3.2984 \(\int \sqrt [3]{a+b x} (c+d x)^{2/3} (e+f x)^2 \, dx\)

Optimal. Leaf size=571 \[ \frac{(b c-a d)^2 \log (a+b x) \left (10 a^2 d^2 f^2-10 a b d f (3 d e-c f)+b^2 \left (7 c^2 f^2-24 c d e f+27 d^2 e^2\right )\right )}{486 b^{11/3} d^{10/3}}+\frac{(b c-a d)^2 \left (10 a^2 d^2 f^2-10 a b d f (3 d e-c f)+b^2 \left (7 c^2 f^2-24 c d e f+27 d^2 e^2\right )\right ) \log \left (\frac{\sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{d} \sqrt [3]{a+b x}}-1\right )}{162 b^{11/3} d^{10/3}}+\frac{(b c-a d)^2 \left (10 a^2 d^2 f^2-10 a b d f (3 d e-c f)+b^2 \left (7 c^2 f^2-24 c d e f+27 d^2 e^2\right )\right ) \tan ^{-1}\left (\frac{2 \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt{3} \sqrt [3]{d} \sqrt [3]{a+b x}}+\frac{1}{\sqrt{3}}\right )}{81 \sqrt{3} b^{11/3} d^{10/3}}+\frac{(a+b x)^{4/3} (c+d x)^{2/3} \left (10 a^2 d^2 f^2-10 a b d f (3 d e-c f)+b^2 \left (7 c^2 f^2-24 c d e f+27 d^2 e^2\right )\right )}{54 b^3 d^2}+\frac{\sqrt [3]{a+b x} (c+d x)^{2/3} (b c-a d) \left (10 a^2 d^2 f^2-10 a b d f (3 d e-c f)+b^2 \left (7 c^2 f^2-24 c d e f+27 d^2 e^2\right )\right )}{81 b^3 d^3}+\frac{f (a+b x)^{4/3} (c+d x)^{5/3} (-8 a d f-7 b c f+15 b d e)}{36 b^2 d^2}+\frac{f (a+b x)^{4/3} (c+d x)^{5/3} (e+f x)}{4 b d} \]

[Out]

((b*c - a*d)*(10*a^2*d^2*f^2 - 10*a*b*d*f*(3*d*e - c*f) + b^2*(27*d^2*e^2 - 24*c
*d*e*f + 7*c^2*f^2))*(a + b*x)^(1/3)*(c + d*x)^(2/3))/(81*b^3*d^3) + ((10*a^2*d^
2*f^2 - 10*a*b*d*f*(3*d*e - c*f) + b^2*(27*d^2*e^2 - 24*c*d*e*f + 7*c^2*f^2))*(a
+ b*x)^(4/3)*(c + d*x)^(2/3))/(54*b^3*d^2) + (f*(15*b*d*e - 7*b*c*f - 8*a*d*f)*
(a + b*x)^(4/3)*(c + d*x)^(5/3))/(36*b^2*d^2) + (f*(a + b*x)^(4/3)*(c + d*x)^(5/
3)*(e + f*x))/(4*b*d) + ((b*c - a*d)^2*(10*a^2*d^2*f^2 - 10*a*b*d*f*(3*d*e - c*f
) + b^2*(27*d^2*e^2 - 24*c*d*e*f + 7*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))])/(81*Sqrt[3]*b^(11/3)*d^(10/3))
+ ((b*c - a*d)^2*(10*a^2*d^2*f^2 - 10*a*b*d*f*(3*d*e - c*f) + b^2*(27*d^2*e^2 -
24*c*d*e*f + 7*c^2*f^2))*Log[a + b*x])/(486*b^(11/3)*d^(10/3)) + ((b*c - a*d)^2
*(10*a^2*d^2*f^2 - 10*a*b*d*f*(3*d*e - c*f) + b^2*(27*d^2*e^2 - 24*c*d*e*f + 7*c
^2*f^2))*Log[-1 + (b^(1/3)*(c + d*x)^(1/3))/(d^(1/3)*(a + b*x)^(1/3))])/(162*b^(
11/3)*d^(10/3))

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Rubi [A]  time = 1.40729, antiderivative size = 571, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{(b c-a d)^2 \log (a+b x) \left (10 a^2 d^2 f^2-10 a b d f (3 d e-c f)+b^2 \left (7 c^2 f^2-24 c d e f+27 d^2 e^2\right )\right )}{486 b^{11/3} d^{10/3}}+\frac{(b c-a d)^2 \left (10 a^2 d^2 f^2-10 a b d f (3 d e-c f)+b^2 \left (7 c^2 f^2-24 c d e f+27 d^2 e^2\right )\right ) \log \left (\frac{\sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{d} \sqrt [3]{a+b x}}-1\right )}{162 b^{11/3} d^{10/3}}+\frac{(b c-a d)^2 \left (10 a^2 d^2 f^2-10 a b d f (3 d e-c f)+b^2 \left (7 c^2 f^2-24 c d e f+27 d^2 e^2\right )\right ) \tan ^{-1}\left (\frac{2 \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt{3} \sqrt [3]{d} \sqrt [3]{a+b x}}+\frac{1}{\sqrt{3}}\right )}{81 \sqrt{3} b^{11/3} d^{10/3}}+\frac{(a+b x)^{4/3} (c+d x)^{2/3} \left (10 a^2 d^2 f^2-10 a b d f (3 d e-c f)+b^2 \left (7 c^2 f^2-24 c d e f+27 d^2 e^2\right )\right )}{54 b^3 d^2}+\frac{\sqrt [3]{a+b x} (c+d x)^{2/3} (b c-a d) \left (10 a^2 d^2 f^2-10 a b d f (3 d e-c f)+b^2 \left (7 c^2 f^2-24 c d e f+27 d^2 e^2\right )\right )}{81 b^3 d^3}+\frac{f (a+b x)^{4/3} (c+d x)^{5/3} (-8 a d f-7 b c f+15 b d e)}{36 b^2 d^2}+\frac{f (a+b x)^{4/3} (c+d x)^{5/3} (e+f x)}{4 b d} \]

Antiderivative was successfully verified.

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

[Out]

((b*c - a*d)*(10*a^2*d^2*f^2 - 10*a*b*d*f*(3*d*e - c*f) + b^2*(27*d^2*e^2 - 24*c
*d*e*f + 7*c^2*f^2))*(a + b*x)^(1/3)*(c + d*x)^(2/3))/(81*b^3*d^3) + ((10*a^2*d^
2*f^2 - 10*a*b*d*f*(3*d*e - c*f) + b^2*(27*d^2*e^2 - 24*c*d*e*f + 7*c^2*f^2))*(a
+ b*x)^(4/3)*(c + d*x)^(2/3))/(54*b^3*d^2) + (f*(15*b*d*e - 7*b*c*f - 8*a*d*f)*
(a + b*x)^(4/3)*(c + d*x)^(5/3))/(36*b^2*d^2) + (f*(a + b*x)^(4/3)*(c + d*x)^(5/
3)*(e + f*x))/(4*b*d) + ((b*c - a*d)^2*(10*a^2*d^2*f^2 - 10*a*b*d*f*(3*d*e - c*f
) + b^2*(27*d^2*e^2 - 24*c*d*e*f + 7*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))])/(81*Sqrt[3]*b^(11/3)*d^(10/3))
+ ((b*c - a*d)^2*(10*a^2*d^2*f^2 - 10*a*b*d*f*(3*d*e - c*f) + b^2*(27*d^2*e^2 -
24*c*d*e*f + 7*c^2*f^2))*Log[a + b*x])/(486*b^(11/3)*d^(10/3)) + ((b*c - a*d)^2
*(10*a^2*d^2*f^2 - 10*a*b*d*f*(3*d*e - c*f) + b^2*(27*d^2*e^2 - 24*c*d*e*f + 7*c
^2*f^2))*Log[-1 + (b^(1/3)*(c + d*x)^(1/3))/(d^(1/3)*(a + b*x)^(1/3))])/(162*b^(
11/3)*d^(10/3))

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Rubi in Sympy [A]  time = 96.5362, size = 612, normalized size = 1.07 \[ \frac{f \left (a + b x\right )^{\frac{4}{3}} \left (c + d x\right )^{\frac{5}{3}} \left (e + f x\right )}{4 b d} - \frac{f \left (a + b x\right )^{\frac{4}{3}} \left (c + d x\right )^{\frac{5}{3}} \left (8 a d f + 7 b c f - 15 b d e\right )}{36 b^{2} d^{2}} - \frac{\sqrt [3]{a + b x} \left (c + d x\right )^{\frac{5}{3}} \left (9 b d \left (- 12 b d e^{2} + f \left (3 a c f + e \left (5 a d + 4 b c\right )\right )\right ) - f \left (5 a d + 4 b c\right ) \left (8 a d f + 7 b c f - 15 b d e\right )\right )}{216 b^{2} d^{3}} - \frac{\sqrt [3]{a + b x} \left (c + d x\right )^{\frac{2}{3}} \left (a d - b c\right ) \left (9 b d \left (- 12 b d e^{2} + f \left (3 a c f + e \left (5 a d + 4 b c\right )\right )\right ) - f \left (5 a d + 4 b c\right ) \left (8 a d f + 7 b c f - 15 b d e\right )\right )}{648 b^{3} d^{3}} - \frac{\left (a d - b c\right )^{2} \left (9 b d \left (- 12 b d e^{2} + f \left (3 a c f + e \left (5 a d + 4 b c\right )\right )\right ) - f \left (5 a d + 4 b c\right ) \left (8 a d f + 7 b c f - 15 b d e\right )\right ) \log{\left (a + b x \right )}}{1944 b^{\frac{11}{3}} d^{\frac{10}{3}}} - \frac{\left (a d - b c\right )^{2} \left (9 b d \left (- 12 b d e^{2} + f \left (3 a c f + e \left (5 a d + 4 b c\right )\right )\right ) - f \left (5 a d + 4 b c\right ) \left (8 a d f + 7 b c f - 15 b d e\right )\right ) \log{\left (\frac{\sqrt [3]{b} \sqrt [3]{c + d x}}{\sqrt [3]{d} \sqrt [3]{a + b x}} - 1 \right )}}{648 b^{\frac{11}{3}} d^{\frac{10}{3}}} - \frac{\sqrt{3} \left (a d - b c\right )^{2} \left (9 b d \left (- 12 b d e^{2} + f \left (3 a c f + e \left (5 a d + 4 b c\right )\right )\right ) - f \left (5 a d + 4 b c\right ) \left (8 a d f + 7 b c f - 15 b d e\right )\right ) \operatorname{atan}{\left (\frac{2 \sqrt{3} \sqrt [3]{b} \sqrt [3]{c + d x}}{3 \sqrt [3]{d} \sqrt [3]{a + b x}} + \frac{\sqrt{3}}{3} \right )}}{972 b^{\frac{11}{3}} d^{\frac{10}{3}}} \]

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)**2,x)

[Out]

f*(a + b*x)**(4/3)*(c + d*x)**(5/3)*(e + f*x)/(4*b*d) - f*(a + b*x)**(4/3)*(c +
d*x)**(5/3)*(8*a*d*f + 7*b*c*f - 15*b*d*e)/(36*b**2*d**2) - (a + b*x)**(1/3)*(c
+ d*x)**(5/3)*(9*b*d*(-12*b*d*e**2 + f*(3*a*c*f + e*(5*a*d + 4*b*c))) - f*(5*a*d
+ 4*b*c)*(8*a*d*f + 7*b*c*f - 15*b*d*e))/(216*b**2*d**3) - (a + b*x)**(1/3)*(c
+ d*x)**(2/3)*(a*d - b*c)*(9*b*d*(-12*b*d*e**2 + f*(3*a*c*f + e*(5*a*d + 4*b*c))
) - f*(5*a*d + 4*b*c)*(8*a*d*f + 7*b*c*f - 15*b*d*e))/(648*b**3*d**3) - (a*d - b
*c)**2*(9*b*d*(-12*b*d*e**2 + f*(3*a*c*f + e*(5*a*d + 4*b*c))) - f*(5*a*d + 4*b*
c)*(8*a*d*f + 7*b*c*f - 15*b*d*e))*log(a + b*x)/(1944*b**(11/3)*d**(10/3)) - (a*
d - b*c)**2*(9*b*d*(-12*b*d*e**2 + f*(3*a*c*f + e*(5*a*d + 4*b*c))) - f*(5*a*d +
4*b*c)*(8*a*d*f + 7*b*c*f - 15*b*d*e))*log(b**(1/3)*(c + d*x)**(1/3)/(d**(1/3)*
(a + b*x)**(1/3)) - 1)/(648*b**(11/3)*d**(10/3)) - sqrt(3)*(a*d - b*c)**2*(9*b*d
*(-12*b*d*e**2 + f*(3*a*c*f + e*(5*a*d + 4*b*c))) - f*(5*a*d + 4*b*c)*(8*a*d*f +
7*b*c*f - 15*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)/(972*b**(11/3)*d**(10/3))

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Mathematica [C]  time = 0.592612, size = 311, normalized size = 0.54 \[ \frac{(c+d x)^{2/3} \left (d (a+b x) \left (20 a^3 d^3 f^2-12 a^2 b d^2 f (c f+5 d e+d f x)+3 a b^2 d \left (-3 c^2 f^2+2 c d f (8 e+f x)+3 d^2 \left (6 e^2+4 e f x+f^2 x^2\right )\right )+b^3 \left (28 c^3 f^2-3 c^2 d f (32 e+7 f x)+18 c d^2 \left (6 e^2+4 e f x+f^2 x^2\right )+27 d^3 x \left (6 e^2+8 e f x+3 f^2 x^2\right )\right )\right )-2 (b c-a d)^2 \left (\frac{d (a+b x)}{a d-b c}\right )^{2/3} \left (10 a^2 d^2 f^2+10 a b d f (c f-3 d e)+b^2 \left (7 c^2 f^2-24 c d e f+27 d^2 e^2\right )\right ) \, _2F_1\left (\frac{2}{3},\frac{2}{3};\frac{5}{3};\frac{b (c+d x)}{b c-a d}\right )\right )}{324 b^3 d^4 (a+b x)^{2/3}} \]

Antiderivative was successfully verified.

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

[Out]

((c + d*x)^(2/3)*(d*(a + b*x)*(20*a^3*d^3*f^2 - 12*a^2*b*d^2*f*(5*d*e + c*f + d*
f*x) + 3*a*b^2*d*(-3*c^2*f^2 + 2*c*d*f*(8*e + f*x) + 3*d^2*(6*e^2 + 4*e*f*x + f^
2*x^2)) + b^3*(28*c^3*f^2 - 3*c^2*d*f*(32*e + 7*f*x) + 18*c*d^2*(6*e^2 + 4*e*f*x
+ f^2*x^2) + 27*d^3*x*(6*e^2 + 8*e*f*x + 3*f^2*x^2))) - 2*(b*c - a*d)^2*(10*a^2
*d^2*f^2 + 10*a*b*d*f*(-3*d*e + c*f) + b^2*(27*d^2*e^2 - 24*c*d*e*f + 7*c^2*f^2)
)*((d*(a + b*x))/(-(b*c) + a*d))^(2/3)*Hypergeometric2F1[2/3, 2/3, 5/3, (b*(c +
d*x))/(b*c - a*d)]))/(324*b^3*d^4*(a + b*x)^(2/3))

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Maple [F]  time = 0.056, size = 0, normalized size = 0. \[ \int \sqrt [3]{bx+a} \left ( dx+c \right ) ^{{\frac{2}{3}}} \left ( fx+e \right ) ^{2}\, dx \]

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)^2,x)

[Out]

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

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

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)^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.370796, size = 1153, normalized size = 2.02 \[ \text{result too large to display} \]

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)^2,x, algorithm="fricas")

[Out]

1/2916*sqrt(3)*(3*sqrt(3)*(81*b^3*d^3*f^2*x^3 + 54*(2*b^3*c*d^2 + a*b^2*d^3)*e^2
- 12*(8*b^3*c^2*d - 4*a*b^2*c*d^2 + 5*a^2*b*d^3)*e*f + (28*b^3*c^3 - 9*a*b^2*c^
2*d - 12*a^2*b*c*d^2 + 20*a^3*d^3)*f^2 + 9*(24*b^3*d^3*e*f + (2*b^3*c*d^2 + a*b^
2*d^3)*f^2)*x^2 + 3*(54*b^3*d^3*e^2 + 12*(2*b^3*c*d^2 + a*b^2*d^3)*e*f - (7*b^3*
c^2*d - 2*a*b^2*c*d^2 + 4*a^2*b*d^3)*f^2)*x)*(b^2*d)^(1/3)*(b*x + a)^(1/3)*(d*x
+ c)^(2/3) - 2*sqrt(3)*(27*(b^4*c^2*d^2 - 2*a*b^3*c*d^3 + a^2*b^2*d^4)*e^2 - 6*(
4*b^4*c^3*d - 3*a*b^3*c^2*d^2 - 6*a^2*b^2*c*d^3 + 5*a^3*b*d^4)*e*f + (7*b^4*c^4
- 4*a*b^3*c^3*d - 3*a^2*b^2*c^2*d^2 - 10*a^3*b*c*d^3 + 10*a^4*d^4)*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)) + 4*sqrt(3)*(27*(b^4*c^2*d^2 - 2*a*b^
3*c*d^3 + a^2*b^2*d^4)*e^2 - 6*(4*b^4*c^3*d - 3*a*b^3*c^2*d^2 - 6*a^2*b^2*c*d^3
+ 5*a^3*b*d^4)*e*f + (7*b^4*c^4 - 4*a*b^3*c^3*d - 3*a^2*b^2*c^2*d^2 - 10*a^3*b*c
*d^3 + 10*a^4*d^4)*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)) - 12*(27*(b^4*c^2*d^2 - 2*a*b^3*c*d^3 + a^2*b^2*d^4)*e^2 -
6*(4*b^4*c^3*d - 3*a*b^3*c^2*d^2 - 6*a^2*b^2*c*d^3 + 5*a^3*b*d^4)*e*f + (7*b^4*
c^4 - 4*a*b^3*c^3*d - 3*a^2*b^2*c^2*d^2 - 10*a^3*b*c*d^3 + 10*a^4*d^4)*f^2)*arct
an(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^3*d^3)

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

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)**2,x)

[Out]

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

[Out]

Exception raised: NotImplementedError