\(\int (e x)^m (c+d x)^3 \sqrt {a+b x^2} \, dx\) [1723]

Optimal result

Integrand size = 24, antiderivative size = 246 \[ \int (e x)^m (c+d x)^3 \sqrt {a+b x^2} \, dx=\frac {3 c d^2 (e x)^{1+m} \left (a+b x^2\right )^{3/2}}{b e (4+m)}+\frac {d^3 (e x)^{2+m} \left (a+b x^2\right )^{3/2}}{b e^2 (5+m)}+\frac {c \left (\frac {c^2}{1+m}-\frac {3 a d^2}{b (4+m)}\right ) (e x)^{1+m} \sqrt {a+b x^2} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {1+m}{2},\frac {3+m}{2},-\frac {b x^2}{a}\right )}{e \sqrt {1+\frac {b x^2}{a}}}+\frac {d \left (\frac {3 c^2}{2+m}-\frac {a d^2}{b (5+m)}\right ) (e x)^{2+m} \sqrt {a+b x^2} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {2+m}{2},\frac {4+m}{2},-\frac {b x^2}{a}\right )}{e^2 \sqrt {1+\frac {b x^2}{a}}} \] Output:

3*c*d^2*(e*x)^(1+m)*(b*x^2+a)^(3/2)/b/e/(4+m)+d^3*(e*x)^(2+m)*(b*x^2+a)^(3 
/2)/b/e^2/(5+m)+c*(c^2/(1+m)-3*a*d^2/b/(4+m))*(e*x)^(1+m)*(b*x^2+a)^(1/2)* 
hypergeom([-1/2, 1/2+1/2*m],[3/2+1/2*m],-b*x^2/a)/e/(1+b*x^2/a)^(1/2)+d*(3 
*c^2/(2+m)-a*d^2/b/(5+m))*(e*x)^(2+m)*(b*x^2+a)^(1/2)*hypergeom([-1/2, 1+1 
/2*m],[2+1/2*m],-b*x^2/a)/e^2/(1+b*x^2/a)^(1/2)
 

Mathematica [A] (verified)

Time = 0.62 (sec) , antiderivative size = 184, normalized size of antiderivative = 0.75 \[ \int (e x)^m (c+d x)^3 \sqrt {a+b x^2} \, dx=\frac {x (e x)^m \sqrt {a+b x^2} \left (\frac {c^3 \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {1+m}{2},\frac {3+m}{2},-\frac {b x^2}{a}\right )}{1+m}+d x \left (\frac {3 c^2 \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {2+m}{2},\frac {4+m}{2},-\frac {b x^2}{a}\right )}{2+m}+d x \left (\frac {3 c \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {3+m}{2},\frac {5+m}{2},-\frac {b x^2}{a}\right )}{3+m}+\frac {d x \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {4+m}{2},\frac {6+m}{2},-\frac {b x^2}{a}\right )}{4+m}\right )\right )\right )}{\sqrt {1+\frac {b x^2}{a}}} \] Input:

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

Output:

(x*(e*x)^m*Sqrt[a + b*x^2]*((c^3*Hypergeometric2F1[-1/2, (1 + m)/2, (3 + m 
)/2, -((b*x^2)/a)])/(1 + m) + d*x*((3*c^2*Hypergeometric2F1[-1/2, (2 + m)/ 
2, (4 + m)/2, -((b*x^2)/a)])/(2 + m) + d*x*((3*c*Hypergeometric2F1[-1/2, ( 
3 + m)/2, (5 + m)/2, -((b*x^2)/a)])/(3 + m) + (d*x*Hypergeometric2F1[-1/2, 
 (4 + m)/2, (6 + m)/2, -((b*x^2)/a)])/(4 + m)))))/Sqrt[1 + (b*x^2)/a]
 

Rubi [A] (verified)

Time = 0.62 (sec) , antiderivative size = 268, normalized size of antiderivative = 1.09, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {559, 2340, 25, 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.

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

\(\Big \downarrow \) 559

\(\displaystyle \frac {\int (e x)^m \sqrt {b x^2+a} \left (b (m+5) c^3+3 b d^2 (m+5) x^2 c-d \left (a d^2 (m+2)-3 b c^2 (m+5)\right ) x\right )dx}{b (m+5)}+\frac {d^3 \left (a+b x^2\right )^{3/2} (e x)^{m+2}}{b e^2 (m+5)}\)

\(\Big \downarrow \) 2340

\(\displaystyle \frac {\frac {\int -b (e x)^m \left (c (m+5) \left (3 a d^2 (m+1)-b c^2 (m+4)\right )+d (m+4) \left (a d^2 (m+2)-3 b c^2 (m+5)\right ) x\right ) \sqrt {b x^2+a}dx}{b (m+4)}+\frac {3 c d^2 (m+5) \left (a+b x^2\right )^{3/2} (e x)^{m+1}}{e (m+4)}}{b (m+5)}+\frac {d^3 \left (a+b x^2\right )^{3/2} (e x)^{m+2}}{b e^2 (m+5)}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {3 c d^2 (m+5) \left (a+b x^2\right )^{3/2} (e x)^{m+1}}{e (m+4)}-\frac {\int b (e x)^m \left (c (m+5) \left (3 a d^2 (m+1)-b c^2 (m+4)\right )+d (m+4) \left (a d^2 (m+2)-3 b c^2 (m+5)\right ) x\right ) \sqrt {b x^2+a}dx}{b (m+4)}}{b (m+5)}+\frac {d^3 \left (a+b x^2\right )^{3/2} (e x)^{m+2}}{b e^2 (m+5)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {3 c d^2 (m+5) \left (a+b x^2\right )^{3/2} (e x)^{m+1}}{e (m+4)}-\frac {\int (e x)^m \left (c (m+5) \left (3 a d^2 (m+1)-b c^2 (m+4)\right )+d (m+4) \left (a d^2 (m+2)-3 b c^2 (m+5)\right ) x\right ) \sqrt {b x^2+a}dx}{m+4}}{b (m+5)}+\frac {d^3 \left (a+b x^2\right )^{3/2} (e x)^{m+2}}{b e^2 (m+5)}\)

\(\Big \downarrow \) 557

\(\displaystyle \frac {\frac {3 c d^2 (m+5) \left (a+b x^2\right )^{3/2} (e x)^{m+1}}{e (m+4)}-\frac {c (m+5) \left (3 a d^2 (m+1)-b c^2 (m+4)\right ) \int (e x)^m \sqrt {b x^2+a}dx+\frac {d (m+4) \left (a d^2 (m+2)-3 b c^2 (m+5)\right ) \int (e x)^{m+1} \sqrt {b x^2+a}dx}{e}}{m+4}}{b (m+5)}+\frac {d^3 \left (a+b x^2\right )^{3/2} (e x)^{m+2}}{b e^2 (m+5)}\)

\(\Big \downarrow \) 279

\(\displaystyle \frac {\frac {3 c d^2 (m+5) \left (a+b x^2\right )^{3/2} (e x)^{m+1}}{e (m+4)}-\frac {\frac {c (m+5) \sqrt {a+b x^2} \left (3 a d^2 (m+1)-b c^2 (m+4)\right ) \int (e x)^m \sqrt {\frac {b x^2}{a}+1}dx}{\sqrt {\frac {b x^2}{a}+1}}+\frac {d (m+4) \sqrt {a+b x^2} \left (a d^2 (m+2)-3 b c^2 (m+5)\right ) \int (e x)^{m+1} \sqrt {\frac {b x^2}{a}+1}dx}{e \sqrt {\frac {b x^2}{a}+1}}}{m+4}}{b (m+5)}+\frac {d^3 \left (a+b x^2\right )^{3/2} (e x)^{m+2}}{b e^2 (m+5)}\)

\(\Big \downarrow \) 278

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

Input:

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

Output:

(d^3*(e*x)^(2 + m)*(a + b*x^2)^(3/2))/(b*e^2*(5 + m)) + ((3*c*d^2*(5 + m)* 
(e*x)^(1 + m)*(a + b*x^2)^(3/2))/(e*(4 + m)) - ((c*(5 + m)*(3*a*d^2*(1 + m 
) - b*c^2*(4 + m))*(e*x)^(1 + m)*Sqrt[a + b*x^2]*Hypergeometric2F1[-1/2, ( 
1 + m)/2, (3 + m)/2, -((b*x^2)/a)])/(e*(1 + m)*Sqrt[1 + (b*x^2)/a]) + (d*( 
4 + m)*(a*d^2*(2 + m) - 3*b*c^2*(5 + m))*(e*x)^(2 + m)*Sqrt[a + b*x^2]*Hyp 
ergeometric2F1[-1/2, (2 + m)/2, (4 + m)/2, -((b*x^2)/a)])/(e^2*(2 + m)*Sqr 
t[1 + (b*x^2)/a]))/(4 + m))/(b*(5 + m))
 

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 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 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]

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

Input:

int((e*x)^m*(d*x+c)^3*(b*x^2+a)^(1/2),x)
 

Output:

int((e*x)^m*(d*x+c)^3*(b*x^2+a)^(1/2),x)
 

Fricas [F]

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

integrate((e*x)^m*(d*x+c)^3*(b*x^2+a)^(1/2),x, algorithm="fricas")
 

Output:

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

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 3.57 (sec) , antiderivative size = 243, normalized size of antiderivative = 0.99 \[ \int (e x)^m (c+d x)^3 \sqrt {a+b x^2} \, dx=\frac {\sqrt {a} c^{3} e^{m} x^{m + 1} \Gamma \left (\frac {m}{2} + \frac {1}{2}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {m}{2} + \frac {1}{2} \\ \frac {m}{2} + \frac {3}{2} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (\frac {m}{2} + \frac {3}{2}\right )} + \frac {3 \sqrt {a} c^{2} d e^{m} x^{m + 2} \Gamma \left (\frac {m}{2} + 1\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {m}{2} + 1 \\ \frac {m}{2} + 2 \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (\frac {m}{2} + 2\right )} + \frac {3 \sqrt {a} c d^{2} e^{m} x^{m + 3} \Gamma \left (\frac {m}{2} + \frac {3}{2}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {m}{2} + \frac {3}{2} \\ \frac {m}{2} + \frac {5}{2} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (\frac {m}{2} + \frac {5}{2}\right )} + \frac {\sqrt {a} d^{3} e^{m} x^{m + 4} \Gamma \left (\frac {m}{2} + 2\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {m}{2} + 2 \\ \frac {m}{2} + 3 \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (\frac {m}{2} + 3\right )} \] Input:

integrate((e*x)**m*(d*x+c)**3*(b*x**2+a)**(1/2),x)
 

Output:

sqrt(a)*c**3*e**m*x**(m + 1)*gamma(m/2 + 1/2)*hyper((-1/2, m/2 + 1/2), (m/ 
2 + 3/2,), b*x**2*exp_polar(I*pi)/a)/(2*gamma(m/2 + 3/2)) + 3*sqrt(a)*c**2 
*d*e**m*x**(m + 2)*gamma(m/2 + 1)*hyper((-1/2, m/2 + 1), (m/2 + 2,), b*x** 
2*exp_polar(I*pi)/a)/(2*gamma(m/2 + 2)) + 3*sqrt(a)*c*d**2*e**m*x**(m + 3) 
*gamma(m/2 + 3/2)*hyper((-1/2, m/2 + 3/2), (m/2 + 5/2,), b*x**2*exp_polar( 
I*pi)/a)/(2*gamma(m/2 + 5/2)) + sqrt(a)*d**3*e**m*x**(m + 4)*gamma(m/2 + 2 
)*hyper((-1/2, m/2 + 2), (m/2 + 3,), b*x**2*exp_polar(I*pi)/a)/(2*gamma(m/ 
2 + 3))
 

Maxima [F]

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

integrate((e*x)^m*(d*x+c)^3*(b*x^2+a)^(1/2),x, algorithm="maxima")
 

Output:

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

Giac [F]

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

integrate((e*x)^m*(d*x+c)^3*(b*x^2+a)^(1/2),x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

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

int((e*x)^m*(a + b*x^2)^(1/2)*(c + d*x)^3,x)
 

Output:

int((e*x)^m*(a + b*x^2)^(1/2)*(c + d*x)^3, x)
 

Reduce [F]

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

int((e*x)^m*(d*x+c)^3*(b*x^2+a)^(1/2),x)
 

Output:

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