3.1584 \(\int \sqrt [3]{a+b x} \sqrt [3]{c+d x} \, dx\)

Optimal. Leaf size=617 \[ -\frac{3^{3/4} \sqrt{2+\sqrt{3}} (b c-a d)^2 ((a+b x) (c+d x))^{2/3} \sqrt{(a d+b c+2 b d x)^2} \left (2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}+(b c-a d)^{2/3}\right ) \sqrt{\frac{2 \sqrt [3]{2} b^{2/3} d^{2/3} ((a+b x) (c+d x))^{2/3}-2^{2/3} \sqrt [3]{b} \sqrt [3]{d} (b c-a d)^{2/3} \sqrt [3]{(a+b x) (c+d x)}+(b c-a d)^{4/3}}{\left (2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}+\left (1+\sqrt{3}\right ) (b c-a d)^{2/3}\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) (b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}}{\left (1+\sqrt{3}\right ) (b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}}\right )|-7-4 \sqrt{3}\right )}{5\ 2^{2/3} b^{4/3} d^{4/3} (a+b x)^{2/3} (c+d x)^{2/3} (a d+b c+2 b d x) \sqrt{\frac{(b c-a d)^{2/3} \left (2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}+(b c-a d)^{2/3}\right )}{\left (2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}+\left (1+\sqrt{3}\right ) (b c-a d)^{2/3}\right )^2}} \sqrt{(a d+b (c+2 d x))^2}}+\frac{3 \sqrt [3]{a+b x} \sqrt [3]{c+d x} (b c-a d)}{10 b d}+\frac{3 (a+b x)^{4/3} \sqrt [3]{c+d x}}{5 b} \]

[Out]

(3*(b*c - a*d)*(a + b*x)^(1/3)*(c + d*x)^(1/3))/(10*b*d) + (3*(a + b*x)^(4/3)*(c
+ d*x)^(1/3))/(5*b) - (3^(3/4)*Sqrt[2 + Sqrt[3]]*(b*c - a*d)^2*((a + b*x)*(c +
d*x))^(2/3)*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[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]])/(5*2^(2/3)*b^(4/3)*d^(4/3)*(a + b*x)^(2/3)*(c + d*x)^(2/3)*(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 = 1.84212, antiderivative size = 617, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21 \[ -\frac{3^{3/4} \sqrt{2+\sqrt{3}} (b c-a d)^2 ((a+b x) (c+d x))^{2/3} \sqrt{(a d+b c+2 b d x)^2} \left (2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}+(b c-a d)^{2/3}\right ) \sqrt{\frac{2 \sqrt [3]{2} b^{2/3} d^{2/3} ((a+b x) (c+d x))^{2/3}-2^{2/3} \sqrt [3]{b} \sqrt [3]{d} (b c-a d)^{2/3} \sqrt [3]{(a+b x) (c+d x)}+(b c-a d)^{4/3}}{\left (2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}+\left (1+\sqrt{3}\right ) (b c-a d)^{2/3}\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) (b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}}{\left (1+\sqrt{3}\right ) (b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}}\right )|-7-4 \sqrt{3}\right )}{5\ 2^{2/3} b^{4/3} d^{4/3} (a+b x)^{2/3} (c+d x)^{2/3} (a d+b c+2 b d x) \sqrt{\frac{(b c-a d)^{2/3} \left (2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}+(b c-a d)^{2/3}\right )}{\left (2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}+\left (1+\sqrt{3}\right ) (b c-a d)^{2/3}\right )^2}} \sqrt{(a d+b (c+2 d x))^2}}+\frac{3 \sqrt [3]{a+b x} \sqrt [3]{c+d x} (b c-a d)}{10 b d}+\frac{3 (a+b x)^{4/3} \sqrt [3]{c+d x}}{5 b} \]

Warning: Unable to verify antiderivative.

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

[Out]

(3*(b*c - a*d)*(a + b*x)^(1/3)*(c + d*x)^(1/3))/(10*b*d) + (3*(a + b*x)^(4/3)*(c
+ d*x)^(1/3))/(5*b) - (3^(3/4)*Sqrt[2 + Sqrt[3]]*(b*c - a*d)^2*((a + b*x)*(c +
d*x))^(2/3)*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[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]])/(5*2^(2/3)*b^(4/3)*d^(4/3)*(a + b*x)^(2/3)*(c + d*x)^(2/3)*(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 = 79.4076, size = 646, normalized size = 1.05 \[ \frac{3 \sqrt [3]{a + b x} \left (c + d x\right )^{\frac{4}{3}}}{5 d} + \frac{3 \sqrt [3]{a + b x} \sqrt [3]{c + d x} \left (a d - b c\right )}{10 b d} - \frac{\sqrt [3]{2} \cdot 3^{\frac{3}{4}} \sqrt{\frac{2 \sqrt [3]{2} b^{\frac{2}{3}} d^{\frac{2}{3}} \left (a c + b d x^{2} + x \left (a d + b c\right )\right )^{\frac{2}{3}} - 2^{\frac{2}{3}} \sqrt [3]{b} \sqrt [3]{d} \left (a d - b c\right )^{\frac{2}{3}} \sqrt [3]{a c + b d x^{2} + x \left (a d + b c\right )} + \left (a d - b c\right )^{\frac{4}{3}}}{\left (2^{\frac{2}{3}} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{a c + b d x^{2} + x \left (a d + b c\right )} + \left (1 + \sqrt{3}\right ) \left (a d - b c\right )^{\frac{2}{3}}\right )^{2}}} \sqrt{\sqrt{3} + 2} \left (a d - b c\right )^{2} \left (2^{\frac{2}{3}} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{a c + b d x^{2} + x \left (a d + b c\right )} + \left (a d - b c\right )^{\frac{2}{3}}\right ) \left (a c + b d x^{2} + x \left (a d + b c\right )\right )^{\frac{2}{3}} \sqrt{\left (a d + b c + 2 b d x\right )^{2}} F\left (\operatorname{asin}{\left (\frac{2^{\frac{2}{3}} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{a c + b d x^{2} + x \left (a d + b c\right )} - \left (-1 + \sqrt{3}\right ) \left (a d - b c\right )^{\frac{2}{3}}}{2^{\frac{2}{3}} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{a c + b d x^{2} + x \left (a d + b c\right )} + \left (1 + \sqrt{3}\right ) \left (a d - b c\right )^{\frac{2}{3}}} \right )}\middle | -7 - 4 \sqrt{3}\right )}{10 b^{\frac{4}{3}} d^{\frac{4}{3}} \sqrt{\frac{\left (a d - b c\right )^{\frac{2}{3}} \left (2^{\frac{2}{3}} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{a c + b d x^{2} + x \left (a d + b c\right )} + \left (a d - b c\right )^{\frac{2}{3}}\right )}{\left (2^{\frac{2}{3}} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{a c + b d x^{2} + x \left (a d + b c\right )} + \left (1 + \sqrt{3}\right ) \left (a d - b c\right )^{\frac{2}{3}}\right )^{2}}} \left (a + b x\right )^{\frac{2}{3}} \left (c + d x\right )^{\frac{2}{3}} \sqrt{b d \left (4 a c + 4 b d x^{2} + x \left (4 a d + 4 b c\right )\right ) + \left (a d - b c\right )^{2}} \left (a d + b c + 2 b d x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3*(a + b*x)**(1/3)*(c + d*x)**(4/3)/(5*d) + 3*(a + b*x)**(1/3)*(c + d*x)**(1/3)*
(a*d - b*c)/(10*b*d) - 2**(1/3)*3**(3/4)*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)*(a*d - b*c)**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))**(2/3)*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))/(10*b**(4/3)*d**(4/3)*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)**(2/3)*(c + d*x
)**(2/3)*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))

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Mathematica [C]  time = 0.184541, size = 108, normalized size = 0.18 \[ \frac{3 \sqrt [3]{c+d x} \left (d (a+b x) (a d+b (c+2 d x))-(b c-a d)^2 \left (\frac{d (a+b x)}{a d-b c}\right )^{2/3} \, _2F_1\left (\frac{1}{3},\frac{2}{3};\frac{4}{3};\frac{b (c+d x)}{b c-a d}\right )\right )}{10 b d^2 (a+b x)^{2/3}} \]

Antiderivative was successfully verified.

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

[Out]

(3*(c + d*x)^(1/3)*(d*(a + b*x)*(a*d + b*(c + 2*d*x)) - (b*c - a*d)^2*((d*(a + b
*x))/(-(b*c) + a*d))^(2/3)*Hypergeometric2F1[1/3, 2/3, 4/3, (b*(c + d*x))/(b*c -
a*d)]))/(10*b*d^2*(a + b*x)^(2/3))

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Maple [F]  time = 0.031, size = 0, normalized size = 0. \[ \int \sqrt [3]{bx+a}\sqrt [3]{dx+c}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \sqrt [3]{a + b x} \sqrt [3]{c + d x}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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