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

Optimal. Leaf size=299 \[ \frac{e^2 \sqrt{e x} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )^2}} (7 a B+2 A b) F\left (\cos ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}{\left (1+\sqrt{3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{27 \sqrt [4]{3} a^{4/3} b^2 \sqrt{\frac{\sqrt [3]{b} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}-\frac{2 e^2 \sqrt{e x} (7 a B+2 A b)}{27 a b^2 \sqrt{a+b x^3}}+\frac{2 (e x)^{7/2} (A b-a B)}{9 a b e \left (a+b x^3\right )^{3/2}} \]

[Out]

(2*(A*b - a*B)*(e*x)^(7/2))/(9*a*b*e*(a + b*x^3)^(3/2)) - (2*(2*A*b + 7*a*B)*e^2
*Sqrt[e*x])/(27*a*b^2*Sqrt[a + b*x^3]) + ((2*A*b + 7*a*B)*e^2*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 + S
qrt[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])/(27*3^(1/4)*a^(4/3)*b^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 = 0.573357, antiderivative size = 299, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{e^2 \sqrt{e x} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )^2}} (7 a B+2 A b) F\left (\cos ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}{\left (1+\sqrt{3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{27 \sqrt [4]{3} a^{4/3} b^2 \sqrt{\frac{\sqrt [3]{b} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}-\frac{2 e^2 \sqrt{e x} (7 a B+2 A b)}{27 a b^2 \sqrt{a+b x^3}}+\frac{2 (e x)^{7/2} (A b-a B)}{9 a b e \left (a+b x^3\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(A*b - a*B)*(e*x)^(7/2))/(9*a*b*e*(a + b*x^3)^(3/2)) - (2*(2*A*b + 7*a*B)*e^2
*Sqrt[e*x])/(27*a*b^2*Sqrt[a + b*x^3]) + ((2*A*b + 7*a*B)*e^2*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 + S
qrt[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])/(27*3^(1/4)*a^(4/3)*b^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 = 33.3658, size = 270, normalized size = 0.9 \[ \frac{2 \left (e x\right )^{\frac{7}{2}} \left (A b - B a\right )}{9 a b e \left (a + b x^{3}\right )^{\frac{3}{2}}} - \frac{4 e^{2} \sqrt{e x} \left (A b + \frac{7 B a}{2}\right )}{27 a b^{2} \sqrt{a + b x^{3}}} + \frac{2 \cdot 3^{\frac{3}{4}} e^{2} \sqrt{e x} \sqrt{\frac{a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2}}{\left (\sqrt [3]{a} + \sqrt [3]{b} x \left (1 + \sqrt{3}\right )\right )^{2}}} \left (\sqrt [3]{a} + \sqrt [3]{b} x\right ) \left (A b + \frac{7 B a}{2}\right ) F\left (\operatorname{acos}{\left (\frac{\sqrt [3]{a} + \sqrt [3]{b} x \left (- \sqrt{3} + 1\right )}{\sqrt [3]{a} + \sqrt [3]{b} x \left (1 + \sqrt{3}\right )} \right )}\middle | \frac{\sqrt{3}}{4} + \frac{1}{2}\right )}{81 a^{\frac{4}{3}} b^{2} \sqrt{\frac{\sqrt [3]{b} x \left (\sqrt [3]{a} + \sqrt [3]{b} x\right )}{\left (\sqrt [3]{a} + \sqrt [3]{b} x \left (1 + \sqrt{3}\right )\right )^{2}}} \sqrt{a + b x^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*(e*x)**(7/2)*(A*b - B*a)/(9*a*b*e*(a + b*x**3)**(3/2)) - 4*e**2*sqrt(e*x)*(A*b
+ 7*B*a/2)/(27*a*b**2*sqrt(a + b*x**3)) + 2*3**(3/4)*e**2*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)*(A*b + 7*B*a/2)*elliptic_f(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)/(81
*a**(4/3)*b**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))

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Mathematica [C]  time = 0.585361, size = 216, normalized size = 0.72 \[ \frac{2 i e^2 \sqrt{e x} \left (3^{3/4} \sqrt [3]{b} x \sqrt{(-1)^{5/6} \left (\frac{\sqrt [3]{-a}}{\sqrt [3]{b} x}-1\right )} \sqrt{\frac{\frac{(-a)^{2/3}}{b^{2/3}}+\frac{\sqrt [3]{-a} x}{\sqrt [3]{b}}+x^2}{x^2}} \left (a+b x^3\right ) (7 a B+2 A b) F\left (\sin ^{-1}\left (\frac{\sqrt{-\frac{i \sqrt [3]{-a}}{\sqrt [3]{b} x}-(-1)^{5/6}}}{\sqrt [4]{3}}\right )|\sqrt [3]{-1}\right )-3 i \sqrt [3]{-a} \left (7 a^2 B+2 a b \left (A+5 B x^3\right )-A b^2 x^3\right )\right )}{81 (-a)^{4/3} b^2 \left (a+b x^3\right )^{3/2}} \]

Warning: Unable to verify antiderivative.

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

[Out]

(((2*I)/81)*e^2*Sqrt[e*x]*((-3*I)*(-a)^(1/3)*(7*a^2*B - A*b^2*x^3 + 2*a*b*(A + 5
*B*x^3)) + 3^(3/4)*b^(1/3)*(2*A*b + 7*a*B)*Sqrt[(-1)^(5/6)*(-1 + (-a)^(1/3)/(b^(
1/3)*x))]*x*Sqrt[((-a)^(2/3)/b^(2/3) + ((-a)^(1/3)*x)/b^(1/3) + x^2)/x^2]*(a + b
*x^3)*EllipticF[ArcSin[Sqrt[-(-1)^(5/6) - (I*(-a)^(1/3))/(b^(1/3)*x)]/3^(1/4)],
(-1)^(1/3)]))/((-a)^(4/3)*b^2*(a + b*x^3)^(3/2))

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Maple [C]  time = 0.086, size = 7083, normalized size = 23.7 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

result too large to display

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (B e^{2} x^{5} + A e^{2} x^{2}\right )} \sqrt{e x}}{{\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )} \sqrt{b x^{3} + a}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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