3.1614 \(\int \frac{1}{(a+b x)^{8/3} (c+d x)^{2/3}} \, dx\)

Optimal. Leaf size=621 \[ \frac{2 \sqrt [3]{2} 3^{3/4} \sqrt{2+\sqrt{3}} d^{5/3} ((a+b x) (c+d x))^{2/3} \sqrt{(a d+b c+2 b d x)^2} \left (2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}+(b c-a d)^{2/3}\right ) \sqrt{\frac{2 \sqrt [3]{2} b^{2/3} d^{2/3} ((a+b x) (c+d x))^{2/3}-2^{2/3} \sqrt [3]{b} \sqrt [3]{d} (b c-a d)^{2/3} \sqrt [3]{(a+b x) (c+d x)}+(b c-a d)^{4/3}}{\left (2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}+\left (1+\sqrt{3}\right ) (b c-a d)^{2/3}\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) (b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}}{\left (1+\sqrt{3}\right ) (b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}}\right )|-7-4 \sqrt{3}\right )}{5 \sqrt [3]{b} (a+b x)^{2/3} (c+d x)^{2/3} (b c-a d)^2 (a d+b c+2 b d x) \sqrt{\frac{(b c-a d)^{2/3} \left (2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}+(b c-a d)^{2/3}\right )}{\left (2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}+\left (1+\sqrt{3}\right ) (b c-a d)^{2/3}\right )^2}} \sqrt{(a d+b (c+2 d x))^2}}+\frac{6 d \sqrt [3]{c+d x}}{5 (a+b x)^{2/3} (b c-a d)^2}-\frac{3 \sqrt [3]{c+d x}}{5 (a+b x)^{5/3} (b c-a d)} \]

[Out]

(-3*(c + d*x)^(1/3))/(5*(b*c - a*d)*(a + b*x)^(5/3)) + (6*d*(c + d*x)^(1/3))/(5*
(b*c - a*d)^2*(a + b*x)^(2/3)) + (2*2^(1/3)*3^(3/4)*Sqrt[2 + Sqrt[3]]*d^(5/3)*((
a + b*x)*(c + d*x))^(2/3)*Sqrt[(b*c + a*d + 2*b*d*x)^2]*((b*c - a*d)^(2/3) + 2^(
2/3)*b^(1/3)*d^(1/3)*((a + b*x)*(c + d*x))^(1/3))*Sqrt[((b*c - a*d)^(4/3) - 2^(2
/3)*b^(1/3)*d^(1/3)*(b*c - a*d)^(2/3)*((a + b*x)*(c + d*x))^(1/3) + 2*2^(1/3)*b^
(2/3)*d^(2/3)*((a + b*x)*(c + d*x))^(2/3))/((1 + Sqrt[3])*(b*c - a*d)^(2/3) + 2^
(2/3)*b^(1/3)*d^(1/3)*((a + b*x)*(c + d*x))^(1/3))^2]*EllipticF[ArcSin[((1 - Sqr
t[3])*(b*c - a*d)^(2/3) + 2^(2/3)*b^(1/3)*d^(1/3)*((a + b*x)*(c + d*x))^(1/3))/(
(1 + Sqrt[3])*(b*c - a*d)^(2/3) + 2^(2/3)*b^(1/3)*d^(1/3)*((a + b*x)*(c + d*x))^
(1/3))], -7 - 4*Sqrt[3]])/(5*b^(1/3)*(b*c - a*d)^2*(a + b*x)^(2/3)*(c + d*x)^(2/
3)*(b*c + a*d + 2*b*d*x)*Sqrt[((b*c - a*d)^(2/3)*((b*c - a*d)^(2/3) + 2^(2/3)*b^
(1/3)*d^(1/3)*((a + b*x)*(c + d*x))^(1/3)))/((1 + Sqrt[3])*(b*c - a*d)^(2/3) + 2
^(2/3)*b^(1/3)*d^(1/3)*((a + b*x)*(c + d*x))^(1/3))^2]*Sqrt[(a*d + b*(c + 2*d*x)
)^2])

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Rubi [A]  time = 1.66537, antiderivative size = 621, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21 \[ \frac{2 \sqrt [3]{2} 3^{3/4} \sqrt{2+\sqrt{3}} d^{5/3} ((a+b x) (c+d x))^{2/3} \sqrt{(a d+b c+2 b d x)^2} \left (2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}+(b c-a d)^{2/3}\right ) \sqrt{\frac{2 \sqrt [3]{2} b^{2/3} d^{2/3} ((a+b x) (c+d x))^{2/3}-2^{2/3} \sqrt [3]{b} \sqrt [3]{d} (b c-a d)^{2/3} \sqrt [3]{(a+b x) (c+d x)}+(b c-a d)^{4/3}}{\left (2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}+\left (1+\sqrt{3}\right ) (b c-a d)^{2/3}\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) (b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}}{\left (1+\sqrt{3}\right ) (b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}}\right )|-7-4 \sqrt{3}\right )}{5 \sqrt [3]{b} (a+b x)^{2/3} (c+d x)^{2/3} (b c-a d)^2 (a d+b c+2 b d x) \sqrt{\frac{(b c-a d)^{2/3} \left (2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}+(b c-a d)^{2/3}\right )}{\left (2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}+\left (1+\sqrt{3}\right ) (b c-a d)^{2/3}\right )^2}} \sqrt{(a d+b (c+2 d x))^2}}+\frac{6 d \sqrt [3]{c+d x}}{5 (a+b x)^{2/3} (b c-a d)^2}-\frac{3 \sqrt [3]{c+d x}}{5 (a+b x)^{5/3} (b c-a d)} \]

Warning: Unable to verify antiderivative.

[In]  Int[1/((a + b*x)^(8/3)*(c + d*x)^(2/3)),x]

[Out]

(-3*(c + d*x)^(1/3))/(5*(b*c - a*d)*(a + b*x)^(5/3)) + (6*d*(c + d*x)^(1/3))/(5*
(b*c - a*d)^2*(a + b*x)^(2/3)) + (2*2^(1/3)*3^(3/4)*Sqrt[2 + Sqrt[3]]*d^(5/3)*((
a + b*x)*(c + d*x))^(2/3)*Sqrt[(b*c + a*d + 2*b*d*x)^2]*((b*c - a*d)^(2/3) + 2^(
2/3)*b^(1/3)*d^(1/3)*((a + b*x)*(c + d*x))^(1/3))*Sqrt[((b*c - a*d)^(4/3) - 2^(2
/3)*b^(1/3)*d^(1/3)*(b*c - a*d)^(2/3)*((a + b*x)*(c + d*x))^(1/3) + 2*2^(1/3)*b^
(2/3)*d^(2/3)*((a + b*x)*(c + d*x))^(2/3))/((1 + Sqrt[3])*(b*c - a*d)^(2/3) + 2^
(2/3)*b^(1/3)*d^(1/3)*((a + b*x)*(c + d*x))^(1/3))^2]*EllipticF[ArcSin[((1 - Sqr
t[3])*(b*c - a*d)^(2/3) + 2^(2/3)*b^(1/3)*d^(1/3)*((a + b*x)*(c + d*x))^(1/3))/(
(1 + Sqrt[3])*(b*c - a*d)^(2/3) + 2^(2/3)*b^(1/3)*d^(1/3)*((a + b*x)*(c + d*x))^
(1/3))], -7 - 4*Sqrt[3]])/(5*b^(1/3)*(b*c - a*d)^2*(a + b*x)^(2/3)*(c + d*x)^(2/
3)*(b*c + a*d + 2*b*d*x)*Sqrt[((b*c - a*d)^(2/3)*((b*c - a*d)^(2/3) + 2^(2/3)*b^
(1/3)*d^(1/3)*((a + b*x)*(c + d*x))^(1/3)))/((1 + Sqrt[3])*(b*c - a*d)^(2/3) + 2
^(2/3)*b^(1/3)*d^(1/3)*((a + b*x)*(c + d*x))^(1/3))^2]*Sqrt[(a*d + b*(c + 2*d*x)
)^2])

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Rubi in Sympy [A]  time = 74.1483, size = 653, normalized size = 1.05 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(b*x+a)**(8/3)/(d*x+c)**(2/3),x)

[Out]

6*d*(c + d*x)**(1/3)/(5*(a + b*x)**(2/3)*(a*d - b*c)**2) + 3*(c + d*x)**(1/3)/(5
*(a + b*x)**(5/3)*(a*d - b*c)) + 2*2**(1/3)*3**(3/4)*d**(5/3)*sqrt((2*2**(1/3)*b
**(2/3)*d**(2/3)*(a*c + b*d*x**2 + x*(a*d + b*c))**(2/3) - 2**(2/3)*b**(1/3)*d**
(1/3)*(a*d - b*c)**(2/3)*(a*c + b*d*x**2 + x*(a*d + b*c))**(1/3) + (a*d - b*c)**
(4/3))/(2**(2/3)*b**(1/3)*d**(1/3)*(a*c + b*d*x**2 + x*(a*d + b*c))**(1/3) + (1
+ sqrt(3))*(a*d - b*c)**(2/3))**2)*sqrt(sqrt(3) + 2)*(2**(2/3)*b**(1/3)*d**(1/3)
*(a*c + b*d*x**2 + x*(a*d + b*c))**(1/3) + (a*d - b*c)**(2/3))*(a*c + b*d*x**2 +
x*(a*d + b*c))**(2/3)*sqrt((a*d + b*c + 2*b*d*x)**2)*elliptic_f(asin((2**(2/3)*
b**(1/3)*d**(1/3)*(a*c + b*d*x**2 + x*(a*d + b*c))**(1/3) - (-1 + sqrt(3))*(a*d
- b*c)**(2/3))/(2**(2/3)*b**(1/3)*d**(1/3)*(a*c + b*d*x**2 + x*(a*d + b*c))**(1/
3) + (1 + sqrt(3))*(a*d - b*c)**(2/3))), -7 - 4*sqrt(3))/(5*b**(1/3)*sqrt((a*d -
b*c)**(2/3)*(2**(2/3)*b**(1/3)*d**(1/3)*(a*c + b*d*x**2 + x*(a*d + b*c))**(1/3)
+ (a*d - b*c)**(2/3))/(2**(2/3)*b**(1/3)*d**(1/3)*(a*c + b*d*x**2 + x*(a*d + b*
c))**(1/3) + (1 + sqrt(3))*(a*d - b*c)**(2/3))**2)*(a + b*x)**(2/3)*(c + d*x)**(
2/3)*(a*d - b*c)**2*sqrt(b*d*(4*a*c + 4*b*d*x**2 + x*(4*a*d + 4*b*c)) + (a*d - b
*c)**2)*(a*d + b*c + 2*b*d*x))

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Mathematica [C]  time = 0.233637, size = 102, normalized size = 0.16 \[ \frac{3 \sqrt [3]{c+d x} \left (2 d (a+b x) \left (\frac{d (a+b x)}{a d-b c}\right )^{2/3} \, _2F_1\left (\frac{1}{3},\frac{2}{3};\frac{4}{3};\frac{b (c+d x)}{b c-a d}\right )+3 a d-b c+2 b d x\right )}{5 (a+b x)^{5/3} (b c-a d)^2} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b*x)^(8/3)*(c + d*x)^(2/3)),x]

[Out]

(3*(c + d*x)^(1/3)*(-(b*c) + 3*a*d + 2*b*d*x + 2*d*(a + b*x)*((d*(a + b*x))/(-(b
*c) + a*d))^(2/3)*Hypergeometric2F1[1/3, 2/3, 4/3, (b*(c + d*x))/(b*c - a*d)]))/
(5*(b*c - a*d)^2*(a + b*x)^(5/3))

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Maple [F]  time = 0.066, size = 0, normalized size = 0. \[ \int{1 \left ( bx+a \right ) ^{-{\frac{8}{3}}} \left ( dx+c \right ) ^{-{\frac{2}{3}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b*x+a)^(8/3)/(d*x+c)^(2/3),x)

[Out]

int(1/(b*x+a)^(8/3)/(d*x+c)^(2/3),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x + a\right )}^{\frac{8}{3}}{\left (d x + c\right )}^{\frac{2}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^(8/3)*(d*x + c)^(2/3)),x, algorithm="maxima")

[Out]

integrate(1/((b*x + a)^(8/3)*(d*x + c)^(2/3)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^(8/3)*(d*x + c)^(2/3)),x, algorithm="fricas")

[Out]

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

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Sympy [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/(b*x+a)**(8/3)/(d*x+c)**(2/3),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x + a\right )}^{\frac{8}{3}}{\left (d x + c\right )}^{\frac{2}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^(8/3)*(d*x + c)^(2/3)),x, algorithm="giac")

[Out]

integrate(1/((b*x + a)^(8/3)*(d*x + c)^(2/3)), x)