3.1609 \(\int \frac{(a+b x)^{7/3}}{(c+d x)^{2/3}} \, dx\)

Optimal. Leaf size=649 \[ -\frac{7\ 3^{3/4} \sqrt{2+\sqrt{3}} (b c-a d)^3 ((a+b x) (c+d x))^{2/3} \sqrt{(a d+b c+2 b d x)^2} \left (2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}+(b c-a d)^{2/3}\right ) \sqrt{\frac{2 \sqrt [3]{2} b^{2/3} d^{2/3} ((a+b x) (c+d x))^{2/3}-2^{2/3} \sqrt [3]{b} \sqrt [3]{d} (b c-a d)^{2/3} \sqrt [3]{(a+b x) (c+d x)}+(b c-a d)^{4/3}}{\left (2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}+\left (1+\sqrt{3}\right ) (b c-a d)^{2/3}\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) (b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}}{\left (1+\sqrt{3}\right ) (b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}}\right )|-7-4 \sqrt{3}\right )}{10\ 2^{2/3} \sqrt [3]{b} d^{10/3} (a+b x)^{2/3} (c+d x)^{2/3} (a d+b c+2 b d x) \sqrt{\frac{(b c-a d)^{2/3} \left (2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}+(b c-a d)^{2/3}\right )}{\left (2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}+\left (1+\sqrt{3}\right ) (b c-a d)^{2/3}\right )^2}} \sqrt{(a d+b (c+2 d x))^2}}+\frac{21 \sqrt [3]{a+b x} \sqrt [3]{c+d x} (b c-a d)^2}{20 d^3}-\frac{21 (a+b x)^{4/3} \sqrt [3]{c+d x} (b c-a d)}{40 d^2}+\frac{3 (a+b x)^{7/3} \sqrt [3]{c+d x}}{8 d} \]

[Out]

(21*(b*c - a*d)^2*(a + b*x)^(1/3)*(c + d*x)^(1/3))/(20*d^3) - (21*(b*c - a*d)*(a
+ b*x)^(4/3)*(c + d*x)^(1/3))/(40*d^2) + (3*(a + b*x)^(7/3)*(c + d*x)^(1/3))/(8
*d) - (7*3^(3/4)*Sqrt[2 + Sqrt[3]]*(b*c - a*d)^3*((a + b*x)*(c + d*x))^(2/3)*Sqr
t[(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 - 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))/((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]])/(10
*2^(2/3)*b^(1/3)*d^(10/3)*(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])

_______________________________________________________________________________________

Rubi [A]  time = 2.41318, antiderivative size = 649, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21 \[ -\frac{7\ 3^{3/4} \sqrt{2+\sqrt{3}} (b c-a d)^3 ((a+b x) (c+d x))^{2/3} \sqrt{(a d+b c+2 b d x)^2} \left (2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}+(b c-a d)^{2/3}\right ) \sqrt{\frac{2 \sqrt [3]{2} b^{2/3} d^{2/3} ((a+b x) (c+d x))^{2/3}-2^{2/3} \sqrt [3]{b} \sqrt [3]{d} (b c-a d)^{2/3} \sqrt [3]{(a+b x) (c+d x)}+(b c-a d)^{4/3}}{\left (2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}+\left (1+\sqrt{3}\right ) (b c-a d)^{2/3}\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) (b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}}{\left (1+\sqrt{3}\right ) (b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}}\right )|-7-4 \sqrt{3}\right )}{10\ 2^{2/3} \sqrt [3]{b} d^{10/3} (a+b x)^{2/3} (c+d x)^{2/3} (a d+b c+2 b d x) \sqrt{\frac{(b c-a d)^{2/3} \left (2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}+(b c-a d)^{2/3}\right )}{\left (2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}+\left (1+\sqrt{3}\right ) (b c-a d)^{2/3}\right )^2}} \sqrt{(a d+b (c+2 d x))^2}}+\frac{21 \sqrt [3]{a+b x} \sqrt [3]{c+d x} (b c-a d)^2}{20 d^3}-\frac{21 (a+b x)^{4/3} \sqrt [3]{c+d x} (b c-a d)}{40 d^2}+\frac{3 (a+b x)^{7/3} \sqrt [3]{c+d x}}{8 d} \]

Warning: Unable to verify antiderivative.

[In]  Int[(a + b*x)^(7/3)/(c + d*x)^(2/3),x]

[Out]

(21*(b*c - a*d)^2*(a + b*x)^(1/3)*(c + d*x)^(1/3))/(20*d^3) - (21*(b*c - a*d)*(a
+ b*x)^(4/3)*(c + d*x)^(1/3))/(40*d^2) + (3*(a + b*x)^(7/3)*(c + d*x)^(1/3))/(8
*d) - (7*3^(3/4)*Sqrt[2 + Sqrt[3]]*(b*c - a*d)^3*((a + b*x)*(c + d*x))^(2/3)*Sqr
t[(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 - 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))/((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]])/(10
*2^(2/3)*b^(1/3)*d^(10/3)*(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])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 92.6699, size = 680, normalized size = 1.05 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**(7/3)/(d*x+c)**(2/3),x)

[Out]

3*(a + b*x)**(7/3)*(c + d*x)**(1/3)/(8*d) + 21*(a + b*x)**(4/3)*(c + d*x)**(1/3)
*(a*d - b*c)/(40*d**2) + 21*(a + b*x)**(1/3)*(c + d*x)**(1/3)*(a*d - b*c)**2/(20
*d**3) + 7*2**(1/3)*3**(3/4)*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)*(a*d - b*c)**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))*(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))/(20*b**(1/3)*d**(10/3)*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)*sq
rt(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))

_______________________________________________________________________________________

Mathematica [C]  time = 0.28733, size = 137, normalized size = 0.21 \[ \frac{3 \sqrt [3]{c+d x} \left (d (a+b x) \left (26 a^2 d^2+a b d (17 d x-35 c)+b^2 \left (14 c^2-7 c d x+5 d^2 x^2\right )\right )-14 (b c-a d)^3 \left (\frac{d (a+b x)}{a d-b c}\right )^{2/3} \, _2F_1\left (\frac{1}{3},\frac{2}{3};\frac{4}{3};\frac{b (c+d x)}{b c-a d}\right )\right )}{40 d^4 (a+b x)^{2/3}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^(7/3)/(c + d*x)^(2/3),x]

[Out]

(3*(c + d*x)^(1/3)*(d*(a + b*x)*(26*a^2*d^2 + a*b*d*(-35*c + 17*d*x) + b^2*(14*c
^2 - 7*c*d*x + 5*d^2*x^2)) - 14*(b*c - a*d)^3*((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)]))/(40*d^4*(a + b*
x)^(2/3))

_______________________________________________________________________________________

Maple [F]  time = 0.042, size = 0, normalized size = 0. \[ \int{1 \left ( bx+a \right ) ^{{\frac{7}{3}}} \left ( dx+c \right ) ^{-{\frac{2}{3}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^(7/3)/(d*x+c)^(2/3),x)

[Out]

int((b*x+a)^(7/3)/(d*x+c)^(2/3),x)

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x + a\right )}^{\frac{7}{3}}}{{\left (d x + c\right )}^{\frac{2}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(7/3)/(d*x + c)^(2/3),x, algorithm="maxima")

[Out]

integrate((b*x + a)^(7/3)/(d*x + c)^(2/3), x)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(7/3)/(d*x + c)^(2/3),x, algorithm="fricas")

[Out]

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

_______________________________________________________________________________________

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((b*x+a)**(7/3)/(d*x+c)**(2/3),x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x + a\right )}^{\frac{7}{3}}}{{\left (d x + c\right )}^{\frac{2}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(7/3)/(d*x + c)^(2/3),x, algorithm="giac")

[Out]

integrate((b*x + a)^(7/3)/(d*x + c)^(2/3), x)