3.3006 \(\int \frac{(a+b x)^2}{\sqrt [3]{c+d x} \sqrt [3]{b c+a d+2 b d x}} \, dx\)

Optimal. Leaf size=1373 \[ \text{result too large to display} \]

[Out]

(-45*(b*c - a*d)*(c + d*x)^(2/3)*(b*c + a*d + 2*b*d*x)^(2/3))/(112*d^3) + (3*(a
+ b*x)*(c + d*x)^(2/3)*(b*c + a*d + 2*b*d*x)^(2/3))/(14*d^2) + (99*(b*c - a*d)^2
*((c + d*x)*(b*c + a*d + 2*b*d*x))^(1/3)*Sqrt[d^2*(3*b*c + a*d + 4*b*d*x)^2]*Sqr
t[(d*(3*b*c + a*d) + 4*b*d^2*x)^2])/(112*b^(2/3)*d^5*(c + d*x)^(1/3)*(b*c + a*d
+ 2*b*d*x)^(1/3)*(3*b*c + a*d + 4*b*d*x)*((1 + Sqrt[3])*(b*c - a*d)^(2/3) + 2*b^
(1/3)*((c + d*x)*(a*d + b*(c + 2*d*x)))^(1/3))) - (99*3^(1/4)*Sqrt[2 - Sqrt[3]]*
(b*c - a*d)^(8/3)*((c + d*x)*(b*c + a*d + 2*b*d*x))^(1/3)*Sqrt[(d*(3*b*c + a*d)
+ 4*b*d^2*x)^2]*((b*c - a*d)^(2/3) + 2*b^(1/3)*((c + d*x)*(a*d + b*(c + 2*d*x)))
^(1/3))*Sqrt[((b*c - a*d)^(4/3) - 2*b^(1/3)*(b*c - a*d)^(2/3)*((c + d*x)*(a*d +
b*(c + 2*d*x)))^(1/3) + 4*b^(2/3)*((c + d*x)*(a*d + b*(c + 2*d*x)))^(2/3))/((1 +
Sqrt[3])*(b*c - a*d)^(2/3) + 2*b^(1/3)*((c + d*x)*(a*d + b*(c + 2*d*x)))^(1/3))
^2]*EllipticE[ArcSin[((1 - Sqrt[3])*(b*c - a*d)^(2/3) + 2*b^(1/3)*((c + d*x)*(a*
d + b*(c + 2*d*x)))^(1/3))/((1 + Sqrt[3])*(b*c - a*d)^(2/3) + 2*b^(1/3)*((c + d*
x)*(a*d + b*(c + 2*d*x)))^(1/3))], -7 - 4*Sqrt[3]])/(224*b^(2/3)*d^3*(c + d*x)^(
1/3)*(b*c + a*d + 2*b*d*x)^(1/3)*(3*b*c + a*d + 4*b*d*x)*Sqrt[d^2*(3*b*c + a*d +
4*b*d*x)^2]*Sqrt[((b*c - a*d)^(2/3)*((b*c - a*d)^(2/3) + 2*b^(1/3)*((c + d*x)*(
a*d + b*(c + 2*d*x)))^(1/3)))/((1 + Sqrt[3])*(b*c - a*d)^(2/3) + 2*b^(1/3)*((c +
d*x)*(a*d + b*(c + 2*d*x)))^(1/3))^2]) + (33*3^(3/4)*(b*c - a*d)^(8/3)*((c + d*
x)*(b*c + a*d + 2*b*d*x))^(1/3)*Sqrt[(d*(3*b*c + a*d) + 4*b*d^2*x)^2]*((b*c - a*
d)^(2/3) + 2*b^(1/3)*((c + d*x)*(a*d + b*(c + 2*d*x)))^(1/3))*Sqrt[((b*c - a*d)^
(4/3) - 2*b^(1/3)*(b*c - a*d)^(2/3)*((c + d*x)*(a*d + b*(c + 2*d*x)))^(1/3) + 4*
b^(2/3)*((c + d*x)*(a*d + b*(c + 2*d*x)))^(2/3))/((1 + Sqrt[3])*(b*c - a*d)^(2/3
) + 2*b^(1/3)*((c + d*x)*(a*d + b*(c + 2*d*x)))^(1/3))^2]*EllipticF[ArcSin[((1 -
Sqrt[3])*(b*c - a*d)^(2/3) + 2*b^(1/3)*((c + d*x)*(a*d + b*(c + 2*d*x)))^(1/3))
/((1 + Sqrt[3])*(b*c - a*d)^(2/3) + 2*b^(1/3)*((c + d*x)*(a*d + b*(c + 2*d*x)))^
(1/3))], -7 - 4*Sqrt[3]])/(56*Sqrt[2]*b^(2/3)*d^3*(c + d*x)^(1/3)*(b*c + a*d + 2
*b*d*x)^(1/3)*(3*b*c + a*d + 4*b*d*x)*Sqrt[d^2*(3*b*c + a*d + 4*b*d*x)^2]*Sqrt[(
(b*c - a*d)^(2/3)*((b*c - a*d)^(2/3) + 2*b^(1/3)*((c + d*x)*(a*d + b*(c + 2*d*x)
))^(1/3)))/((1 + Sqrt[3])*(b*c - a*d)^(2/3) + 2*b^(1/3)*((c + d*x)*(a*d + b*(c +
2*d*x)))^(1/3))^2])

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Rubi [A]  time = 4.42352, antiderivative size = 1373, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.212 \[ -\frac{99 \sqrt [4]{3} \sqrt{2-\sqrt{3}} \sqrt [3]{(c+d x) (b c+a d+2 b d x)} \sqrt{\left (4 b x d^2+(3 b c+a d) d\right )^2} \left ((b c-a d)^{2/3}+2 \sqrt [3]{b} \sqrt [3]{(c+d x) (a d+b (c+2 d x))}\right ) \sqrt{\frac{(b c-a d)^{4/3}-2 \sqrt [3]{b} \sqrt [3]{(c+d x) (a d+b (c+2 d x))} (b c-a d)^{2/3}+4 b^{2/3} ((c+d x) (a d+b (c+2 d x)))^{2/3}}{\left (\left (1+\sqrt{3}\right ) (b c-a d)^{2/3}+2 \sqrt [3]{b} \sqrt [3]{(c+d x) (a d+b (c+2 d x))}\right )^2}} E\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) (b c-a d)^{2/3}+2 \sqrt [3]{b} \sqrt [3]{(c+d x) (a d+b (c+2 d x))}}{\left (1+\sqrt{3}\right ) (b c-a d)^{2/3}+2 \sqrt [3]{b} \sqrt [3]{(c+d x) (a d+b (c+2 d x))}}\right )|-7-4 \sqrt{3}\right ) (b c-a d)^{8/3}}{224 b^{2/3} d^3 \sqrt [3]{c+d x} \sqrt [3]{b c+a d+2 b d x} (3 b c+a d+4 b d x) \sqrt{d^2 (3 b c+a d+4 b d x)^2} \sqrt{\frac{(b c-a d)^{2/3} \left ((b c-a d)^{2/3}+2 \sqrt [3]{b} \sqrt [3]{(c+d x) (a d+b (c+2 d x))}\right )}{\left (\left (1+\sqrt{3}\right ) (b c-a d)^{2/3}+2 \sqrt [3]{b} \sqrt [3]{(c+d x) (a d+b (c+2 d x))}\right )^2}}}+\frac{33\ 3^{3/4} \sqrt [3]{(c+d x) (b c+a d+2 b d x)} \sqrt{\left (4 b x d^2+(3 b c+a d) d\right )^2} \left ((b c-a d)^{2/3}+2 \sqrt [3]{b} \sqrt [3]{(c+d x) (a d+b (c+2 d x))}\right ) \sqrt{\frac{(b c-a d)^{4/3}-2 \sqrt [3]{b} \sqrt [3]{(c+d x) (a d+b (c+2 d x))} (b c-a d)^{2/3}+4 b^{2/3} ((c+d x) (a d+b (c+2 d x)))^{2/3}}{\left (\left (1+\sqrt{3}\right ) (b c-a d)^{2/3}+2 \sqrt [3]{b} \sqrt [3]{(c+d x) (a d+b (c+2 d x))}\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) (b c-a d)^{2/3}+2 \sqrt [3]{b} \sqrt [3]{(c+d x) (a d+b (c+2 d x))}}{\left (1+\sqrt{3}\right ) (b c-a d)^{2/3}+2 \sqrt [3]{b} \sqrt [3]{(c+d x) (a d+b (c+2 d x))}}\right )|-7-4 \sqrt{3}\right ) (b c-a d)^{8/3}}{56 \sqrt{2} b^{2/3} d^3 \sqrt [3]{c+d x} \sqrt [3]{b c+a d+2 b d x} (3 b c+a d+4 b d x) \sqrt{d^2 (3 b c+a d+4 b d x)^2} \sqrt{\frac{(b c-a d)^{2/3} \left ((b c-a d)^{2/3}+2 \sqrt [3]{b} \sqrt [3]{(c+d x) (a d+b (c+2 d x))}\right )}{\left (\left (1+\sqrt{3}\right ) (b c-a d)^{2/3}+2 \sqrt [3]{b} \sqrt [3]{(c+d x) (a d+b (c+2 d x))}\right )^2}}}+\frac{99 \sqrt [3]{(c+d x) (b c+a d+2 b d x)} \sqrt{d^2 (3 b c+a d+4 b d x)^2} \sqrt{\left (4 b x d^2+(3 b c+a d) d\right )^2} (b c-a d)^2}{112 b^{2/3} d^5 \sqrt [3]{c+d x} \sqrt [3]{b c+a d+2 b d x} (3 b c+a d+4 b d x) \left (\left (1+\sqrt{3}\right ) (b c-a d)^{2/3}+2 \sqrt [3]{b} \sqrt [3]{(c+d x) (a d+b (c+2 d x))}\right )}-\frac{45 (c+d x)^{2/3} (b c+a d+2 b d x)^{2/3} (b c-a d)}{112 d^3}+\frac{3 (a+b x) (c+d x)^{2/3} (b c+a d+2 b d x)^{2/3}}{14 d^2} \]

Warning: Unable to verify antiderivative.

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

[Out]

(-45*(b*c - a*d)*(c + d*x)^(2/3)*(b*c + a*d + 2*b*d*x)^(2/3))/(112*d^3) + (3*(a
+ b*x)*(c + d*x)^(2/3)*(b*c + a*d + 2*b*d*x)^(2/3))/(14*d^2) + (99*(b*c - a*d)^2
*((c + d*x)*(b*c + a*d + 2*b*d*x))^(1/3)*Sqrt[d^2*(3*b*c + a*d + 4*b*d*x)^2]*Sqr
t[(d*(3*b*c + a*d) + 4*b*d^2*x)^2])/(112*b^(2/3)*d^5*(c + d*x)^(1/3)*(b*c + a*d
+ 2*b*d*x)^(1/3)*(3*b*c + a*d + 4*b*d*x)*((1 + Sqrt[3])*(b*c - a*d)^(2/3) + 2*b^
(1/3)*((c + d*x)*(a*d + b*(c + 2*d*x)))^(1/3))) - (99*3^(1/4)*Sqrt[2 - Sqrt[3]]*
(b*c - a*d)^(8/3)*((c + d*x)*(b*c + a*d + 2*b*d*x))^(1/3)*Sqrt[(d*(3*b*c + a*d)
+ 4*b*d^2*x)^2]*((b*c - a*d)^(2/3) + 2*b^(1/3)*((c + d*x)*(a*d + b*(c + 2*d*x)))
^(1/3))*Sqrt[((b*c - a*d)^(4/3) - 2*b^(1/3)*(b*c - a*d)^(2/3)*((c + d*x)*(a*d +
b*(c + 2*d*x)))^(1/3) + 4*b^(2/3)*((c + d*x)*(a*d + b*(c + 2*d*x)))^(2/3))/((1 +
Sqrt[3])*(b*c - a*d)^(2/3) + 2*b^(1/3)*((c + d*x)*(a*d + b*(c + 2*d*x)))^(1/3))
^2]*EllipticE[ArcSin[((1 - Sqrt[3])*(b*c - a*d)^(2/3) + 2*b^(1/3)*((c + d*x)*(a*
d + b*(c + 2*d*x)))^(1/3))/((1 + Sqrt[3])*(b*c - a*d)^(2/3) + 2*b^(1/3)*((c + d*
x)*(a*d + b*(c + 2*d*x)))^(1/3))], -7 - 4*Sqrt[3]])/(224*b^(2/3)*d^3*(c + d*x)^(
1/3)*(b*c + a*d + 2*b*d*x)^(1/3)*(3*b*c + a*d + 4*b*d*x)*Sqrt[d^2*(3*b*c + a*d +
4*b*d*x)^2]*Sqrt[((b*c - a*d)^(2/3)*((b*c - a*d)^(2/3) + 2*b^(1/3)*((c + d*x)*(
a*d + b*(c + 2*d*x)))^(1/3)))/((1 + Sqrt[3])*(b*c - a*d)^(2/3) + 2*b^(1/3)*((c +
d*x)*(a*d + b*(c + 2*d*x)))^(1/3))^2]) + (33*3^(3/4)*(b*c - a*d)^(8/3)*((c + d*
x)*(b*c + a*d + 2*b*d*x))^(1/3)*Sqrt[(d*(3*b*c + a*d) + 4*b*d^2*x)^2]*((b*c - a*
d)^(2/3) + 2*b^(1/3)*((c + d*x)*(a*d + b*(c + 2*d*x)))^(1/3))*Sqrt[((b*c - a*d)^
(4/3) - 2*b^(1/3)*(b*c - a*d)^(2/3)*((c + d*x)*(a*d + b*(c + 2*d*x)))^(1/3) + 4*
b^(2/3)*((c + d*x)*(a*d + b*(c + 2*d*x)))^(2/3))/((1 + Sqrt[3])*(b*c - a*d)^(2/3
) + 2*b^(1/3)*((c + d*x)*(a*d + b*(c + 2*d*x)))^(1/3))^2]*EllipticF[ArcSin[((1 -
Sqrt[3])*(b*c - a*d)^(2/3) + 2*b^(1/3)*((c + d*x)*(a*d + b*(c + 2*d*x)))^(1/3))
/((1 + Sqrt[3])*(b*c - a*d)^(2/3) + 2*b^(1/3)*((c + d*x)*(a*d + b*(c + 2*d*x)))^
(1/3))], -7 - 4*Sqrt[3]])/(56*Sqrt[2]*b^(2/3)*d^3*(c + d*x)^(1/3)*(b*c + a*d + 2
*b*d*x)^(1/3)*(3*b*c + a*d + 4*b*d*x)*Sqrt[d^2*(3*b*c + a*d + 4*b*d*x)^2]*Sqrt[(
(b*c - a*d)^(2/3)*((b*c - a*d)^(2/3) + 2*b^(1/3)*((c + d*x)*(a*d + b*(c + 2*d*x)
))^(1/3)))/((1 + Sqrt[3])*(b*c - a*d)^(2/3) + 2*b^(1/3)*((c + d*x)*(a*d + b*(c +
2*d*x)))^(1/3))^2])

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

[Out]

Timed out

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Mathematica [C]  time = 0.33746, size = 129, normalized size = 0.09 \[ \frac{3 (a d+b (c+2 d x))^{2/3} \left (33 \sqrt [3]{2} (b c-a d)^2 \sqrt [3]{\frac{b (c+d x)}{b c-a d}} \, _2F_1\left (\frac{1}{3},\frac{2}{3};\frac{5}{3};\frac{a d+b (c+2 d x)}{a d-b c}\right )-2 b (c+d x) (-23 a d+15 b c-8 b d x)\right )}{224 b d^3 \sqrt [3]{c+d x}} \]

Antiderivative was successfully verified.

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

[Out]

(3*(a*d + b*(c + 2*d*x))^(2/3)*(-2*b*(c + d*x)*(15*b*c - 23*a*d - 8*b*d*x) + 33*
2^(1/3)*(b*c - a*d)^2*((b*(c + d*x))/(b*c - a*d))^(1/3)*Hypergeometric2F1[1/3, 2
/3, 5/3, (a*d + b*(c + 2*d*x))/(-(b*c) + a*d)]))/(224*b*d^3*(c + d*x)^(1/3))

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Maple [F]  time = 0.089, size = 0, normalized size = 0. \[ \int{ \left ( bx+a \right ) ^{2}{\frac{1}{\sqrt [3]{dx+c}}}{\frac{1}{\sqrt [3]{2\,bdx+ad+bc}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((b^2*x^2 + 2*a*b*x + a^2)/((2*b*d*x + b*c + a*d)^(1/3)*(d*x + c)^(1/3))
, x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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