3.1623 \(\int \frac{(a+b x)^{8/3}}{(c+d x)^{4/3}} \, dx\)

Optimal. Leaf size=1355 \[ \text{result too large to display} \]

[Out]

(-3*(a + b*x)^(8/3))/(d*(c + d*x)^(1/3)) - (30*b*(b*c - a*d)*(a + b*x)^(2/3)*(c
+ d*x)^(2/3))/(7*d^3) + (24*b*(a + b*x)^(5/3)*(c + d*x)^(2/3))/(7*d^2) + (30*2^(
2/3)*b^(1/3)*(b*c - a*d)^2*((a + b*x)*(c + d*x))^(1/3)*Sqrt[(b*c + a*d + 2*b*d*x
)^2]*Sqrt[(a*d + b*(c + 2*d*x))^2])/(7*d^(11/3)*(a + b*x)^(1/3)*(c + d*x)^(1/3)*
(b*c + a*d + 2*b*d*x)*((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))) - (15*2^(2/3)*3^(1/4)*Sqrt[2 - Sqrt[3]]*b^(1/3)*(
b*c - a*d)^(8/3)*((a + b*x)*(c + d*x))^(1/3)*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]*Ellipti
cE[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]])/(7*d^(11/3)*(a + b*x)^(1/3)*(c + d*
x)^(1/3)*(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]) + (20*2^(1/6)*3^(3/4)*b^(1/3)*(b*c - a*d)^(8/3)*((a + b*x)*(c + d*x)
)^(1/3)*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 - 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*Sq
rt[3]])/(7*d^(11/3)*(a + b*x)^(1/3)*(c + d*x)^(1/3)*(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 = 4.75571, antiderivative size = 1355, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.368 \[ -\frac{15\ 2^{2/3} \sqrt [4]{3} \sqrt{2-\sqrt{3}} \sqrt [3]{b} \sqrt [3]{(a+b x) (c+d x)} \sqrt{(b c+a d+2 b d x)^2} \left ((b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}\right ) \sqrt{\frac{(b c-a d)^{4/3}-2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)} (b c-a d)^{2/3}+2 \sqrt [3]{2} b^{2/3} d^{2/3} ((a+b x) (c+d x))^{2/3}}{\left (\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 )^2}} E\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 ) (b c-a d)^{8/3}}{7 d^{11/3} \sqrt [3]{a+b x} \sqrt [3]{c+d x} (b c+a d+2 b d x) \sqrt{\frac{(b c-a d)^{2/3} \left ((b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}\right )}{\left (\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 )^2}} \sqrt{(a d+b (c+2 d x))^2}}+\frac{20 \sqrt [6]{2} 3^{3/4} \sqrt [3]{b} \sqrt [3]{(a+b x) (c+d x)} \sqrt{(b c+a d+2 b d x)^2} \left ((b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}\right ) \sqrt{\frac{(b c-a d)^{4/3}-2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)} (b c-a d)^{2/3}+2 \sqrt [3]{2} b^{2/3} d^{2/3} ((a+b x) (c+d x))^{2/3}}{\left (\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 )^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 ) (b c-a d)^{8/3}}{7 d^{11/3} \sqrt [3]{a+b x} \sqrt [3]{c+d x} (b c+a d+2 b d x) \sqrt{\frac{(b c-a d)^{2/3} \left ((b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}\right )}{\left (\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 )^2}} \sqrt{(a d+b (c+2 d x))^2}}+\frac{30\ 2^{2/3} \sqrt [3]{b} \sqrt [3]{(a+b x) (c+d x)} \sqrt{(b c+a d+2 b d x)^2} \sqrt{(a d+b (c+2 d x))^2} (b c-a d)^2}{7 d^{11/3} \sqrt [3]{a+b x} \sqrt [3]{c+d x} (b c+a d+2 b d x) \left (\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 )}-\frac{30 b (a+b x)^{2/3} (c+d x)^{2/3} (b c-a d)}{7 d^3}+\frac{24 b (a+b x)^{5/3} (c+d x)^{2/3}}{7 d^2}-\frac{3 (a+b x)^{8/3}}{d \sqrt [3]{c+d x}} \]

Warning: Unable to verify antiderivative.

[In]  Int[(a + b*x)^(8/3)/(c + d*x)^(4/3),x]

[Out]

(-3*(a + b*x)^(8/3))/(d*(c + d*x)^(1/3)) - (30*b*(b*c - a*d)*(a + b*x)^(2/3)*(c
+ d*x)^(2/3))/(7*d^3) + (24*b*(a + b*x)^(5/3)*(c + d*x)^(2/3))/(7*d^2) + (30*2^(
2/3)*b^(1/3)*(b*c - a*d)^2*((a + b*x)*(c + d*x))^(1/3)*Sqrt[(b*c + a*d + 2*b*d*x
)^2]*Sqrt[(a*d + b*(c + 2*d*x))^2])/(7*d^(11/3)*(a + b*x)^(1/3)*(c + d*x)^(1/3)*
(b*c + a*d + 2*b*d*x)*((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))) - (15*2^(2/3)*3^(1/4)*Sqrt[2 - Sqrt[3]]*b^(1/3)*(
b*c - a*d)^(8/3)*((a + b*x)*(c + d*x))^(1/3)*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]*Ellipti
cE[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]])/(7*d^(11/3)*(a + b*x)^(1/3)*(c + d*
x)^(1/3)*(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]) + (20*2^(1/6)*3^(3/4)*b^(1/3)*(b*c - a*d)^(8/3)*((a + b*x)*(c + d*x)
)^(1/3)*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 - 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*Sq
rt[3]])/(7*d^(11/3)*(a + b*x)^(1/3)*(c + d*x)^(1/3)*(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 [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)**(8/3)/(d*x+c)**(4/3),x)

[Out]

Timed out

_______________________________________________________________________________________

Mathematica [C]  time = 0.274742, size = 131, normalized size = 0.1 \[ \frac{3 (c+d x)^{2/3} \left (d (a+b x) \left (-\frac{7 (b c-a d)^2}{c+d x}+b (4 a d-3 b c)+b^2 d x\right )+10 b (b c-a d)^2 \sqrt [3]{\frac{d (a+b x)}{a d-b c}} \, _2F_1\left (\frac{1}{3},\frac{2}{3};\frac{5}{3};\frac{b (c+d x)}{b c-a d}\right )\right )}{7 d^4 \sqrt [3]{a+b x}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^(8/3)/(c + d*x)^(4/3),x]

[Out]

(3*(c + d*x)^(2/3)*(d*(a + b*x)*(b*(-3*b*c + 4*a*d) + b^2*d*x - (7*(b*c - a*d)^2
)/(c + d*x)) + 10*b*(b*c - a*d)^2*((d*(a + b*x))/(-(b*c) + a*d))^(1/3)*Hypergeom
etric2F1[1/3, 2/3, 5/3, (b*(c + d*x))/(b*c - a*d)]))/(7*d^4*(a + b*x)^(1/3))

_______________________________________________________________________________________

Maple [F]  time = 0.055, size = 0, normalized size = 0. \[ \int{1 \left ( bx+a \right ) ^{{\frac{8}{3}}} \left ( dx+c \right ) ^{-{\frac{4}{3}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^(8/3)/(d*x+c)^(4/3),x)

[Out]

int((b*x+a)^(8/3)/(d*x+c)^(4/3),x)

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x + a\right )}^{\frac{8}{3}}}{{\left (d x + c\right )}^{\frac{4}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(8/3)/(d*x + c)^(4/3),x, algorithm="maxima")

[Out]

integrate((b*x + a)^(8/3)/(d*x + c)^(4/3), x)

_______________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}{\left (b x + a\right )}^{\frac{2}{3}}}{{\left (d x + c\right )}^{\frac{4}{3}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(8/3)/(d*x + c)^(4/3),x, algorithm="fricas")

[Out]

integral((b^2*x^2 + 2*a*b*x + a^2)*(b*x + a)^(2/3)/(d*x + c)^(4/3), 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)**(8/3)/(d*x+c)**(4/3),x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x + a\right )}^{\frac{8}{3}}}{{\left (d x + c\right )}^{\frac{4}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(8/3)/(d*x + c)^(4/3),x, algorithm="giac")

[Out]

integrate((b*x + a)^(8/3)/(d*x + c)^(4/3), x)