\(\int (d+e x)^m \sqrt {a+c x^2} \, dx\) [164]

Optimal result

Integrand size = 19, antiderivative size = 154 \[ \int (d+e x)^m \sqrt {a+c x^2} \, dx=\frac {(d+e x)^{1+m} \sqrt {a+c x^2} \operatorname {AppellF1}\left (1+m,-\frac {1}{2},-\frac {1}{2},2+m,\frac {d+e x}{d-\frac {\sqrt {-a} e}{\sqrt {c}}},\frac {d+e x}{d+\frac {\sqrt {-a} e}{\sqrt {c}}}\right )}{e (1+m) \sqrt {1-\frac {d+e x}{d-\frac {\sqrt {-a} e}{\sqrt {c}}}} \sqrt {1-\frac {d+e x}{d+\frac {\sqrt {-a} e}{\sqrt {c}}}}} \] Output:

(e*x+d)^(1+m)*(c*x^2+a)^(1/2)*AppellF1(1+m,-1/2,-1/2,2+m,(e*x+d)/(d-(-a)^( 
1/2)*e/c^(1/2)),(e*x+d)/(d+(-a)^(1/2)*e/c^(1/2)))/e/(1+m)/(1-(e*x+d)/(d-(- 
a)^(1/2)*e/c^(1/2)))^(1/2)/(1-(e*x+d)/(d+(-a)^(1/2)*e/c^(1/2)))^(1/2)
 

Mathematica [A] (warning: unable to verify)

Time = 0.27 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.03 \[ \int (d+e x)^m \sqrt {a+c x^2} \, dx=\frac {(d+e x)^{1+m} \sqrt {a+c x^2} \operatorname {AppellF1}\left (1+m,-\frac {1}{2},-\frac {1}{2},2+m,\frac {d+e x}{d-\sqrt {-\frac {a}{c}} e},\frac {d+e x}{d+\sqrt {-\frac {a}{c}} e}\right )}{e (1+m) \sqrt {\frac {e \left (\sqrt {-\frac {a}{c}}-x\right )}{d+\sqrt {-\frac {a}{c}} e}} \sqrt {\frac {e \left (\sqrt {-\frac {a}{c}}+x\right )}{-d+\sqrt {-\frac {a}{c}} e}}} \] Input:

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

Output:

((d + e*x)^(1 + m)*Sqrt[a + c*x^2]*AppellF1[1 + m, -1/2, -1/2, 2 + m, (d + 
 e*x)/(d - Sqrt[-(a/c)]*e), (d + e*x)/(d + Sqrt[-(a/c)]*e)])/(e*(1 + m)*Sq 
rt[(e*(Sqrt[-(a/c)] - x))/(d + Sqrt[-(a/c)]*e)]*Sqrt[(e*(Sqrt[-(a/c)] + x) 
)/(-d + Sqrt[-(a/c)]*e)])
 

Rubi [A] (verified)

Time = 0.40 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {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.

\(\displaystyle \int \sqrt {a+c x^2} (d+e x)^m \, dx\)

\(\Big \downarrow \) 514

\(\displaystyle \frac {\sqrt {a+c x^2} \int (d+e x)^m \sqrt {1-\frac {d+e x}{d-\frac {\sqrt {-a} e}{\sqrt {c}}}} \sqrt {1-\frac {d+e x}{d+\frac {\sqrt {-a} e}{\sqrt {c}}}}d(d+e x)}{e \sqrt {1-\frac {d+e x}{d-\frac {\sqrt {-a} e}{\sqrt {c}}}} \sqrt {1-\frac {d+e x}{\frac {\sqrt {-a} e}{\sqrt {c}}+d}}}\)

\(\Big \downarrow \) 150

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

Input:

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

Output:

((d + e*x)^(1 + m)*Sqrt[a + c*x^2]*AppellF1[1 + m, -1/2, -1/2, 2 + m, (d + 
 e*x)/(d - (Sqrt[-a]*e)/Sqrt[c]), (d + e*x)/(d + (Sqrt[-a]*e)/Sqrt[c])])/( 
e*(1 + m)*Sqrt[1 - (d + e*x)/(d - (Sqrt[-a]*e)/Sqrt[c])]*Sqrt[1 - (d + e*x 
)/(d + (Sqrt[-a]*e)/Sqrt[c])])
 

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]
 

Maple [F]

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

Input:

int((e*x+d)^m*(c*x^2+a)^(1/2),x)
 

Output:

int((e*x+d)^m*(c*x^2+a)^(1/2),x)
 

Fricas [F]

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

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

Output:

integral(sqrt(c*x^2 + a)*(e*x + d)^m, x)
 

Sympy [F]

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

integrate((e*x+d)**m*(c*x**2+a)**(1/2),x)
 

Output:

Integral(sqrt(a + c*x**2)*(d + e*x)**m, x)
 

Maxima [F]

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

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

Output:

integrate(sqrt(c*x^2 + a)*(e*x + d)^m, x)
 

Giac [F]

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

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

Output:

integrate(sqrt(c*x^2 + a)*(e*x + d)^m, x)
 

Mupad [F(-1)]

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

int((a + c*x^2)^(1/2)*(d + e*x)^m,x)
 

Output:

int((a + c*x^2)^(1/2)*(d + e*x)^m, x)
 

Reduce [F]

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

int((e*x+d)^m*(c*x^2+a)^(1/2),x)
 

Output:

int((d + e*x)**m*sqrt(a + c*x**2),x)