3.1622 \(\int \frac{(a+b x)^{8/3}}{(c+d x)^{4/3}} \, dx\)

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*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])

________________________________________________________________________________________

Rubi [A]  time = 2.51064, 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, Rules used = {47, 50, 62, 623, 303, 218, 1877} \[ -\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}} \]

Antiderivative was successfully verified.

[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*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])

Rule 47

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + 1)), x] - Dist[(d*n)/(b*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d},
x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && LtQ[m, -1] &&  !(IntegerQ[n] &&  !IntegerQ[m]) &&  !(ILeQ[m + n + 2, 0
] && (FractionQ[m] || GeQ[2*n + m + 1, 0])) && IntLinearQ[a, b, c, d, m, n, x]

Rule 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 62

Int[((a_.) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(m_), x_Symbol] :> Dist[((a + b*x)^m*(c + d*x)^m)/((a + b*x)
*(c + d*x))^m, Int[(a*c + (b*c + a*d)*x + b*d*x^2)^m, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] &&
LtQ[-1, m, 0] && LeQ[3, Denominator[m], 4]

Rule 623

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{d = Denominator[p]}, Dist[(d*Sqrt[(b + 2*c*x)
^2])/(b + 2*c*x), Subst[Int[x^(d*(p + 1) - 1)/Sqrt[b^2 - 4*a*c + 4*c*x^d], x], x, (a + b*x + c*x^2)^(1/d)], x]
/; 3 <= d <= 4] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && RationalQ[p]

Rule 303

Int[(x_)/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Dist[(Sq
rt[2]*s)/(Sqrt[2 + Sqrt[3]]*r), Int[1/Sqrt[a + b*x^3], x], x] + Dist[1/r, Int[((1 - Sqrt[3])*s + r*x)/Sqrt[a +
b*x^3], x], x]] /; FreeQ[{a, b}, x] && PosQ[a]

Rule 218

Int[1/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Simp[(2*Sqr
t[2 + Sqrt[3]]*(s + r*x)*Sqrt[(s^2 - r*s*x + r^2*x^2)/((1 + Sqrt[3])*s + r*x)^2]*EllipticF[ArcSin[((1 - Sqrt[3
])*s + r*x)/((1 + Sqrt[3])*s + r*x)], -7 - 4*Sqrt[3]])/(3^(1/4)*r*Sqrt[a + b*x^3]*Sqrt[(s*(s + r*x))/((1 + Sqr
t[3])*s + r*x)^2]), x]] /; FreeQ[{a, b}, x] && PosQ[a]

Rule 1877

Int[((c_) + (d_.)*(x_))/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Simplify[((1 - Sqrt[3])*d)/c]]
, s = Denom[Simplify[((1 - Sqrt[3])*d)/c]]}, Simp[(2*d*s^3*Sqrt[a + b*x^3])/(a*r^2*((1 + Sqrt[3])*s + r*x)), x
] - Simp[(3^(1/4)*Sqrt[2 - Sqrt[3]]*d*s*(s + r*x)*Sqrt[(s^2 - r*s*x + r^2*x^2)/((1 + Sqrt[3])*s + r*x)^2]*Elli
pticE[ArcSin[((1 - Sqrt[3])*s + r*x)/((1 + Sqrt[3])*s + r*x)], -7 - 4*Sqrt[3]])/(r^2*Sqrt[a + b*x^3]*Sqrt[(s*(
s + r*x))/((1 + Sqrt[3])*s + r*x)^2]), x]] /; FreeQ[{a, b, c, d}, x] && PosQ[a] && EqQ[b*c^3 - 2*(5 - 3*Sqrt[3
])*a*d^3, 0]

Rubi steps

\begin{align*} \int \frac{(a+b x)^{8/3}}{(c+d x)^{4/3}} \, dx &=-\frac{3 (a+b x)^{8/3}}{d \sqrt [3]{c+d x}}+\frac{(8 b) \int \frac{(a+b x)^{5/3}}{\sqrt [3]{c+d x}} \, dx}{d}\\ &=-\frac{3 (a+b x)^{8/3}}{d \sqrt [3]{c+d x}}+\frac{24 b (a+b x)^{5/3} (c+d x)^{2/3}}{7 d^2}-\frac{(40 b (b c-a d)) \int \frac{(a+b x)^{2/3}}{\sqrt [3]{c+d x}} \, dx}{7 d^2}\\ &=-\frac{3 (a+b x)^{8/3}}{d \sqrt [3]{c+d x}}-\frac{30 b (b c-a d) (a+b x)^{2/3} (c+d x)^{2/3}}{7 d^3}+\frac{24 b (a+b x)^{5/3} (c+d x)^{2/3}}{7 d^2}+\frac{\left (20 b (b c-a d)^2\right ) \int \frac{1}{\sqrt [3]{a+b x} \sqrt [3]{c+d x}} \, dx}{7 d^3}\\ &=-\frac{3 (a+b x)^{8/3}}{d \sqrt [3]{c+d x}}-\frac{30 b (b c-a d) (a+b x)^{2/3} (c+d x)^{2/3}}{7 d^3}+\frac{24 b (a+b x)^{5/3} (c+d x)^{2/3}}{7 d^2}+\frac{\left (20 b (b c-a d)^2 \sqrt [3]{(a+b x) (c+d x)}\right ) \int \frac{1}{\sqrt [3]{a c+(b c+a d) x+b d x^2}} \, dx}{7 d^3 \sqrt [3]{a+b x} \sqrt [3]{c+d x}}\\ &=-\frac{3 (a+b x)^{8/3}}{d \sqrt [3]{c+d x}}-\frac{30 b (b c-a d) (a+b x)^{2/3} (c+d x)^{2/3}}{7 d^3}+\frac{24 b (a+b x)^{5/3} (c+d x)^{2/3}}{7 d^2}+\frac{\left (60 b (b c-a d)^2 \sqrt [3]{(a+b x) (c+d x)} \sqrt{(b c+a d+2 b d x)^2}\right ) \operatorname{Subst}\left (\int \frac{x}{\sqrt{-4 a b c d+(b c+a d)^2+4 b d x^3}} \, dx,x,\sqrt [3]{(a+b x) (c+d x)}\right )}{7 d^3 \sqrt [3]{a+b x} \sqrt [3]{c+d x} (b c+a d+2 b d x)}\\ &=-\frac{3 (a+b x)^{8/3}}{d \sqrt [3]{c+d x}}-\frac{30 b (b c-a d) (a+b x)^{2/3} (c+d x)^{2/3}}{7 d^3}+\frac{24 b (a+b x)^{5/3} (c+d x)^{2/3}}{7 d^2}+\frac{\left (30 \sqrt [3]{2} b^{2/3} (b c-a d)^2 \sqrt [3]{(a+b x) (c+d x)} \sqrt{(b c+a d+2 b d x)^2}\right ) \operatorname{Subst}\left (\int \frac{\left (1-\sqrt{3}\right ) (b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} x}{\sqrt{-4 a b c d+(b c+a d)^2+4 b d x^3}} \, dx,x,\sqrt [3]{(a+b x) (c+d x)}\right )}{7 d^{10/3} \sqrt [3]{a+b x} \sqrt [3]{c+d x} (b c+a d+2 b d x)}+\frac{\left (30\ 2^{5/6} b^{2/3} (b c-a d)^{8/3} \sqrt [3]{(a+b x) (c+d x)} \sqrt{(b c+a d+2 b d x)^2}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{-4 a b c d+(b c+a d)^2+4 b d x^3}} \, dx,x,\sqrt [3]{(a+b x) (c+d x)}\right )}{7 \sqrt{2+\sqrt{3}} d^{10/3} \sqrt [3]{a+b x} \sqrt [3]{c+d x} (b c+a d+2 b d x)}\\ &=-\frac{3 (a+b x)^{8/3}}{d \sqrt [3]{c+d x}}-\frac{30 b (b c-a d) (a+b x)^{2/3} (c+d x)^{2/3}}{7 d^3}+\frac{24 b (a+b x)^{5/3} (c+d x)^{2/3}}{7 d^2}+\frac{30\ 2^{2/3} \sqrt [3]{b} (b c-a d)^2 \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}}{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{15\ 2^{2/3} \sqrt [4]{3} \sqrt{2-\sqrt{3}} \sqrt [3]{b} (b c-a d)^{8/3} \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} (b c-a d)^{2/3} \sqrt [3]{(a+b x) (c+d x)}+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 )}{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} (b c-a d)^{8/3} \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} (b c-a d)^{2/3} \sqrt [3]{(a+b x) (c+d x)}+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 )}{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}}\\ \end{align*}

Mathematica [C]  time = 0.0579846, size = 73, normalized size = 0.05 \[ \frac{3 (a+b x)^{11/3} \left (\frac{b (c+d x)}{b c-a d}\right )^{4/3} \, _2F_1\left (\frac{4}{3},\frac{11}{3};\frac{14}{3};\frac{d (a+b x)}{a d-b c}\right )}{11 b (c+d x)^{4/3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3*(a + b*x)^(11/3)*((b*(c + d*x))/(b*c - a*d))^(4/3)*Hypergeometric2F1[4/3, 11/3, 14/3, (d*(a + b*x))/(-(b*c)
+ a*d)])/(11*b*(c + d*x)^(4/3))

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Maple [F]  time = 0.037, size = 0, normalized size = 0. \begin{align*} \int{ \left ( bx+a \right ) ^{{\frac{8}{3}}} \left ( dx+c \right ) ^{-{\frac{4}{3}}}}\, dx \end{align*}

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)

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

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)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\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{2}{3}}}{d^{2} x^{2} + 2 \, c d x + c^{2}}, x\right ) \end{align*}

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)^(2/3)/(d^2*x^2 + 2*c*d*x + c^2), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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

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

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)