3.2985 \(\int \sqrt [3]{a+b x} (c+d x)^{2/3} (e+f x) \, dx\)

Optimal. Leaf size=331 \[ \frac{(b c-a d)^2 \log (a+b x) (-5 a d f-4 b c f+9 b d e)}{162 b^{8/3} d^{7/3}}+\frac{(b c-a d)^2 (-5 a d f-4 b c f+9 b d e) \log \left (\frac{\sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{d} \sqrt [3]{a+b x}}-1\right )}{54 b^{8/3} d^{7/3}}+\frac{(b c-a d)^2 (-5 a d f-4 b c f+9 b d e) \tan ^{-1}\left (\frac{2 \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt{3} \sqrt [3]{d} \sqrt [3]{a+b x}}+\frac{1}{\sqrt{3}}\right )}{27 \sqrt{3} b^{8/3} d^{7/3}}+\frac{\sqrt [3]{a+b x} (c+d x)^{2/3} (b c-a d) (-5 a d f-4 b c f+9 b d e)}{27 b^2 d^2}+\frac{(a+b x)^{4/3} (c+d x)^{2/3} (-5 a d f-4 b c f+9 b d e)}{18 b^2 d}+\frac{f (a+b x)^{4/3} (c+d x)^{5/3}}{3 b d} \]

[Out]

((b*c - a*d)*(9*b*d*e - 4*b*c*f - 5*a*d*f)*(a + b*x)^(1/3)*(c + d*x)^(2/3))/(27*
b^2*d^2) + ((9*b*d*e - 4*b*c*f - 5*a*d*f)*(a + b*x)^(4/3)*(c + d*x)^(2/3))/(18*b
^2*d) + (f*(a + b*x)^(4/3)*(c + d*x)^(5/3))/(3*b*d) + ((b*c - a*d)^2*(9*b*d*e -
4*b*c*f - 5*a*d*f)*ArcTan[1/Sqrt[3] + (2*b^(1/3)*(c + d*x)^(1/3))/(Sqrt[3]*d^(1/
3)*(a + b*x)^(1/3))])/(27*Sqrt[3]*b^(8/3)*d^(7/3)) + ((b*c - a*d)^2*(9*b*d*e - 4
*b*c*f - 5*a*d*f)*Log[a + b*x])/(162*b^(8/3)*d^(7/3)) + ((b*c - a*d)^2*(9*b*d*e
- 4*b*c*f - 5*a*d*f)*Log[-1 + (b^(1/3)*(c + d*x)^(1/3))/(d^(1/3)*(a + b*x)^(1/3)
)])/(54*b^(8/3)*d^(7/3))

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Rubi [A]  time = 0.595837, antiderivative size = 331, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ \frac{(b c-a d)^2 \log (a+b x) (-5 a d f-4 b c f+9 b d e)}{162 b^{8/3} d^{7/3}}+\frac{(b c-a d)^2 (-5 a d f-4 b c f+9 b d e) \log \left (\frac{\sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{d} \sqrt [3]{a+b x}}-1\right )}{54 b^{8/3} d^{7/3}}+\frac{(b c-a d)^2 (-5 a d f-4 b c f+9 b d e) \tan ^{-1}\left (\frac{2 \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt{3} \sqrt [3]{d} \sqrt [3]{a+b x}}+\frac{1}{\sqrt{3}}\right )}{27 \sqrt{3} b^{8/3} d^{7/3}}+\frac{\sqrt [3]{a+b x} (c+d x)^{2/3} (b c-a d) (-5 a d f-4 b c f+9 b d e)}{27 b^2 d^2}+\frac{(a+b x)^{4/3} (c+d x)^{2/3} (-5 a d f-4 b c f+9 b d e)}{18 b^2 d}+\frac{f (a+b x)^{4/3} (c+d x)^{5/3}}{3 b d} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^(1/3)*(c + d*x)^(2/3)*(e + f*x),x]

[Out]

((b*c - a*d)*(9*b*d*e - 4*b*c*f - 5*a*d*f)*(a + b*x)^(1/3)*(c + d*x)^(2/3))/(27*
b^2*d^2) + ((9*b*d*e - 4*b*c*f - 5*a*d*f)*(a + b*x)^(4/3)*(c + d*x)^(2/3))/(18*b
^2*d) + (f*(a + b*x)^(4/3)*(c + d*x)^(5/3))/(3*b*d) + ((b*c - a*d)^2*(9*b*d*e -
4*b*c*f - 5*a*d*f)*ArcTan[1/Sqrt[3] + (2*b^(1/3)*(c + d*x)^(1/3))/(Sqrt[3]*d^(1/
3)*(a + b*x)^(1/3))])/(27*Sqrt[3]*b^(8/3)*d^(7/3)) + ((b*c - a*d)^2*(9*b*d*e - 4
*b*c*f - 5*a*d*f)*Log[a + b*x])/(162*b^(8/3)*d^(7/3)) + ((b*c - a*d)^2*(9*b*d*e
- 4*b*c*f - 5*a*d*f)*Log[-1 + (b^(1/3)*(c + d*x)^(1/3))/(d^(1/3)*(a + b*x)^(1/3)
)])/(54*b^(8/3)*d^(7/3))

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Rubi in Sympy [A]  time = 44.7123, size = 328, normalized size = 0.99 \[ \frac{f \left (a + b x\right )^{\frac{4}{3}} \left (c + d x\right )^{\frac{5}{3}}}{3 b d} - \frac{\sqrt [3]{a + b x} \left (c + d x\right )^{\frac{5}{3}} \left (5 a d f + 4 b c f - 9 b d e\right )}{18 b d^{2}} - \frac{\sqrt [3]{a + b x} \left (c + d x\right )^{\frac{2}{3}} \left (a d - b c\right ) \left (5 a d f + 4 b c f - 9 b d e\right )}{54 b^{2} d^{2}} - \frac{\left (a d - b c\right )^{2} \left (5 a d f + 4 b c f - 9 b d e\right ) \log{\left (a + b x \right )}}{162 b^{\frac{8}{3}} d^{\frac{7}{3}}} - \frac{\left (a d - b c\right )^{2} \left (5 a d f + 4 b c f - 9 b d e\right ) \log{\left (\frac{\sqrt [3]{b} \sqrt [3]{c + d x}}{\sqrt [3]{d} \sqrt [3]{a + b x}} - 1 \right )}}{54 b^{\frac{8}{3}} d^{\frac{7}{3}}} - \frac{\sqrt{3} \left (a d - b c\right )^{2} \left (5 a d f + 4 b c f - 9 b d e\right ) \operatorname{atan}{\left (\frac{2 \sqrt{3} \sqrt [3]{b} \sqrt [3]{c + d x}}{3 \sqrt [3]{d} \sqrt [3]{a + b x}} + \frac{\sqrt{3}}{3} \right )}}{81 b^{\frac{8}{3}} d^{\frac{7}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**(1/3)*(d*x+c)**(2/3)*(f*x+e),x)

[Out]

f*(a + b*x)**(4/3)*(c + d*x)**(5/3)/(3*b*d) - (a + b*x)**(1/3)*(c + d*x)**(5/3)*
(5*a*d*f + 4*b*c*f - 9*b*d*e)/(18*b*d**2) - (a + b*x)**(1/3)*(c + d*x)**(2/3)*(a
*d - b*c)*(5*a*d*f + 4*b*c*f - 9*b*d*e)/(54*b**2*d**2) - (a*d - b*c)**2*(5*a*d*f
+ 4*b*c*f - 9*b*d*e)*log(a + b*x)/(162*b**(8/3)*d**(7/3)) - (a*d - b*c)**2*(5*a
*d*f + 4*b*c*f - 9*b*d*e)*log(b**(1/3)*(c + d*x)**(1/3)/(d**(1/3)*(a + b*x)**(1/
3)) - 1)/(54*b**(8/3)*d**(7/3)) - sqrt(3)*(a*d - b*c)**2*(5*a*d*f + 4*b*c*f - 9*
b*d*e)*atan(2*sqrt(3)*b**(1/3)*(c + d*x)**(1/3)/(3*d**(1/3)*(a + b*x)**(1/3)) +
sqrt(3)/3)/(81*b**(8/3)*d**(7/3))

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Mathematica [C]  time = 0.284676, size = 175, normalized size = 0.53 \[ \frac{(c+d x)^{2/3} \left (d (a+b x) \left (-5 a^2 d^2 f+a b d (4 c f+9 d e+3 d f x)+b^2 \left (-8 c^2 f+6 c d (3 e+f x)+9 d^2 x (3 e+2 f x)\right )\right )+(b c-a d)^2 \left (\frac{d (a+b x)}{a d-b c}\right )^{2/3} (5 a d f+4 b c f-9 b d e) \, _2F_1\left (\frac{2}{3},\frac{2}{3};\frac{5}{3};\frac{b (c+d x)}{b c-a d}\right )\right )}{54 b^2 d^3 (a+b x)^{2/3}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^(1/3)*(c + d*x)^(2/3)*(e + f*x),x]

[Out]

((c + d*x)^(2/3)*(d*(a + b*x)*(-5*a^2*d^2*f + a*b*d*(9*d*e + 4*c*f + 3*d*f*x) +
b^2*(-8*c^2*f + 6*c*d*(3*e + f*x) + 9*d^2*x*(3*e + 2*f*x))) + (b*c - a*d)^2*(-9*
b*d*e + 4*b*c*f + 5*a*d*f)*((d*(a + b*x))/(-(b*c) + a*d))^(2/3)*Hypergeometric2F
1[2/3, 2/3, 5/3, (b*(c + d*x))/(b*c - a*d)]))/(54*b^2*d^3*(a + b*x)^(2/3))

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Maple [F]  time = 0.04, size = 0, normalized size = 0. \[ \int \sqrt [3]{bx+a} \left ( dx+c \right ) ^{{\frac{2}{3}}} \left ( fx+e \right ) \, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^(1/3)*(d*x+c)^(2/3)*(f*x+e),x)

[Out]

int((b*x+a)^(1/3)*(d*x+c)^(2/3)*(f*x+e),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (b x + a\right )}^{\frac{1}{3}}{\left (d x + c\right )}^{\frac{2}{3}}{\left (f x + e\right )}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(1/3)*(d*x + c)^(2/3)*(f*x + e),x, algorithm="maxima")

[Out]

integrate((b*x + a)^(1/3)*(d*x + c)^(2/3)*(f*x + e), x)

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Fricas [A]  time = 0.2433, size = 701, normalized size = 2.12 \[ \frac{\sqrt{3}{\left (3 \, \sqrt{3}{\left (18 \, b^{2} d^{2} f x^{2} + 9 \,{\left (2 \, b^{2} c d + a b d^{2}\right )} e -{\left (8 \, b^{2} c^{2} - 4 \, a b c d + 5 \, a^{2} d^{2}\right )} f + 3 \,{\left (9 \, b^{2} d^{2} e +{\left (2 \, b^{2} c d + a b d^{2}\right )} f\right )} x\right )} \left (-b^{2} d\right )^{\frac{1}{3}}{\left (b x + a\right )}^{\frac{1}{3}}{\left (d x + c\right )}^{\frac{2}{3}} + \sqrt{3}{\left (9 \,{\left (b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} e -{\left (4 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - 6 \, a^{2} b c d^{2} + 5 \, a^{3} d^{3}\right )} f\right )} \log \left (\frac{b^{2} d x + b^{2} c - \left (-b^{2} d\right )^{\frac{1}{3}}{\left (b x + a\right )}^{\frac{1}{3}}{\left (d x + c\right )}^{\frac{2}{3}} b + \left (-b^{2} d\right )^{\frac{2}{3}}{\left (b x + a\right )}^{\frac{2}{3}}{\left (d x + c\right )}^{\frac{1}{3}}}{d x + c}\right ) - 2 \, \sqrt{3}{\left (9 \,{\left (b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} e -{\left (4 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - 6 \, a^{2} b c d^{2} + 5 \, a^{3} d^{3}\right )} f\right )} \log \left (\frac{b d x + b c + \left (-b^{2} d\right )^{\frac{1}{3}}{\left (b x + a\right )}^{\frac{1}{3}}{\left (d x + c\right )}^{\frac{2}{3}}}{d x + c}\right ) - 6 \,{\left (9 \,{\left (b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} e -{\left (4 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - 6 \, a^{2} b c d^{2} + 5 \, a^{3} d^{3}\right )} f\right )} \arctan \left (\frac{2 \, \sqrt{3} \left (-b^{2} d\right )^{\frac{1}{3}}{\left (b x + a\right )}^{\frac{1}{3}}{\left (d x + c\right )}^{\frac{2}{3}} - \sqrt{3}{\left (b d x + b c\right )}}{3 \,{\left (b d x + b c\right )}}\right )\right )}}{486 \, \left (-b^{2} d\right )^{\frac{1}{3}} b^{2} d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(1/3)*(d*x + c)^(2/3)*(f*x + e),x, algorithm="fricas")

[Out]

1/486*sqrt(3)*(3*sqrt(3)*(18*b^2*d^2*f*x^2 + 9*(2*b^2*c*d + a*b*d^2)*e - (8*b^2*
c^2 - 4*a*b*c*d + 5*a^2*d^2)*f + 3*(9*b^2*d^2*e + (2*b^2*c*d + a*b*d^2)*f)*x)*(-
b^2*d)^(1/3)*(b*x + a)^(1/3)*(d*x + c)^(2/3) + sqrt(3)*(9*(b^3*c^2*d - 2*a*b^2*c
*d^2 + a^2*b*d^3)*e - (4*b^3*c^3 - 3*a*b^2*c^2*d - 6*a^2*b*c*d^2 + 5*a^3*d^3)*f)
*log((b^2*d*x + b^2*c - (-b^2*d)^(1/3)*(b*x + a)^(1/3)*(d*x + c)^(2/3)*b + (-b^2
*d)^(2/3)*(b*x + a)^(2/3)*(d*x + c)^(1/3))/(d*x + c)) - 2*sqrt(3)*(9*(b^3*c^2*d
- 2*a*b^2*c*d^2 + a^2*b*d^3)*e - (4*b^3*c^3 - 3*a*b^2*c^2*d - 6*a^2*b*c*d^2 + 5*
a^3*d^3)*f)*log((b*d*x + b*c + (-b^2*d)^(1/3)*(b*x + a)^(1/3)*(d*x + c)^(2/3))/(
d*x + c)) - 6*(9*(b^3*c^2*d - 2*a*b^2*c*d^2 + a^2*b*d^3)*e - (4*b^3*c^3 - 3*a*b^
2*c^2*d - 6*a^2*b*c*d^2 + 5*a^3*d^3)*f)*arctan(1/3*(2*sqrt(3)*(-b^2*d)^(1/3)*(b*
x + a)^(1/3)*(d*x + c)^(2/3) - sqrt(3)*(b*d*x + b*c))/(b*d*x + b*c)))/((-b^2*d)^
(1/3)*b^2*d^2)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \sqrt [3]{a + b x} \left (c + d x\right )^{\frac{2}{3}} \left (e + f x\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**(1/3)*(d*x+c)**(2/3)*(f*x+e),x)

[Out]

Integral((a + b*x)**(1/3)*(c + d*x)**(2/3)*(e + f*x), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (b x + a\right )}^{\frac{1}{3}}{\left (d x + c\right )}^{\frac{2}{3}}{\left (f x + e\right )}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(1/3)*(d*x + c)^(2/3)*(f*x + e),x, algorithm="giac")

[Out]

integrate((b*x + a)^(1/3)*(d*x + c)^(2/3)*(f*x + e), x)