3.1.39 \(\int \frac {1}{(b x+c x^2)^{7/3}} \, dx\) [39]

Optimal. Leaf size=838 \[ \frac {3 (b+2 c x) \left (-\frac {c \left (b x+c x^2\right )}{b^2}\right )^{7/3}}{4 c \left (-\frac {c x (b+c x)}{b^2}\right )^{4/3} \left (b x+c x^2\right )^{7/3}}+\frac {15 (b+2 c x) \left (-\frac {c \left (b x+c x^2\right )}{b^2}\right )^{7/3}}{2 c \sqrt [3]{-\frac {c x (b+c x)}{b^2}} \left (b x+c x^2\right )^{7/3}}+\frac {15 (b+2 c x) \left (-\frac {c \left (b x+c x^2\right )}{b^2}\right )^{7/3}}{\sqrt [3]{2} c \left (b x+c x^2\right )^{7/3} \left (1-\sqrt {3}-2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}\right )}+\frac {15 \sqrt [4]{3} \sqrt {2+\sqrt {3}} b^2 \left (-\frac {c \left (b x+c x^2\right )}{b^2}\right )^{7/3} \left (1-2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}\right ) \sqrt {\frac {1+2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}+2 \sqrt [3]{2} \left (-\frac {c x (b+c x)}{b^2}\right )^{2/3}}{\left (1-\sqrt {3}-2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}\right )^2}} E\left (\sin ^{-1}\left (\frac {1+\sqrt {3}-2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}}{1-\sqrt {3}-2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}}\right )|-7+4 \sqrt {3}\right )}{2 \sqrt [3]{2} c (b+2 c x) \left (b x+c x^2\right )^{7/3} \sqrt {-\frac {1-2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}}{\left (1-\sqrt {3}-2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}\right )^2}}}-\frac {5 \sqrt [6]{2} 3^{3/4} b^2 \left (-\frac {c \left (b x+c x^2\right )}{b^2}\right )^{7/3} \left (1-2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}\right ) \sqrt {\frac {1+2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}+2 \sqrt [3]{2} \left (-\frac {c x (b+c x)}{b^2}\right )^{2/3}}{\left (1-\sqrt {3}-2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}\right )^2}} F\left (\sin ^{-1}\left (\frac {1+\sqrt {3}-2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}}{1-\sqrt {3}-2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}}\right )|-7+4 \sqrt {3}\right )}{c (b+2 c x) \left (b x+c x^2\right )^{7/3} \sqrt {-\frac {1-2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}}{\left (1-\sqrt {3}-2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}\right )^2}}} \]

[Out]

3/4*(2*c*x+b)*(-c*(c*x^2+b*x)/b^2)^(7/3)/c/(-c*x*(c*x+b)/b^2)^(4/3)/(c*x^2+b*x)^(7/3)+15/2*(2*c*x+b)*(-c*(c*x^
2+b*x)/b^2)^(7/3)/c/(-c*x*(c*x+b)/b^2)^(1/3)/(c*x^2+b*x)^(7/3)+15/2*(2*c*x+b)*(-c*(c*x^2+b*x)/b^2)^(7/3)*2^(2/
3)/c/(c*x^2+b*x)^(7/3)/(1-2^(2/3)*(-c*x*(c*x+b)/b^2)^(1/3)-3^(1/2))-5*2^(1/6)*3^(3/4)*b^2*(-c*(c*x^2+b*x)/b^2)
^(7/3)*(1-2^(2/3)*(-c*x*(c*x+b)/b^2)^(1/3))*EllipticF((1-2^(2/3)*(-c*x*(c*x+b)/b^2)^(1/3)+3^(1/2))/(1-2^(2/3)*
(-c*x*(c*x+b)/b^2)^(1/3)-3^(1/2)),2*I-I*3^(1/2))*((1+2^(2/3)*(-c*x*(c*x+b)/b^2)^(1/3)+2*2^(1/3)*(-c*x*(c*x+b)/
b^2)^(2/3))/(1-2^(2/3)*(-c*x*(c*x+b)/b^2)^(1/3)-3^(1/2))^2)^(1/2)/c/(2*c*x+b)/(c*x^2+b*x)^(7/3)/((-1+2^(2/3)*(
-c*x*(c*x+b)/b^2)^(1/3))/(1-2^(2/3)*(-c*x*(c*x+b)/b^2)^(1/3)-3^(1/2))^2)^(1/2)+15/4*3^(1/4)*b^2*(-c*(c*x^2+b*x
)/b^2)^(7/3)*(1-2^(2/3)*(-c*x*(c*x+b)/b^2)^(1/3))*EllipticE((1-2^(2/3)*(-c*x*(c*x+b)/b^2)^(1/3)+3^(1/2))/(1-2^
(2/3)*(-c*x*(c*x+b)/b^2)^(1/3)-3^(1/2)),2*I-I*3^(1/2))*((1+2^(2/3)*(-c*x*(c*x+b)/b^2)^(1/3)+2*2^(1/3)*(-c*x*(c
*x+b)/b^2)^(2/3))/(1-2^(2/3)*(-c*x*(c*x+b)/b^2)^(1/3)-3^(1/2))^2)^(1/2)*(1/2*6^(1/2)+1/2*2^(1/2))*2^(2/3)/c/(2
*c*x+b)/(c*x^2+b*x)^(7/3)/((-1+2^(2/3)*(-c*x*(c*x+b)/b^2)^(1/3))/(1-2^(2/3)*(-c*x*(c*x+b)/b^2)^(1/3)-3^(1/2))^
2)^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 1.12, antiderivative size = 838, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.538, Rules used = {636, 633, 205, 241, 310, 225, 1893} \begin {gather*} \frac {15 \sqrt [4]{3} \sqrt {2+\sqrt {3}} b^2 \left (1-2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}\right ) \sqrt {\frac {2 \sqrt [3]{2} \left (-\frac {c x (b+c x)}{b^2}\right )^{2/3}+2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}+1}{\left (-2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}-\sqrt {3}+1\right )^2}} E\left (\text {ArcSin}\left (\frac {-2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}+\sqrt {3}+1}{-2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}-\sqrt {3}+1}\right )|-7+4 \sqrt {3}\right ) \left (-\frac {c \left (c x^2+b x\right )}{b^2}\right )^{7/3}}{2 \sqrt [3]{2} c (b+2 c x) \left (c x^2+b x\right )^{7/3} \sqrt {-\frac {1-2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}}{\left (-2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}-\sqrt {3}+1\right )^2}}}-\frac {5 \sqrt [6]{2} 3^{3/4} b^2 \left (1-2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}\right ) \sqrt {\frac {2 \sqrt [3]{2} \left (-\frac {c x (b+c x)}{b^2}\right )^{2/3}+2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}+1}{\left (-2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}-\sqrt {3}+1\right )^2}} F\left (\text {ArcSin}\left (\frac {-2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}+\sqrt {3}+1}{-2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}-\sqrt {3}+1}\right )|-7+4 \sqrt {3}\right ) \left (-\frac {c \left (c x^2+b x\right )}{b^2}\right )^{7/3}}{c (b+2 c x) \left (c x^2+b x\right )^{7/3} \sqrt {-\frac {1-2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}}{\left (-2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}-\sqrt {3}+1\right )^2}}}+\frac {15 (b+2 c x) \left (-\frac {c \left (c x^2+b x\right )}{b^2}\right )^{7/3}}{\sqrt [3]{2} c \left (c x^2+b x\right )^{7/3} \left (-2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}-\sqrt {3}+1\right )}+\frac {15 (b+2 c x) \left (-\frac {c \left (c x^2+b x\right )}{b^2}\right )^{7/3}}{2 c \sqrt [3]{-\frac {c x (b+c x)}{b^2}} \left (c x^2+b x\right )^{7/3}}+\frac {3 (b+2 c x) \left (-\frac {c \left (c x^2+b x\right )}{b^2}\right )^{7/3}}{4 c \left (-\frac {c x (b+c x)}{b^2}\right )^{4/3} \left (c x^2+b x\right )^{7/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(b*x + c*x^2)^(-7/3),x]

[Out]

(3*(b + 2*c*x)*(-((c*(b*x + c*x^2))/b^2))^(7/3))/(4*c*(-((c*x*(b + c*x))/b^2))^(4/3)*(b*x + c*x^2)^(7/3)) + (1
5*(b + 2*c*x)*(-((c*(b*x + c*x^2))/b^2))^(7/3))/(2*c*(-((c*x*(b + c*x))/b^2))^(1/3)*(b*x + c*x^2)^(7/3)) + (15
*(b + 2*c*x)*(-((c*(b*x + c*x^2))/b^2))^(7/3))/(2^(1/3)*c*(b*x + c*x^2)^(7/3)*(1 - Sqrt[3] - 2^(2/3)*(-((c*x*(
b + c*x))/b^2))^(1/3))) + (15*3^(1/4)*Sqrt[2 + Sqrt[3]]*b^2*(-((c*(b*x + c*x^2))/b^2))^(7/3)*(1 - 2^(2/3)*(-((
c*x*(b + c*x))/b^2))^(1/3))*Sqrt[(1 + 2^(2/3)*(-((c*x*(b + c*x))/b^2))^(1/3) + 2*2^(1/3)*(-((c*x*(b + c*x))/b^
2))^(2/3))/(1 - Sqrt[3] - 2^(2/3)*(-((c*x*(b + c*x))/b^2))^(1/3))^2]*EllipticE[ArcSin[(1 + Sqrt[3] - 2^(2/3)*(
-((c*x*(b + c*x))/b^2))^(1/3))/(1 - Sqrt[3] - 2^(2/3)*(-((c*x*(b + c*x))/b^2))^(1/3))], -7 + 4*Sqrt[3]])/(2*2^
(1/3)*c*(b + 2*c*x)*(b*x + c*x^2)^(7/3)*Sqrt[-((1 - 2^(2/3)*(-((c*x*(b + c*x))/b^2))^(1/3))/(1 - Sqrt[3] - 2^(
2/3)*(-((c*x*(b + c*x))/b^2))^(1/3))^2)]) - (5*2^(1/6)*3^(3/4)*b^2*(-((c*(b*x + c*x^2))/b^2))^(7/3)*(1 - 2^(2/
3)*(-((c*x*(b + c*x))/b^2))^(1/3))*Sqrt[(1 + 2^(2/3)*(-((c*x*(b + c*x))/b^2))^(1/3) + 2*2^(1/3)*(-((c*x*(b + c
*x))/b^2))^(2/3))/(1 - Sqrt[3] - 2^(2/3)*(-((c*x*(b + c*x))/b^2))^(1/3))^2]*EllipticF[ArcSin[(1 + Sqrt[3] - 2^
(2/3)*(-((c*x*(b + c*x))/b^2))^(1/3))/(1 - Sqrt[3] - 2^(2/3)*(-((c*x*(b + c*x))/b^2))^(1/3))], -7 + 4*Sqrt[3]]
)/(c*(b + 2*c*x)*(b*x + c*x^2)^(7/3)*Sqrt[-((1 - 2^(2/3)*(-((c*x*(b + c*x))/b^2))^(1/3))/(1 - Sqrt[3] - 2^(2/3
)*(-((c*x*(b + c*x))/b^2))^(1/3))^2)])

Rule 205

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^n)^(p + 1)/(a*n*(p + 1))), x] + Dist[(n*(p
+ 1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (
IntegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[
p])

Rule 225

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

Rule 241

Int[((a_) + (b_.)*(x_)^2)^(-1/3), x_Symbol] :> Dist[3*(Sqrt[b*x^2]/(2*b*x)), Subst[Int[x/Sqrt[-a + x^3], x], x
, (a + b*x^2)^(1/3)], x] /; FreeQ[{a, b}, x]

Rule 310

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

Rule 633

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[1/(2*c*(-4*(c/(b^2 - 4*a*c)))^p), Subst[Int[Si
mp[1 - x^2/(b^2 - 4*a*c), x]^p, x], x, b + 2*c*x], x] /; FreeQ[{a, b, c, p}, x] && GtQ[4*a - b^2/c, 0]

Rule 636

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

Rule 1893

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]/(r^2
*Sqrt[a + b*x^3]*Sqrt[(-s)*((s + r*x)/((1 - Sqrt[3])*s + r*x)^2)]))*EllipticE[ArcSin[((1 + Sqrt[3])*s + r*x)/(
(1 - Sqrt[3])*s + r*x)], -7 + 4*Sqrt[3]], x]] /; FreeQ[{a, b, c, d}, x] && NegQ[a] && EqQ[b*c^3 - 2*(5 + 3*Sqr
t[3])*a*d^3, 0]

Rubi steps

\begin {align*} \int \frac {1}{\left (b x+c x^2\right )^{7/3}} \, dx &=\frac {\left (-\frac {c \left (b x+c x^2\right )}{b^2}\right )^{7/3} \int \frac {1}{\left (-\frac {c x}{b}-\frac {c^2 x^2}{b^2}\right )^{7/3}} \, dx}{\left (b x+c x^2\right )^{7/3}}\\ &=-\frac {\left (8\ 2^{2/3} b^2 \left (-\frac {c \left (b x+c x^2\right )}{b^2}\right )^{7/3}\right ) \text {Subst}\left (\int \frac {1}{\left (1-\frac {b^2 x^2}{c^2}\right )^{7/3}} \, dx,x,-\frac {c}{b}-\frac {2 c^2 x}{b^2}\right )}{c^2 \left (b x+c x^2\right )^{7/3}}\\ &=\frac {3 (b+2 c x) \left (-\frac {c \left (b x+c x^2\right )}{b^2}\right )^{7/3}}{4 c \left (-\frac {c x (b+c x)}{b^2}\right )^{4/3} \left (b x+c x^2\right )^{7/3}}-\frac {\left (5\ 2^{2/3} b^2 \left (-\frac {c \left (b x+c x^2\right )}{b^2}\right )^{7/3}\right ) \text {Subst}\left (\int \frac {1}{\left (1-\frac {b^2 x^2}{c^2}\right )^{4/3}} \, dx,x,-\frac {c}{b}-\frac {2 c^2 x}{b^2}\right )}{c^2 \left (b x+c x^2\right )^{7/3}}\\ &=\frac {3 (b+2 c x) \left (-\frac {c \left (b x+c x^2\right )}{b^2}\right )^{7/3}}{4 c \left (-\frac {c x (b+c x)}{b^2}\right )^{4/3} \left (b x+c x^2\right )^{7/3}}+\frac {15 (b+2 c x) \left (-\frac {c \left (b x+c x^2\right )}{b^2}\right )^{7/3}}{2 c \sqrt [3]{-\frac {c x (b+c x)}{b^2}} \left (b x+c x^2\right )^{7/3}}+\frac {\left (5 b^2 \left (-\frac {c \left (b x+c x^2\right )}{b^2}\right )^{7/3}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{1-\frac {b^2 x^2}{c^2}}} \, dx,x,-\frac {c}{b}-\frac {2 c^2 x}{b^2}\right )}{\sqrt [3]{2} c^2 \left (b x+c x^2\right )^{7/3}}\\ &=\frac {3 (b+2 c x) \left (-\frac {c \left (b x+c x^2\right )}{b^2}\right )^{7/3}}{4 c \left (-\frac {c x (b+c x)}{b^2}\right )^{4/3} \left (b x+c x^2\right )^{7/3}}+\frac {15 (b+2 c x) \left (-\frac {c \left (b x+c x^2\right )}{b^2}\right )^{7/3}}{2 c \sqrt [3]{-\frac {c x (b+c x)}{b^2}} \left (b x+c x^2\right )^{7/3}}-\frac {\left (15 \left (-\frac {c \left (b x+c x^2\right )}{b^2}\right )^{7/3} \sqrt {-1-\frac {4 c x}{b}-\frac {4 c^2 x^2}{b^2}}\right ) \text {Subst}\left (\int \frac {x}{\sqrt {-1+x^3}} \, dx,x,2^{2/3} \sqrt [3]{-\frac {c x \left (1+\frac {c x}{b}\right )}{b}}\right )}{2 \sqrt [3]{2} \left (-\frac {c}{b}-\frac {2 c^2 x}{b^2}\right ) \left (b x+c x^2\right )^{7/3}}\\ &=\frac {3 (b+2 c x) \left (-\frac {c \left (b x+c x^2\right )}{b^2}\right )^{7/3}}{4 c \left (-\frac {c x (b+c x)}{b^2}\right )^{4/3} \left (b x+c x^2\right )^{7/3}}+\frac {15 (b+2 c x) \left (-\frac {c \left (b x+c x^2\right )}{b^2}\right )^{7/3}}{2 c \sqrt [3]{-\frac {c x (b+c x)}{b^2}} \left (b x+c x^2\right )^{7/3}}+\frac {\left (15 \left (-\frac {c \left (b x+c x^2\right )}{b^2}\right )^{7/3} \sqrt {-1-\frac {4 c x}{b}-\frac {4 c^2 x^2}{b^2}}\right ) \text {Subst}\left (\int \frac {1+\sqrt {3}-x}{\sqrt {-1+x^3}} \, dx,x,2^{2/3} \sqrt [3]{-\frac {c x \left (1+\frac {c x}{b}\right )}{b}}\right )}{2 \sqrt [3]{2} \left (-\frac {c}{b}-\frac {2 c^2 x}{b^2}\right ) \left (b x+c x^2\right )^{7/3}}-\frac {\left (15 \sqrt {2+\sqrt {3}} \left (-\frac {c \left (b x+c x^2\right )}{b^2}\right )^{7/3} \sqrt {-1-\frac {4 c x}{b}-\frac {4 c^2 x^2}{b^2}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-1+x^3}} \, dx,x,2^{2/3} \sqrt [3]{-\frac {c x \left (1+\frac {c x}{b}\right )}{b}}\right )}{2^{5/6} \left (-\frac {c}{b}-\frac {2 c^2 x}{b^2}\right ) \left (b x+c x^2\right )^{7/3}}\\ &=\frac {3 (b+2 c x) \left (-\frac {c \left (b x+c x^2\right )}{b^2}\right )^{7/3}}{4 c \left (-\frac {c x (b+c x)}{b^2}\right )^{4/3} \left (b x+c x^2\right )^{7/3}}+\frac {15 (b+2 c x) \left (-\frac {c \left (b x+c x^2\right )}{b^2}\right )^{7/3}}{2 c \sqrt [3]{-\frac {c x (b+c x)}{b^2}} \left (b x+c x^2\right )^{7/3}}-\frac {15 b^2 \left (-\frac {c \left (b x+c x^2\right )}{b^2}\right )^{7/3} \sqrt {-1-\frac {4 c x}{b}-\frac {4 c^2 x^2}{b^2}} \sqrt {-1-\frac {4 c x (b+c x)}{b^2}}}{\sqrt [3]{2} c (b+2 c x) \left (b x+c x^2\right )^{7/3} \left (1-\sqrt {3}-2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}\right )}+\frac {15 \sqrt [4]{3} \sqrt {2+\sqrt {3}} b^2 \left (-\frac {c \left (b x+c x^2\right )}{b^2}\right )^{7/3} \sqrt {-1-\frac {4 c x}{b}-\frac {4 c^2 x^2}{b^2}} \left (1-2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}\right ) \sqrt {\frac {1+2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}+2 \sqrt [3]{2} \left (-\frac {c x (b+c x)}{b^2}\right )^{2/3}}{\left (1-\sqrt {3}-2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}\right )^2}} E\left (\sin ^{-1}\left (\frac {1+\sqrt {3}-2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}}{1-\sqrt {3}-2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}}\right )|-7+4 \sqrt {3}\right )}{2 \sqrt [3]{2} c (b+2 c x) \left (b x+c x^2\right )^{7/3} \sqrt {-1-\frac {4 c x (b+c x)}{b^2}} \sqrt {-\frac {1-2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}}{\left (1-\sqrt {3}-2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}\right )^2}}}-\frac {5 \sqrt [6]{2} 3^{3/4} b^2 \left (-\frac {c \left (b x+c x^2\right )}{b^2}\right )^{7/3} \sqrt {-1-\frac {4 c x}{b}-\frac {4 c^2 x^2}{b^2}} \left (1-2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}\right ) \sqrt {\frac {1+2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}+2 \sqrt [3]{2} \left (-\frac {c x (b+c x)}{b^2}\right )^{2/3}}{\left (1-\sqrt {3}-2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}\right )^2}} F\left (\sin ^{-1}\left (\frac {1+\sqrt {3}-2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}}{1-\sqrt {3}-2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}}\right )|-7+4 \sqrt {3}\right )}{c (b+2 c x) \left (b x+c x^2\right )^{7/3} \sqrt {-1-\frac {4 c x (b+c x)}{b^2}} \sqrt {-\frac {1-2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}}{\left (1-\sqrt {3}-2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}\right )^2}}}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
time = 10.03, size = 50, normalized size = 0.06 \begin {gather*} -\frac {3 \sqrt [3]{1+\frac {c x}{b}} \, _2F_1\left (-\frac {4}{3},\frac {7}{3};-\frac {1}{3};-\frac {c x}{b}\right )}{4 b^2 x \sqrt [3]{x (b+c x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(b*x + c*x^2)^(-7/3),x]

[Out]

(-3*(1 + (c*x)/b)^(1/3)*Hypergeometric2F1[-4/3, 7/3, -1/3, -((c*x)/b)])/(4*b^2*x*(x*(b + c*x))^(1/3))

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Maple [F]
time = 0.05, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (c \,x^{2}+b x \right )^{\frac {7}{3}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(c*x^2+b*x)^(7/3),x)

[Out]

int(1/(c*x^2+b*x)^(7/3),x)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*x^2+b*x)^(7/3),x, algorithm="maxima")

[Out]

integrate((c*x^2 + b*x)^(-7/3), x)

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*x^2+b*x)^(7/3),x, algorithm="fricas")

[Out]

integral((c*x^2 + b*x)^(2/3)/(c^3*x^6 + 3*b*c^2*x^5 + 3*b^2*c*x^4 + b^3*x^3), x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (b x + c x^{2}\right )^{\frac {7}{3}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*x**2+b*x)**(7/3),x)

[Out]

Integral((b*x + c*x**2)**(-7/3), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*x^2+b*x)^(7/3),x, algorithm="giac")

[Out]

integrate((c*x^2 + b*x)^(-7/3), x)

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Mupad [B]
time = 0.25, size = 36, normalized size = 0.04 \begin {gather*} -\frac {3\,x\,{\left (\frac {c\,x}{b}+1\right )}^{7/3}\,{{}}_2{\mathrm {F}}_1\left (-\frac {4}{3},\frac {7}{3};\ -\frac {1}{3};\ -\frac {c\,x}{b}\right )}{4\,{\left (c\,x^2+b\,x\right )}^{7/3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b*x + c*x^2)^(7/3),x)

[Out]

-(3*x*((c*x)/b + 1)^(7/3)*hypergeom([-4/3, 7/3], -1/3, -(c*x)/b))/(4*(b*x + c*x^2)^(7/3))

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