\(\int \frac {(d+e x^3) (a+b x^3+c x^6)^p}{x^4} \, dx\) [76]

Optimal result

Integrand size = 25, antiderivative size = 336 \[ \int \frac {\left (d+e x^3\right ) \left (a+b x^3+c x^6\right )^p}{x^4} \, dx=-\frac {d \left (a+b x^3+c x^6\right )^{1+p}}{3 a x^3}+\frac {2^{-1+2 p} (a e+b d p) \left (\frac {b-\sqrt {b^2-4 a c}+2 c x^3}{c x^3}\right )^{-p} \left (\frac {b+\sqrt {b^2-4 a c}+2 c x^3}{c x^3}\right )^{-p} \left (a+b x^3+c x^6\right )^p \operatorname {AppellF1}\left (-2 p,-p,-p,1-2 p,-\frac {b-\sqrt {b^2-4 a c}}{2 c x^3},-\frac {b+\sqrt {b^2-4 a c}}{2 c x^3}\right )}{3 a p}-\frac {2^{1+p} c d (1+2 p) \left (-\frac {b-\sqrt {b^2-4 a c}+2 c x^3}{\sqrt {b^2-4 a c}}\right )^{-1-p} \left (a+b x^3+c x^6\right )^{1+p} \operatorname {Hypergeometric2F1}\left (-p,1+p,2+p,\frac {b+\sqrt {b^2-4 a c}+2 c x^3}{2 \sqrt {b^2-4 a c}}\right )}{3 a \sqrt {b^2-4 a c} (1+p)} \] Output:

-1/3*d*(c*x^6+b*x^3+a)^(p+1)/a/x^3+1/3*2^(-1+2*p)*(b*d*p+a*e)*(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)/a/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)*c*d*(1+2*p)*(-(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))/a/(-4*a*c+b^2)^(1/2)/(p+1)
 

Mathematica [A] (warning: unable to verify)

Time = 0.70 (sec) , antiderivative size = 300, normalized size of antiderivative = 0.89 \[ \int \frac {\left (d+e x^3\right ) \left (a+b x^3+c x^6\right )^p}{x^4} \, dx=\frac {\left (1+\frac {b-\sqrt {b^2-4 a c}}{2 c x^3}\right )^{-p} \left (\frac {b-\sqrt {b^2-4 a c}}{2 c}+x^3\right )^{-p} \left (\frac {b-\sqrt {b^2-4 a c}+2 c x^3}{c}\right )^p \left (\frac {b+\sqrt {b^2-4 a c}+2 c x^3}{c x^3}\right )^{-p} \left (a+b x^3+c x^6\right )^p \left (2 d p \operatorname {AppellF1}\left (1-2 p,-p,-p,2-2 p,-\frac {b+\sqrt {b^2-4 a c}}{2 c x^3},\frac {-b+\sqrt {b^2-4 a c}}{2 c x^3}\right )+e (-1+2 p) x^3 \operatorname {AppellF1}\left (-2 p,-p,-p,1-2 p,-\frac {b+\sqrt {b^2-4 a c}}{2 c x^3},\frac {-b+\sqrt {b^2-4 a c}}{2 c x^3}\right )\right )}{6 p (-1+2 p) x^3} \] Input:

Integrate[((d + e*x^3)*(a + b*x^3 + c*x^6)^p)/x^4,x]
 

Output:

(((b - Sqrt[b^2 - 4*a*c] + 2*c*x^3)/c)^p*(a + b*x^3 + c*x^6)^p*(2*d*p*Appe 
llF1[1 - 2*p, -p, -p, 2 - 2*p, -1/2*(b + Sqrt[b^2 - 4*a*c])/(c*x^3), (-b + 
 Sqrt[b^2 - 4*a*c])/(2*c*x^3)] + e*(-1 + 2*p)*x^3*AppellF1[-2*p, -p, -p, 1 
 - 2*p, -1/2*(b + Sqrt[b^2 - 4*a*c])/(c*x^3), (-b + Sqrt[b^2 - 4*a*c])/(2* 
c*x^3)]))/(6*p*(-1 + 2*p)*(1 + (b - Sqrt[b^2 - 4*a*c])/(2*c*x^3))^p*x^3*(( 
b - Sqrt[b^2 - 4*a*c])/(2*c) + x^3)^p*((b + Sqrt[b^2 - 4*a*c] + 2*c*x^3)/( 
c*x^3))^p)
 

Rubi [A] (verified)

Time = 0.49 (sec) , antiderivative size = 332, normalized size of antiderivative = 0.99, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {1802, 1237, 25, 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.

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

\(\Big \downarrow \) 1802

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

\(\Big \downarrow \) 1237

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 1269

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

\(\Big \downarrow \) 1096

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

\(\Big \downarrow \) 1178

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

\(\Big \downarrow \) 150

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

Input:

Int[((d + e*x^3)*(a + b*x^3 + c*x^6)^p)/x^4,x]
 

Output:

(-((d*(a + b*x^3 + c*x^6)^(1 + p))/(a*x^3)) + ((2^(-1 + 2*p)*(a*e + b*d*p) 
*(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)*c*d*(1 + 2*p)*(-((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)))/a)/3
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 1237

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(e*f - d*g)*(d + e*x)^(m + 1)*((a + b* 
x + c*x^2)^(p + 1)/((m + 1)*(c*d^2 - b*d*e + a*e^2))), x] + Simp[1/((m + 1) 
*(c*d^2 - b*d*e + a*e^2))   Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p*Simp[ 
(c*d*f - f*b*e + a*e*g)*(m + 1) + b*(d*g - e*f)*(p + 1) - c*(e*f - d*g)*(m 
+ 2*p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && LtQ[m, -1 
] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
 

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]

\[\int \frac {\left (e \,x^{3}+d \right ) \left (c \,x^{6}+b \,x^{3}+a \right )^{p}}{x^{4}}d x\]

Input:

int((e*x^3+d)*(c*x^6+b*x^3+a)^p/x^4,x)
 

Output:

int((e*x^3+d)*(c*x^6+b*x^3+a)^p/x^4,x)
 

Fricas [F]

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

integrate((e*x^3+d)*(c*x^6+b*x^3+a)^p/x^4,x, algorithm="fricas")
 

Output:

integral((e*x^3 + d)*(c*x^6 + b*x^3 + a)^p/x^4, x)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (d+e x^3\right ) \left (a+b x^3+c x^6\right )^p}{x^4} \, dx=\text {Timed out} \] Input:

integrate((e*x**3+d)*(c*x**6+b*x**3+a)**p/x**4,x)
 

Output:

Timed out
 

Maxima [F]

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

integrate((e*x^3+d)*(c*x^6+b*x^3+a)^p/x^4,x, algorithm="maxima")
 

Output:

integrate((e*x^3 + d)*(c*x^6 + b*x^3 + a)^p/x^4, x)
 

Giac [F]

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

integrate((e*x^3+d)*(c*x^6+b*x^3+a)^p/x^4,x, algorithm="giac")
 

Output:

integrate((e*x^3 + d)*(c*x^6 + b*x^3 + a)^p/x^4, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (d+e x^3\right ) \left (a+b x^3+c x^6\right )^p}{x^4} \, dx=\int \frac {\left (e\,x^3+d\right )\,{\left (c\,x^6+b\,x^3+a\right )}^p}{x^4} \,d x \] Input:

int(((d + e*x^3)*(a + b*x^3 + c*x^6)^p)/x^4,x)
 

Output:

int(((d + e*x^3)*(a + b*x^3 + c*x^6)^p)/x^4, x)
 

Reduce [F]

\[ \int \frac {\left (d+e x^3\right ) \left (a+b x^3+c x^6\right )^p}{x^4} \, dx =\text {Too large to display} \] Input:

int((e*x^3+d)*(c*x^6+b*x^3+a)^p/x^4,x)
 

Output:

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