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

Optimal. Leaf size=376 \[ \frac{8 a^{5/4} c^{3/4} \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} \left (25 \sqrt{a} B+63 A \sqrt{c}\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{105 e^3 \sqrt{e x} \sqrt{a+c x^2}}-\frac{48 a^{5/4} A c^{5/4} \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{5 e^3 \sqrt{e x} \sqrt{a+c x^2}}-\frac{8 a c \sqrt{a+c x^2} (63 A-25 B x)}{105 e^3 \sqrt{e x}}-\frac{4 \left (a+c x^2\right )^{3/2} (25 a B-21 A c x)}{105 e^2 (e x)^{3/2}}-\frac{2 \left (a+c x^2\right )^{5/2} (7 A-5 B x)}{35 e (e x)^{5/2}}+\frac{48 a A c^{3/2} x \sqrt{a+c x^2}}{5 e^3 \sqrt{e x} \left (\sqrt{a}+\sqrt{c} x\right )} \]

[Out]

(-8*a*c*(63*A - 25*B*x)*Sqrt[a + c*x^2])/(105*e^3*Sqrt[e*x]) + (48*a*A*c^(3/2)*x
*Sqrt[a + c*x^2])/(5*e^3*Sqrt[e*x]*(Sqrt[a] + Sqrt[c]*x)) - (4*(25*a*B - 21*A*c*
x)*(a + c*x^2)^(3/2))/(105*e^2*(e*x)^(3/2)) - (2*(7*A - 5*B*x)*(a + c*x^2)^(5/2)
)/(35*e*(e*x)^(5/2)) - (48*a^(5/4)*A*c^(5/4)*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])/(5*e^3*Sqrt[e*x]*Sqrt[a + c*x^2]) + (8*a^(5/4)*(25*Sqrt[a]*B + 63*A*Sq
rt[c])*c^(3/4)*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])/(105*e^3*Sqrt[e*x]*S
qrt[a + c*x^2])

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Rubi [A]  time = 0.962693, antiderivative size = 376, 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 \[ \frac{8 a^{5/4} c^{3/4} \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} \left (25 \sqrt{a} B+63 A \sqrt{c}\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{105 e^3 \sqrt{e x} \sqrt{a+c x^2}}-\frac{48 a^{5/4} A c^{5/4} \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{5 e^3 \sqrt{e x} \sqrt{a+c x^2}}-\frac{8 a c \sqrt{a+c x^2} (63 A-25 B x)}{105 e^3 \sqrt{e x}}-\frac{4 \left (a+c x^2\right )^{3/2} (25 a B-21 A c x)}{105 e^2 (e x)^{3/2}}-\frac{2 \left (a+c x^2\right )^{5/2} (7 A-5 B x)}{35 e (e x)^{5/2}}+\frac{48 a A c^{3/2} x \sqrt{a+c x^2}}{5 e^3 \sqrt{e x} \left (\sqrt{a}+\sqrt{c} x\right )} \]

Antiderivative was successfully verified.

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

[Out]

(-8*a*c*(63*A - 25*B*x)*Sqrt[a + c*x^2])/(105*e^3*Sqrt[e*x]) + (48*a*A*c^(3/2)*x
*Sqrt[a + c*x^2])/(5*e^3*Sqrt[e*x]*(Sqrt[a] + Sqrt[c]*x)) - (4*(25*a*B - 21*A*c*
x)*(a + c*x^2)^(3/2))/(105*e^2*(e*x)^(3/2)) - (2*(7*A - 5*B*x)*(a + c*x^2)^(5/2)
)/(35*e*(e*x)^(5/2)) - (48*a^(5/4)*A*c^(5/4)*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])/(5*e^3*Sqrt[e*x]*Sqrt[a + c*x^2]) + (8*a^(5/4)*(25*Sqrt[a]*B + 63*A*Sq
rt[c])*c^(3/4)*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])/(105*e^3*Sqrt[e*x]*S
qrt[a + c*x^2])

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Rubi in Sympy [A]  time = 123.844, size = 369, normalized size = 0.98 \[ - \frac{48 A a^{\frac{5}{4}} c^{\frac{5}{4}} \sqrt{x} \sqrt{\frac{a + c x^{2}}{\left (\sqrt{a} + \sqrt{c} x\right )^{2}}} \left (\sqrt{a} + \sqrt{c} x\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{5 e^{3} \sqrt{e x} \sqrt{a + c x^{2}}} + \frac{48 A a c^{\frac{3}{2}} x \sqrt{a + c x^{2}}}{5 e^{3} \sqrt{e x} \left (\sqrt{a} + \sqrt{c} x\right )} + \frac{8 a^{\frac{5}{4}} c^{\frac{3}{4}} \sqrt{x} \sqrt{\frac{a + c x^{2}}{\left (\sqrt{a} + \sqrt{c} x\right )^{2}}} \left (\sqrt{a} + \sqrt{c} x\right ) \left (63 A \sqrt{c} + 25 B \sqrt{a}\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{105 e^{3} \sqrt{e x} \sqrt{a + c x^{2}}} - \frac{16 a c \left (\frac{63 A}{2} - \frac{25 B x}{2}\right ) \sqrt{a + c x^{2}}}{105 e^{3} \sqrt{e x}} - \frac{4 \left (\frac{7 A}{2} - \frac{5 B x}{2}\right ) \left (a + c x^{2}\right )^{\frac{5}{2}}}{35 e \left (e x\right )^{\frac{5}{2}}} - \frac{8 \left (a + c x^{2}\right )^{\frac{3}{2}} \left (- \frac{21 A c x}{2} + \frac{25 B a}{2}\right )}{105 e^{2} \left (e x\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-48*A*a**(5/4)*c**(5/4)*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**3*sqr
t(e*x)*sqrt(a + c*x**2)) + 48*A*a*c**(3/2)*x*sqrt(a + c*x**2)/(5*e**3*sqrt(e*x)*
(sqrt(a) + sqrt(c)*x)) + 8*a**(5/4)*c**(3/4)*sqrt(x)*sqrt((a + c*x**2)/(sqrt(a)
+ sqrt(c)*x)**2)*(sqrt(a) + sqrt(c)*x)*(63*A*sqrt(c) + 25*B*sqrt(a))*elliptic_f(
2*atan(c**(1/4)*sqrt(x)/a**(1/4)), 1/2)/(105*e**3*sqrt(e*x)*sqrt(a + c*x**2)) -
16*a*c*(63*A/2 - 25*B*x/2)*sqrt(a + c*x**2)/(105*e**3*sqrt(e*x)) - 4*(7*A/2 - 5*
B*x/2)*(a + c*x**2)**(5/2)/(35*e*(e*x)**(5/2)) - 8*(a + c*x**2)**(3/2)*(-21*A*c*
x/2 + 25*B*a/2)/(105*e**2*(e*x)**(3/2))

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Mathematica [C]  time = 1.19484, size = 254, normalized size = 0.68 \[ \frac{x \left (16 a^{3/2} c x^{7/2} \sqrt{\frac{a}{c x^2}+1} \left (63 A \sqrt{c}+25 i \sqrt{a} B\right ) F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{a}}{\sqrt{c}}}}{\sqrt{x}}\right )\right |-1\right )-1008 a^{3/2} A c^{3/2} x^{7/2} \sqrt{\frac{a}{c x^2}+1} E\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{a}}{\sqrt{c}}}}{\sqrt{x}}\right )\right |-1\right )-2 \sqrt{\frac{i \sqrt{a}}{\sqrt{c}}} \left (a+c x^2\right ) \left (7 a^2 (3 A+5 B x)-4 a c x^2 (63 A+20 B x)-3 c^2 x^4 (7 A+5 B x)\right )\right )}{105 \sqrt{\frac{i \sqrt{a}}{\sqrt{c}}} (e x)^{7/2} \sqrt{a+c x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(x*(-2*Sqrt[(I*Sqrt[a])/Sqrt[c]]*(a + c*x^2)*(7*a^2*(3*A + 5*B*x) - 3*c^2*x^4*(7
*A + 5*B*x) - 4*a*c*x^2*(63*A + 20*B*x)) - 1008*a^(3/2)*A*c^(3/2)*Sqrt[1 + a/(c*
x^2)]*x^(7/2)*EllipticE[I*ArcSinh[Sqrt[(I*Sqrt[a])/Sqrt[c]]/Sqrt[x]], -1] + 16*a
^(3/2)*((25*I)*Sqrt[a]*B + 63*A*Sqrt[c])*c*Sqrt[1 + a/(c*x^2)]*x^(7/2)*EllipticF
[I*ArcSinh[Sqrt[(I*Sqrt[a])/Sqrt[c]]/Sqrt[x]], -1]))/(105*Sqrt[(I*Sqrt[a])/Sqrt[
c]]*(e*x)^(7/2)*Sqrt[a + c*x^2])

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Maple [A]  time = 0.029, size = 367, normalized size = 1. \[{\frac{2}{105\,{x}^{2}{e}^{3}} \left ( 504\,A\sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{-{\frac{cx}{\sqrt{-ac}}}}{\it EllipticE} \left ( \sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}},1/2\,\sqrt{2} \right ){x}^{2}{a}^{2}c-252\,A\sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{-{\frac{cx}{\sqrt{-ac}}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}},1/2\,\sqrt{2} \right ){x}^{2}{a}^{2}c+100\,B\sqrt{-ac}\sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{-{\frac{cx}{\sqrt{-ac}}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}},1/2\,\sqrt{2} \right ){x}^{2}{a}^{2}+15\,B{c}^{3}{x}^{7}+21\,A{c}^{3}{x}^{6}+95\,aB{c}^{2}{x}^{5}-231\,aA{c}^{2}{x}^{4}+45\,{a}^{2}Bc{x}^{3}-273\,{a}^{2}Ac{x}^{2}-35\,{a}^{3}Bx-21\,A{a}^{3} \right ){\frac{1}{\sqrt{c{x}^{2}+a}}}{\frac{1}{\sqrt{ex}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/105/x^2*(504*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))*x^2*a^2*c-252*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))*x^2*a^2*c+1
00*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))*x^2*a^2+15*B*c^3*x^7+21*A*c^3*x^6+95*a*B*c^2
*x^5-231*a*A*c^2*x^4+45*a^2*B*c*x^3-273*a^2*A*c*x^2-35*a^3*B*x-21*A*a^3)/(c*x^2+
a)^(1/2)/e^3/(e*x)^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^(5/2)*(B*x + A)/(e*x)^(7/2),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)*e^3*x^3), 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((B*x+A)*(c*x**2+a)**(5/2)/(e*x)**(7/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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