\(\int (d+e x)^{1-2 p} (e+f x) (a+c x^2)^p \, dx\) [309]

Optimal result

Integrand size = 26, antiderivative size = 333 \[ \int (d+e x)^{1-2 p} (e+f x) \left (a+c x^2\right )^p \, dx=\frac {\left (e^2-d f\right ) (d+e x)^{2-2 p} \left (a+c x^2\right )^p \left (1-\frac {d+e x}{d-\frac {\sqrt {-a} e}{\sqrt {c}}}\right )^{-p} \left (1-\frac {d+e x}{d+\frac {\sqrt {-a} e}{\sqrt {c}}}\right )^{-p} \operatorname {AppellF1}\left (2-2 p,-p,-p,3-2 p,\frac {d+e x}{d-\frac {\sqrt {-a} e}{\sqrt {c}}},\frac {d+e x}{d+\frac {\sqrt {-a} e}{\sqrt {c}}}\right )}{2 e^2 (1-p)}+\frac {f (d+e x)^{3-2 p} \left (a+c x^2\right )^p \left (1-\frac {d+e x}{d-\frac {\sqrt {-a} e}{\sqrt {c}}}\right )^{-p} \left (1-\frac {d+e x}{d+\frac {\sqrt {-a} e}{\sqrt {c}}}\right )^{-p} \operatorname {AppellF1}\left (3-2 p,-p,-p,4-2 p,\frac {d+e x}{d-\frac {\sqrt {-a} e}{\sqrt {c}}},\frac {d+e x}{d+\frac {\sqrt {-a} e}{\sqrt {c}}}\right )}{e^2 (3-2 p)} \] Output:

1/2*(-d*f+e^2)*(e*x+d)^(2-2*p)*(c*x^2+a)^p*AppellF1(2-2*p,-p,-p,3-2*p,(e*x 
+d)/(d-(-a)^(1/2)*e/c^(1/2)),(e*x+d)/(d+(-a)^(1/2)*e/c^(1/2)))/e^2/(1-p)/( 
(1-(e*x+d)/(d-(-a)^(1/2)*e/c^(1/2)))^p)/((1-(e*x+d)/(d+(-a)^(1/2)*e/c^(1/2 
)))^p)+f*(e*x+d)^(3-2*p)*(c*x^2+a)^p*AppellF1(3-2*p,-p,-p,4-2*p,(e*x+d)/(d 
-(-a)^(1/2)*e/c^(1/2)),(e*x+d)/(d+(-a)^(1/2)*e/c^(1/2)))/e^2/(3-2*p)/((1-( 
e*x+d)/(d-(-a)^(1/2)*e/c^(1/2)))^p)/((1-(e*x+d)/(d+(-a)^(1/2)*e/c^(1/2)))^ 
p)
 

Mathematica [F]

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

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

Output:

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

Rubi [A] (verified)

Time = 0.38 (sec) , antiderivative size = 333, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {719, 514, 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.

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

\(\Big \downarrow \) 719

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

\(\Big \downarrow \) 514

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

\(\Big \downarrow \) 150

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

Input:

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

Output:

((e^2 - d*f)*(d + e*x)^(2 - 2*p)*(a + c*x^2)^p*AppellF1[2 - 2*p, -p, -p, 3 
 - 2*p, (d + e*x)/(d - (Sqrt[-a]*e)/Sqrt[c]), (d + e*x)/(d + (Sqrt[-a]*e)/ 
Sqrt[c])])/(2*e^2*(1 - p)*(1 - (d + e*x)/(d - (Sqrt[-a]*e)/Sqrt[c]))^p*(1 
- (d + e*x)/(d + (Sqrt[-a]*e)/Sqrt[c]))^p) + (f*(d + e*x)^(3 - 2*p)*(a + c 
*x^2)^p*AppellF1[3 - 2*p, -p, -p, 4 - 2*p, (d + e*x)/(d - (Sqrt[-a]*e)/Sqr 
t[c]), (d + e*x)/(d + (Sqrt[-a]*e)/Sqrt[c])])/(e^2*(3 - 2*p)*(1 - (d + e*x 
)/(d - (Sqrt[-a]*e)/Sqrt[c]))^p*(1 - (d + e*x)/(d + (Sqrt[-a]*e)/Sqrt[c])) 
^p)
 

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 514

Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[ 
{q = Rt[-a/b, 2]}, Simp[(a + b*x^2)^p/(d*(1 - (c + d*x)/(c - d*q))^p*(1 - ( 
c + d*x)/(c + d*q))^p)   Subst[Int[x^n*Simp[1 - x/(c + d*q), x]^p*Simp[1 - 
x/(c - d*q), x]^p, x], x, c + d*x], x]] /; FreeQ[{a, b, c, d, n, p}, x] && 
NeQ[b*c^2 + a*d^2, 0]
 

rule 719

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

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

int((a + c*x**2)**p/(d + e*x)**(2*p),x)*d*e + int(((a + c*x**2)**p*x**2)/( 
d + e*x)**(2*p),x)*e*f + int(((a + c*x**2)**p*x)/(d + e*x)**(2*p),x)*d*f + 
 int(((a + c*x**2)**p*x)/(d + e*x)**(2*p),x)*e**2