\(\int \frac {(e x)^{1-2 p} (a+b x^2)^p}{c+d x} \, dx\) [1932]

Optimal result

Integrand size = 26, antiderivative size = 225 \[ \int \frac {(e x)^{1-2 p} \left (a+b x^2\right )^p}{c+d x} \, dx=-\frac {d (e x)^{3-2 p} \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \operatorname {AppellF1}\left (\frac {3}{2}-p,-p,1,\frac {5}{2}-p,-\frac {b x^2}{a},\frac {d^2 x^2}{c^2}\right )}{c^2 e^2 (3-2 p)}-\frac {c (e x)^{2-2 p} \left (a+b x^2\right )^p \operatorname {Hypergeometric2F1}\left (1,-p,1-p,\frac {\left (b+\frac {a d^2}{c^2}\right ) x^2}{a+b x^2}\right )}{2 d^2 e p x^2}+\frac {c (e x)^{2-2 p} \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (-p,-p,1-p,-\frac {b x^2}{a}\right )}{2 d^2 e p x^2} \] Output:

-d*(e*x)^(3-2*p)*(b*x^2+a)^p*AppellF1(3/2-p,1,-p,5/2-p,d^2*x^2/c^2,-b*x^2/ 
a)/c^2/e^2/(3-2*p)/((1+b*x^2/a)^p)-1/2*c*(e*x)^(2-2*p)*(b*x^2+a)^p*hyperge 
om([1, -p],[1-p],(b+a*d^2/c^2)*x^2/(b*x^2+a))/d^2/e/p/x^2+1/2*c*(e*x)^(2-2 
*p)*(b*x^2+a)^p*hypergeom([-p, -p],[1-p],-b*x^2/a)/d^2/e/p/x^2/((1+b*x^2/a 
)^p)
 

Mathematica [F]

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

Integrate[((e*x)^(1 - 2*p)*(a + b*x^2)^p)/(c + d*x),x]
 

Output:

Integrate[((e*x)^(1 - 2*p)*(a + b*x^2)^p)/(c + d*x), x]
 

Rubi [A] (verified)

Time = 0.68 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.02, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {623, 621, 393, 140, 27, 76, 74, 141, 152, 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.

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

\(\Big \downarrow \) 623

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

\(\Big \downarrow \) 621

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

\(\Big \downarrow \) 393

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

\(\Big \downarrow \) 140

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 76

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

\(\Big \downarrow \) 74

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

\(\Big \downarrow \) 141

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

\(\Big \downarrow \) 152

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

\(\Big \downarrow \) 150

\(\displaystyle x^{2 p-1} (e x)^{1-2 p} \left (\frac {1}{2} c x^{-2 p} \left (x^2\right )^p \left (\frac {\left (x^2\right )^{-p} \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \operatorname {Hypergeometric2F1}\left (-p,-p,1-p,-\frac {b x^2}{a}\right )}{d^2 p}-\frac {\left (x^2\right )^{-p} \left (a+b x^2\right )^p \operatorname {Hypergeometric2F1}\left (1,-p,1-p,\frac {\left (\frac {a d^2}{c^2}+b\right ) x^2}{b x^2+a}\right )}{d^2 p}\right )-\frac {d x^{3-2 p} \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \operatorname {AppellF1}\left (\frac {3}{2}-p,-p,1,\frac {5}{2}-p,-\frac {b x^2}{a},\frac {d^2 x^2}{c^2}\right )}{c^2 (3-2 p)}\right )\)

Input:

Int[((e*x)^(1 - 2*p)*(a + b*x^2)^p)/(c + d*x),x]
 

Output:

x^(-1 + 2*p)*(e*x)^(1 - 2*p)*(-((d*x^(3 - 2*p)*(a + b*x^2)^p*AppellF1[3/2 
- p, -p, 1, 5/2 - p, -((b*x^2)/a), (d^2*x^2)/c^2])/(c^2*(3 - 2*p)*(1 + (b* 
x^2)/a)^p)) + (c*(x^2)^p*(-(((a + b*x^2)^p*Hypergeometric2F1[1, -p, 1 - p, 
 ((b + (a*d^2)/c^2)*x^2)/(a + b*x^2)])/(d^2*p*(x^2)^p)) + ((a + b*x^2)^p*H 
ypergeometric2F1[-p, -p, 1 - p, -((b*x^2)/a)])/(d^2*p*(x^2)^p*(1 + (b*x^2) 
/a)^p)))/(2*x^(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 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 76

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

rule 140

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

rule 141

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Simp[(b*c - a*d)^n*((a + b*x)^(m + 1)/((m + 1)*(b*e - a*f)^( 
n + 1)*(e + f*x)^(m + 1)))*Hypergeometric2F1[m + 1, -n, m + 2, (-(d*e - c*f 
))*((a + b*x)/((b*c - a*d)*(e + f*x)))], x] /; FreeQ[{a, b, c, d, e, f, m, 
p}, x] && EqQ[m + n + p + 2, 0] && ILtQ[n, 0] && (SumSimplerQ[m, 1] ||  !Su 
mSimplerQ[p, 1]) &&  !ILtQ[m, 0]
 

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 152

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

rule 393

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

rule 621

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

rule 623

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

Maple [F]

\[\int \frac {\left (e x \right )^{1-2 p} \left (b \,x^{2}+a \right )^{p}}{d x +c}d x\]

Input:

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

Output:

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

Fricas [F]

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

integrate((e*x)^(1-2*p)*(b*x^2+a)^p/(d*x+c),x, algorithm="fricas")
 

Output:

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

Sympy [F(-1)]

Timed out. \[ \int \frac {(e x)^{1-2 p} \left (a+b x^2\right )^p}{c+d x} \, dx=\text {Timed out} \] Input:

integrate((e*x)**(1-2*p)*(b*x**2+a)**p/(d*x+c),x)
 

Output:

Timed out
 

Maxima [F]

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

integrate((e*x)^(1-2*p)*(b*x^2+a)^p/(d*x+c),x, algorithm="maxima")
 

Output:

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

Giac [F]

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

integrate((e*x)^(1-2*p)*(b*x^2+a)^p/(d*x+c),x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {(e x)^{1-2 p} \left (a+b x^2\right )^p}{c+d x} \, dx=\int \frac {{\left (e\,x\right )}^{1-2\,p}\,{\left (b\,x^2+a\right )}^p}{c+d\,x} \,d x \] Input:

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

Output:

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

Reduce [F]

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

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

Output:

(e*( - 2*(a + b*x**2)**p*c*p + (a + b*x**2)**p*c + 2*(a + b*x**2)**p*d*p*x 
 - 4*x**(2*p)*int((a + b*x**2)**p/(x**(2*p)*a*c*x + x**(2*p)*a*d*x**2 + x* 
*(2*p)*b*c*x**3 + x**(2*p)*b*d*x**4),x)*a*c**2*p**2 + 2*x**(2*p)*int((a + 
b*x**2)**p/(x**(2*p)*a*c*x + x**(2*p)*a*d*x**2 + x**(2*p)*b*c*x**3 + x**(2 
*p)*b*d*x**4),x)*a*c**2*p - 2*x**(2*p)*int(((a + b*x**2)**p*x**2)/(x**(2*p 
)*a*c + x**(2*p)*a*d*x + x**(2*p)*b*c*x**2 + x**(2*p)*b*d*x**3),x)*b*c*d*p 
 + 4*x**(2*p)*int(((a + b*x**2)**p*x)/(x**(2*p)*a*c + x**(2*p)*a*d*x + x** 
(2*p)*b*c*x**2 + x**(2*p)*b*d*x**3),x)*a*d**2*p**2))/(2*x**(2*p)*e**(2*p)* 
d**2*p)