\(\int \frac {(e x)^m (a+b x^2)}{(c+d x)^3} \, dx\) [106]

Optimal result

Integrand size = 20, antiderivative size = 133 \[ \int \frac {(e x)^m \left (a+b x^2\right )}{(c+d x)^3} \, dx=\frac {\left (b c^2+a d^2\right ) (e x)^{1+m}}{2 c d^2 e (c+d x)^2}+\frac {b (e x)^{1+m}}{d^2 e m (c+d x)}+\frac {\left (a d^2 (1-m) m-b c^2 \left (2+3 m+m^2\right )\right ) (e x)^{1+m} \operatorname {Hypergeometric2F1}\left (2,1+m,2+m,-\frac {d x}{c}\right )}{2 c^3 d^2 e m (1+m)} \] Output:

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

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.65 \[ \int \frac {(e x)^m \left (a+b x^2\right )}{(c+d x)^3} \, dx=\frac {x (e x)^m \left (b c^2 \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,-\frac {d x}{c}\right )-2 b c^2 \operatorname {Hypergeometric2F1}\left (2,1+m,2+m,-\frac {d x}{c}\right )+\left (b c^2+a d^2\right ) \operatorname {Hypergeometric2F1}\left (3,1+m,2+m,-\frac {d x}{c}\right )\right )}{c^3 d^2 (1+m)} \] Input:

Integrate[((e*x)^m*(a + b*x^2))/(c + d*x)^3,x]
 

Output:

(x*(e*x)^m*(b*c^2*Hypergeometric2F1[1, 1 + m, 2 + m, -((d*x)/c)] - 2*b*c^2 
*Hypergeometric2F1[2, 1 + m, 2 + m, -((d*x)/c)] + (b*c^2 + a*d^2)*Hypergeo 
metric2F1[3, 1 + m, 2 + m, -((d*x)/c)]))/(c^3*d^2*(1 + m))
 

Rubi [A] (verified)

Time = 0.50 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.16, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {519, 25, 27, 87, 74}

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.

\(\displaystyle \int \frac {\left (a+b x^2\right ) (e x)^m}{(c+d x)^3} \, dx\)

\(\Big \downarrow \) 519

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 87

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

\(\Big \downarrow \) 74

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

Input:

Int[((e*x)^m*(a + b*x^2))/(c + d*x)^3,x]
 

Output:

((a + (b*c^2)/d^2)*(e*x)^(1 + m))/(2*c*e*(c + d*x)^2) + (((a*d^2*(1 - m) - 
 b*c^2*(3 + m))*(e*x)^(1 + m))/(c*d*e*(c + d*x)) - ((a*d^2*(1 - m)*m - b*c 
^2*(2 + 3*m + m^2))*(e*x)^(1 + m)*Hypergeometric2F1[1, 1 + m, 2 + m, -((d* 
x)/c)])/(c^2*d*e*(1 + m)))/(2*c*d)
 

Defintions of rubi rules used

rule 25

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

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 74

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[c^n*((b*x 
)^(m + 1)/(b*(m + 1)))*Hypergeometric2F1[-n, m + 1, m + 2, (-d)*(x/c)], x] 
/; FreeQ[{b, c, d, m, n}, x] &&  !IntegerQ[m] && (IntegerQ[n] || (GtQ[c, 0] 
 &&  !(EqQ[n, -2^(-1)] && EqQ[c^2 - d^2, 0] && GtQ[-d/(b*c), 0])))
 

rule 87

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p 
_.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p 
+ 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p 
+ 1)))/(f*(p + 1)*(c*f - d*e))   Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] 
/; FreeQ[{a, b, c, d, e, f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || Intege 
rQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || LtQ[p, n]))))
 

rule 519

Int[((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_.), 
 x_Symbol] :> With[{Qx = PolynomialQuotient[(a + b*x^2)^p, c + d*x, x], R = 
 PolynomialRemainder[(a + b*x^2)^p, c + d*x, x]}, Simp[(-R)*(e*x)^(m + 1)*( 
(c + d*x)^(n + 1)/(c*e*(n + 1))), x] + Simp[1/(c*(n + 1))   Int[(e*x)^m*(c 
+ d*x)^(n + 1)*ExpandToSum[c*(n + 1)*Qx + R*(m + n + 2), x], x], x]] /; Fre 
eQ[{a, b, c, d, e, m}, x] && IGtQ[p, 0] && LtQ[n, -1] &&  !IntegerQ[m]
 

Maple [F]

\[\int \frac {\left (e x \right )^{m} \left (b \,x^{2}+a \right )}{\left (d x +c \right )^{3}}d x\]

Input:

int((e*x)^m*(b*x^2+a)/(d*x+c)^3,x)
 

Output:

int((e*x)^m*(b*x^2+a)/(d*x+c)^3,x)
 

Fricas [F]

\[ \int \frac {(e x)^m \left (a+b x^2\right )}{(c+d x)^3} \, dx=\int { \frac {{\left (b x^{2} + a\right )} \left (e x\right )^{m}}{{\left (d x + c\right )}^{3}} \,d x } \] Input:

integrate((e*x)^m*(b*x^2+a)/(d*x+c)^3,x, algorithm="fricas")
 

Output:

integral((b*x^2 + a)*(e*x)^m/(d^3*x^3 + 3*c*d^2*x^2 + 3*c^2*d*x + c^3), x)
 

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 3.48 (sec) , antiderivative size = 1991, normalized size of antiderivative = 14.97 \[ \int \frac {(e x)^m \left (a+b x^2\right )}{(c+d x)^3} \, dx=\text {Too large to display} \] Input:

integrate((e*x)**m*(b*x**2+a)/(d*x+c)**3,x)
 

Output:

a*(c**2*e**m*m**3*x**(m + 1)*lerchphi(d*x*exp_polar(I*pi)/c, 1, m + 1)*gam 
ma(m + 1)/(2*c**5*gamma(m + 2) + 4*c**4*d*x*gamma(m + 2) + 2*c**3*d**2*x** 
2*gamma(m + 2)) - c**2*e**m*m**2*x**(m + 1)*gamma(m + 1)/(2*c**5*gamma(m + 
 2) + 4*c**4*d*x*gamma(m + 2) + 2*c**3*d**2*x**2*gamma(m + 2)) - c**2*e**m 
*m*x**(m + 1)*lerchphi(d*x*exp_polar(I*pi)/c, 1, m + 1)*gamma(m + 1)/(2*c* 
*5*gamma(m + 2) + 4*c**4*d*x*gamma(m + 2) + 2*c**3*d**2*x**2*gamma(m + 2)) 
 + c**2*e**m*m*x**(m + 1)*gamma(m + 1)/(2*c**5*gamma(m + 2) + 4*c**4*d*x*g 
amma(m + 2) + 2*c**3*d**2*x**2*gamma(m + 2)) + 2*c**2*e**m*x**(m + 1)*gamm 
a(m + 1)/(2*c**5*gamma(m + 2) + 4*c**4*d*x*gamma(m + 2) + 2*c**3*d**2*x**2 
*gamma(m + 2)) + 2*c*d*e**m*m**3*x*x**(m + 1)*lerchphi(d*x*exp_polar(I*pi) 
/c, 1, m + 1)*gamma(m + 1)/(2*c**5*gamma(m + 2) + 4*c**4*d*x*gamma(m + 2) 
+ 2*c**3*d**2*x**2*gamma(m + 2)) - c*d*e**m*m**2*x*x**(m + 1)*gamma(m + 1) 
/(2*c**5*gamma(m + 2) + 4*c**4*d*x*gamma(m + 2) + 2*c**3*d**2*x**2*gamma(m 
 + 2)) - 2*c*d*e**m*m*x*x**(m + 1)*lerchphi(d*x*exp_polar(I*pi)/c, 1, m + 
1)*gamma(m + 1)/(2*c**5*gamma(m + 2) + 4*c**4*d*x*gamma(m + 2) + 2*c**3*d* 
*2*x**2*gamma(m + 2)) + c*d*e**m*x*x**(m + 1)*gamma(m + 1)/(2*c**5*gamma(m 
 + 2) + 4*c**4*d*x*gamma(m + 2) + 2*c**3*d**2*x**2*gamma(m + 2)) + d**2*e* 
*m*m**3*x**2*x**(m + 1)*lerchphi(d*x*exp_polar(I*pi)/c, 1, m + 1)*gamma(m 
+ 1)/(2*c**5*gamma(m + 2) + 4*c**4*d*x*gamma(m + 2) + 2*c**3*d**2*x**2*gam 
ma(m + 2)) - d**2*e**m*m*x**2*x**(m + 1)*lerchphi(d*x*exp_polar(I*pi)/c...
 

Maxima [F]

\[ \int \frac {(e x)^m \left (a+b x^2\right )}{(c+d x)^3} \, dx=\int { \frac {{\left (b x^{2} + a\right )} \left (e x\right )^{m}}{{\left (d x + c\right )}^{3}} \,d x } \] Input:

integrate((e*x)^m*(b*x^2+a)/(d*x+c)^3,x, algorithm="maxima")
 

Output:

integrate((b*x^2 + a)*(e*x)^m/(d*x + c)^3, x)
 

Giac [F]

\[ \int \frac {(e x)^m \left (a+b x^2\right )}{(c+d x)^3} \, dx=\int { \frac {{\left (b x^{2} + a\right )} \left (e x\right )^{m}}{{\left (d x + c\right )}^{3}} \,d x } \] Input:

integrate((e*x)^m*(b*x^2+a)/(d*x+c)^3,x, algorithm="giac")
                                                                                    
                                                                                    
 

Output:

integrate((b*x^2 + a)*(e*x)^m/(d*x + c)^3, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(e x)^m \left (a+b x^2\right )}{(c+d x)^3} \, dx=\int \frac {{\left (e\,x\right )}^m\,\left (b\,x^2+a\right )}{{\left (c+d\,x\right )}^3} \,d x \] Input:

int(((e*x)^m*(a + b*x^2))/(c + d*x)^3,x)
 

Output:

int(((e*x)^m*(a + b*x^2))/(c + d*x)^3, x)
 

Reduce [F]

\[ \int \frac {(e x)^m \left (a+b x^2\right )}{(c+d x)^3} \, dx=\text {too large to display} \] Input:

int((e*x)^m*(b*x^2+a)/(d*x+c)^3,x)
 

Output:

(e**m*(x**m*a*d**2*m**2 - x**m*a*d**2*m + x**m*b*c**2*m**2 + 3*x**m*b*c**2 
*m + 2*x**m*b*c**2 - x**m*b*c*d*m**2*x + 4*x**m*b*c*d*x + x**m*b*d**2*m**2 
*x**2 - 3*x**m*b*d**2*m*x**2 + 2*x**m*b*d**2*x**2 - int(x**m/(c**3*m**2*x 
- 3*c**3*m*x + 2*c**3*x + 3*c**2*d*m**2*x**2 - 9*c**2*d*m*x**2 + 6*c**2*d* 
x**2 + 3*c*d**2*m**2*x**3 - 9*c*d**2*m*x**3 + 6*c*d**2*x**3 + d**3*m**2*x* 
*4 - 3*d**3*m*x**4 + 2*d**3*x**4),x)*a*c**3*d**2*m**5 + 4*int(x**m/(c**3*m 
**2*x - 3*c**3*m*x + 2*c**3*x + 3*c**2*d*m**2*x**2 - 9*c**2*d*m*x**2 + 6*c 
**2*d*x**2 + 3*c*d**2*m**2*x**3 - 9*c*d**2*m*x**3 + 6*c*d**2*x**3 + d**3*m 
**2*x**4 - 3*d**3*m*x**4 + 2*d**3*x**4),x)*a*c**3*d**2*m**4 - 5*int(x**m/( 
c**3*m**2*x - 3*c**3*m*x + 2*c**3*x + 3*c**2*d*m**2*x**2 - 9*c**2*d*m*x**2 
 + 6*c**2*d*x**2 + 3*c*d**2*m**2*x**3 - 9*c*d**2*m*x**3 + 6*c*d**2*x**3 + 
d**3*m**2*x**4 - 3*d**3*m*x**4 + 2*d**3*x**4),x)*a*c**3*d**2*m**3 + 2*int( 
x**m/(c**3*m**2*x - 3*c**3*m*x + 2*c**3*x + 3*c**2*d*m**2*x**2 - 9*c**2*d* 
m*x**2 + 6*c**2*d*x**2 + 3*c*d**2*m**2*x**3 - 9*c*d**2*m*x**3 + 6*c*d**2*x 
**3 + d**3*m**2*x**4 - 3*d**3*m*x**4 + 2*d**3*x**4),x)*a*c**3*d**2*m**2 - 
2*int(x**m/(c**3*m**2*x - 3*c**3*m*x + 2*c**3*x + 3*c**2*d*m**2*x**2 - 9*c 
**2*d*m*x**2 + 6*c**2*d*x**2 + 3*c*d**2*m**2*x**3 - 9*c*d**2*m*x**3 + 6*c* 
d**2*x**3 + d**3*m**2*x**4 - 3*d**3*m*x**4 + 2*d**3*x**4),x)*a*c**2*d**3*m 
**5*x + 8*int(x**m/(c**3*m**2*x - 3*c**3*m*x + 2*c**3*x + 3*c**2*d*m**2*x* 
*2 - 9*c**2*d*m*x**2 + 6*c**2*d*x**2 + 3*c*d**2*m**2*x**3 - 9*c*d**2*m*...