\(\int \frac {(d+e x^3) (a+b x^3+c x^6)^p}{x} \, dx\) [75]

Optimal result

Integrand size = 25, antiderivative size = 290 \[ \int \frac {\left (d+e x^3\right ) \left (a+b x^3+c x^6\right )^p}{x} \, dx=\frac {2^{-1+2 p} d \left (\frac {b-\sqrt {b^2-4 a c}+2 c x^3}{c x^3}\right )^{-p} \left (\frac {b+\sqrt {b^2-4 a c}+2 c x^3}{c x^3}\right )^{-p} \left (a+b x^3+c x^6\right )^p \operatorname {AppellF1}\left (-2 p,-p,-p,1-2 p,-\frac {b-\sqrt {b^2-4 a c}}{2 c x^3},-\frac {b+\sqrt {b^2-4 a c}}{2 c x^3}\right )}{3 p}-\frac {2^{1+p} e \left (-\frac {b-\sqrt {b^2-4 a c}+2 c x^3}{\sqrt {b^2-4 a c}}\right )^{-1-p} \left (a+b x^3+c x^6\right )^{1+p} \operatorname {Hypergeometric2F1}\left (-p,1+p,2+p,\frac {b+\sqrt {b^2-4 a c}+2 c x^3}{2 \sqrt {b^2-4 a c}}\right )}{3 \sqrt {b^2-4 a c} (1+p)} \] Output:

1/3*2^(-1+2*p)*d*(c*x^6+b*x^3+a)^p*AppellF1(-2*p,-p,-p,1-2*p,-1/2*(b-(-4*a 
*c+b^2)^(1/2))/c/x^3,-1/2*(b+(-4*a*c+b^2)^(1/2))/c/x^3)/p/(((b-(-4*a*c+b^2 
)^(1/2)+2*c*x^3)/c/x^3)^p)/(((b+(-4*a*c+b^2)^(1/2)+2*c*x^3)/c/x^3)^p)-1/3* 
2^(p+1)*e*(-(b-(-4*a*c+b^2)^(1/2)+2*c*x^3)/(-4*a*c+b^2)^(1/2))^(-1-p)*(c*x 
^6+b*x^3+a)^(p+1)*hypergeom([-p, p+1],[2+p],1/2*(b+(-4*a*c+b^2)^(1/2)+2*c* 
x^3)/(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2)/(p+1)
 

Mathematica [A] (warning: unable to verify)

Time = 0.74 (sec) , antiderivative size = 276, normalized size of antiderivative = 0.95 \[ \int \frac {\left (d+e x^3\right ) \left (a+b x^3+c x^6\right )^p}{x} \, dx=\frac {1}{6} \left (a+b x^3+c x^6\right )^p \left (\frac {4^p d \left (\frac {b-\sqrt {b^2-4 a c}+2 c x^3}{c x^3}\right )^{-p} \left (\frac {b+\sqrt {b^2-4 a c}+2 c x^3}{c x^3}\right )^{-p} \operatorname {AppellF1}\left (-2 p,-p,-p,1-2 p,-\frac {b+\sqrt {b^2-4 a c}}{2 c x^3},\frac {-b+\sqrt {b^2-4 a c}}{2 c x^3}\right )}{p}+\frac {2^p e \left (b-\sqrt {b^2-4 a c}+2 c x^3\right ) \left (\frac {b+\sqrt {b^2-4 a c}+2 c x^3}{\sqrt {b^2-4 a c}}\right )^{-p} \operatorname {Hypergeometric2F1}\left (-p,1+p,2+p,\frac {-b+\sqrt {b^2-4 a c}-2 c x^3}{2 \sqrt {b^2-4 a c}}\right )}{c (1+p)}\right ) \] Input:

Integrate[((d + e*x^3)*(a + b*x^3 + c*x^6)^p)/x,x]
 

Output:

((a + b*x^3 + c*x^6)^p*((4^p*d*AppellF1[-2*p, -p, -p, 1 - 2*p, -1/2*(b + S 
qrt[b^2 - 4*a*c])/(c*x^3), (-b + Sqrt[b^2 - 4*a*c])/(2*c*x^3)])/(p*((b - S 
qrt[b^2 - 4*a*c] + 2*c*x^3)/(c*x^3))^p*((b + Sqrt[b^2 - 4*a*c] + 2*c*x^3)/ 
(c*x^3))^p) + (2^p*e*(b - Sqrt[b^2 - 4*a*c] + 2*c*x^3)*Hypergeometric2F1[- 
p, 1 + p, 2 + p, (-b + Sqrt[b^2 - 4*a*c] - 2*c*x^3)/(2*Sqrt[b^2 - 4*a*c])] 
)/(c*(1 + p)*((b + Sqrt[b^2 - 4*a*c] + 2*c*x^3)/Sqrt[b^2 - 4*a*c])^p)))/6
 

Rubi [A] (verified)

Time = 0.43 (sec) , antiderivative size = 289, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {1802, 1269, 1096, 1178, 150}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {\left (d+e x^3\right ) \left (a+b x^3+c x^6\right )^p}{x} \, dx\)

\(\Big \downarrow \) 1802

\(\displaystyle \frac {1}{3} \int \frac {\left (e x^3+d\right ) \left (c x^6+b x^3+a\right )^p}{x^3}dx^3\)

\(\Big \downarrow \) 1269

\(\displaystyle \frac {1}{3} \left (d \int \frac {\left (c x^6+b x^3+a\right )^p}{x^3}dx^3+e \int \left (c x^6+b x^3+a\right )^pdx^3\right )\)

\(\Big \downarrow \) 1096

\(\displaystyle \frac {1}{3} \left (d \int \frac {\left (c x^6+b x^3+a\right )^p}{x^3}dx^3-\frac {e 2^{p+1} \left (-\frac {-\sqrt {b^2-4 a c}+b+2 c x^3}{\sqrt {b^2-4 a c}}\right )^{-p-1} \left (a+b x^3+c x^6\right )^{p+1} \operatorname {Hypergeometric2F1}\left (-p,p+1,p+2,\frac {2 c x^3+b+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}\right )}{(p+1) \sqrt {b^2-4 a c}}\right )\)

\(\Big \downarrow \) 1178

\(\displaystyle \frac {1}{3} \left (-d 4^p \left (\frac {1}{x^3}\right )^{2 p} \left (\frac {-\sqrt {b^2-4 a c}+b+2 c x^3}{c x^3}\right )^{-p} \left (\frac {\sqrt {b^2-4 a c}+b+2 c x^3}{c x^3}\right )^{-p} \left (a+b x^3+c x^6\right )^p \int \left (\frac {b-\sqrt {b^2-4 a c}}{2 c x^3}+1\right )^p \left (\frac {b+\sqrt {b^2-4 a c}}{2 c x^3}+1\right )^p \left (\frac {1}{x^3}\right )^{-2 p-1}d\frac {1}{x^3}-\frac {e 2^{p+1} \left (a+b x^3+c x^6\right )^{p+1} \left (-\frac {-\sqrt {b^2-4 a c}+b+2 c x^3}{\sqrt {b^2-4 a c}}\right )^{-p-1} \operatorname {Hypergeometric2F1}\left (-p,p+1,p+2,\frac {2 c x^3+b+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}\right )}{(p+1) \sqrt {b^2-4 a c}}\right )\)

\(\Big \downarrow \) 150

\(\displaystyle \frac {1}{3} \left (\frac {d 2^{2 p-1} \left (\frac {-\sqrt {b^2-4 a c}+b+2 c x^3}{c x^3}\right )^{-p} \left (\frac {\sqrt {b^2-4 a c}+b+2 c x^3}{c x^3}\right )^{-p} \left (a+b x^3+c x^6\right )^p \operatorname {AppellF1}\left (-2 p,-p,-p,1-2 p,-\frac {b-\sqrt {b^2-4 a c}}{2 c x^3},-\frac {b+\sqrt {b^2-4 a c}}{2 c x^3}\right )}{p}-\frac {e 2^{p+1} \left (-\frac {-\sqrt {b^2-4 a c}+b+2 c x^3}{\sqrt {b^2-4 a c}}\right )^{-p-1} \left (a+b x^3+c x^6\right )^{p+1} \operatorname {Hypergeometric2F1}\left (-p,p+1,p+2,\frac {2 c x^3+b+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}\right )}{(p+1) \sqrt {b^2-4 a c}}\right )\)

Input:

Int[((d + e*x^3)*(a + b*x^3 + c*x^6)^p)/x,x]
 

Output:

((2^(-1 + 2*p)*d*(a + b*x^3 + c*x^6)^p*AppellF1[-2*p, -p, -p, 1 - 2*p, -1/ 
2*(b - Sqrt[b^2 - 4*a*c])/(c*x^3), -1/2*(b + Sqrt[b^2 - 4*a*c])/(c*x^3)])/ 
(p*((b - Sqrt[b^2 - 4*a*c] + 2*c*x^3)/(c*x^3))^p*((b + Sqrt[b^2 - 4*a*c] + 
 2*c*x^3)/(c*x^3))^p) - (2^(1 + p)*e*(-((b - Sqrt[b^2 - 4*a*c] + 2*c*x^3)/ 
Sqrt[b^2 - 4*a*c]))^(-1 - p)*(a + b*x^3 + c*x^6)^(1 + p)*Hypergeometric2F1 
[-p, 1 + p, 2 + p, (b + Sqrt[b^2 - 4*a*c] + 2*c*x^3)/(2*Sqrt[b^2 - 4*a*c]) 
])/(Sqrt[b^2 - 4*a*c]*(1 + p)))/3
 

Defintions of rubi rules used

rule 150

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((e_) + (f_.)*(x_))^(p_), x_ 
] :> Simp[c^n*e^p*((b*x)^(m + 1)/(b*(m + 1)))*AppellF1[m + 1, -n, -p, m + 2 
, (-d)*(x/c), (-f)*(x/e)], x] /; FreeQ[{b, c, d, e, f, m, n, p}, x] &&  !In 
tegerQ[m] &&  !IntegerQ[n] && GtQ[c, 0] && (IntegerQ[p] || GtQ[e, 0])
 

rule 1096

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Rt[b^2 
 - 4*a*c, 2]}, Simp[(-(a + b*x + c*x^2)^(p + 1)/(q*(p + 1)*((q - b - 2*c*x) 
/(2*q))^(p + 1)))*Hypergeometric2F1[-p, p + 1, p + 2, (b + q + 2*c*x)/(2*q) 
], x]] /; FreeQ[{a, b, c, p}, x] &&  !IntegerQ[4*p] &&  !IntegerQ[3*p]
 

rule 1178

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S 
ymbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(-(1/(d + e*x))^(2*p))*((a + 
b*x + c*x^2)^p/(e*(e*((b - q + 2*c*x)/(2*c*(d + e*x))))^p*(e*((b + q + 2*c* 
x)/(2*c*(d + e*x))))^p))   Subst[Int[x^(-m - 2*(p + 1))*Simp[1 - (d - e*((b 
 - q)/(2*c)))*x, x]^p*Simp[1 - (d - e*((b + q)/(2*c)))*x, x]^p, x], x, 1/(d 
 + e*x)], x]] /; FreeQ[{a, b, c, d, e, p}, x] && ILtQ[m, 0]
 

rule 1269

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[g/e   Int[(d + e*x)^(m + 1)*(a + b*x + 
 c*x^2)^p, x], x] + Simp[(e*f - d*g)/e   Int[(d + e*x)^m*(a + b*x + c*x^2)^ 
p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] &&  !IGtQ[m, 0]
 

rule 1802

Int[(x_)^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.)*((d_) + ( 
e_.)*(x_)^(n_))^(q_.), x_Symbol] :> Simp[1/n   Subst[Int[x^(Simplify[(m + 1 
)/n] - 1)*(d + e*x)^q*(a + b*x + c*x^2)^p, x], x, x^n], x] /; FreeQ[{a, b, 
c, d, e, m, n, p, q}, x] && EqQ[n2, 2*n] && IntegerQ[Simplify[(m + 1)/n]]
 

Maple [F]

\[\int \frac {\left (e \,x^{3}+d \right ) \left (c \,x^{6}+b \,x^{3}+a \right )^{p}}{x}d x\]

Input:

int((e*x^3+d)*(c*x^6+b*x^3+a)^p/x,x)
 

Output:

int((e*x^3+d)*(c*x^6+b*x^3+a)^p/x,x)
 

Fricas [F]

\[ \int \frac {\left (d+e x^3\right ) \left (a+b x^3+c x^6\right )^p}{x} \, dx=\int { \frac {{\left (e x^{3} + d\right )} {\left (c x^{6} + b x^{3} + a\right )}^{p}}{x} \,d x } \] Input:

integrate((e*x^3+d)*(c*x^6+b*x^3+a)^p/x,x, algorithm="fricas")
 

Output:

integral((e*x^3 + d)*(c*x^6 + b*x^3 + a)^p/x, x)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (d+e x^3\right ) \left (a+b x^3+c x^6\right )^p}{x} \, dx=\text {Timed out} \] Input:

integrate((e*x**3+d)*(c*x**6+b*x**3+a)**p/x,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {\left (d+e x^3\right ) \left (a+b x^3+c x^6\right )^p}{x} \, dx=\int { \frac {{\left (e x^{3} + d\right )} {\left (c x^{6} + b x^{3} + a\right )}^{p}}{x} \,d x } \] Input:

integrate((e*x^3+d)*(c*x^6+b*x^3+a)^p/x,x, algorithm="maxima")
 

Output:

integrate((e*x^3 + d)*(c*x^6 + b*x^3 + a)^p/x, x)
 

Giac [F]

\[ \int \frac {\left (d+e x^3\right ) \left (a+b x^3+c x^6\right )^p}{x} \, dx=\int { \frac {{\left (e x^{3} + d\right )} {\left (c x^{6} + b x^{3} + a\right )}^{p}}{x} \,d x } \] Input:

integrate((e*x^3+d)*(c*x^6+b*x^3+a)^p/x,x, algorithm="giac")
 

Output:

integrate((e*x^3 + d)*(c*x^6 + b*x^3 + a)^p/x, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (d+e x^3\right ) \left (a+b x^3+c x^6\right )^p}{x} \, dx=\int \frac {\left (e\,x^3+d\right )\,{\left (c\,x^6+b\,x^3+a\right )}^p}{x} \,d x \] Input:

int(((d + e*x^3)*(a + b*x^3 + c*x^6)^p)/x,x)
 

Output:

int(((d + e*x^3)*(a + b*x^3 + c*x^6)^p)/x, x)
 

Reduce [F]

\[ \int \frac {\left (d+e x^3\right ) \left (a+b x^3+c x^6\right )^p}{x} \, dx=\frac {2 \left (c \,x^{6}+b \,x^{3}+a \right )^{p} a e p +2 \left (c \,x^{6}+b \,x^{3}+a \right )^{p} b d p +\left (c \,x^{6}+b \,x^{3}+a \right )^{p} b d +\left (c \,x^{6}+b \,x^{3}+a \right )^{p} b e p \,x^{3}+12 \left (\int \frac {\left (c \,x^{6}+b \,x^{3}+a \right )^{p}}{2 c p \,x^{7}+c \,x^{7}+2 b p \,x^{4}+b \,x^{4}+2 a p x +a x}d x \right ) a b d \,p^{3}+12 \left (\int \frac {\left (c \,x^{6}+b \,x^{3}+a \right )^{p}}{2 c p \,x^{7}+c \,x^{7}+2 b p \,x^{4}+b \,x^{4}+2 a p x +a x}d x \right ) a b d \,p^{2}+3 \left (\int \frac {\left (c \,x^{6}+b \,x^{3}+a \right )^{p}}{2 c p \,x^{7}+c \,x^{7}+2 b p \,x^{4}+b \,x^{4}+2 a p x +a x}d x \right ) a b d p -24 \left (\int \frac {\left (c \,x^{6}+b \,x^{3}+a \right )^{p} x^{5}}{2 c p \,x^{6}+c \,x^{6}+2 b p \,x^{3}+b \,x^{3}+2 a p +a}d x \right ) a c e \,p^{3}-12 \left (\int \frac {\left (c \,x^{6}+b \,x^{3}+a \right )^{p} x^{5}}{2 c p \,x^{6}+c \,x^{6}+2 b p \,x^{3}+b \,x^{3}+2 a p +a}d x \right ) a c e \,p^{2}+6 \left (\int \frac {\left (c \,x^{6}+b \,x^{3}+a \right )^{p} x^{5}}{2 c p \,x^{6}+c \,x^{6}+2 b p \,x^{3}+b \,x^{3}+2 a p +a}d x \right ) b^{2} e \,p^{3}+3 \left (\int \frac {\left (c \,x^{6}+b \,x^{3}+a \right )^{p} x^{5}}{2 c p \,x^{6}+c \,x^{6}+2 b p \,x^{3}+b \,x^{3}+2 a p +a}d x \right ) b^{2} e \,p^{2}-12 \left (\int \frac {\left (c \,x^{6}+b \,x^{3}+a \right )^{p} x^{5}}{2 c p \,x^{6}+c \,x^{6}+2 b p \,x^{3}+b \,x^{3}+2 a p +a}d x \right ) b c d \,p^{3}-12 \left (\int \frac {\left (c \,x^{6}+b \,x^{3}+a \right )^{p} x^{5}}{2 c p \,x^{6}+c \,x^{6}+2 b p \,x^{3}+b \,x^{3}+2 a p +a}d x \right ) b c d \,p^{2}-3 \left (\int \frac {\left (c \,x^{6}+b \,x^{3}+a \right )^{p} x^{5}}{2 c p \,x^{6}+c \,x^{6}+2 b p \,x^{3}+b \,x^{3}+2 a p +a}d x \right ) b c d p}{3 b p \left (2 p +1\right )} \] Input:

int((e*x^3+d)*(c*x^6+b*x^3+a)^p/x,x)
 

Output:

(2*(a + b*x**3 + c*x**6)**p*a*e*p + 2*(a + b*x**3 + c*x**6)**p*b*d*p + (a 
+ b*x**3 + c*x**6)**p*b*d + (a + b*x**3 + c*x**6)**p*b*e*p*x**3 + 12*int(( 
a + b*x**3 + c*x**6)**p/(2*a*p*x + a*x + 2*b*p*x**4 + b*x**4 + 2*c*p*x**7 
+ c*x**7),x)*a*b*d*p**3 + 12*int((a + b*x**3 + c*x**6)**p/(2*a*p*x + a*x + 
 2*b*p*x**4 + b*x**4 + 2*c*p*x**7 + c*x**7),x)*a*b*d*p**2 + 3*int((a + b*x 
**3 + c*x**6)**p/(2*a*p*x + a*x + 2*b*p*x**4 + b*x**4 + 2*c*p*x**7 + c*x** 
7),x)*a*b*d*p - 24*int(((a + b*x**3 + c*x**6)**p*x**5)/(2*a*p + a + 2*b*p* 
x**3 + b*x**3 + 2*c*p*x**6 + c*x**6),x)*a*c*e*p**3 - 12*int(((a + b*x**3 + 
 c*x**6)**p*x**5)/(2*a*p + a + 2*b*p*x**3 + b*x**3 + 2*c*p*x**6 + c*x**6), 
x)*a*c*e*p**2 + 6*int(((a + b*x**3 + c*x**6)**p*x**5)/(2*a*p + a + 2*b*p*x 
**3 + b*x**3 + 2*c*p*x**6 + c*x**6),x)*b**2*e*p**3 + 3*int(((a + b*x**3 + 
c*x**6)**p*x**5)/(2*a*p + a + 2*b*p*x**3 + b*x**3 + 2*c*p*x**6 + c*x**6),x 
)*b**2*e*p**2 - 12*int(((a + b*x**3 + c*x**6)**p*x**5)/(2*a*p + a + 2*b*p* 
x**3 + b*x**3 + 2*c*p*x**6 + c*x**6),x)*b*c*d*p**3 - 12*int(((a + b*x**3 + 
 c*x**6)**p*x**5)/(2*a*p + a + 2*b*p*x**3 + b*x**3 + 2*c*p*x**6 + c*x**6), 
x)*b*c*d*p**2 - 3*int(((a + b*x**3 + c*x**6)**p*x**5)/(2*a*p + a + 2*b*p*x 
**3 + b*x**3 + 2*c*p*x**6 + c*x**6),x)*b*c*d*p)/(3*b*p*(2*p + 1))