3.1599 \(\int \frac{1}{(a+b x)^{4/3} \sqrt [3]{c+d x}} \, dx\)

Optimal. Leaf size=1297 \[ \text{result too large to display} \]

[Out]

(-3*(c + d*x)^(2/3))/((b*c - a*d)*(a + b*x)^(1/3)) + (3*d^(1/3)*((a + b*x)*(c +
d*x))^(1/3)*Sqrt[(b*c + a*d + 2*b*d*x)^2]*Sqrt[(a*d + b*(c + 2*d*x))^2])/(2^(1/3
)*b^(2/3)*(b*c - a*d)*(a + b*x)^(1/3)*(c + d*x)^(1/3)*(b*c + a*d + 2*b*d*x)*((1
+ Sqrt[3])*(b*c - a*d)^(2/3) + 2^(2/3)*b^(1/3)*d^(1/3)*((a + b*x)*(c + d*x))^(1/
3))) - (3*3^(1/4)*Sqrt[2 - Sqrt[3]]*d^(1/3)*((a + b*x)*(c + d*x))^(1/3)*Sqrt[(b*
c + a*d + 2*b*d*x)^2]*((b*c - a*d)^(2/3) + 2^(2/3)*b^(1/3)*d^(1/3)*((a + b*x)*(c
+ d*x))^(1/3))*Sqrt[((b*c - a*d)^(4/3) - 2^(2/3)*b^(1/3)*d^(1/3)*(b*c - a*d)^(2
/3)*((a + b*x)*(c + d*x))^(1/3) + 2*2^(1/3)*b^(2/3)*d^(2/3)*((a + b*x)*(c + d*x)
)^(2/3))/((1 + Sqrt[3])*(b*c - a*d)^(2/3) + 2^(2/3)*b^(1/3)*d^(1/3)*((a + b*x)*(
c + d*x))^(1/3))^2]*EllipticE[ArcSin[((1 - Sqrt[3])*(b*c - a*d)^(2/3) + 2^(2/3)*
b^(1/3)*d^(1/3)*((a + b*x)*(c + d*x))^(1/3))/((1 + Sqrt[3])*(b*c - a*d)^(2/3) +
2^(2/3)*b^(1/3)*d^(1/3)*((a + b*x)*(c + d*x))^(1/3))], -7 - 4*Sqrt[3]])/(2*2^(1/
3)*b^(2/3)*(b*c - a*d)^(1/3)*(a + b*x)^(1/3)*(c + d*x)^(1/3)*(b*c + a*d + 2*b*d*
x)*Sqrt[((b*c - a*d)^(2/3)*((b*c - a*d)^(2/3) + 2^(2/3)*b^(1/3)*d^(1/3)*((a + b*
x)*(c + d*x))^(1/3)))/((1 + Sqrt[3])*(b*c - a*d)^(2/3) + 2^(2/3)*b^(1/3)*d^(1/3)
*((a + b*x)*(c + d*x))^(1/3))^2]*Sqrt[(a*d + b*(c + 2*d*x))^2]) + (2^(1/6)*3^(3/
4)*d^(1/3)*((a + b*x)*(c + d*x))^(1/3)*Sqrt[(b*c + a*d + 2*b*d*x)^2]*((b*c - a*d
)^(2/3) + 2^(2/3)*b^(1/3)*d^(1/3)*((a + b*x)*(c + d*x))^(1/3))*Sqrt[((b*c - a*d)
^(4/3) - 2^(2/3)*b^(1/3)*d^(1/3)*(b*c - a*d)^(2/3)*((a + b*x)*(c + d*x))^(1/3) +
2*2^(1/3)*b^(2/3)*d^(2/3)*((a + b*x)*(c + d*x))^(2/3))/((1 + Sqrt[3])*(b*c - a*
d)^(2/3) + 2^(2/3)*b^(1/3)*d^(1/3)*((a + b*x)*(c + d*x))^(1/3))^2]*EllipticF[Arc
Sin[((1 - Sqrt[3])*(b*c - a*d)^(2/3) + 2^(2/3)*b^(1/3)*d^(1/3)*((a + b*x)*(c + d
*x))^(1/3))/((1 + Sqrt[3])*(b*c - a*d)^(2/3) + 2^(2/3)*b^(1/3)*d^(1/3)*((a + b*x
)*(c + d*x))^(1/3))], -7 - 4*Sqrt[3]])/(b^(2/3)*(b*c - a*d)^(1/3)*(a + b*x)^(1/3
)*(c + d*x)^(1/3)*(b*c + a*d + 2*b*d*x)*Sqrt[((b*c - a*d)^(2/3)*((b*c - a*d)^(2/
3) + 2^(2/3)*b^(1/3)*d^(1/3)*((a + b*x)*(c + d*x))^(1/3)))/((1 + Sqrt[3])*(b*c -
a*d)^(2/3) + 2^(2/3)*b^(1/3)*d^(1/3)*((a + b*x)*(c + d*x))^(1/3))^2]*Sqrt[(a*d
+ b*(c + 2*d*x))^2])

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Rubi [A]  time = 3.40174, antiderivative size = 1297, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.316 \[ -\frac{3 \sqrt [4]{3} \sqrt{2-\sqrt{3}} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)} \sqrt{(b c+a d+2 b d x)^2} \left ((b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}\right ) \sqrt{\frac{(b c-a d)^{4/3}-2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)} (b c-a d)^{2/3}+2 \sqrt [3]{2} b^{2/3} d^{2/3} ((a+b x) (c+d x))^{2/3}}{\left (\left (1+\sqrt{3}\right ) (b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}\right )^2}} E\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) (b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}}{\left (1+\sqrt{3}\right ) (b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}}\right )|-7-4 \sqrt{3}\right )}{2 \sqrt [3]{2} b^{2/3} \sqrt [3]{b c-a d} \sqrt [3]{a+b x} \sqrt [3]{c+d x} (b c+a d+2 b d x) \sqrt{\frac{(b c-a d)^{2/3} \left ((b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}\right )}{\left (\left (1+\sqrt{3}\right ) (b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}\right )^2}} \sqrt{(a d+b (c+2 d x))^2}}+\frac{\sqrt [6]{2} 3^{3/4} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)} \sqrt{(b c+a d+2 b d x)^2} \left ((b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}\right ) \sqrt{\frac{(b c-a d)^{4/3}-2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)} (b c-a d)^{2/3}+2 \sqrt [3]{2} b^{2/3} d^{2/3} ((a+b x) (c+d x))^{2/3}}{\left (\left (1+\sqrt{3}\right ) (b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) (b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}}{\left (1+\sqrt{3}\right ) (b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}}\right )|-7-4 \sqrt{3}\right )}{b^{2/3} \sqrt [3]{b c-a d} \sqrt [3]{a+b x} \sqrt [3]{c+d x} (b c+a d+2 b d x) \sqrt{\frac{(b c-a d)^{2/3} \left ((b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}\right )}{\left (\left (1+\sqrt{3}\right ) (b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}\right )^2}} \sqrt{(a d+b (c+2 d x))^2}}-\frac{3 (c+d x)^{2/3}}{(b c-a d) \sqrt [3]{a+b x}}+\frac{3 \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)} \sqrt{(b c+a d+2 b d x)^2} \sqrt{(a d+b (c+2 d x))^2}}{\sqrt [3]{2} b^{2/3} (b c-a d) \sqrt [3]{a+b x} \sqrt [3]{c+d x} (b c+a d+2 b d x) \left (\left (1+\sqrt{3}\right ) (b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}\right )} \]

Warning: Unable to verify antiderivative.

[In]  Int[1/((a + b*x)^(4/3)*(c + d*x)^(1/3)),x]

[Out]

(-3*(c + d*x)^(2/3))/((b*c - a*d)*(a + b*x)^(1/3)) + (3*d^(1/3)*((a + b*x)*(c +
d*x))^(1/3)*Sqrt[(b*c + a*d + 2*b*d*x)^2]*Sqrt[(a*d + b*(c + 2*d*x))^2])/(2^(1/3
)*b^(2/3)*(b*c - a*d)*(a + b*x)^(1/3)*(c + d*x)^(1/3)*(b*c + a*d + 2*b*d*x)*((1
+ Sqrt[3])*(b*c - a*d)^(2/3) + 2^(2/3)*b^(1/3)*d^(1/3)*((a + b*x)*(c + d*x))^(1/
3))) - (3*3^(1/4)*Sqrt[2 - Sqrt[3]]*d^(1/3)*((a + b*x)*(c + d*x))^(1/3)*Sqrt[(b*
c + a*d + 2*b*d*x)^2]*((b*c - a*d)^(2/3) + 2^(2/3)*b^(1/3)*d^(1/3)*((a + b*x)*(c
+ d*x))^(1/3))*Sqrt[((b*c - a*d)^(4/3) - 2^(2/3)*b^(1/3)*d^(1/3)*(b*c - a*d)^(2
/3)*((a + b*x)*(c + d*x))^(1/3) + 2*2^(1/3)*b^(2/3)*d^(2/3)*((a + b*x)*(c + d*x)
)^(2/3))/((1 + Sqrt[3])*(b*c - a*d)^(2/3) + 2^(2/3)*b^(1/3)*d^(1/3)*((a + b*x)*(
c + d*x))^(1/3))^2]*EllipticE[ArcSin[((1 - Sqrt[3])*(b*c - a*d)^(2/3) + 2^(2/3)*
b^(1/3)*d^(1/3)*((a + b*x)*(c + d*x))^(1/3))/((1 + Sqrt[3])*(b*c - a*d)^(2/3) +
2^(2/3)*b^(1/3)*d^(1/3)*((a + b*x)*(c + d*x))^(1/3))], -7 - 4*Sqrt[3]])/(2*2^(1/
3)*b^(2/3)*(b*c - a*d)^(1/3)*(a + b*x)^(1/3)*(c + d*x)^(1/3)*(b*c + a*d + 2*b*d*
x)*Sqrt[((b*c - a*d)^(2/3)*((b*c - a*d)^(2/3) + 2^(2/3)*b^(1/3)*d^(1/3)*((a + b*
x)*(c + d*x))^(1/3)))/((1 + Sqrt[3])*(b*c - a*d)^(2/3) + 2^(2/3)*b^(1/3)*d^(1/3)
*((a + b*x)*(c + d*x))^(1/3))^2]*Sqrt[(a*d + b*(c + 2*d*x))^2]) + (2^(1/6)*3^(3/
4)*d^(1/3)*((a + b*x)*(c + d*x))^(1/3)*Sqrt[(b*c + a*d + 2*b*d*x)^2]*((b*c - a*d
)^(2/3) + 2^(2/3)*b^(1/3)*d^(1/3)*((a + b*x)*(c + d*x))^(1/3))*Sqrt[((b*c - a*d)
^(4/3) - 2^(2/3)*b^(1/3)*d^(1/3)*(b*c - a*d)^(2/3)*((a + b*x)*(c + d*x))^(1/3) +
2*2^(1/3)*b^(2/3)*d^(2/3)*((a + b*x)*(c + d*x))^(2/3))/((1 + Sqrt[3])*(b*c - a*
d)^(2/3) + 2^(2/3)*b^(1/3)*d^(1/3)*((a + b*x)*(c + d*x))^(1/3))^2]*EllipticF[Arc
Sin[((1 - Sqrt[3])*(b*c - a*d)^(2/3) + 2^(2/3)*b^(1/3)*d^(1/3)*((a + b*x)*(c + d
*x))^(1/3))/((1 + Sqrt[3])*(b*c - a*d)^(2/3) + 2^(2/3)*b^(1/3)*d^(1/3)*((a + b*x
)*(c + d*x))^(1/3))], -7 - 4*Sqrt[3]])/(b^(2/3)*(b*c - a*d)^(1/3)*(a + b*x)^(1/3
)*(c + d*x)^(1/3)*(b*c + a*d + 2*b*d*x)*Sqrt[((b*c - a*d)^(2/3)*((b*c - a*d)^(2/
3) + 2^(2/3)*b^(1/3)*d^(1/3)*((a + b*x)*(c + d*x))^(1/3)))/((1 + Sqrt[3])*(b*c -
a*d)^(2/3) + 2^(2/3)*b^(1/3)*d^(1/3)*((a + b*x)*(c + d*x))^(1/3))^2]*Sqrt[(a*d
+ b*(c + 2*d*x))^2])

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Rubi in Sympy [A]  time = 163.213, size = 1399, normalized size = 1.08 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(b*x+a)**(4/3)/(d*x+c)**(1/3),x)

[Out]

3*(c + d*x)**(2/3)/((a + b*x)**(1/3)*(a*d - b*c)) + 3*2**(2/3)*3**(1/4)*d**(1/3)
*sqrt((2*2**(1/3)*b**(2/3)*d**(2/3)*(a*c + b*d*x**2 + x*(a*d + b*c))**(2/3) - 2*
*(2/3)*b**(1/3)*d**(1/3)*(a*d - b*c)**(2/3)*(a*c + b*d*x**2 + x*(a*d + b*c))**(1
/3) + (a*d - b*c)**(4/3))/(2**(2/3)*b**(1/3)*d**(1/3)*(a*c + b*d*x**2 + x*(a*d +
b*c))**(1/3) + (1 + sqrt(3))*(a*d - b*c)**(2/3))**2)*sqrt(-sqrt(3) + 2)*(2**(2/
3)*b**(1/3)*d**(1/3)*(a*c + b*d*x**2 + x*(a*d + b*c))**(1/3) + (a*d - b*c)**(2/3
))*(a*c + b*d*x**2 + x*(a*d + b*c))**(1/3)*sqrt((a*d + b*c + 2*b*d*x)**2)*ellipt
ic_e(asin((2**(2/3)*b**(1/3)*d**(1/3)*(a*c + b*d*x**2 + x*(a*d + b*c))**(1/3) -
(-1 + sqrt(3))*(a*d - b*c)**(2/3))/(2**(2/3)*b**(1/3)*d**(1/3)*(a*c + b*d*x**2 +
x*(a*d + b*c))**(1/3) + (1 + sqrt(3))*(a*d - b*c)**(2/3))), -7 - 4*sqrt(3))/(4*
b**(2/3)*sqrt((a*d - b*c)**(2/3)*(2**(2/3)*b**(1/3)*d**(1/3)*(a*c + b*d*x**2 + x
*(a*d + b*c))**(1/3) + (a*d - b*c)**(2/3))/(2**(2/3)*b**(1/3)*d**(1/3)*(a*c + b*
d*x**2 + x*(a*d + b*c))**(1/3) + (1 + sqrt(3))*(a*d - b*c)**(2/3))**2)*(a + b*x)
**(1/3)*(c + d*x)**(1/3)*(a*d - b*c)**(1/3)*sqrt(b*d*(4*a*c + 4*b*d*x**2 + x*(4*
a*d + 4*b*c)) + (a*d - b*c)**2)*(a*d + b*c + 2*b*d*x)) - 2**(1/6)*3**(3/4)*d**(1
/3)*sqrt((2*2**(1/3)*b**(2/3)*d**(2/3)*(a*c + b*d*x**2 + x*(a*d + b*c))**(2/3) -
2**(2/3)*b**(1/3)*d**(1/3)*(a*d - b*c)**(2/3)*(a*c + b*d*x**2 + x*(a*d + b*c))*
*(1/3) + (a*d - b*c)**(4/3))/(2**(2/3)*b**(1/3)*d**(1/3)*(a*c + b*d*x**2 + x*(a*
d + b*c))**(1/3) + (1 + sqrt(3))*(a*d - b*c)**(2/3))**2)*(2**(2/3)*b**(1/3)*d**(
1/3)*(a*c + b*d*x**2 + x*(a*d + b*c))**(1/3) + (a*d - b*c)**(2/3))*(a*c + b*d*x*
*2 + x*(a*d + b*c))**(1/3)*sqrt((a*d + b*c + 2*b*d*x)**2)*elliptic_f(asin((2**(2
/3)*b**(1/3)*d**(1/3)*(a*c + b*d*x**2 + x*(a*d + b*c))**(1/3) - (-1 + sqrt(3))*(
a*d - b*c)**(2/3))/(2**(2/3)*b**(1/3)*d**(1/3)*(a*c + b*d*x**2 + x*(a*d + b*c))*
*(1/3) + (1 + sqrt(3))*(a*d - b*c)**(2/3))), -7 - 4*sqrt(3))/(b**(2/3)*sqrt((a*d
- b*c)**(2/3)*(2**(2/3)*b**(1/3)*d**(1/3)*(a*c + b*d*x**2 + x*(a*d + b*c))**(1/
3) + (a*d - b*c)**(2/3))/(2**(2/3)*b**(1/3)*d**(1/3)*(a*c + b*d*x**2 + x*(a*d +
b*c))**(1/3) + (1 + sqrt(3))*(a*d - b*c)**(2/3))**2)*(a + b*x)**(1/3)*(c + d*x)*
*(1/3)*(a*d - b*c)**(1/3)*sqrt(b*d*(4*a*c + 4*b*d*x**2 + x*(4*a*d + 4*b*c)) + (a
*d - b*c)**2)*(a*d + b*c + 2*b*d*x)) - 3*2**(2/3)*d**(1/3)*sqrt(b*d*(4*a*c + 4*b
*d*x**2 + x*(4*a*d + 4*b*c)) + (a*d - b*c)**2)*(a*c + b*d*x**2 + x*(a*d + b*c))*
*(1/3)*sqrt((a*d + b*c + 2*b*d*x)**2)/(2*b**(2/3)*(a + b*x)**(1/3)*(c + d*x)**(1
/3)*(a*d - b*c)*(2**(2/3)*b**(1/3)*d**(1/3)*(a*c + b*d*x**2 + x*(a*d + b*c))**(1
/3) + (1 + sqrt(3))*(a*d - b*c)**(2/3))*(a*d + b*c + 2*b*d*x))

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Mathematica [C]  time = 0.102696, size = 83, normalized size = 0.06 \[ \frac{3 (c+d x)^{2/3} \left (\sqrt [3]{\frac{d (a+b x)}{a d-b c}} \, _2F_1\left (\frac{1}{3},\frac{2}{3};\frac{5}{3};\frac{b (c+d x)}{b c-a d}\right )-2\right )}{2 \sqrt [3]{a+b x} (b c-a d)} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b*x)^(4/3)*(c + d*x)^(1/3)),x]

[Out]

(3*(c + d*x)^(2/3)*(-2 + ((d*(a + b*x))/(-(b*c) + a*d))^(1/3)*Hypergeometric2F1[
1/3, 2/3, 5/3, (b*(c + d*x))/(b*c - a*d)]))/(2*(b*c - a*d)*(a + b*x)^(1/3))

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Maple [F]  time = 0.056, size = 0, normalized size = 0. \[ \int{1 \left ( bx+a \right ) ^{-{\frac{4}{3}}}{\frac{1}{\sqrt [3]{dx+c}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b*x+a)^(4/3)/(d*x+c)^(1/3),x)

[Out]

int(1/(b*x+a)^(4/3)/(d*x+c)^(1/3),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x + a\right )}^{\frac{4}{3}}{\left (d x + c\right )}^{\frac{1}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^(4/3)*(d*x + c)^(1/3)),x, algorithm="maxima")

[Out]

integrate(1/((b*x + a)^(4/3)*(d*x + c)^(1/3)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (b x + a\right )}^{\frac{4}{3}}{\left (d x + c\right )}^{\frac{1}{3}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^(4/3)*(d*x + c)^(1/3)),x, algorithm="fricas")

[Out]

integral(1/((b*x + a)^(4/3)*(d*x + c)^(1/3)), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (a + b x\right )^{\frac{4}{3}} \sqrt [3]{c + d x}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(b*x+a)**(4/3)/(d*x+c)**(1/3),x)

[Out]

Integral(1/((a + b*x)**(4/3)*(c + d*x)**(1/3)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x + a\right )}^{\frac{4}{3}}{\left (d x + c\right )}^{\frac{1}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^(4/3)*(d*x + c)^(1/3)),x, algorithm="giac")

[Out]

integrate(1/((b*x + a)^(4/3)*(d*x + c)^(1/3)), x)