3.2998 \(\int \frac{(e+f x)^3}{\sqrt [3]{a+b x} (c+d x)^{2/3}} \, dx\)

Optimal. Leaf size=587 \[ \frac{f (a+b x)^{2/3} \sqrt [3]{c+d x} \left (28 a^2 d^2 f^2+3 b d f x (-7 a d f-8 b c f+15 b d e)-a b d f (108 d e-31 c f)+b^2 \left (40 c^2 f^2-135 c d e f+144 d^2 e^2\right )\right )}{54 b^3 d^3}+\frac{\log (c+d x) \left (14 a^3 d^3 f^3-6 a^2 b d^2 f^2 (9 d e-2 c f)+3 a b^2 d f \left (5 c^2 f^2-18 c d e f+27 d^2 e^2\right )+b^3 \left (-\left (-40 c^3 f^3+135 c^2 d e f^2-162 c d^2 e^2 f+81 d^3 e^3\right )\right )\right )}{162 b^{10/3} d^{11/3}}+\frac{\left (14 a^3 d^3 f^3-6 a^2 b d^2 f^2 (9 d e-2 c f)+3 a b^2 d f \left (5 c^2 f^2-18 c d e f+27 d^2 e^2\right )+b^3 \left (-\left (-40 c^3 f^3+135 c^2 d e f^2-162 c d^2 e^2 f+81 d^3 e^3\right )\right )\right ) \log \left (\frac{\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{b} \sqrt [3]{c+d x}}-1\right )}{54 b^{10/3} d^{11/3}}+\frac{\left (14 a^3 d^3 f^3-6 a^2 b d^2 f^2 (9 d e-2 c f)+3 a b^2 d f \left (5 c^2 f^2-18 c d e f+27 d^2 e^2\right )+b^3 \left (-\left (-40 c^3 f^3+135 c^2 d e f^2-162 c d^2 e^2 f+81 d^3 e^3\right )\right )\right ) \tan ^{-1}\left (\frac{2 \sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt{3} \sqrt [3]{b} \sqrt [3]{c+d x}}+\frac{1}{\sqrt{3}}\right )}{27 \sqrt{3} b^{10/3} d^{11/3}}+\frac{f (a+b x)^{2/3} \sqrt [3]{c+d x} (e+f x)^2}{3 b d} \]

[Out]

(f*(a + b*x)^(2/3)*(c + d*x)^(1/3)*(e + f*x)^2)/(3*b*d) + (f*(a + b*x)^(2/3)*(c
+ d*x)^(1/3)*(28*a^2*d^2*f^2 - a*b*d*f*(108*d*e - 31*c*f) + b^2*(144*d^2*e^2 - 1
35*c*d*e*f + 40*c^2*f^2) + 3*b*d*f*(15*b*d*e - 8*b*c*f - 7*a*d*f)*x))/(54*b^3*d^
3) + ((14*a^3*d^3*f^3 - 6*a^2*b*d^2*f^2*(9*d*e - 2*c*f) + 3*a*b^2*d*f*(27*d^2*e^
2 - 18*c*d*e*f + 5*c^2*f^2) - b^3*(81*d^3*e^3 - 162*c*d^2*e^2*f + 135*c^2*d*e*f^
2 - 40*c^3*f^3))*ArcTan[1/Sqrt[3] + (2*d^(1/3)*(a + b*x)^(1/3))/(Sqrt[3]*b^(1/3)
*(c + d*x)^(1/3))])/(27*Sqrt[3]*b^(10/3)*d^(11/3)) + ((14*a^3*d^3*f^3 - 6*a^2*b*
d^2*f^2*(9*d*e - 2*c*f) + 3*a*b^2*d*f*(27*d^2*e^2 - 18*c*d*e*f + 5*c^2*f^2) - b^
3*(81*d^3*e^3 - 162*c*d^2*e^2*f + 135*c^2*d*e*f^2 - 40*c^3*f^3))*Log[c + d*x])/(
162*b^(10/3)*d^(11/3)) + ((14*a^3*d^3*f^3 - 6*a^2*b*d^2*f^2*(9*d*e - 2*c*f) + 3*
a*b^2*d*f*(27*d^2*e^2 - 18*c*d*e*f + 5*c^2*f^2) - b^3*(81*d^3*e^3 - 162*c*d^2*e^
2*f + 135*c^2*d*e*f^2 - 40*c^3*f^3))*Log[-1 + (d^(1/3)*(a + b*x)^(1/3))/(b^(1/3)
*(c + d*x)^(1/3))])/(54*b^(10/3)*d^(11/3))

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Rubi [A]  time = 1.17511, antiderivative size = 587, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115 \[ \frac{f (a+b x)^{2/3} \sqrt [3]{c+d x} \left (28 a^2 d^2 f^2+3 b d f x (-7 a d f-8 b c f+15 b d e)-a b d f (108 d e-31 c f)+b^2 \left (40 c^2 f^2-135 c d e f+144 d^2 e^2\right )\right )}{54 b^3 d^3}+\frac{\log (c+d x) \left (14 a^3 d^3 f^3-6 a^2 b d^2 f^2 (9 d e-2 c f)+3 a b^2 d f \left (5 c^2 f^2-18 c d e f+27 d^2 e^2\right )+b^3 \left (-\left (-40 c^3 f^3+135 c^2 d e f^2-162 c d^2 e^2 f+81 d^3 e^3\right )\right )\right )}{162 b^{10/3} d^{11/3}}+\frac{\left (14 a^3 d^3 f^3-6 a^2 b d^2 f^2 (9 d e-2 c f)+3 a b^2 d f \left (5 c^2 f^2-18 c d e f+27 d^2 e^2\right )+b^3 \left (-\left (-40 c^3 f^3+135 c^2 d e f^2-162 c d^2 e^2 f+81 d^3 e^3\right )\right )\right ) \log \left (\frac{\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{b} \sqrt [3]{c+d x}}-1\right )}{54 b^{10/3} d^{11/3}}+\frac{\left (14 a^3 d^3 f^3-6 a^2 b d^2 f^2 (9 d e-2 c f)+3 a b^2 d f \left (5 c^2 f^2-18 c d e f+27 d^2 e^2\right )+b^3 \left (-\left (-40 c^3 f^3+135 c^2 d e f^2-162 c d^2 e^2 f+81 d^3 e^3\right )\right )\right ) \tan ^{-1}\left (\frac{2 \sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt{3} \sqrt [3]{b} \sqrt [3]{c+d x}}+\frac{1}{\sqrt{3}}\right )}{27 \sqrt{3} b^{10/3} d^{11/3}}+\frac{f (a+b x)^{2/3} \sqrt [3]{c+d x} (e+f x)^2}{3 b d} \]

Antiderivative was successfully verified.

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

[Out]

(f*(a + b*x)^(2/3)*(c + d*x)^(1/3)*(e + f*x)^2)/(3*b*d) + (f*(a + b*x)^(2/3)*(c
+ d*x)^(1/3)*(28*a^2*d^2*f^2 - a*b*d*f*(108*d*e - 31*c*f) + b^2*(144*d^2*e^2 - 1
35*c*d*e*f + 40*c^2*f^2) + 3*b*d*f*(15*b*d*e - 8*b*c*f - 7*a*d*f)*x))/(54*b^3*d^
3) + ((14*a^3*d^3*f^3 - 6*a^2*b*d^2*f^2*(9*d*e - 2*c*f) + 3*a*b^2*d*f*(27*d^2*e^
2 - 18*c*d*e*f + 5*c^2*f^2) - b^3*(81*d^3*e^3 - 162*c*d^2*e^2*f + 135*c^2*d*e*f^
2 - 40*c^3*f^3))*ArcTan[1/Sqrt[3] + (2*d^(1/3)*(a + b*x)^(1/3))/(Sqrt[3]*b^(1/3)
*(c + d*x)^(1/3))])/(27*Sqrt[3]*b^(10/3)*d^(11/3)) + ((14*a^3*d^3*f^3 - 6*a^2*b*
d^2*f^2*(9*d*e - 2*c*f) + 3*a*b^2*d*f*(27*d^2*e^2 - 18*c*d*e*f + 5*c^2*f^2) - b^
3*(81*d^3*e^3 - 162*c*d^2*e^2*f + 135*c^2*d*e*f^2 - 40*c^3*f^3))*Log[c + d*x])/(
162*b^(10/3)*d^(11/3)) + ((14*a^3*d^3*f^3 - 6*a^2*b*d^2*f^2*(9*d*e - 2*c*f) + 3*
a*b^2*d*f*(27*d^2*e^2 - 18*c*d*e*f + 5*c^2*f^2) - b^3*(81*d^3*e^3 - 162*c*d^2*e^
2*f + 135*c^2*d*e*f^2 - 40*c^3*f^3))*Log[-1 + (d^(1/3)*(a + b*x)^(1/3))/(b^(1/3)
*(c + d*x)^(1/3))])/(54*b^(10/3)*d^(11/3))

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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Mathematica [C]  time = 0.54581, size = 275, normalized size = 0.47 \[ \frac{\sqrt [3]{c+d x} \left (d f (a+b x) \left (28 a^2 d^2 f^2+a b d f (31 c f-3 d (36 e+7 f x))+b^2 \left (40 c^2 f^2-3 c d f (45 e+8 f x)+9 d^2 \left (18 e^2+9 e f x+2 f^2 x^2\right )\right )\right )+2 \sqrt [3]{\frac{d (a+b x)}{a d-b c}} \left (-14 a^3 d^3 f^3+6 a^2 b d^2 f^2 (9 d e-2 c f)-3 a b^2 d f \left (5 c^2 f^2-18 c d e f+27 d^2 e^2\right )+b^3 \left (-40 c^3 f^3+135 c^2 d e f^2-162 c d^2 e^2 f+81 d^3 e^3\right )\right ) \, _2F_1\left (\frac{1}{3},\frac{1}{3};\frac{4}{3};\frac{b (c+d x)}{b c-a d}\right )\right )}{54 b^3 d^4 \sqrt [3]{a+b x}} \]

Antiderivative was successfully verified.

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

[Out]

((c + d*x)^(1/3)*(d*f*(a + b*x)*(28*a^2*d^2*f^2 + a*b*d*f*(31*c*f - 3*d*(36*e +
7*f*x)) + b^2*(40*c^2*f^2 - 3*c*d*f*(45*e + 8*f*x) + 9*d^2*(18*e^2 + 9*e*f*x + 2
*f^2*x^2))) + 2*(-14*a^3*d^3*f^3 + 6*a^2*b*d^2*f^2*(9*d*e - 2*c*f) - 3*a*b^2*d*f
*(27*d^2*e^2 - 18*c*d*e*f + 5*c^2*f^2) + b^3*(81*d^3*e^3 - 162*c*d^2*e^2*f + 135
*c^2*d*e*f^2 - 40*c^3*f^3))*((d*(a + b*x))/(-(b*c) + a*d))^(1/3)*Hypergeometric2
F1[1/3, 1/3, 4/3, (b*(c + d*x))/(b*c - a*d)]))/(54*b^3*d^4*(a + b*x)^(1/3))

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Maple [F]  time = 0.063, size = 0, normalized size = 0. \[ \int{ \left ( fx+e \right ) ^{3}{\frac{1}{\sqrt [3]{bx+a}}} \left ( dx+c \right ) ^{-{\frac{2}{3}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.579854, size = 898, normalized size = 1.53 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/486*sqrt(3)*(3*sqrt(3)*(18*b^2*d^2*f^3*x^2 + 162*b^2*d^2*e^2*f - 27*(5*b^2*c*d
+ 4*a*b*d^2)*e*f^2 + (40*b^2*c^2 + 31*a*b*c*d + 28*a^2*d^2)*f^3 + 3*(27*b^2*d^2
*e*f^2 - (8*b^2*c*d + 7*a*b*d^2)*f^3)*x)*(b*d^2)^(1/3)*(b*x + a)^(2/3)*(d*x + c)
^(1/3) + sqrt(3)*(81*b^3*d^3*e^3 - 81*(2*b^3*c*d^2 + a*b^2*d^3)*e^2*f + 27*(5*b^
3*c^2*d + 2*a*b^2*c*d^2 + 2*a^2*b*d^3)*e*f^2 - (40*b^3*c^3 + 15*a*b^2*c^2*d + 12
*a^2*b*c*d^2 + 14*a^3*d^3)*f^3)*log((b*d^2*x + a*d^2 + (b*d^2)^(1/3)*(b*x + a)^(
2/3)*(d*x + c)^(1/3)*d + (b*d^2)^(2/3)*(b*x + a)^(1/3)*(d*x + c)^(2/3))/(b*x + a
)) - 2*sqrt(3)*(81*b^3*d^3*e^3 - 81*(2*b^3*c*d^2 + a*b^2*d^3)*e^2*f + 27*(5*b^3*
c^2*d + 2*a*b^2*c*d^2 + 2*a^2*b*d^3)*e*f^2 - (40*b^3*c^3 + 15*a*b^2*c^2*d + 12*a
^2*b*c*d^2 + 14*a^3*d^3)*f^3)*log(-(b*d*x + a*d - (b*d^2)^(1/3)*(b*x + a)^(2/3)*
(d*x + c)^(1/3))/(b*x + a)) + 6*(81*b^3*d^3*e^3 - 81*(2*b^3*c*d^2 + a*b^2*d^3)*e
^2*f + 27*(5*b^3*c^2*d + 2*a*b^2*c*d^2 + 2*a^2*b*d^3)*e*f^2 - (40*b^3*c^3 + 15*a
*b^2*c^2*d + 12*a^2*b*c*d^2 + 14*a^3*d^3)*f^3)*arctan(1/3*(2*sqrt(3)*(b*d^2)^(1/
3)*(b*x + a)^(2/3)*(d*x + c)^(1/3) + sqrt(3)*(b*d*x + a*d))/(b*d*x + a*d)))/((b*
d^2)^(1/3)*b^3*d^3)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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