(1/4)*Sqrt[x])/a^
(1/4)], 1/2])/(5*e*Sqrt[e*x]*Sqrt[a + c*x^2]) + (4*a^(5/4)*(5*Sqrt[a]*B + 21*A*S
qrt[c])*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])/(35*c^(1/4)*e*Sqrt[e*x]*Sqr
t[a + c*x^2])

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Rubi in Sympy [A]  time = 103.346, size = 323, normalized size = 0.95 \[ - \frac{24 A a^{\frac{5}{4}} \sqrt [4]{c} \sqrt{x} \sqrt{\frac{a + c x^{2}}{\left (\sqrt{a} + \sqrt{c} x\right )^{2}}} \left (\sqrt{a} + \sqrt{c} x\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{5 e \sqrt{e x} \sqrt{a + c x^{2}}} + \frac{24 A a \sqrt{c} x \sqrt{a + c x^{2}}}{5 e \sqrt{e x} \left (\sqrt{a} + \sqrt{c} x\right )} + \frac{4 a^{\frac{5}{4}} \sqrt{x} \sqrt{\frac{a + c x^{2}}{\left (\sqrt{a} + \sqrt{c} x\right )^{2}}} \left (\sqrt{a} + \sqrt{c} x\right ) \left (21 A \sqrt{c} + 5 B \sqrt{a}\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{35 \sqrt [4]{c} e \sqrt{e x} \sqrt{a + c x^{2}}} - \frac{4 \left (\frac{7 A}{2} - \frac{B x}{2}\right ) \left (a + c x^{2}\right )^{\frac{3}{2}}}{7 e \sqrt{e x}} + \frac{8 \sqrt{e x} \sqrt{a + c x^{2}} \left (\frac{21 A c x}{2} + \frac{5 B a}{2}\right )}{35 e^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-24*A*a**(5/4)*c**(1/4)*sqrt(x)*sqrt((a + c*x**2)/(sqrt(a) + sqrt(c)*x)**2)*(sqr
t(a) + sqrt(c)*x)*elliptic_e(2*atan(c**(1/4)*sqrt(x)/a**(1/4)), 1/2)/(5*e*sqrt(e
*x)*sqrt(a + c*x**2)) + 24*A*a*sqrt(c)*x*sqrt(a + c*x**2)/(5*e*sqrt(e*x)*(sqrt(a
) + sqrt(c)*x)) + 4*a**(5/4)*sqrt(x)*sqrt((a + c*x**2)/(sqrt(a) + sqrt(c)*x)**2)
*(sqrt(a) + sqrt(c)*x)*(21*A*sqrt(c) + 5*B*sqrt(a))*elliptic_f(2*atan(c**(1/4)*s
qrt(x)/a**(1/4)), 1/2)/(35*c**(1/4)*e*sqrt(e*x)*sqrt(a + c*x**2)) - 4*(7*A/2 - B
*x/2)*(a + c*x**2)**(3/2)/(7*e*sqrt(e*x)) + 8*sqrt(e*x)*sqrt(a + c*x**2)*(21*A*c
*x/2 + 5*B*a/2)/(35*e**2)

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Mathematica [C]  time = 0.96291, size = 232, normalized size = 0.68 \[ \frac{x \left (8 a^{3/2} x^{3/2} \sqrt{\frac{a}{c x^2}+1} \left (21 A \sqrt{c}+5 i \sqrt{a} B\right ) F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{a}}{\sqrt{c}}}}{\sqrt{x}}\right )\right |-1\right )-168 a^{3/2} A \sqrt{c} x^{3/2} \sqrt{\frac{a}{c x^2}+1} E\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{a}}{\sqrt{c}}}}{\sqrt{x}}\right )\right |-1\right )+2 \sqrt{\frac{i \sqrt{a}}{\sqrt{c}}} \left (a+c x^2\right ) \left (a (49 A+15 B x)+c x^2 (7 A+5 B x)\right )\right )}{35 \sqrt{\frac{i \sqrt{a}}{\sqrt{c}}} (e x)^{3/2} \sqrt{a+c x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(x*(2*Sqrt[(I*Sqrt[a])/Sqrt[c]]*(a + c*x^2)*(c*x^2*(7*A + 5*B*x) + a*(49*A + 15*
B*x)) - 168*a^(3/2)*A*Sqrt[c]*Sqrt[1 + a/(c*x^2)]*x^(3/2)*EllipticE[I*ArcSinh[Sq
rt[(I*Sqrt[a])/Sqrt[c]]/Sqrt[x]], -1] + 8*a^(3/2)*((5*I)*Sqrt[a]*B + 21*A*Sqrt[c
])*Sqrt[1 + a/(c*x^2)]*x^(3/2)*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[a])/Sqrt[c]]/Sqr
t[x]], -1]))/(35*Sqrt[(I*Sqrt[a])/Sqrt[c]]*(e*x)^(3/2)*Sqrt[a + c*x^2])

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Maple [A]  time = 0.027, size = 340, normalized size = 1. \[{\frac{2}{35\,ce} \left ( 84\,A\sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{-{\frac{cx}{\sqrt{-ac}}}}{\it EllipticE} \left ( \sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}},1/2\,\sqrt{2} \right ){a}^{2}c-42\,A\sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{-{\frac{cx}{\sqrt{-ac}}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}},1/2\,\sqrt{2} \right ){a}^{2}c+10\,B\sqrt{-ac}\sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{-{\frac{cx}{\sqrt{-ac}}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}},1/2\,\sqrt{2} \right ){a}^{2}+5\,B{c}^{3}{x}^{5}+7\,A{c}^{3}{x}^{4}+20\,aB{c}^{2}{x}^{3}-28\,aA{c}^{2}{x}^{2}+15\,{a}^{2}Bcx-35\,A{a}^{2}c \right ){\frac{1}{\sqrt{c{x}^{2}+a}}}{\frac{1}{\sqrt{ex}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/35*(84*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^2*c-42*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)*Ellip
ticF(((c*x+(-a*c)^(1/2))/(-a*c)^(1/2))^(1/2),1/2*2^(1/2))*a^2*c+10*B*(-a*c)^(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)*EllipticF(((c*x+(-a*c)^(1/2))/(-a*c)^(1/2))
^(1/2),1/2*2^(1/2))*a^2+5*B*c^3*x^5+7*A*c^3*x^4+20*a*B*c^2*x^3-28*a*A*c^2*x^2+15
*a^2*B*c*x-35*A*a^2*c)/(c*x^2+a)^(1/2)/c/e/(e*x)^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 32.3509, size = 202, normalized size = 0.59 \[ \frac{A a^{\frac{3}{2}} \Gamma \left (- \frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, - \frac{1}{4} \\ \frac{3}{4} \end{matrix}\middle |{\frac{c x^{2} e^{i \pi }}{a}} \right )}}{2 e^{\frac{3}{2}} \sqrt{x} \Gamma \left (\frac{3}{4}\right )} + \frac{A \sqrt{a} c x^{\frac{3}{2}} \Gamma \left (\frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{3}{4} \\ \frac{7}{4} \end{matrix}\middle |{\frac{c x^{2} e^{i \pi }}{a}} \right )}}{2 e^{\frac{3}{2}} \Gamma \left (\frac{7}{4}\right )} + \frac{B a^{\frac{3}{2}} \sqrt{x} \Gamma \left (\frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{1}{4} \\ \frac{5}{4} \end{matrix}\middle |{\frac{c x^{2} e^{i \pi }}{a}} \right )}}{2 e^{\frac{3}{2}} \Gamma \left (\frac{5}{4}\right )} + \frac{B \sqrt{a} c x^{\frac{5}{2}} \Gamma \left (\frac{5}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{5}{4} \\ \frac{9}{4} \end{matrix}\middle |{\frac{c x^{2} e^{i \pi }}{a}} \right )}}{2 e^{\frac{3}{2}} \Gamma \left (\frac{9}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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