3.3010 \(\int \frac{1}{(a+b x)^2 \sqrt [3]{c+d x} \sqrt [3]{b c+a d+2 b d x}} \, dx\)

Optimal. Leaf size=1510 \[ \text{result too large to display} \]

[Out]

-(((c + d*x)^(2/3)*(b*c + a*d + 2*b*d*x)^(2/3))/((b*c - a*d)^2*(a + b*x))) + (((
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]*Sqrt[(
d*(3*b*c + a*d) + 4*b*d^2*x)^2])/(b^(2/3)*d*(b*c - a*d)^2*(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))) + (Sqrt[3]*d*ArcTan[1/Sqrt[
3] + (2*b^(2/3)*(c + d*x)^(2/3))/(Sqrt[3]*(b*c - a*d)^(1/3)*(b*c + a*d + 2*b*d*x
)^(1/3))])/(2*b^(2/3)*(b*c - a*d)^(5/3)) - (3^(1/4)*Sqrt[2 - Sqrt[3]]*d*((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]])/(2*b^(2/3)*(b*c - a*d)^(4/3)*(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]*S
qrt[((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]) + (Sqrt[2]*d*((c + d*x)*(b*c + a*d + 2*b*d*x))^(1/3)*S
qrt[(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]])/(3^(1/4
)*b^(2/3)*(b*c - a*d)^(4/3)*(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]) +
(d*Log[a + b*x])/(2*b^(2/3)*(b*c - a*d)^(5/3)) - (3*d*Log[(b^(2/3)*(c + d*x)^(2
/3))/(b*c - a*d)^(1/3) - (b*c + a*d + 2*b*d*x)^(1/3)])/(4*b^(2/3)*(b*c - a*d)^(5
/3))

_______________________________________________________________________________________

Rubi [A]  time = 4.48089, antiderivative size = 1510, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.242 \[ \frac{\sqrt{3} d \tan ^{-1}\left (\frac{2 b^{2/3} (c+d x)^{2/3}}{\sqrt{3} \sqrt [3]{b c-a d} \sqrt [3]{b c+a d+2 b d x}}+\frac{1}{\sqrt{3}}\right )}{2 b^{2/3} (b c-a d)^{5/3}}-\frac{\sqrt [4]{3} \sqrt{2-\sqrt{3}} d \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 )}{2 b^{2/3} (b c-a d)^{4/3} \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{\sqrt{2} d \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 [4]{3} b^{2/3} (b c-a d)^{4/3} \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{d \log (a+b x)}{2 b^{2/3} (b c-a d)^{5/3}}-\frac{3 d \log \left (\frac{b^{2/3} (c+d x)^{2/3}}{\sqrt [3]{b c-a d}}-\sqrt [3]{b c+a d+2 b d x}\right )}{4 b^{2/3} (b c-a d)^{5/3}}-\frac{(c+d x)^{2/3} (b c+a d+2 b d x)^{2/3}}{(b c-a d)^2 (a+b x)}+\frac{\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^{2/3} d (b c-a d)^2 \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/((a + b*x)^2*(c + d*x)^(1/3)*(b*c + a*d + 2*b*d*x)^(1/3)),x]

[Out]

-(((c + d*x)^(2/3)*(b*c + a*d + 2*b*d*x)^(2/3))/((b*c - a*d)^2*(a + b*x))) + (((
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]*Sqrt[(
d*(3*b*c + a*d) + 4*b*d^2*x)^2])/(b^(2/3)*d*(b*c - a*d)^2*(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))) + (Sqrt[3]*d*ArcTan[1/Sqrt[
3] + (2*b^(2/3)*(c + d*x)^(2/3))/(Sqrt[3]*(b*c - a*d)^(1/3)*(b*c + a*d + 2*b*d*x
)^(1/3))])/(2*b^(2/3)*(b*c - a*d)^(5/3)) - (3^(1/4)*Sqrt[2 - Sqrt[3]]*d*((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]])/(2*b^(2/3)*(b*c - a*d)^(4/3)*(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]*S
qrt[((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]) + (Sqrt[2]*d*((c + d*x)*(b*c + a*d + 2*b*d*x))^(1/3)*S
qrt[(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]])/(3^(1/4
)*b^(2/3)*(b*c - a*d)^(4/3)*(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]) +
(d*Log[a + b*x])/(2*b^(2/3)*(b*c - a*d)^(5/3)) - (3*d*Log[(b^(2/3)*(c + d*x)^(2
/3))/(b*c - a*d)^(1/3) - (b*c + a*d + 2*b*d*x)^(1/3)])/(4*b^(2/3)*(b*c - a*d)^(5
/3))

_______________________________________________________________________________________

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

[Out]

Timed out

_______________________________________________________________________________________

Mathematica [C]  time = 4.38205, size = 593, normalized size = 0.39 \[ \frac{(c+d x)^{2/3} (a d+b (c+2 d x))^{2/3} \left (\frac{d \left (-\frac{16 (b c-a d)^2 F_1\left (\frac{5}{3};\frac{1}{3},1;\frac{8}{3};\frac{b c-a d}{2 b c+2 b d x},\frac{b c-a d}{b c+b d x}\right )}{d (a+b x) \left (16 b (c+d x) F_1\left (\frac{5}{3};\frac{1}{3},1;\frac{8}{3};\frac{b c-a d}{2 b c+2 b d x},\frac{b c-a d}{b c+b d x}\right )+(b c-a d) \left (6 F_1\left (\frac{8}{3};\frac{1}{3},2;\frac{11}{3};\frac{b c-a d}{2 b c+2 b d x},\frac{b c-a d}{b c+b d x}\right )+F_1\left (\frac{8}{3};\frac{4}{3},1;\frac{11}{3};\frac{b c-a d}{2 b c+2 b d x},\frac{b c-a d}{b c+b d x}\right )\right )\right )}+\frac{100 b (c+d x) (b c-a d) F_1\left (\frac{2}{3};\frac{1}{3},1;\frac{5}{3};\frac{b c-a d}{2 b c+2 b d x},\frac{b c-a d}{b c+b d x}\right )}{d (a+b x) \left (10 b (c+d x) F_1\left (\frac{2}{3};\frac{1}{3},1;\frac{5}{3};\frac{b c-a d}{2 b c+2 b d x},\frac{b c-a d}{b c+b d x}\right )+(b c-a d) \left (6 F_1\left (\frac{5}{3};\frac{1}{3},2;\frac{8}{3};\frac{b c-a d}{2 b c+2 b d x},\frac{b c-a d}{b c+b d x}\right )+F_1\left (\frac{5}{3};\frac{4}{3},1;\frac{8}{3};\frac{b c-a d}{2 b c+2 b d x},\frac{b c-a d}{b c+b d x}\right )\right )\right )}+\frac{5 a d}{b c+b d x}-\frac{5 c}{c+d x}+10\right )}{a d+b c+2 b d x}-\frac{5}{a+b x}\right )}{5 (b c-a d)^2} \]

Warning: Unable to verify antiderivative.

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

[Out]

((c + d*x)^(2/3)*(a*d + b*(c + 2*d*x))^(2/3)*(-5/(a + b*x) + (d*(10 - (5*c)/(c +
d*x) + (5*a*d)/(b*c + b*d*x) + (100*b*(b*c - a*d)*(c + d*x)*AppellF1[2/3, 1/3,
1, 5/3, (b*c - a*d)/(2*b*c + 2*b*d*x), (b*c - a*d)/(b*c + b*d*x)])/(d*(a + b*x)*
(10*b*(c + d*x)*AppellF1[2/3, 1/3, 1, 5/3, (b*c - a*d)/(2*b*c + 2*b*d*x), (b*c -
a*d)/(b*c + b*d*x)] + (b*c - a*d)*(6*AppellF1[5/3, 1/3, 2, 8/3, (b*c - a*d)/(2*
b*c + 2*b*d*x), (b*c - a*d)/(b*c + b*d*x)] + AppellF1[5/3, 4/3, 1, 8/3, (b*c - a
*d)/(2*b*c + 2*b*d*x), (b*c - a*d)/(b*c + b*d*x)]))) - (16*(b*c - a*d)^2*AppellF
1[5/3, 1/3, 1, 8/3, (b*c - a*d)/(2*b*c + 2*b*d*x), (b*c - a*d)/(b*c + b*d*x)])/(
d*(a + b*x)*(16*b*(c + d*x)*AppellF1[5/3, 1/3, 1, 8/3, (b*c - a*d)/(2*b*c + 2*b*
d*x), (b*c - a*d)/(b*c + b*d*x)] + (b*c - a*d)*(6*AppellF1[8/3, 1/3, 2, 11/3, (b
*c - a*d)/(2*b*c + 2*b*d*x), (b*c - a*d)/(b*c + b*d*x)] + AppellF1[8/3, 4/3, 1,
11/3, (b*c - a*d)/(2*b*c + 2*b*d*x), (b*c - a*d)/(b*c + b*d*x)])))))/(b*c + a*d
+ 2*b*d*x)))/(5*(b*c - a*d)^2)

_______________________________________________________________________________________

Maple [F]  time = 0.083, size = 0, normalized size = 0. \[ \int{\frac{1}{ \left ( bx+a \right ) ^{2}}{\frac{1}{\sqrt [3]{dx+c}}}{\frac{1}{\sqrt [3]{2\,bdx+ad+bc}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

_______________________________________________________________________________________

Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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