3.448 \(\int \frac{(A+B x) \left (a+c x^2\right )^{3/2}}{(e x)^{7/2}} \, dx\)

Optimal. Leaf size=339 \[ \frac{4 \sqrt [4]{a} c^{3/4} \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} \left (5 \sqrt{a} B+9 A \sqrt{c}\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{15 e^3 \sqrt{e x} \sqrt{a+c x^2}}-\frac{4 c \sqrt{a+c x^2} (9 A-5 B x)}{15 e^3 \sqrt{e x}}-\frac{2 \left (a+c x^2\right )^{3/2} (3 A+5 B x)}{15 e (e x)^{5/2}}+\frac{24 A c^{3/2} x \sqrt{a+c x^2}}{5 e^3 \sqrt{e x} \left (\sqrt{a}+\sqrt{c} x\right )}-\frac{24 \sqrt [4]{a} A c^{5/4} \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{5 e^3 \sqrt{e x} \sqrt{a+c x^2}} \]

[Out]

(-4*c*(9*A - 5*B*x)*Sqrt[a + c*x^2])/(15*e^3*Sqrt[e*x]) + (24*A*c^(3/2)*x*Sqrt[a
+ c*x^2])/(5*e^3*Sqrt[e*x]*(Sqrt[a] + Sqrt[c]*x)) - (2*(3*A + 5*B*x)*(a + c*x^2
)^(3/2))/(15*e*(e*x)^(5/2)) - (24*a^(1/4)*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]) + (4*a^(1/4)*(5*Sqrt[a]*B + 9
*A*Sqrt[c])*c^(3/4)*Sqrt[x]*(Sqrt[a] + Sqrt[c]*x)*Sqrt[(a + c*x^2)/(Sqrt[a] + Sq
rt[c]*x)^2]*EllipticF[2*ArcTan[(c^(1/4)*Sqrt[x])/a^(1/4)], 1/2])/(15*e^3*Sqrt[e*
x]*Sqrt[a + c*x^2])

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Rubi [A]  time = 0.807945, antiderivative size = 339, 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 \[ \frac{4 \sqrt [4]{a} c^{3/4} \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} \left (5 \sqrt{a} B+9 A \sqrt{c}\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{15 e^3 \sqrt{e x} \sqrt{a+c x^2}}-\frac{4 c \sqrt{a+c x^2} (9 A-5 B x)}{15 e^3 \sqrt{e x}}-\frac{2 \left (a+c x^2\right )^{3/2} (3 A+5 B x)}{15 e (e x)^{5/2}}+\frac{24 A c^{3/2} x \sqrt{a+c x^2}}{5 e^3 \sqrt{e x} \left (\sqrt{a}+\sqrt{c} x\right )}-\frac{24 \sqrt [4]{a} A c^{5/4} \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{5 e^3 \sqrt{e x} \sqrt{a+c x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(-4*c*(9*A - 5*B*x)*Sqrt[a + c*x^2])/(15*e^3*Sqrt[e*x]) + (24*A*c^(3/2)*x*Sqrt[a
+ c*x^2])/(5*e^3*Sqrt[e*x]*(Sqrt[a] + Sqrt[c]*x)) - (2*(3*A + 5*B*x)*(a + c*x^2
)^(3/2))/(15*e*(e*x)^(5/2)) - (24*a^(1/4)*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]) + (4*a^(1/4)*(5*Sqrt[a]*B + 9
*A*Sqrt[c])*c^(3/4)*Sqrt[x]*(Sqrt[a] + Sqrt[c]*x)*Sqrt[(a + c*x^2)/(Sqrt[a] + Sq
rt[c]*x)^2]*EllipticF[2*ArcTan[(c^(1/4)*Sqrt[x])/a^(1/4)], 1/2])/(15*e^3*Sqrt[e*
x]*Sqrt[a + c*x^2])

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Rubi in Sympy [A]  time = 105.053, size = 326, normalized size = 0.96 \[ - \frac{24 A \sqrt [4]{a} c^{\frac{5}{4}} \sqrt{x} \sqrt{\frac{a + c x^{2}}{\left (\sqrt{a} + \sqrt{c} x\right )^{2}}} \left (\sqrt{a} + \sqrt{c} x\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{5 e^{3} \sqrt{e x} \sqrt{a + c x^{2}}} + \frac{24 A c^{\frac{3}{2}} x \sqrt{a + c x^{2}}}{5 e^{3} \sqrt{e x} \left (\sqrt{a} + \sqrt{c} x\right )} + \frac{4 \sqrt [4]{a} c^{\frac{3}{4}} \sqrt{x} \sqrt{\frac{a + c x^{2}}{\left (\sqrt{a} + \sqrt{c} x\right )^{2}}} \left (\sqrt{a} + \sqrt{c} x\right ) \left (9 A \sqrt{c} + 5 B \sqrt{a}\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{15 e^{3} \sqrt{e x} \sqrt{a + c x^{2}}} - \frac{8 c \left (\frac{9 A}{2} - \frac{5 B x}{2}\right ) \sqrt{a + c x^{2}}}{15 e^{3} \sqrt{e x}} - \frac{4 \left (\frac{3 A}{2} + \frac{5 B x}{2}\right ) \left (a + c x^{2}\right )^{\frac{3}{2}}}{15 e \left (e x\right )^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-24*A*a**(1/4)*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)) + 24*A*c**(3/2)*x*sqrt(a + c*x**2)/(5*e**3*sqrt(e*x)*(s
qrt(a) + sqrt(c)*x)) + 4*a**(1/4)*c**(3/4)*sqrt(x)*sqrt((a + c*x**2)/(sqrt(a) +
sqrt(c)*x)**2)*(sqrt(a) + sqrt(c)*x)*(9*A*sqrt(c) + 5*B*sqrt(a))*elliptic_f(2*at
an(c**(1/4)*sqrt(x)/a**(1/4)), 1/2)/(15*e**3*sqrt(e*x)*sqrt(a + c*x**2)) - 8*c*(
9*A/2 - 5*B*x/2)*sqrt(a + c*x**2)/(15*e**3*sqrt(e*x)) - 4*(3*A/2 + 5*B*x/2)*(a +
c*x**2)**(3/2)/(15*e*(e*x)**(5/2))

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Mathematica [C]  time = 0.933037, size = 233, normalized size = 0.69 \[ \frac{x \left (-2 \sqrt{\frac{i \sqrt{a}}{\sqrt{c}}} \left (a+c x^2\right ) \left (a (3 A+5 B x)-5 c x^2 (3 A+B x)\right )+8 \sqrt{a} c x^{7/2} \sqrt{\frac{a}{c x^2}+1} \left (9 A \sqrt{c}+5 i \sqrt{a} B\right ) F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{a}}{\sqrt{c}}}}{\sqrt{x}}\right )\right |-1\right )-72 \sqrt{a} A c^{3/2} x^{7/2} \sqrt{\frac{a}{c x^2}+1} E\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{a}}{\sqrt{c}}}}{\sqrt{x}}\right )\right |-1\right )\right )}{15 \sqrt{\frac{i \sqrt{a}}{\sqrt{c}}} (e x)^{7/2} \sqrt{a+c x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(x*(-2*Sqrt[(I*Sqrt[a])/Sqrt[c]]*(a + c*x^2)*(-5*c*x^2*(3*A + B*x) + a*(3*A + 5*
B*x)) - 72*Sqrt[a]*A*c^(3/2)*Sqrt[1 + a/(c*x^2)]*x^(7/2)*EllipticE[I*ArcSinh[Sqr
t[(I*Sqrt[a])/Sqrt[c]]/Sqrt[x]], -1] + 8*Sqrt[a]*((5*I)*Sqrt[a]*B + 9*A*Sqrt[c])
*c*Sqrt[1 + a/(c*x^2)]*x^(7/2)*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[a])/Sqrt[c]]/Sqr
t[x]], -1]))/(15*Sqrt[(I*Sqrt[a])/Sqrt[c]]*(e*x)^(7/2)*Sqrt[a + c*x^2])

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Maple [A]  time = 0.028, size = 329, normalized size = 1. \[{\frac{2}{15\,{x}^{2}{e}^{3}} \left ( 36\,A\sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{-{\frac{cx}{\sqrt{-ac}}}}{\it EllipticE} \left ( \sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}},1/2\,\sqrt{2} \right ){x}^{2}ac-18\,A\sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{-{\frac{cx}{\sqrt{-ac}}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}},1/2\,\sqrt{2} \right ){x}^{2}ac+10\,B\sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{-{\frac{cx}{\sqrt{-ac}}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}},1/2\,\sqrt{2} \right ) \sqrt{-ac}{x}^{2}a+5\,B{c}^{2}{x}^{5}-21\,A{c}^{2}{x}^{4}-24\,aAc{x}^{2}-5\,{a}^{2}Bx-3\,A{a}^{2} \right ){\frac{1}{\sqrt{c{x}^{2}+a}}}{\frac{1}{\sqrt{ex}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/15/x^2*(36*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*c-18*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*c+10*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)*x^2*a+5*B*c^2*x^5-21*A*c^2*x^4-24*a*A*c*x^2-5*a^2*B*
x-3*A*a^2)/(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. \[ \int \frac{{\left (c x^{2} + a\right )}^{\frac{3}{2}}{\left (B x + A\right )}}{\left (e x\right )^{\frac{7}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (c x^{2} + a\right )}^{\frac{3}{2}}{\left (B x + A\right )}}{\left (e x\right )^{\frac{7}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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