\(\int (c x)^m (a+b x^2)^p (A+B x^2+C x^4+D x^6) \, dx\) [265]

Optimal result

Integrand size = 32, antiderivative size = 346 \[ \int (c x)^m \left (a+b x^2\right )^p \left (A+B x^2+C x^4+D x^6\right ) \, dx=\frac {\left (a^2 D \left (15+8 m+m^2\right )-a b C (3+m) (7+m+2 p)+b^2 B \left (35+m^2+24 p+4 p^2+4 m (3+p)\right )\right ) (c x)^{1+m} \left (a+b x^2\right )^{1+p}}{b^3 c (3+m+2 p) (5+m+2 p) (7+m+2 p)}-\frac {(a D (5+m)-b C (7+m+2 p)) (c x)^{3+m} \left (a+b x^2\right )^{1+p}}{b^2 c^3 (5+m+2 p) (7+m+2 p)}+\frac {D (c x)^{5+m} \left (a+b x^2\right )^{1+p}}{b c^5 (7+m+2 p)}+\frac {\left (\frac {A}{1+m}-\frac {a \left (a^2 D \left (15+8 m+m^2\right )-a b C (3+m) (7+m+2 p)+b^2 B \left (35+m^2+24 p+4 p^2+4 m (3+p)\right )\right )}{b^3 (3+m+2 p) (5+m+2 p) (7+m+2 p)}\right ) (c x)^{1+m} \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1+m}{2},-p,\frac {3+m}{2},-\frac {b x^2}{a}\right )}{c} \] Output:

(a^2*D*(m^2+8*m+15)-a*b*C*(3+m)*(7+m+2*p)+b^2*B*(35+m^2+24*p+4*p^2+4*m*(3+ 
p)))*(c*x)^(1+m)*(b*x^2+a)^(p+1)/b^3/c/(3+m+2*p)/(5+m+2*p)/(7+m+2*p)-(a*D* 
(5+m)-b*C*(7+m+2*p))*(c*x)^(3+m)*(b*x^2+a)^(p+1)/b^2/c^3/(5+m+2*p)/(7+m+2* 
p)+D*(c*x)^(5+m)*(b*x^2+a)^(p+1)/b/c^5/(7+m+2*p)+(A/(1+m)-a*(a^2*D*(m^2+8* 
m+15)-a*b*C*(3+m)*(7+m+2*p)+b^2*B*(35+m^2+24*p+4*p^2+4*m*(3+p)))/b^3/(3+m+ 
2*p)/(5+m+2*p)/(7+m+2*p))*(c*x)^(1+m)*(b*x^2+a)^p*hypergeom([-p, 1/2+1/2*m 
],[3/2+1/2*m],-b*x^2/a)/c/((1+b*x^2/a)^p)
 

Mathematica [A] (verified)

Time = 0.32 (sec) , antiderivative size = 176, normalized size of antiderivative = 0.51 \[ \int (c x)^m \left (a+b x^2\right )^p \left (A+B x^2+C x^4+D x^6\right ) \, dx=x (c x)^m \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \left (\frac {A \operatorname {Hypergeometric2F1}\left (\frac {1+m}{2},-p,\frac {3+m}{2},-\frac {b x^2}{a}\right )}{1+m}+\frac {B x^2 \operatorname {Hypergeometric2F1}\left (\frac {3+m}{2},-p,\frac {5+m}{2},-\frac {b x^2}{a}\right )}{3+m}+\frac {C x^4 \operatorname {Hypergeometric2F1}\left (\frac {5+m}{2},-p,\frac {7+m}{2},-\frac {b x^2}{a}\right )}{5+m}+\frac {D x^6 \operatorname {Hypergeometric2F1}\left (\frac {7+m}{2},-p,\frac {9+m}{2},-\frac {b x^2}{a}\right )}{7+m}\right ) \] Input:

Integrate[(c*x)^m*(a + b*x^2)^p*(A + B*x^2 + C*x^4 + D*x^6),x]
 

Output:

(x*(c*x)^m*(a + b*x^2)^p*((A*Hypergeometric2F1[(1 + m)/2, -p, (3 + m)/2, - 
((b*x^2)/a)])/(1 + m) + (B*x^2*Hypergeometric2F1[(3 + m)/2, -p, (5 + m)/2, 
 -((b*x^2)/a)])/(3 + m) + (C*x^4*Hypergeometric2F1[(5 + m)/2, -p, (7 + m)/ 
2, -((b*x^2)/a)])/(5 + m) + (D*x^6*Hypergeometric2F1[(7 + m)/2, -p, (9 + m 
)/2, -((b*x^2)/a)])/(7 + m)))/(1 + (b*x^2)/a)^p
 

Rubi [A] (verified)

Time = 0.71 (sec) , antiderivative size = 350, normalized size of antiderivative = 1.01, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.156, Rules used = {2340, 1590, 363, 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.

\(\displaystyle \int (c x)^m \left (a+b x^2\right )^p \left (A+B x^2+C x^4+D x^6\right ) \, dx\)

\(\Big \downarrow \) 2340

\(\displaystyle \frac {\int (c x)^m \left (b x^2+a\right )^p \left (-\left ((a D (m+5)-b C (m+2 p+7)) x^4\right )+b B (m+2 p+7) x^2+A b (m+2 p+7)\right )dx}{b (m+2 p+7)}+\frac {D (c x)^{m+5} \left (a+b x^2\right )^{p+1}}{b c^5 (m+2 p+7)}\)

\(\Big \downarrow \) 1590

\(\displaystyle \frac {\frac {\int (c x)^m \left (b x^2+a\right )^p \left (A (m+2 p+5) (m+2 p+7) b^2+\left (D \left (m^2+8 m+15\right ) a^2-b C (m+3) (m+2 p+7) a+b^2 B \left (m^2+4 (p+3) m+4 p^2+24 p+35\right )\right ) x^2\right )dx}{b (m+2 p+5)}-\frac {(c x)^{m+3} \left (a+b x^2\right )^{p+1} (a D (m+5)-b C (m+2 p+7))}{b c^3 (m+2 p+5)}}{b (m+2 p+7)}+\frac {D (c x)^{m+5} \left (a+b x^2\right )^{p+1}}{b c^5 (m+2 p+7)}\)

\(\Big \downarrow \) 363

\(\displaystyle \frac {\frac {\frac {\left (A b^3 (m+2 p+5) (m+2 p+7)-\frac {a (m+1) \left (a^2 D \left (m^2+8 m+15\right )-a b C (m+3) (m+2 p+7)+b^2 B \left (m^2+4 m (p+3)+4 p^2+24 p+35\right )\right )}{m+2 p+3}\right ) \int (c x)^m \left (b x^2+a\right )^pdx}{b}+\frac {(c x)^{m+1} \left (a+b x^2\right )^{p+1} \left (a^2 D \left (m^2+8 m+15\right )-a b C (m+3) (m+2 p+7)+b^2 B \left (m^2+4 m (p+3)+4 p^2+24 p+35\right )\right )}{b c (m+2 p+3)}}{b (m+2 p+5)}-\frac {(c x)^{m+3} \left (a+b x^2\right )^{p+1} (a D (m+5)-b C (m+2 p+7))}{b c^3 (m+2 p+5)}}{b (m+2 p+7)}+\frac {D (c x)^{m+5} \left (a+b x^2\right )^{p+1}}{b c^5 (m+2 p+7)}\)

\(\Big \downarrow \) 279

\(\displaystyle \frac {\frac {\frac {\left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \left (A b^3 (m+2 p+5) (m+2 p+7)-\frac {a (m+1) \left (a^2 D \left (m^2+8 m+15\right )-a b C (m+3) (m+2 p+7)+b^2 B \left (m^2+4 m (p+3)+4 p^2+24 p+35\right )\right )}{m+2 p+3}\right ) \int (c x)^m \left (\frac {b x^2}{a}+1\right )^pdx}{b}+\frac {(c x)^{m+1} \left (a+b x^2\right )^{p+1} \left (a^2 D \left (m^2+8 m+15\right )-a b C (m+3) (m+2 p+7)+b^2 B \left (m^2+4 m (p+3)+4 p^2+24 p+35\right )\right )}{b c (m+2 p+3)}}{b (m+2 p+5)}-\frac {(c x)^{m+3} \left (a+b x^2\right )^{p+1} (a D (m+5)-b C (m+2 p+7))}{b c^3 (m+2 p+5)}}{b (m+2 p+7)}+\frac {D (c x)^{m+5} \left (a+b x^2\right )^{p+1}}{b c^5 (m+2 p+7)}\)

\(\Big \downarrow \) 278

\(\displaystyle \frac {\frac {\frac {(c x)^{m+1} \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {m+1}{2},-p,\frac {m+3}{2},-\frac {b x^2}{a}\right ) \left (A b^3 (m+2 p+5) (m+2 p+7)-\frac {a (m+1) \left (a^2 D \left (m^2+8 m+15\right )-a b C (m+3) (m+2 p+7)+b^2 B \left (m^2+4 m (p+3)+4 p^2+24 p+35\right )\right )}{m+2 p+3}\right )}{b c (m+1)}+\frac {(c x)^{m+1} \left (a+b x^2\right )^{p+1} \left (a^2 D \left (m^2+8 m+15\right )-a b C (m+3) (m+2 p+7)+b^2 B \left (m^2+4 m (p+3)+4 p^2+24 p+35\right )\right )}{b c (m+2 p+3)}}{b (m+2 p+5)}-\frac {(c x)^{m+3} \left (a+b x^2\right )^{p+1} (a D (m+5)-b C (m+2 p+7))}{b c^3 (m+2 p+5)}}{b (m+2 p+7)}+\frac {D (c x)^{m+5} \left (a+b x^2\right )^{p+1}}{b c^5 (m+2 p+7)}\)

Input:

Int[(c*x)^m*(a + b*x^2)^p*(A + B*x^2 + C*x^4 + D*x^6),x]
 

Output:

(D*(c*x)^(5 + m)*(a + b*x^2)^(1 + p))/(b*c^5*(7 + m + 2*p)) + (-(((a*D*(5 
+ m) - b*C*(7 + m + 2*p))*(c*x)^(3 + m)*(a + b*x^2)^(1 + p))/(b*c^3*(5 + m 
 + 2*p))) + (((a^2*D*(15 + 8*m + m^2) - a*b*C*(3 + m)*(7 + m + 2*p) + b^2* 
B*(35 + m^2 + 24*p + 4*p^2 + 4*m*(3 + p)))*(c*x)^(1 + m)*(a + b*x^2)^(1 + 
p))/(b*c*(3 + m + 2*p)) + ((A*b^3*(5 + m + 2*p)*(7 + m + 2*p) - (a*(1 + m) 
*(a^2*D*(15 + 8*m + m^2) - a*b*C*(3 + m)*(7 + m + 2*p) + b^2*B*(35 + m^2 + 
 24*p + 4*p^2 + 4*m*(3 + p))))/(3 + m + 2*p))*(c*x)^(1 + m)*(a + b*x^2)^p* 
Hypergeometric2F1[(1 + m)/2, -p, (3 + m)/2, -((b*x^2)/a)])/(b*c*(1 + m)*(1 
 + (b*x^2)/a)^p))/(b*(5 + m + 2*p)))/(b*(7 + m + 2*p))
 

Defintions of rubi rules used

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 363

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2), x 
_Symbol] :> Simp[d*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(b*e*(m + 2*p + 3))), 
 x] - Simp[(a*d*(m + 1) - b*c*(m + 2*p + 3))/(b*(m + 2*p + 3))   Int[(e*x)^ 
m*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b*c - a*d 
, 0] && NeQ[m + 2*p + 3, 0]
 

rule 1590

Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + ( 
c_.)*(x_)^4)^(p_.), x_Symbol] :> Simp[c^p*(f*x)^(m + 4*p - 1)*((d + e*x^2)^ 
(q + 1)/(e*f^(4*p - 1)*(m + 4*p + 2*q + 1))), x] + Simp[1/(e*(m + 4*p + 2*q 
 + 1))   Int[(f*x)^m*(d + e*x^2)^q*ExpandToSum[e*(m + 4*p + 2*q + 1)*((a + 
b*x^2 + c*x^4)^p - c^p*x^(4*p)) - d*c^p*(m + 4*p - 1)*x^(4*p - 2), x], x], 
x] /; FreeQ[{a, b, c, d, e, f, m, q}, x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[p, 
0] &&  !IntegerQ[q] && NeQ[m + 4*p + 2*q + 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]

\[\int \left (c x \right )^{m} \left (b \,x^{2}+a \right )^{p} \left (D x^{6}+C \,x^{4}+x^{2} B +A \right )d x\]

Input:

int((c*x)^m*(b*x^2+a)^p*(D*x^6+C*x^4+B*x^2+A),x)
 

Output:

int((c*x)^m*(b*x^2+a)^p*(D*x^6+C*x^4+B*x^2+A),x)
 

Fricas [F]

\[ \int (c x)^m \left (a+b x^2\right )^p \left (A+B x^2+C x^4+D x^6\right ) \, dx=\int { {\left (D x^{6} + C x^{4} + B x^{2} + A\right )} {\left (b x^{2} + a\right )}^{p} \left (c x\right )^{m} \,d x } \] Input:

integrate((c*x)^m*(b*x^2+a)^p*(D*x^6+C*x^4+B*x^2+A),x, algorithm="fricas")
 

Output:

integral((D*x^6 + C*x^4 + B*x^2 + A)*(b*x^2 + a)^p*(c*x)^m, x)
 

Sympy [F(-1)]

Timed out. \[ \int (c x)^m \left (a+b x^2\right )^p \left (A+B x^2+C x^4+D x^6\right ) \, dx=\text {Timed out} \] Input:

integrate((c*x)**m*(b*x**2+a)**p*(D*x**6+C*x**4+B*x**2+A),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int (c x)^m \left (a+b x^2\right )^p \left (A+B x^2+C x^4+D x^6\right ) \, dx=\int { {\left (D x^{6} + C x^{4} + B x^{2} + A\right )} {\left (b x^{2} + a\right )}^{p} \left (c x\right )^{m} \,d x } \] Input:

integrate((c*x)^m*(b*x^2+a)^p*(D*x^6+C*x^4+B*x^2+A),x, algorithm="maxima")
 

Output:

integrate((D*x^6 + C*x^4 + B*x^2 + A)*(b*x^2 + a)^p*(c*x)^m, x)
 

Giac [F]

\[ \int (c x)^m \left (a+b x^2\right )^p \left (A+B x^2+C x^4+D x^6\right ) \, dx=\int { {\left (D x^{6} + C x^{4} + B x^{2} + A\right )} {\left (b x^{2} + a\right )}^{p} \left (c x\right )^{m} \,d x } \] Input:

integrate((c*x)^m*(b*x^2+a)^p*(D*x^6+C*x^4+B*x^2+A),x, algorithm="giac")
 

Output:

integrate((D*x^6 + C*x^4 + B*x^2 + A)*(b*x^2 + a)^p*(c*x)^m, x)
 

Mupad [F(-1)]

Timed out. \[ \int (c x)^m \left (a+b x^2\right )^p \left (A+B x^2+C x^4+D x^6\right ) \, dx=\int {\left (c\,x\right )}^m\,{\left (b\,x^2+a\right )}^p\,\left (A+B\,x^2+C\,x^4+x^6\,D\right ) \,d x \] Input:

int((c*x)^m*(a + b*x^2)^p*(A + B*x^2 + C*x^4 + x^6*D),x)
 

Output:

int((c*x)^m*(a + b*x^2)^p*(A + B*x^2 + C*x^4 + x^6*D), x)
 

Reduce [F]

\[ \int (c x)^m \left (a+b x^2\right )^p \left (A+B x^2+C x^4+D x^6\right ) \, dx=\text {too large to display} \] Input:

int((c*x)^m*(b*x^2+a)^p*(D*x^6+C*x^4+B*x^2+A),x)
 

Output:

(c**m*(2*x**m*(a + b*x**2)**p*a**3*d*m**2*p*x + 16*x**m*(a + b*x**2)**p*a* 
*3*d*m*p*x + 30*x**m*(a + b*x**2)**p*a**3*d*p*x - 2*x**m*(a + b*x**2)**p*a 
**2*b*c*m**2*p*x - 4*x**m*(a + b*x**2)**p*a**2*b*c*m*p**2*x - 20*x**m*(a + 
 b*x**2)**p*a**2*b*c*m*p*x - 12*x**m*(a + b*x**2)**p*a**2*b*c*p**2*x - 42* 
x**m*(a + b*x**2)**p*a**2*b*c*p*x - 2*x**m*(a + b*x**2)**p*a**2*b*d*m**2*p 
*x**3 - 4*x**m*(a + b*x**2)**p*a**2*b*d*m*p**2*x**3 - 12*x**m*(a + b*x**2) 
**p*a**2*b*d*m*p*x**3 - 20*x**m*(a + b*x**2)**p*a**2*b*d*p**2*x**3 - 10*x* 
*m*(a + b*x**2)**p*a**2*b*d*p*x**3 + x**m*(a + b*x**2)**p*a*b**3*m**3*x + 
8*x**m*(a + b*x**2)**p*a*b**3*m**2*p*x + 15*x**m*(a + b*x**2)**p*a*b**3*m* 
*2*x + 20*x**m*(a + b*x**2)**p*a*b**3*m*p**2*x + 84*x**m*(a + b*x**2)**p*a 
*b**3*m*p*x + 71*x**m*(a + b*x**2)**p*a*b**3*m*x + 16*x**m*(a + b*x**2)**p 
*a*b**3*p**3*x + 108*x**m*(a + b*x**2)**p*a*b**3*p**2*x + 212*x**m*(a + b* 
x**2)**p*a*b**3*p*x + 105*x**m*(a + b*x**2)**p*a*b**3*x + 2*x**m*(a + b*x* 
*2)**p*a*b**2*c*m**2*p*x**3 + 8*x**m*(a + b*x**2)**p*a*b**2*c*m*p**2*x**3 
+ 16*x**m*(a + b*x**2)**p*a*b**2*c*m*p*x**3 + 8*x**m*(a + b*x**2)**p*a*b** 
2*c*p**3*x**3 + 32*x**m*(a + b*x**2)**p*a*b**2*c*p**2*x**3 + 14*x**m*(a + 
b*x**2)**p*a*b**2*c*p*x**3 + 2*x**m*(a + b*x**2)**p*a*b**2*d*m**2*p*x**5 + 
 8*x**m*(a + b*x**2)**p*a*b**2*d*m*p**2*x**5 + 8*x**m*(a + b*x**2)**p*a*b* 
*2*d*m*p*x**5 + 8*x**m*(a + b*x**2)**p*a*b**2*d*p**3*x**5 + 16*x**m*(a + b 
*x**2)**p*a*b**2*d*p**2*x**5 + 6*x**m*(a + b*x**2)**p*a*b**2*d*p*x**5 +...