\(\int \frac {(e x)^m (c^2-d^2 x^2)^p}{(c+d x)^3} \, dx\) [1276]

Optimal result

Integrand size = 27, antiderivative size = 233 \[ \int \frac {(e x)^m \left (c^2-d^2 x^2\right )^p}{(c+d x)^3} \, dx=\frac {2 (e x)^{1+m} \left (c^2-d^2 x^2\right )^{-1+p}}{e (2-p) (c+d x)}-\frac {(2 m+p) (e x)^{1+m} \left (c^2-d^2 x^2\right )^p \left (1-\frac {d^2 x^2}{c^2}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1+m}{2},2-p,\frac {3+m}{2},\frac {d^2 x^2}{c^2}\right )}{c^3 e (1+m) (2-p)}-\frac {d (2-2 m-3 p) (e x)^{2+m} \left (c^2-d^2 x^2\right )^p \left (1-\frac {d^2 x^2}{c^2}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {2+m}{2},2-p,\frac {4+m}{2},\frac {d^2 x^2}{c^2}\right )}{c^4 e^2 (2+m) (2-p)} \] Output:

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

Mathematica [A] (verified)

Time = 0.29 (sec) , antiderivative size = 206, normalized size of antiderivative = 0.88 \[ \int \frac {(e x)^m \left (c^2-d^2 x^2\right )^p}{(c+d x)^3} \, dx=\frac {x (e x)^m \left (c^2-d^2 x^2\right )^p \left (1-\frac {d^2 x^2}{c^2}\right )^{-p} \left (\frac {c^3 \operatorname {Hypergeometric2F1}\left (\frac {1+m}{2},3-p,\frac {3+m}{2},\frac {d^2 x^2}{c^2}\right )}{1+m}+d x \left (-\frac {3 c^2 \operatorname {Hypergeometric2F1}\left (\frac {2+m}{2},3-p,\frac {4+m}{2},\frac {d^2 x^2}{c^2}\right )}{2+m}+d x \left (\frac {3 c \operatorname {Hypergeometric2F1}\left (\frac {3+m}{2},3-p,\frac {5+m}{2},\frac {d^2 x^2}{c^2}\right )}{3+m}-\frac {d x \operatorname {Hypergeometric2F1}\left (\frac {4+m}{2},3-p,\frac {6+m}{2},\frac {d^2 x^2}{c^2}\right )}{4+m}\right )\right )\right )}{c^6} \] Input:

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

Output:

(x*(e*x)^m*(c^2 - d^2*x^2)^p*((c^3*Hypergeometric2F1[(1 + m)/2, 3 - p, (3 
+ m)/2, (d^2*x^2)/c^2])/(1 + m) + d*x*((-3*c^2*Hypergeometric2F1[(2 + m)/2 
, 3 - p, (4 + m)/2, (d^2*x^2)/c^2])/(2 + m) + d*x*((3*c*Hypergeometric2F1[ 
(3 + m)/2, 3 - p, (5 + m)/2, (d^2*x^2)/c^2])/(3 + m) - (d*x*Hypergeometric 
2F1[(4 + m)/2, 3 - p, (6 + m)/2, (d^2*x^2)/c^2])/(4 + m)))))/(c^6*(1 - (d^ 
2*x^2)/c^2)^p)
 

Rubi [A] (verified)

Time = 1.01 (sec) , antiderivative size = 314, normalized size of antiderivative = 1.35, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {570, 559, 2340, 27, 557, 279, 278}

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.

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

\(\Big \downarrow \) 570

\(\displaystyle \int (c-d x)^3 (e x)^m \left (c^2-d^2 x^2\right )^{p-3}dx\)

\(\Big \downarrow \) 559

\(\displaystyle \frac {\int (e x)^m \left (c^2-d^2 x^2\right )^{p-3} \left (3 c (-m-2 p+2) x^2 d^4-2 c^2 (-2 m-3 p+2) x d^3+c^3 (-m-2 p+2) d^2\right )dx}{d^2 (-m-2 p+2)}-\frac {d (e x)^{m+2} \left (c^2-d^2 x^2\right )^{p-2}}{e^2 (-m-2 p+2)}\)

\(\Big \downarrow \) 2340

\(\displaystyle \frac {\frac {\int -2 c^2 d^4 (e x)^m (c (-m-2 p+2) (2 m+p)+d (-2 m-3 p+2) (-m-2 p+3) x) \left (c^2-d^2 x^2\right )^{p-3}dx}{d^2 (-m-2 p+3)}+\frac {3 c d^2 (-m-2 p+2) (e x)^{m+1} \left (c^2-d^2 x^2\right )^{p-2}}{e (-m-2 p+3)}}{d^2 (-m-2 p+2)}-\frac {d (e x)^{m+2} \left (c^2-d^2 x^2\right )^{p-2}}{e^2 (-m-2 p+2)}\)

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 557

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

\(\Big \downarrow \) 279

\(\displaystyle \frac {\frac {3 c d^2 (-m-2 p+2) (e x)^{m+1} \left (c^2-d^2 x^2\right )^{p-2}}{e (-m-2 p+3)}-\frac {2 c^2 d^2 \left (\frac {d (-2 m-3 p+2) (-m-2 p+3) \left (c^2-d^2 x^2\right )^p \left (1-\frac {d^2 x^2}{c^2}\right )^{-p} \int (e x)^{m+1} \left (1-\frac {d^2 x^2}{c^2}\right )^{p-3}dx}{c^6 e}+\frac {(-m-2 p+2) (2 m+p) \left (c^2-d^2 x^2\right )^p \left (1-\frac {d^2 x^2}{c^2}\right )^{-p} \int (e x)^m \left (1-\frac {d^2 x^2}{c^2}\right )^{p-3}dx}{c^5}\right )}{-m-2 p+3}}{d^2 (-m-2 p+2)}-\frac {d (e x)^{m+2} \left (c^2-d^2 x^2\right )^{p-2}}{e^2 (-m-2 p+2)}\)

\(\Big \downarrow \) 278

\(\displaystyle \frac {\frac {3 c d^2 (-m-2 p+2) (e x)^{m+1} \left (c^2-d^2 x^2\right )^{p-2}}{e (-m-2 p+3)}-\frac {2 c^2 d^2 \left (\frac {d (-2 m-3 p+2) (-m-2 p+3) (e x)^{m+2} \left (c^2-d^2 x^2\right )^p \left (1-\frac {d^2 x^2}{c^2}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {m+2}{2},3-p,\frac {m+4}{2},\frac {d^2 x^2}{c^2}\right )}{c^6 e^2 (m+2)}+\frac {(-m-2 p+2) (2 m+p) (e x)^{m+1} \left (c^2-d^2 x^2\right )^p \left (1-\frac {d^2 x^2}{c^2}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {m+1}{2},3-p,\frac {m+3}{2},\frac {d^2 x^2}{c^2}\right )}{c^5 e (m+1)}\right )}{-m-2 p+3}}{d^2 (-m-2 p+2)}-\frac {d (e x)^{m+2} \left (c^2-d^2 x^2\right )^{p-2}}{e^2 (-m-2 p+2)}\)

Input:

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

Output:

-((d*(e*x)^(2 + m)*(c^2 - d^2*x^2)^(-2 + p))/(e^2*(2 - m - 2*p))) + ((3*c* 
d^2*(2 - m - 2*p)*(e*x)^(1 + m)*(c^2 - d^2*x^2)^(-2 + p))/(e*(3 - m - 2*p) 
) - (2*c^2*d^2*(((2 - m - 2*p)*(2*m + p)*(e*x)^(1 + m)*(c^2 - d^2*x^2)^p*H 
ypergeometric2F1[(1 + m)/2, 3 - p, (3 + m)/2, (d^2*x^2)/c^2])/(c^5*e*(1 + 
m)*(1 - (d^2*x^2)/c^2)^p) + (d*(2 - 2*m - 3*p)*(3 - m - 2*p)*(e*x)^(2 + m) 
*(c^2 - d^2*x^2)^p*Hypergeometric2F1[(2 + m)/2, 3 - p, (4 + m)/2, (d^2*x^2 
)/c^2])/(c^6*e^2*(2 + m)*(1 - (d^2*x^2)/c^2)^p)))/(3 - m - 2*p))/(d^2*(2 - 
 m - 2*p))
 

Defintions of rubi rules used

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 278

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[a^p*(( 
c*x)^(m + 1)/(c*(m + 1)))*Hypergeometric2F1[-p, (m + 1)/2, (m + 1)/2 + 1, ( 
-b)*(x^2/a)], x] /; FreeQ[{a, b, c, m, p}, x] &&  !IGtQ[p, 0] && (ILtQ[p, 0 
] || GtQ[a, 0])
 

rule 279

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[a^IntP 
art[p]*((a + b*x^2)^FracPart[p]/(1 + b*(x^2/a))^FracPart[p])   Int[(c*x)^m* 
(1 + b*(x^2/a))^p, x], x] /; FreeQ[{a, b, c, m, p}, x] &&  !IGtQ[p, 0] && 
!(ILtQ[p, 0] || GtQ[a, 0])
 

rule 557

Int[((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_), x_Sym 
bol] :> Simp[c   Int[(e*x)^m*(a + b*x^2)^p, x], x] + Simp[d/e   Int[(e*x)^( 
m + 1)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x]
 

rule 559

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

rule 570

Int[((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), 
x_Symbol] :> Simp[c^(2*n)/a^n   Int[(e*x)^m*((a + b*x^2)^(n + p)/(c - d*x)^ 
n), x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && EqQ[b*c^2 + a*d^2, 0] && I 
LtQ[n, -1] &&  !(IGtQ[m, 0] && ILtQ[m + n, 0] &&  !GtQ[p, 1])
 

rule 2340

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[ 
{q = Expon[Pq, x], f = Coeff[Pq, x, Expon[Pq, x]]}, Simp[f*(c*x)^(m + q - 1 
)*((a + b*x^2)^(p + 1)/(b*c^(q - 1)*(m + q + 2*p + 1))), x] + Simp[1/(b*(m 
+ q + 2*p + 1))   Int[(c*x)^m*(a + b*x^2)^p*ExpandToSum[b*(m + q + 2*p + 1) 
*Pq - b*f*(m + q + 2*p + 1)*x^q - a*f*(m + q - 1)*x^(q - 2), x], x], x] /; 
GtQ[q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, b, c, m, p}, x] && PolyQ 
[Pq, x] && ( !IGtQ[m, 0] || IGtQ[p + 1/2, -1])
 

Maple [F]

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

Input:

int((e*x)^m*(-d^2*x^2+c^2)^p/(d*x+c)^3,x)
 

Output:

int((e*x)^m*(-d^2*x^2+c^2)^p/(d*x+c)^3,x)
 

Fricas [F]

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

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

Output:

integral((-d^2*x^2 + c^2)^p*(e*x)^m/(d^3*x^3 + 3*c*d^2*x^2 + 3*c^2*d*x + c 
^3), x)
 

Sympy [F]

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

integrate((e*x)**m*(-d**2*x**2+c**2)**p/(d*x+c)**3,x)
 

Output:

Integral((e*x)**m*(-(-c + d*x)*(c + d*x))**p/(c + d*x)**3, x)
 

Maxima [F]

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

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

Output:

integrate((-d^2*x^2 + c^2)^p*(e*x)^m/(d*x + c)^3, x)
 

Giac [F]

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

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

Output:

integrate((-d^2*x^2 + c^2)^p*(e*x)^m/(d*x + c)^3, x)
 

Mupad [F(-1)]

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

int(((c^2 - d^2*x^2)^p*(e*x)^m)/(c + d*x)^3,x)
 

Output:

int(((c^2 - d^2*x^2)^p*(e*x)^m)/(c + d*x)^3, x)
 

Reduce [F]

\[ \int \frac {(e x)^m \left (c^2-d^2 x^2\right )^p}{(c+d x)^3} \, dx=\frac {e^{m} \left (-x^{m} \left (-d^{2} x^{2}+c^{2}\right )^{p}-\left (\int \frac {x^{m} \left (-d^{2} x^{2}+c^{2}\right )^{p} x}{-d^{4} x^{4}-2 c \,d^{3} x^{3}+2 c^{3} d x +c^{4}}d x \right ) c^{2} d^{2} m -2 \left (\int \frac {x^{m} \left (-d^{2} x^{2}+c^{2}\right )^{p} x}{-d^{4} x^{4}-2 c \,d^{3} x^{3}+2 c^{3} d x +c^{4}}d x \right ) c^{2} d^{2} p -2 \left (\int \frac {x^{m} \left (-d^{2} x^{2}+c^{2}\right )^{p} x}{-d^{4} x^{4}-2 c \,d^{3} x^{3}+2 c^{3} d x +c^{4}}d x \right ) c \,d^{3} m x -4 \left (\int \frac {x^{m} \left (-d^{2} x^{2}+c^{2}\right )^{p} x}{-d^{4} x^{4}-2 c \,d^{3} x^{3}+2 c^{3} d x +c^{4}}d x \right ) c \,d^{3} p x -\left (\int \frac {x^{m} \left (-d^{2} x^{2}+c^{2}\right )^{p} x}{-d^{4} x^{4}-2 c \,d^{3} x^{3}+2 c^{3} d x +c^{4}}d x \right ) d^{4} m \,x^{2}-2 \left (\int \frac {x^{m} \left (-d^{2} x^{2}+c^{2}\right )^{p} x}{-d^{4} x^{4}-2 c \,d^{3} x^{3}+2 c^{3} d x +c^{4}}d x \right ) d^{4} p \,x^{2}+\left (\int \frac {x^{m} \left (-d^{2} x^{2}+c^{2}\right )^{p}}{-d^{4} x^{5}-2 c \,d^{3} x^{4}+2 c^{3} d \,x^{2}+c^{4} x}d x \right ) c^{4} m +2 \left (\int \frac {x^{m} \left (-d^{2} x^{2}+c^{2}\right )^{p}}{-d^{4} x^{5}-2 c \,d^{3} x^{4}+2 c^{3} d \,x^{2}+c^{4} x}d x \right ) c^{3} d m x +\left (\int \frac {x^{m} \left (-d^{2} x^{2}+c^{2}\right )^{p}}{-d^{4} x^{5}-2 c \,d^{3} x^{4}+2 c^{3} d \,x^{2}+c^{4} x}d x \right ) c^{2} d^{2} m \,x^{2}\right )}{2 d \left (d^{2} x^{2}+2 c d x +c^{2}\right )} \] Input:

int((e*x)^m*(-d^2*x^2+c^2)^p/(d*x+c)^3,x)
 

Output:

(e**m*( - x**m*(c**2 - d**2*x**2)**p - int((x**m*(c**2 - d**2*x**2)**p*x)/ 
(c**4 + 2*c**3*d*x - 2*c*d**3*x**3 - d**4*x**4),x)*c**2*d**2*m - 2*int((x* 
*m*(c**2 - d**2*x**2)**p*x)/(c**4 + 2*c**3*d*x - 2*c*d**3*x**3 - d**4*x**4 
),x)*c**2*d**2*p - 2*int((x**m*(c**2 - d**2*x**2)**p*x)/(c**4 + 2*c**3*d*x 
 - 2*c*d**3*x**3 - d**4*x**4),x)*c*d**3*m*x - 4*int((x**m*(c**2 - d**2*x** 
2)**p*x)/(c**4 + 2*c**3*d*x - 2*c*d**3*x**3 - d**4*x**4),x)*c*d**3*p*x - i 
nt((x**m*(c**2 - d**2*x**2)**p*x)/(c**4 + 2*c**3*d*x - 2*c*d**3*x**3 - d** 
4*x**4),x)*d**4*m*x**2 - 2*int((x**m*(c**2 - d**2*x**2)**p*x)/(c**4 + 2*c* 
*3*d*x - 2*c*d**3*x**3 - d**4*x**4),x)*d**4*p*x**2 + int((x**m*(c**2 - d** 
2*x**2)**p)/(c**4*x + 2*c**3*d*x**2 - 2*c*d**3*x**4 - d**4*x**5),x)*c**4*m 
 + 2*int((x**m*(c**2 - d**2*x**2)**p)/(c**4*x + 2*c**3*d*x**2 - 2*c*d**3*x 
**4 - d**4*x**5),x)*c**3*d*m*x + int((x**m*(c**2 - d**2*x**2)**p)/(c**4*x 
+ 2*c**3*d*x**2 - 2*c*d**3*x**4 - d**4*x**5),x)*c**2*d**2*m*x**2))/(2*d*(c 
**2 + 2*c*d*x + d**2*x**2))