3.456 \(\int \frac{(A+B x) \left (a+c x^2\right )^{5/2}}{(e x)^{9/2}} \, dx\)

Optimal. Leaf size=377 \[ \frac{8 a^{3/4} c^{5/4} \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} \left (63 \sqrt{a} B+25 A \sqrt{c}\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{105 e^4 \sqrt{e x} \sqrt{a+c x^2}}-\frac{48 a^{5/4} B c^{5/4} \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{5 e^4 \sqrt{e x} \sqrt{a+c x^2}}-\frac{8 c \sqrt{a+c x^2} (63 a B-25 A c x)}{105 e^4 \sqrt{e x}}-\frac{4 \left (a+c x^2\right )^{3/2} (21 a B+25 A c x)}{105 e^2 (e x)^{5/2}}-\frac{2 \left (a+c x^2\right )^{5/2} (5 A-7 B x)}{35 e (e x)^{7/2}}+\frac{48 a B c^{3/2} x \sqrt{a+c x^2}}{5 e^4 \sqrt{e x} \left (\sqrt{a}+\sqrt{c} x\right )} \]

[Out]

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

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Rubi [A]  time = 0.977449, antiderivative size = 377, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.292 \[ \frac{8 a^{3/4} c^{5/4} \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} \left (63 \sqrt{a} B+25 A \sqrt{c}\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{105 e^4 \sqrt{e x} \sqrt{a+c x^2}}-\frac{48 a^{5/4} B c^{5/4} \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{5 e^4 \sqrt{e x} \sqrt{a+c x^2}}-\frac{8 c \sqrt{a+c x^2} (63 a B-25 A c x)}{105 e^4 \sqrt{e x}}-\frac{4 \left (a+c x^2\right )^{3/2} (21 a B+25 A c x)}{105 e^2 (e x)^{5/2}}-\frac{2 \left (a+c x^2\right )^{5/2} (5 A-7 B x)}{35 e (e x)^{7/2}}+\frac{48 a B c^{3/2} x \sqrt{a+c x^2}}{5 e^4 \sqrt{e x} \left (\sqrt{a}+\sqrt{c} x\right )} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*(a + c*x^2)^(5/2))/(e*x)^(9/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 129.779, size = 371, normalized size = 0.98 \[ - \frac{48 B a^{\frac{5}{4}} c^{\frac{5}{4}} \sqrt{x} \sqrt{\frac{a + c x^{2}}{\left (\sqrt{a} + \sqrt{c} x\right )^{2}}} \left (\sqrt{a} + \sqrt{c} x\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{5 e^{4} \sqrt{e x} \sqrt{a + c x^{2}}} + \frac{48 B a c^{\frac{3}{2}} x \sqrt{a + c x^{2}}}{5 e^{4} \sqrt{e x} \left (\sqrt{a} + \sqrt{c} x\right )} + \frac{8 a^{\frac{3}{4}} c^{\frac{5}{4}} \sqrt{x} \sqrt{\frac{a + c x^{2}}{\left (\sqrt{a} + \sqrt{c} x\right )^{2}}} \left (\sqrt{a} + \sqrt{c} x\right ) \left (25 A \sqrt{c} + 63 B \sqrt{a}\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{105 e^{4} \sqrt{e x} \sqrt{a + c x^{2}}} - \frac{16 c \sqrt{a + c x^{2}} \left (- \frac{25 A c x}{2} + \frac{63 B a}{2}\right )}{105 e^{4} \sqrt{e x}} - \frac{4 \left (\frac{5 A}{2} - \frac{7 B x}{2}\right ) \left (a + c x^{2}\right )^{\frac{5}{2}}}{35 e \left (e x\right )^{\frac{7}{2}}} - \frac{8 \left (a + c x^{2}\right )^{\frac{3}{2}} \left (\frac{25 A c x}{2} + \frac{21 B a}{2}\right )}{105 e^{2} \left (e x\right )^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(c*x**2+a)**(5/2)/(e*x)**(9/2),x)

[Out]

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

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Mathematica [C]  time = 1.30294, size = 259, normalized size = 0.69 \[ \frac{2 \sqrt{e x} \left (-504 a^{3/2} B c^{3/2} x^{9/2} \sqrt{\frac{a}{c x^2}+1} E\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{a}}{\sqrt{c}}}}{\sqrt{x}}\right )\right |-1\right )-\sqrt{\frac{i \sqrt{a}}{\sqrt{c}}} \left (a+c x^2\right ) \left (3 a^2 (5 A+7 B x)+4 a c x^2 (20 A-63 B x)-7 c^2 x^4 (5 A+3 B x)\right )+8 a c^{3/2} x^{9/2} \sqrt{\frac{a}{c x^2}+1} \left (63 \sqrt{a} B+25 i A \sqrt{c}\right ) F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{a}}{\sqrt{c}}}}{\sqrt{x}}\right )\right |-1\right )\right )}{105 e^5 x^4 \sqrt{\frac{i \sqrt{a}}{\sqrt{c}}} \sqrt{a+c x^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*(a + c*x^2)^(5/2))/(e*x)^(9/2),x]

[Out]

(2*Sqrt[e*x]*(-(Sqrt[(I*Sqrt[a])/Sqrt[c]]*(a + c*x^2)*(4*a*c*x^2*(20*A - 63*B*x)
- 7*c^2*x^4*(5*A + 3*B*x) + 3*a^2*(5*A + 7*B*x))) - 504*a^(3/2)*B*c^(3/2)*Sqrt[
1 + a/(c*x^2)]*x^(9/2)*EllipticE[I*ArcSinh[Sqrt[(I*Sqrt[a])/Sqrt[c]]/Sqrt[x]], -
1] + 8*a*(63*Sqrt[a]*B + (25*I)*A*Sqrt[c])*c^(3/2)*Sqrt[1 + a/(c*x^2)]*x^(9/2)*E
llipticF[I*ArcSinh[Sqrt[(I*Sqrt[a])/Sqrt[c]]/Sqrt[x]], -1]))/(105*Sqrt[(I*Sqrt[a
])/Sqrt[c]]*e^5*x^4*Sqrt[a + c*x^2])

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Maple [A]  time = 0.028, size = 366, normalized size = 1. \[{\frac{2}{105\,{x}^{3}{e}^{4}} \left ( 100\,A\sqrt{-ac}\sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{-{\frac{cx}{\sqrt{-ac}}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}},1/2\,\sqrt{2} \right ){x}^{3}ac+504\,B\sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{-{\frac{cx}{\sqrt{-ac}}}}{\it EllipticE} \left ( \sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}},1/2\,\sqrt{2} \right ){x}^{3}{a}^{2}c-252\,B\sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{-{\frac{cx}{\sqrt{-ac}}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}},1/2\,\sqrt{2} \right ){x}^{3}{a}^{2}c+21\,B{c}^{3}{x}^{7}+35\,A{c}^{3}{x}^{6}-231\,aB{c}^{2}{x}^{5}-45\,aA{c}^{2}{x}^{4}-273\,{a}^{2}Bc{x}^{3}-95\,{a}^{2}Ac{x}^{2}-21\,{a}^{3}Bx-15\,A{a}^{3} \right ){\frac{1}{\sqrt{c{x}^{2}+a}}}{\frac{1}{\sqrt{ex}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(c*x^2+a)^(5/2)/(e*x)^(9/2),x)

[Out]

2/105/x^3*(100*A*(-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^3*a*c+504*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^3*a^2*c-252*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^3*a^2*c+21*B*c^3*x^7+35*A*c^3*x^6-231*a*B*c^
2*x^5-45*a*A*c^2*x^4-273*a^2*B*c*x^3-95*a^2*A*c*x^2-21*a^3*B*x-15*A*a^3)/(c*x^2+
a)^(1/2)/e^4/(e*x)^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (c x^{2} + a\right )}^{\frac{5}{2}}{\left (B x + A\right )}}{\left (e x\right )^{\frac{9}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^(5/2)*(B*x + A)/(e*x)^(9/2),x, algorithm="maxima")

[Out]

integrate((c*x^2 + a)^(5/2)*(B*x + A)/(e*x)^(9/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^(5/2)*(B*x + A)/(e*x)^(9/2),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^4*x^4), 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)**(9/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (c x^{2} + a\right )}^{\frac{5}{2}}{\left (B x + A\right )}}{\left (e x\right )^{\frac{9}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^(5/2)*(B*x + A)/(e*x)^(9/2),x, algorithm="giac")

[Out]

integrate((c*x^2 + a)^(5/2)*(B*x + A)/(e*x)^(9/2), x)