3.3008 \(\int \frac{1}{\sqrt [3]{c+d x} \sqrt [3]{b c+a d+2 b d x}} \, dx\)

Optimal. Leaf size=1283 \[ \text{result too large to display} \]

[Out]

(3*((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]*S
qrt[(d*(3*b*c + a*d) + 4*b*d^2*x)^2])/(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)*((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))) - (3*3^(1/4)*Sqrt[2 - Sqrt[3]]*(
b*c - a*d)^(2/3)*((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]])/(4*b^(2/3)*d*(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]) + (3^(3/4)*(b*c - a*d)^(2/3)*((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 + S
qrt[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]])/(Sqrt[2]*b^(2/3)*d*(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 = 2.57359, antiderivative size = 1283, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192 \[ -\frac{3 \sqrt [4]{3} \sqrt{2-\sqrt{3}} (b c-a d)^{2/3} \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 )}{4 b^{2/3} d \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{3^{3/4} (b c-a d)^{2/3} \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}} 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 )}{\sqrt{2} b^{2/3} d \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{3 \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}}{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) \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 )} \]

Warning: Unable to verify antiderivative.

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

[Out]

(3*((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]*S
qrt[(d*(3*b*c + a*d) + 4*b*d^2*x)^2])/(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)*((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))) - (3*3^(1/4)*Sqrt[2 - Sqrt[3]]*(
b*c - a*d)^(2/3)*((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]])/(4*b^(2/3)*d*(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]) + (3^(3/4)*(b*c - a*d)^(2/3)*((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 + S
qrt[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]])/(Sqrt[2]*b^(2/3)*d*(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 [A]  time = 138.341, size = 1527, normalized size = 1.19 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3*3**(1/4)*sqrt((4*b**(2/3)*(2*b*d**2*x**2 + c*(a*d + b*c) + d*x*(a*d + 3*b*c))
**(2/3) - 2*b**(1/3)*(a*d - b*c)**(2/3)*(2*b*d**2*x**2 + c*(a*d + b*c) + d*x*(a*
d + 3*b*c))**(1/3) + (a*d - b*c)**(4/3))/(2*b**(1/3)*(2*b*d**2*x**2 + c*(a*d + b
*c) + d*x*(a*d + 3*b*c))**(1/3) + (1 + sqrt(3))*(a*d - b*c)**(2/3))**2)*sqrt(-sq
rt(3) + 2)*(a*d - b*c)**(2/3)*(2*b**(1/3)*(2*b*d**2*x**2 + c*(a*d + b*c) + d*x*(
a*d + 3*b*c))**(1/3) + (a*d - b*c)**(2/3))*(2*b*d**2*x**2 + c*(a*d + b*c) + d*x*
(a*d + 3*b*c))**(1/3)*sqrt((4*b*d**2*x + d*(a*d + 3*b*c))**2)*elliptic_e(asin((2
*b**(1/3)*(2*b*d**2*x**2 + c*(a*d + b*c) + d*x*(a*d + 3*b*c))**(1/3) - (-1 + sqr
t(3))*(a*d - b*c)**(2/3))/(2*b**(1/3)*(2*b*d**2*x**2 + c*(a*d + b*c) + d*x*(a*d
+ 3*b*c))**(1/3) + (1 + sqrt(3))*(a*d - b*c)**(2/3))), -7 - 4*sqrt(3))/(4*b**(2/
3)*d*sqrt((a*d - b*c)**(2/3)*(2*b**(1/3)*(2*b*d**2*x**2 + c*(a*d + b*c) + d*x*(a
*d + 3*b*c))**(1/3) + (a*d - b*c)**(2/3))/(2*b**(1/3)*(2*b*d**2*x**2 + c*(a*d +
b*c) + d*x*(a*d + 3*b*c))**(1/3) + (1 + sqrt(3))*(a*d - b*c)**(2/3))**2)*(c + d*
x)**(1/3)*sqrt(b*d**2*(16*b*d**2*x**2 + 8*c*(a*d + b*c) + 8*d*x*(a*d + 3*b*c)) +
d**2*(a*d - b*c)**2)*(a*d + b*c + 2*b*d*x)**(1/3)*(a*d + 3*b*c + 4*b*d*x)) + sq
rt(2)*3**(3/4)*sqrt((4*b**(2/3)*(2*b*d**2*x**2 + c*(a*d + b*c) + d*x*(a*d + 3*b*
c))**(2/3) - 2*b**(1/3)*(a*d - b*c)**(2/3)*(2*b*d**2*x**2 + c*(a*d + b*c) + d*x*
(a*d + 3*b*c))**(1/3) + (a*d - b*c)**(4/3))/(2*b**(1/3)*(2*b*d**2*x**2 + c*(a*d
+ b*c) + d*x*(a*d + 3*b*c))**(1/3) + (1 + sqrt(3))*(a*d - b*c)**(2/3))**2)*(a*d
- b*c)**(2/3)*(2*b**(1/3)*(2*b*d**2*x**2 + c*(a*d + b*c) + d*x*(a*d + 3*b*c))**(
1/3) + (a*d - b*c)**(2/3))*(2*b*d**2*x**2 + c*(a*d + b*c) + d*x*(a*d + 3*b*c))**
(1/3)*sqrt((4*b*d**2*x + d*(a*d + 3*b*c))**2)*elliptic_f(asin((2*b**(1/3)*(2*b*d
**2*x**2 + c*(a*d + b*c) + d*x*(a*d + 3*b*c))**(1/3) - (-1 + sqrt(3))*(a*d - b*c
)**(2/3))/(2*b**(1/3)*(2*b*d**2*x**2 + c*(a*d + b*c) + d*x*(a*d + 3*b*c))**(1/3)
+ (1 + sqrt(3))*(a*d - b*c)**(2/3))), -7 - 4*sqrt(3))/(2*b**(2/3)*d*sqrt((a*d -
b*c)**(2/3)*(2*b**(1/3)*(2*b*d**2*x**2 + c*(a*d + b*c) + d*x*(a*d + 3*b*c))**(1
/3) + (a*d - b*c)**(2/3))/(2*b**(1/3)*(2*b*d**2*x**2 + c*(a*d + b*c) + d*x*(a*d
+ 3*b*c))**(1/3) + (1 + sqrt(3))*(a*d - b*c)**(2/3))**2)*(c + d*x)**(1/3)*sqrt(b
*d**2*(16*b*d**2*x**2 + 8*c*(a*d + b*c) + 8*d*x*(a*d + 3*b*c)) + d**2*(a*d - b*c
)**2)*(a*d + b*c + 2*b*d*x)**(1/3)*(a*d + 3*b*c + 4*b*d*x)) + 3*sqrt(b*d**2*(16*
b*d**2*x**2 + 8*c*(a*d + b*c) + 8*d*x*(a*d + 3*b*c)) + d**2*(a*d - b*c)**2)*(2*b
*d**2*x**2 + c*(a*d + b*c) + d*x*(a*d + 3*b*c))**(1/3)*sqrt((4*b*d**2*x + d*(a*d
+ 3*b*c))**2)/(2*b**(2/3)*d**3*(c + d*x)**(1/3)*(2*b**(1/3)*(2*b*d**2*x**2 + c*
(a*d + b*c) + d*x*(a*d + 3*b*c))**(1/3) + (1 + sqrt(3))*(a*d - b*c)**(2/3))*(a*d
+ b*c + 2*b*d*x)**(1/3)*(a*d + 3*b*c + 4*b*d*x))

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Mathematica [C]  time = 0.109817, size = 94, normalized size = 0.07 \[ \frac{3 \sqrt [3]{\frac{b (c+d x)}{b c-a d}} (a d+b (c+2 d x))^{2/3} \, _2F_1\left (\frac{1}{3},\frac{2}{3};\frac{5}{3};\frac{a d+b (c+2 d x)}{a d-b c}\right )}{2\ 2^{2/3} b d \sqrt [3]{c+d x}} \]

Antiderivative was successfully verified.

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

[Out]

(3*((b*(c + d*x))/(b*c - a*d))^(1/3)*(a*d + b*(c + 2*d*x))^(2/3)*Hypergeometric2
F1[1/3, 2/3, 5/3, (a*d + b*(c + 2*d*x))/(-(b*c) + a*d)])/(2*2^(2/3)*b*d*(c + d*x
)^(1/3))

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Maple [F]  time = 0.056, size = 0, normalized size = 0. \[ \int{1{\frac{1}{\sqrt [3]{dx+c}}}{\frac{1}{\sqrt [3]{2\,bdx+ad+bc}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

int(1/(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. \[ \int \frac{1}{{\left (2 \, b d x + b c + a d\right )}^{\frac{1}{3}}{\left (d x + c\right )}^{\frac{1}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/((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. \[{\rm integral}\left (\frac{1}{{\left (2 \, b d x + b c + a d\right )}^{\frac{1}{3}}{\left (d x + c\right )}^{\frac{1}{3}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(1/((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. \[ \int \frac{1}{\sqrt [3]{c + d x} \sqrt [3]{a d + b c + 2 b d x}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/((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. \[ \int \frac{1}{{\left (2 \, b d x + b c + a d\right )}^{\frac{1}{3}}{\left (d x + c\right )}^{\frac{1}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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