3.540 \(\int \frac{\left (a+b x^3\right )^{5/2} \left (A+B x^3\right )}{(e x)^{3/2}} \, dx\)

Optimal. Leaf size=650 \[ -\frac{27\ 3^{3/4} \left (1-\sqrt{3}\right ) a^{7/3} \sqrt{e x} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )^2}} (a B+20 A b) F\left (\cos ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}{\left (1+\sqrt{3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{896 b^{2/3} e^2 \sqrt{\frac{\sqrt [3]{b} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}-\frac{81 \sqrt [4]{3} a^{7/3} \sqrt{e x} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )^2}} (a B+20 A b) E\left (\cos ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}{\left (1+\sqrt{3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{448 b^{2/3} e^2 \sqrt{\frac{\sqrt [3]{b} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}+\frac{81 \left (1+\sqrt{3}\right ) a^2 \sqrt{e x} \sqrt{a+b x^3} (a B+20 A b)}{448 b^{2/3} e^2 \left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )}+\frac{(e x)^{5/2} \left (a+b x^3\right )^{5/2} (a B+20 A b)}{10 a e^4}+\frac{3 (e x)^{5/2} \left (a+b x^3\right )^{3/2} (a B+20 A b)}{28 e^4}+\frac{27 a (e x)^{5/2} \sqrt{a+b x^3} (a B+20 A b)}{224 e^4}-\frac{2 A \left (a+b x^3\right )^{7/2}}{a e \sqrt{e x}} \]

[Out]

(27*a*(20*A*b + a*B)*(e*x)^(5/2)*Sqrt[a + b*x^3])/(224*e^4) + (81*(1 + Sqrt[3])*
a^2*(20*A*b + a*B)*Sqrt[e*x]*Sqrt[a + b*x^3])/(448*b^(2/3)*e^2*(a^(1/3) + (1 + S
qrt[3])*b^(1/3)*x)) + (3*(20*A*b + a*B)*(e*x)^(5/2)*(a + b*x^3)^(3/2))/(28*e^4)
+ ((20*A*b + a*B)*(e*x)^(5/2)*(a + b*x^3)^(5/2))/(10*a*e^4) - (2*A*(a + b*x^3)^(
7/2))/(a*e*Sqrt[e*x]) - (81*3^(1/4)*a^(7/3)*(20*A*b + a*B)*Sqrt[e*x]*(a^(1/3) +
b^(1/3)*x)*Sqrt[(a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2)/(a^(1/3) + (1 + Sqrt
[3])*b^(1/3)*x)^2]*EllipticE[ArcCos[(a^(1/3) + (1 - Sqrt[3])*b^(1/3)*x)/(a^(1/3)
+ (1 + Sqrt[3])*b^(1/3)*x)], (2 + Sqrt[3])/4])/(448*b^(2/3)*e^2*Sqrt[(b^(1/3)*x
*(a^(1/3) + b^(1/3)*x))/(a^(1/3) + (1 + Sqrt[3])*b^(1/3)*x)^2]*Sqrt[a + b*x^3])
- (27*3^(3/4)*(1 - Sqrt[3])*a^(7/3)*(20*A*b + a*B)*Sqrt[e*x]*(a^(1/3) + b^(1/3)*
x)*Sqrt[(a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2)/(a^(1/3) + (1 + Sqrt[3])*b^(
1/3)*x)^2]*EllipticF[ArcCos[(a^(1/3) + (1 - Sqrt[3])*b^(1/3)*x)/(a^(1/3) + (1 +
Sqrt[3])*b^(1/3)*x)], (2 + Sqrt[3])/4])/(896*b^(2/3)*e^2*Sqrt[(b^(1/3)*x*(a^(1/3
) + b^(1/3)*x))/(a^(1/3) + (1 + Sqrt[3])*b^(1/3)*x)^2]*Sqrt[a + b*x^3])

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Rubi [A]  time = 1.67717, antiderivative size = 650, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ -\frac{27\ 3^{3/4} \left (1-\sqrt{3}\right ) a^{7/3} \sqrt{e x} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )^2}} (a B+20 A b) F\left (\cos ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}{\left (1+\sqrt{3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{896 b^{2/3} e^2 \sqrt{\frac{\sqrt [3]{b} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}-\frac{81 \sqrt [4]{3} a^{7/3} \sqrt{e x} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )^2}} (a B+20 A b) E\left (\cos ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}{\left (1+\sqrt{3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{448 b^{2/3} e^2 \sqrt{\frac{\sqrt [3]{b} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}+\frac{81 \left (1+\sqrt{3}\right ) a^2 \sqrt{e x} \sqrt{a+b x^3} (a B+20 A b)}{448 b^{2/3} e^2 \left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )}+\frac{(e x)^{5/2} \left (a+b x^3\right )^{5/2} (a B+20 A b)}{10 a e^4}+\frac{3 (e x)^{5/2} \left (a+b x^3\right )^{3/2} (a B+20 A b)}{28 e^4}+\frac{27 a (e x)^{5/2} \sqrt{a+b x^3} (a B+20 A b)}{224 e^4}-\frac{2 A \left (a+b x^3\right )^{7/2}}{a e \sqrt{e x}} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x^3)^(5/2)*(A + B*x^3))/(e*x)^(3/2),x]

[Out]

(27*a*(20*A*b + a*B)*(e*x)^(5/2)*Sqrt[a + b*x^3])/(224*e^4) + (81*(1 + Sqrt[3])*
a^2*(20*A*b + a*B)*Sqrt[e*x]*Sqrt[a + b*x^3])/(448*b^(2/3)*e^2*(a^(1/3) + (1 + S
qrt[3])*b^(1/3)*x)) + (3*(20*A*b + a*B)*(e*x)^(5/2)*(a + b*x^3)^(3/2))/(28*e^4)
+ ((20*A*b + a*B)*(e*x)^(5/2)*(a + b*x^3)^(5/2))/(10*a*e^4) - (2*A*(a + b*x^3)^(
7/2))/(a*e*Sqrt[e*x]) - (81*3^(1/4)*a^(7/3)*(20*A*b + a*B)*Sqrt[e*x]*(a^(1/3) +
b^(1/3)*x)*Sqrt[(a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2)/(a^(1/3) + (1 + Sqrt
[3])*b^(1/3)*x)^2]*EllipticE[ArcCos[(a^(1/3) + (1 - Sqrt[3])*b^(1/3)*x)/(a^(1/3)
+ (1 + Sqrt[3])*b^(1/3)*x)], (2 + Sqrt[3])/4])/(448*b^(2/3)*e^2*Sqrt[(b^(1/3)*x
*(a^(1/3) + b^(1/3)*x))/(a^(1/3) + (1 + Sqrt[3])*b^(1/3)*x)^2]*Sqrt[a + b*x^3])
- (27*3^(3/4)*(1 - Sqrt[3])*a^(7/3)*(20*A*b + a*B)*Sqrt[e*x]*(a^(1/3) + b^(1/3)*
x)*Sqrt[(a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2)/(a^(1/3) + (1 + Sqrt[3])*b^(
1/3)*x)^2]*EllipticF[ArcCos[(a^(1/3) + (1 - Sqrt[3])*b^(1/3)*x)/(a^(1/3) + (1 +
Sqrt[3])*b^(1/3)*x)], (2 + Sqrt[3])/4])/(896*b^(2/3)*e^2*Sqrt[(b^(1/3)*x*(a^(1/3
) + b^(1/3)*x))/(a^(1/3) + (1 + Sqrt[3])*b^(1/3)*x)^2]*Sqrt[a + b*x^3])

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Rubi in Sympy [A]  time = 94.6629, size = 598, normalized size = 0.92 \[ - \frac{2 A \left (a + b x^{3}\right )^{\frac{7}{2}}}{a e \sqrt{e x}} - \frac{81 \sqrt [4]{3} a^{\frac{7}{3}} \sqrt{e x} \sqrt{\frac{a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2}}{\left (\sqrt [3]{a} + \sqrt [3]{b} x \left (1 + \sqrt{3}\right )\right )^{2}}} \left (\sqrt [3]{a} + \sqrt [3]{b} x\right ) \left (20 A b + B a\right ) E\left (\operatorname{acos}{\left (\frac{\sqrt [3]{a} + \sqrt [3]{b} x \left (- \sqrt{3} + 1\right )}{\sqrt [3]{a} + \sqrt [3]{b} x \left (1 + \sqrt{3}\right )} \right )}\middle | \frac{\sqrt{3}}{4} + \frac{1}{2}\right )}{448 b^{\frac{2}{3}} e^{2} \sqrt{\frac{\sqrt [3]{b} x \left (\sqrt [3]{a} + \sqrt [3]{b} x\right )}{\left (\sqrt [3]{a} + \sqrt [3]{b} x \left (1 + \sqrt{3}\right )\right )^{2}}} \sqrt{a + b x^{3}}} - \frac{27 \cdot 3^{\frac{3}{4}} a^{\frac{7}{3}} \sqrt{e x} \sqrt{\frac{a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2}}{\left (\sqrt [3]{a} + \sqrt [3]{b} x \left (1 + \sqrt{3}\right )\right )^{2}}} \left (- \sqrt{3} + 1\right ) \left (\sqrt [3]{a} + \sqrt [3]{b} x\right ) \left (20 A b + B a\right ) F\left (\operatorname{acos}{\left (\frac{\sqrt [3]{a} + \sqrt [3]{b} x \left (- \sqrt{3} + 1\right )}{\sqrt [3]{a} + \sqrt [3]{b} x \left (1 + \sqrt{3}\right )} \right )}\middle | \frac{\sqrt{3}}{4} + \frac{1}{2}\right )}{896 b^{\frac{2}{3}} e^{2} \sqrt{\frac{\sqrt [3]{b} x \left (\sqrt [3]{a} + \sqrt [3]{b} x\right )}{\left (\sqrt [3]{a} + \sqrt [3]{b} x \left (1 + \sqrt{3}\right )\right )^{2}}} \sqrt{a + b x^{3}}} + \frac{a^{2} \sqrt{e x} \left (\frac{81}{224} + \frac{81 \sqrt{3}}{224}\right ) \sqrt{a + b x^{3}} \left (20 A b + B a\right )}{2 b^{\frac{2}{3}} e^{2} \left (\sqrt [3]{a} + \sqrt [3]{b} x \left (1 + \sqrt{3}\right )\right )} + \frac{27 a \left (e x\right )^{\frac{5}{2}} \sqrt{a + b x^{3}} \left (20 A b + B a\right )}{224 e^{4}} + \frac{3 \left (e x\right )^{\frac{5}{2}} \left (a + b x^{3}\right )^{\frac{3}{2}} \left (20 A b + B a\right )}{28 e^{4}} + \frac{\left (e x\right )^{\frac{5}{2}} \left (a + b x^{3}\right )^{\frac{5}{2}} \left (20 A b + B a\right )}{10 a e^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**3+a)**(5/2)*(B*x**3+A)/(e*x)**(3/2),x)

[Out]

-2*A*(a + b*x**3)**(7/2)/(a*e*sqrt(e*x)) - 81*3**(1/4)*a**(7/3)*sqrt(e*x)*sqrt((
a**(2/3) - a**(1/3)*b**(1/3)*x + b**(2/3)*x**2)/(a**(1/3) + b**(1/3)*x*(1 + sqrt
(3)))**2)*(a**(1/3) + b**(1/3)*x)*(20*A*b + B*a)*elliptic_e(acos((a**(1/3) + b**
(1/3)*x*(-sqrt(3) + 1))/(a**(1/3) + b**(1/3)*x*(1 + sqrt(3)))), sqrt(3)/4 + 1/2)
/(448*b**(2/3)*e**2*sqrt(b**(1/3)*x*(a**(1/3) + b**(1/3)*x)/(a**(1/3) + b**(1/3)
*x*(1 + sqrt(3)))**2)*sqrt(a + b*x**3)) - 27*3**(3/4)*a**(7/3)*sqrt(e*x)*sqrt((a
**(2/3) - a**(1/3)*b**(1/3)*x + b**(2/3)*x**2)/(a**(1/3) + b**(1/3)*x*(1 + sqrt(
3)))**2)*(-sqrt(3) + 1)*(a**(1/3) + b**(1/3)*x)*(20*A*b + B*a)*elliptic_f(acos((
a**(1/3) + b**(1/3)*x*(-sqrt(3) + 1))/(a**(1/3) + b**(1/3)*x*(1 + sqrt(3)))), sq
rt(3)/4 + 1/2)/(896*b**(2/3)*e**2*sqrt(b**(1/3)*x*(a**(1/3) + b**(1/3)*x)/(a**(1
/3) + b**(1/3)*x*(1 + sqrt(3)))**2)*sqrt(a + b*x**3)) + a**2*sqrt(e*x)*(81/224 +
81*sqrt(3)/224)*sqrt(a + b*x**3)*(20*A*b + B*a)/(2*b**(2/3)*e**2*(a**(1/3) + b*
*(1/3)*x*(1 + sqrt(3)))) + 27*a*(e*x)**(5/2)*sqrt(a + b*x**3)*(20*A*b + B*a)/(22
4*e**4) + 3*(e*x)**(5/2)*(a + b*x**3)**(3/2)*(20*A*b + B*a)/(28*e**4) + (e*x)**(
5/2)*(a + b*x**3)**(5/2)*(20*A*b + B*a)/(10*a*e**4)

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Mathematica [C]  time = 4.40926, size = 329, normalized size = 0.51 \[ \frac{x^{3/2} \left (\frac{2 \left (a+b x^3\right ) \left (a^2 \left (367 B x^3-2240 A\right )+4 a b x^3 \left (155 A+86 B x^3\right )+16 b^2 x^6 \left (10 A+7 B x^3\right )\right )}{5 \sqrt{x}}-\frac{27 a^2 x^{5/2} (a B+20 A b) \left (-3 \left (\frac{a}{x^3}+b\right )+\frac{\sqrt [6]{-1} 3^{3/4} a b^{2/3} \sqrt{\frac{(-1)^{5/6} \left (\sqrt [3]{-a}-\sqrt [3]{b} x\right )}{\sqrt [3]{b} x}} \sqrt{\frac{\frac{(-a)^{2/3}}{b^{2/3}}+\frac{\sqrt [3]{-a} x}{\sqrt [3]{b}}+x^2}{x^2}} \left (\sqrt [3]{-1} F\left (\sin ^{-1}\left (\frac{\sqrt{-\frac{i \sqrt [3]{-a}}{\sqrt [3]{b} x}-(-1)^{5/6}}}{\sqrt [4]{3}}\right )|\sqrt [3]{-1}\right )-i \sqrt{3} E\left (\sin ^{-1}\left (\frac{\sqrt{-\frac{i \sqrt [3]{-a}}{\sqrt [3]{b} x}-(-1)^{5/6}}}{\sqrt [4]{3}}\right )|\sqrt [3]{-1}\right )\right )}{(-a)^{2/3} x}\right )}{b}\right )}{448 (e x)^{3/2} \sqrt{a+b x^3}} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[((a + b*x^3)^(5/2)*(A + B*x^3))/(e*x)^(3/2),x]

[Out]

(x^(3/2)*((2*(a + b*x^3)*(16*b^2*x^6*(10*A + 7*B*x^3) + 4*a*b*x^3*(155*A + 86*B*
x^3) + a^2*(-2240*A + 367*B*x^3)))/(5*Sqrt[x]) - (27*a^2*(20*A*b + a*B)*x^(5/2)*
(-3*(b + a/x^3) + ((-1)^(1/6)*3^(3/4)*a*b^(2/3)*Sqrt[((-1)^(5/6)*((-a)^(1/3) - b
^(1/3)*x))/(b^(1/3)*x)]*Sqrt[((-a)^(2/3)/b^(2/3) + ((-a)^(1/3)*x)/b^(1/3) + x^2)
/x^2]*((-I)*Sqrt[3]*EllipticE[ArcSin[Sqrt[-(-1)^(5/6) - (I*(-a)^(1/3))/(b^(1/3)*
x)]/3^(1/4)], (-1)^(1/3)] + (-1)^(1/3)*EllipticF[ArcSin[Sqrt[-(-1)^(5/6) - (I*(-
a)^(1/3))/(b^(1/3)*x)]/3^(1/4)], (-1)^(1/3)]))/((-a)^(2/3)*x)))/b))/(448*(e*x)^(
3/2)*Sqrt[a + b*x^3])

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Maple [C]  time = 0.058, size = 6530, normalized size = 10.1 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^3+a)^(5/2)*(B*x^3+A)/(e*x)^(3/2),x)

[Out]

result too large to display

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (B x^{3} + A\right )}{\left (b x^{3} + a\right )}^{\frac{5}{2}}}{\left (e x\right )^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^3 + A)*(b*x^3 + a)^(5/2)/(e*x)^(3/2),x, algorithm="maxima")

[Out]

integrate((B*x^3 + A)*(b*x^3 + a)^(5/2)/(e*x)^(3/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (B b^{2} x^{9} +{\left (2 \, B a b + A b^{2}\right )} x^{6} +{\left (B a^{2} + 2 \, A a b\right )} x^{3} + A a^{2}\right )} \sqrt{b x^{3} + a}}{\sqrt{e x} e x}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^3 + A)*(b*x^3 + a)^(5/2)/(e*x)^(3/2),x, algorithm="fricas")

[Out]

integral((B*b^2*x^9 + (2*B*a*b + A*b^2)*x^6 + (B*a^2 + 2*A*a*b)*x^3 + A*a^2)*sqr
t(b*x^3 + a)/(sqrt(e*x)*e*x), 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**3+a)**(5/2)*(B*x**3+A)/(e*x)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (B x^{3} + A\right )}{\left (b x^{3} + a\right )}^{\frac{5}{2}}}{\left (e x\right )^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^3 + A)*(b*x^3 + a)^(5/2)/(e*x)^(3/2),x, algorithm="giac")

[Out]

integrate((B*x^3 + A)*(b*x^3 + a)^(5/2)/(e*x)^(3/2), x)