3.2991 \(\int \frac{\sqrt [3]{a+b x} (e+f x)^2}{\sqrt [3]{c+d x}} \, dx\)

Optimal. Leaf size=475 \[ \frac{\sqrt [3]{a+b x} (c+d x)^{2/3} \left (5 a^2 d^2 f^2-2 a b d f (9 d e-4 c f)+b^2 \left (14 c^2 f^2-36 c d e f+27 d^2 e^2\right )\right )}{27 b^2 d^3}+\frac{(b c-a d) \log (a+b x) \left (5 a^2 d^2 f^2-2 a b d f (9 d e-4 c f)+b^2 \left (14 c^2 f^2-36 c d e f+27 d^2 e^2\right )\right )}{162 b^{8/3} d^{10/3}}+\frac{(b c-a d) \left (5 a^2 d^2 f^2-2 a b d f (9 d e-4 c f)+b^2 \left (14 c^2 f^2-36 c d e f+27 d^2 e^2\right )\right ) \log \left (\frac{\sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{d} \sqrt [3]{a+b x}}-1\right )}{54 b^{8/3} d^{10/3}}+\frac{(b c-a d) \left (5 a^2 d^2 f^2-2 a b d f (9 d e-4 c f)+b^2 \left (14 c^2 f^2-36 c d e f+27 d^2 e^2\right )\right ) \tan ^{-1}\left (\frac{2 \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt{3} \sqrt [3]{d} \sqrt [3]{a+b x}}+\frac{1}{\sqrt{3}}\right )}{27 \sqrt{3} b^{8/3} d^{10/3}}+\frac{f (a+b x)^{4/3} (c+d x)^{2/3} (-5 a d f-7 b c f+12 b d e)}{18 b^2 d^2}+\frac{f (a+b x)^{4/3} (c+d x)^{2/3} (e+f x)}{3 b d} \]

[Out]

((5*a^2*d^2*f^2 - 2*a*b*d*f*(9*d*e - 4*c*f) + b^2*(27*d^2*e^2 - 36*c*d*e*f + 14*
c^2*f^2))*(a + b*x)^(1/3)*(c + d*x)^(2/3))/(27*b^2*d^3) + (f*(12*b*d*e - 7*b*c*f
- 5*a*d*f)*(a + b*x)^(4/3)*(c + d*x)^(2/3))/(18*b^2*d^2) + (f*(a + b*x)^(4/3)*(
c + d*x)^(2/3)*(e + f*x))/(3*b*d) + ((b*c - a*d)*(5*a^2*d^2*f^2 - 2*a*b*d*f*(9*d
*e - 4*c*f) + b^2*(27*d^2*e^2 - 36*c*d*e*f + 14*c^2*f^2))*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^(10/3)) + ((b*c - a*d)*(5*a^2*d^2*f^2 - 2*a*b*d*f*(9*d*e - 4*c*f) + b^2*(27*d
^2*e^2 - 36*c*d*e*f + 14*c^2*f^2))*Log[a + b*x])/(162*b^(8/3)*d^(10/3)) + ((b*c
- a*d)*(5*a^2*d^2*f^2 - 2*a*b*d*f*(9*d*e - 4*c*f) + b^2*(27*d^2*e^2 - 36*c*d*e*f
+ 14*c^2*f^2))*Log[-1 + (b^(1/3)*(c + d*x)^(1/3))/(d^(1/3)*(a + b*x)^(1/3))])/(
54*b^(8/3)*d^(10/3))

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Rubi [A]  time = 1.0738, antiderivative size = 475, 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 \[ \frac{\sqrt [3]{a+b x} (c+d x)^{2/3} \left (5 a^2 d^2 f^2-2 a b d f (9 d e-4 c f)+b^2 \left (14 c^2 f^2-36 c d e f+27 d^2 e^2\right )\right )}{27 b^2 d^3}+\frac{(b c-a d) \log (a+b x) \left (5 a^2 d^2 f^2-2 a b d f (9 d e-4 c f)+b^2 \left (14 c^2 f^2-36 c d e f+27 d^2 e^2\right )\right )}{162 b^{8/3} d^{10/3}}+\frac{(b c-a d) \left (5 a^2 d^2 f^2-2 a b d f (9 d e-4 c f)+b^2 \left (14 c^2 f^2-36 c d e f+27 d^2 e^2\right )\right ) \log \left (\frac{\sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{d} \sqrt [3]{a+b x}}-1\right )}{54 b^{8/3} d^{10/3}}+\frac{(b c-a d) \left (5 a^2 d^2 f^2-2 a b d f (9 d e-4 c f)+b^2 \left (14 c^2 f^2-36 c d e f+27 d^2 e^2\right )\right ) \tan ^{-1}\left (\frac{2 \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt{3} \sqrt [3]{d} \sqrt [3]{a+b x}}+\frac{1}{\sqrt{3}}\right )}{27 \sqrt{3} b^{8/3} d^{10/3}}+\frac{f (a+b x)^{4/3} (c+d x)^{2/3} (-5 a d f-7 b c f+12 b d e)}{18 b^2 d^2}+\frac{f (a+b x)^{4/3} (c+d x)^{2/3} (e+f x)}{3 b d} \]

Antiderivative was successfully verified.

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

[Out]

((5*a^2*d^2*f^2 - 2*a*b*d*f*(9*d*e - 4*c*f) + b^2*(27*d^2*e^2 - 36*c*d*e*f + 14*
c^2*f^2))*(a + b*x)^(1/3)*(c + d*x)^(2/3))/(27*b^2*d^3) + (f*(12*b*d*e - 7*b*c*f
- 5*a*d*f)*(a + b*x)^(4/3)*(c + d*x)^(2/3))/(18*b^2*d^2) + (f*(a + b*x)^(4/3)*(
c + d*x)^(2/3)*(e + f*x))/(3*b*d) + ((b*c - a*d)*(5*a^2*d^2*f^2 - 2*a*b*d*f*(9*d
*e - 4*c*f) + b^2*(27*d^2*e^2 - 36*c*d*e*f + 14*c^2*f^2))*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^(10/3)) + ((b*c - a*d)*(5*a^2*d^2*f^2 - 2*a*b*d*f*(9*d*e - 4*c*f) + b^2*(27*d
^2*e^2 - 36*c*d*e*f + 14*c^2*f^2))*Log[a + b*x])/(162*b^(8/3)*d^(10/3)) + ((b*c
- a*d)*(5*a^2*d^2*f^2 - 2*a*b*d*f*(9*d*e - 4*c*f) + b^2*(27*d^2*e^2 - 36*c*d*e*f
+ 14*c^2*f^2))*Log[-1 + (b^(1/3)*(c + d*x)^(1/3))/(d^(1/3)*(a + b*x)^(1/3))])/(
54*b^(8/3)*d^(10/3))

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Rubi in Sympy [A]  time = 62.6763, size = 502, normalized size = 1.06 \[ \frac{f \left (a + b x\right )^{\frac{4}{3}} \left (c + d x\right )^{\frac{2}{3}} \left (e + f x\right )}{3 b d} - \frac{f \left (a + b x\right )^{\frac{4}{3}} \left (c + d x\right )^{\frac{2}{3}} \left (5 a d f + 7 b c f - 12 b d e\right )}{18 b^{2} d^{2}} - \frac{\sqrt [3]{a + b x} \left (c + d x\right )^{\frac{2}{3}} \left (3 b d \left (- 9 b d e^{2} + f \left (3 a c f + 2 e \left (a d + 2 b c\right )\right )\right ) - f \left (a d + 2 b c\right ) \left (5 a d f + 7 b c f - 12 b d e\right )\right )}{27 b^{2} d^{3}} + \frac{\left (a d - b c\right ) \left (3 b d \left (- 9 b d e^{2} + f \left (3 a c f + 2 e \left (a d + 2 b c\right )\right )\right ) - f \left (a d + 2 b c\right ) \left (5 a d f + 7 b c f - 12 b d e\right )\right ) \log{\left (a + b x \right )}}{162 b^{\frac{8}{3}} d^{\frac{10}{3}}} + \frac{\left (a d - b c\right ) \left (3 b d \left (- 9 b d e^{2} + f \left (3 a c f + 2 e \left (a d + 2 b c\right )\right )\right ) - f \left (a d + 2 b c\right ) \left (5 a d f + 7 b c f - 12 b d e\right )\right ) \log{\left (\frac{\sqrt [3]{b} \sqrt [3]{c + d x}}{\sqrt [3]{d} \sqrt [3]{a + b x}} - 1 \right )}}{54 b^{\frac{8}{3}} d^{\frac{10}{3}}} + \frac{\sqrt{3} \left (a d - b c\right ) \left (3 b d \left (- 9 b d e^{2} + f \left (3 a c f + 2 e \left (a d + 2 b c\right )\right )\right ) - f \left (a d + 2 b c\right ) \left (5 a d f + 7 b c f - 12 b d e\right )\right ) \operatorname{atan}{\left (\frac{2 \sqrt{3} \sqrt [3]{b} \sqrt [3]{c + d x}}{3 \sqrt [3]{d} \sqrt [3]{a + b x}} + \frac{\sqrt{3}}{3} \right )}}{81 b^{\frac{8}{3}} d^{\frac{10}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

f*(a + b*x)**(4/3)*(c + d*x)**(2/3)*(e + f*x)/(3*b*d) - f*(a + b*x)**(4/3)*(c +
d*x)**(2/3)*(5*a*d*f + 7*b*c*f - 12*b*d*e)/(18*b**2*d**2) - (a + b*x)**(1/3)*(c
+ d*x)**(2/3)*(3*b*d*(-9*b*d*e**2 + f*(3*a*c*f + 2*e*(a*d + 2*b*c))) - f*(a*d +
2*b*c)*(5*a*d*f + 7*b*c*f - 12*b*d*e))/(27*b**2*d**3) + (a*d - b*c)*(3*b*d*(-9*b
*d*e**2 + f*(3*a*c*f + 2*e*(a*d + 2*b*c))) - f*(a*d + 2*b*c)*(5*a*d*f + 7*b*c*f
- 12*b*d*e))*log(a + b*x)/(162*b**(8/3)*d**(10/3)) + (a*d - b*c)*(3*b*d*(-9*b*d*
e**2 + f*(3*a*c*f + 2*e*(a*d + 2*b*c))) - f*(a*d + 2*b*c)*(5*a*d*f + 7*b*c*f - 1
2*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**(10/3)) + sqrt(3)*(a*d - b*c)*(3*b*d*(-9*b*d*e**2 + f*(3*a*c*f + 2*e*(a
*d + 2*b*c))) - f*(a*d + 2*b*c)*(5*a*d*f + 7*b*c*f - 12*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**(10/3))

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Mathematica [C]  time = 0.451907, size = 229, normalized size = 0.48 \[ \frac{(c+d x)^{2/3} \left (d (a+b x) \left (-5 a^2 d^2 f^2+a b d f (3 d (6 e+f x)-5 c f)+b^2 \left (28 c^2 f^2-3 c d f (24 e+7 f x)+18 d^2 \left (3 e^2+3 e f x+f^2 x^2\right )\right )\right )-(b c-a d) \left (\frac{d (a+b x)}{a d-b c}\right )^{2/3} \left (5 a^2 d^2 f^2+2 a b d f (4 c f-9 d e)+b^2 \left (14 c^2 f^2-36 c d e f+27 d^2 e^2\right )\right ) \, _2F_1\left (\frac{2}{3},\frac{2}{3};\frac{5}{3};\frac{b (c+d x)}{b c-a d}\right )\right )}{54 b^2 d^4 (a+b x)^{2/3}} \]

Antiderivative was successfully verified.

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

[Out]

((c + d*x)^(2/3)*(d*(a + b*x)*(-5*a^2*d^2*f^2 + a*b*d*f*(-5*c*f + 3*d*(6*e + f*x
)) + b^2*(28*c^2*f^2 - 3*c*d*f*(24*e + 7*f*x) + 18*d^2*(3*e^2 + 3*e*f*x + f^2*x^
2))) - (b*c - a*d)*(5*a^2*d^2*f^2 + 2*a*b*d*f*(-9*d*e + 4*c*f) + b^2*(27*d^2*e^2
- 36*c*d*e*f + 14*c^2*f^2))*((d*(a + b*x))/(-(b*c) + a*d))^(2/3)*Hypergeometric
2F1[2/3, 2/3, 5/3, (b*(c + d*x))/(b*c - a*d)]))/(54*b^2*d^4*(a + b*x)^(2/3))

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Maple [F]  time = 0.048, size = 0, normalized size = 0. \[ \int{ \left ( fx+e \right ) ^{2}\sqrt [3]{bx+a}{\frac{1}{\sqrt [3]{dx+c}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.287787, size = 838, normalized size = 1.76 \[ \frac{\sqrt{3}{\left (3 \, \sqrt{3}{\left (18 \, b^{2} d^{2} f^{2} x^{2} + 54 \, b^{2} d^{2} e^{2} - 18 \,{\left (4 \, b^{2} c d - a b d^{2}\right )} e f +{\left (28 \, b^{2} c^{2} - 5 \, a b c d - 5 \, a^{2} d^{2}\right )} f^{2} + 3 \,{\left (18 \, b^{2} d^{2} e f -{\left (7 \, b^{2} c d - a b d^{2}\right )} f^{2}\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 (27 \,{\left (b^{3} c d^{2} - a b^{2} d^{3}\right )} e^{2} - 18 \,{\left (2 \, b^{3} c^{2} d - a b^{2} c d^{2} - a^{2} b d^{3}\right )} e f +{\left (14 \, b^{3} c^{3} - 6 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} - 5 \, a^{3} d^{3}\right )} f^{2}\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 (27 \,{\left (b^{3} c d^{2} - a b^{2} d^{3}\right )} e^{2} - 18 \,{\left (2 \, b^{3} c^{2} d - a b^{2} c d^{2} - a^{2} b d^{3}\right )} e f +{\left (14 \, b^{3} c^{3} - 6 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} - 5 \, a^{3} d^{3}\right )} f^{2}\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 (27 \,{\left (b^{3} c d^{2} - a b^{2} d^{3}\right )} e^{2} - 18 \,{\left (2 \, b^{3} c^{2} d - a b^{2} c d^{2} - a^{2} b d^{3}\right )} e f +{\left (14 \, b^{3} c^{3} - 6 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} - 5 \, a^{3} d^{3}\right )} f^{2}\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^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/486*sqrt(3)*(3*sqrt(3)*(18*b^2*d^2*f^2*x^2 + 54*b^2*d^2*e^2 - 18*(4*b^2*c*d -
a*b*d^2)*e*f + (28*b^2*c^2 - 5*a*b*c*d - 5*a^2*d^2)*f^2 + 3*(18*b^2*d^2*e*f - (7
*b^2*c*d - a*b*d^2)*f^2)*x)*(-b^2*d)^(1/3)*(b*x + a)^(1/3)*(d*x + c)^(2/3) + sqr
t(3)*(27*(b^3*c*d^2 - a*b^2*d^3)*e^2 - 18*(2*b^3*c^2*d - a*b^2*c*d^2 - a^2*b*d^3
)*e*f + (14*b^3*c^3 - 6*a*b^2*c^2*d - 3*a^2*b*c*d^2 - 5*a^3*d^3)*f^2)*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)*(27*(b^3*c*d^2 - a*b^2*d^
3)*e^2 - 18*(2*b^3*c^2*d - a*b^2*c*d^2 - a^2*b*d^3)*e*f + (14*b^3*c^3 - 6*a*b^2*
c^2*d - 3*a^2*b*c*d^2 - 5*a^3*d^3)*f^2)*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*(27*(b^3*c*d^2 - a*b^2*d^3)*e^2 - 18*(
2*b^3*c^2*d - a*b^2*c*d^2 - a^2*b*d^3)*e*f + (14*b^3*c^3 - 6*a*b^2*c^2*d - 3*a^2
*b*c*d^2 - 5*a^3*d^3)*f^2)*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^3
)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: NotImplementedError