3.524 \(\int \frac{\sqrt{a+b x^3} \left (A+B x^3\right )}{x^{9/2}} \, dx\)

Optimal. Leaf size=564 \[ -\frac{3^{3/4} \left (1-\sqrt{3}\right ) \sqrt [3]{b} \sqrt{x} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )^2}} (7 a B+2 A b) F\left (\cos ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}{\left (1+\sqrt{3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{14 a^{2/3} \sqrt{\frac{\sqrt [3]{b} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}-\frac{3 \sqrt [4]{3} \sqrt [3]{b} \sqrt{x} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )^2}} (7 a B+2 A b) E\left (\cos ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}{\left (1+\sqrt{3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{7 a^{2/3} \sqrt{\frac{\sqrt [3]{b} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}-\frac{2 \sqrt{a+b x^3} (7 a B+2 A b)}{7 a \sqrt{x}}+\frac{3 \left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt{x} \sqrt{a+b x^3} (7 a B+2 A b)}{7 a \left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )}-\frac{2 A \left (a+b x^3\right )^{3/2}}{7 a x^{7/2}} \]

[Out]

(-2*(2*A*b + 7*a*B)*Sqrt[a + b*x^3])/(7*a*Sqrt[x]) + (3*(1 + Sqrt[3])*b^(1/3)*(2
*A*b + 7*a*B)*Sqrt[x]*Sqrt[a + b*x^3])/(7*a*(a^(1/3) + (1 + Sqrt[3])*b^(1/3)*x))
- (2*A*(a + b*x^3)^(3/2))/(7*a*x^(7/2)) - (3*3^(1/4)*b^(1/3)*(2*A*b + 7*a*B)*Sq
rt[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])/(7*a^(2/3)*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]) - (3^(3/4)*(1 - Sqrt[3])*b^(1/3)*(2*A*b + 7*a*B)*Sqrt[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])/(14*a^(2/3)*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 = 1.16434, antiderivative size = 564, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{3^{3/4} \left (1-\sqrt{3}\right ) \sqrt [3]{b} \sqrt{x} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )^2}} (7 a B+2 A b) F\left (\cos ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}{\left (1+\sqrt{3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{14 a^{2/3} \sqrt{\frac{\sqrt [3]{b} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}-\frac{3 \sqrt [4]{3} \sqrt [3]{b} \sqrt{x} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )^2}} (7 a B+2 A b) E\left (\cos ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}{\left (1+\sqrt{3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{7 a^{2/3} \sqrt{\frac{\sqrt [3]{b} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}-\frac{2 \sqrt{a+b x^3} (7 a B+2 A b)}{7 a \sqrt{x}}+\frac{3 \left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt{x} \sqrt{a+b x^3} (7 a B+2 A b)}{7 a \left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )}-\frac{2 A \left (a+b x^3\right )^{3/2}}{7 a x^{7/2}} \]

Antiderivative was successfully verified.

[In]  Int[(Sqrt[a + b*x^3]*(A + B*x^3))/x^(9/2),x]

[Out]

(-2*(2*A*b + 7*a*B)*Sqrt[a + b*x^3])/(7*a*Sqrt[x]) + (3*(1 + Sqrt[3])*b^(1/3)*(2
*A*b + 7*a*B)*Sqrt[x]*Sqrt[a + b*x^3])/(7*a*(a^(1/3) + (1 + Sqrt[3])*b^(1/3)*x))
- (2*A*(a + b*x^3)^(3/2))/(7*a*x^(7/2)) - (3*3^(1/4)*b^(1/3)*(2*A*b + 7*a*B)*Sq
rt[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])/(7*a^(2/3)*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]) - (3^(3/4)*(1 - Sqrt[3])*b^(1/3)*(2*A*b + 7*a*B)*Sqrt[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])/(14*a^(2/3)*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 = 52.3209, size = 513, normalized size = 0.91 \[ - \frac{2 A \left (a + b x^{3}\right )^{\frac{3}{2}}}{7 a x^{\frac{7}{2}}} + \frac{\sqrt [3]{b} \sqrt{x} \left (\frac{6}{7} + \frac{6 \sqrt{3}}{7}\right ) \sqrt{a + b x^{3}} \left (A b + \frac{7 B a}{2}\right )}{a \left (\sqrt [3]{a} + \sqrt [3]{b} x \left (1 + \sqrt{3}\right )\right )} - \frac{4 \sqrt{a + b x^{3}} \left (A b + \frac{7 B a}{2}\right )}{7 a \sqrt{x}} - \frac{6 \sqrt [4]{3} \sqrt [3]{b} \sqrt{x} \sqrt{\frac{a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2}}{\left (\sqrt [3]{a} + \sqrt [3]{b} x \left (1 + \sqrt{3}\right )\right )^{2}}} \left (\sqrt [3]{a} + \sqrt [3]{b} x\right ) \left (A b + \frac{7 B a}{2}\right ) E\left (\operatorname{acos}{\left (\frac{\sqrt [3]{a} + \sqrt [3]{b} x \left (- \sqrt{3} + 1\right )}{\sqrt [3]{a} + \sqrt [3]{b} x \left (1 + \sqrt{3}\right )} \right )}\middle | \frac{\sqrt{3}}{4} + \frac{1}{2}\right )}{7 a^{\frac{2}{3}} \sqrt{\frac{\sqrt [3]{b} x \left (\sqrt [3]{a} + \sqrt [3]{b} x\right )}{\left (\sqrt [3]{a} + \sqrt [3]{b} x \left (1 + \sqrt{3}\right )\right )^{2}}} \sqrt{a + b x^{3}}} - \frac{3^{\frac{3}{4}} \sqrt [3]{b} \sqrt{x} \sqrt{\frac{a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2}}{\left (\sqrt [3]{a} + \sqrt [3]{b} x \left (1 + \sqrt{3}\right )\right )^{2}}} \left (- \sqrt{3} + 1\right ) \left (\sqrt [3]{a} + \sqrt [3]{b} x\right ) \left (A b + \frac{7 B a}{2}\right ) F\left (\operatorname{acos}{\left (\frac{\sqrt [3]{a} + \sqrt [3]{b} x \left (- \sqrt{3} + 1\right )}{\sqrt [3]{a} + \sqrt [3]{b} x \left (1 + \sqrt{3}\right )} \right )}\middle | \frac{\sqrt{3}}{4} + \frac{1}{2}\right )}{7 a^{\frac{2}{3}} \sqrt{\frac{\sqrt [3]{b} x \left (\sqrt [3]{a} + \sqrt [3]{b} x\right )}{\left (\sqrt [3]{a} + \sqrt [3]{b} x \left (1 + \sqrt{3}\right )\right )^{2}}} \sqrt{a + b x^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x**3+A)*(b*x**3+a)**(1/2)/x**(9/2),x)

[Out]

-2*A*(a + b*x**3)**(3/2)/(7*a*x**(7/2)) + b**(1/3)*sqrt(x)*(6/7 + 6*sqrt(3)/7)*s
qrt(a + b*x**3)*(A*b + 7*B*a/2)/(a*(a**(1/3) + b**(1/3)*x*(1 + sqrt(3)))) - 4*sq
rt(a + b*x**3)*(A*b + 7*B*a/2)/(7*a*sqrt(x)) - 6*3**(1/4)*b**(1/3)*sqrt(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 + sqr
t(3)))**2)*(a**(1/3) + b**(1/3)*x)*(A*b + 7*B*a/2)*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)/(7*a**(2/3)*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)) - 3**(3/4)*b**(1/3)*sqrt(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)*(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)/(7*a**(2/3)*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 = 2.57209, size = 285, normalized size = 0.51 \[ -\frac{-2 (-a)^{2/3} \left (a+b x^3\right ) \left (x^3 (7 a B+3 A b)+a A\right )+x^3 (7 a B+2 A b) \left (3 (-a)^{2/3} \left (a+b x^3\right )+(-1)^{2/3} 3^{3/4} a b^{2/3} x^2 \sqrt{\frac{(-1)^{5/6} \left (\sqrt [3]{-a}-\sqrt [3]{b} x\right )}{\sqrt [3]{b} x}} \sqrt{\frac{\frac{(-a)^{2/3}}{b^{2/3}}+\frac{\sqrt [3]{-a} x}{\sqrt [3]{b}}+x^2}{x^2}} \left ((-1)^{5/6} F\left (\sin ^{-1}\left (\frac{\sqrt{-\frac{i \sqrt [3]{-a}}{\sqrt [3]{b} x}-(-1)^{5/6}}}{\sqrt [4]{3}}\right )|\sqrt [3]{-1}\right )+\sqrt{3} E\left (\sin ^{-1}\left (\frac{\sqrt{-\frac{i \sqrt [3]{-a}}{\sqrt [3]{b} x}-(-1)^{5/6}}}{\sqrt [4]{3}}\right )|\sqrt [3]{-1}\right )\right )\right )}{7 (-a)^{5/3} x^{7/2} \sqrt{a+b x^3}} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[(Sqrt[a + b*x^3]*(A + B*x^3))/x^(9/2),x]

[Out]

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

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Maple [C]  time = 0.12, size = 5911, normalized size = 10.5 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x^3+A)*(b*x^3+a)^(1/2)/x^(9/2),x)

[Out]

result too large to display

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (B x^{3} + A\right )} \sqrt{b x^{3} + a}}{x^{\frac{9}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^3 + A)*sqrt(b*x^3 + a)/x^(9/2),x, algorithm="maxima")

[Out]

integrate((B*x^3 + A)*sqrt(b*x^3 + a)/x^(9/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^3 + A)*sqrt(b*x^3 + a)/x^(9/2),x, algorithm="fricas")

[Out]

integral((B*x^3 + A)*sqrt(b*x^3 + a)/x^(9/2), 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)*(b*x**3+a)**(1/2)/x**(9/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (B x^{3} + A\right )} \sqrt{b x^{3} + a}}{x^{\frac{9}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^3 + A)*sqrt(b*x^3 + a)/x^(9/2),x, algorithm="giac")

[Out]

integrate((B*x^3 + A)*sqrt(b*x^3 + a)/x^(9/2), x)