Integrand size = 20, antiderivative size = 199 \[ \int \frac {x^4 \left (a+b x^2\right )^p}{c+d x} \, dx=\frac {\left (b c^2-a d^2\right ) \left (a+b x^2\right )^{1+p}}{2 b^2 d^3 (1+p)}+\frac {\left (a+b x^2\right )^{2+p}}{2 b^2 d (2+p)}+\frac {x^5 \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \operatorname {AppellF1}\left (\frac {5}{2},-p,1,\frac {7}{2},-\frac {b x^2}{a},\frac {d^2 x^2}{c^2}\right )}{5 c}-\frac {c^4 \left (a+b x^2\right )^{1+p} \operatorname {Hypergeometric2F1}\left (1,1+p,2+p,\frac {d^2 \left (a+b x^2\right )}{b c^2+a d^2}\right )}{2 d^3 \left (b c^2+a d^2\right ) (1+p)} \] Output:
1/2*(-a*d^2+b*c^2)*(b*x^2+a)^(p+1)/b^2/d^3/(p+1)+1/2*(b*x^2+a)^(2+p)/b^2/d /(2+p)+1/5*x^5*(b*x^2+a)^p*AppellF1(5/2,1,-p,7/2,d^2*x^2/c^2,-b*x^2/a)/c/( (1+b*x^2/a)^p)-1/2*c^4*(b*x^2+a)^(p+1)*hypergeom([1, p+1],[2+p],d^2*(b*x^2 +a)/(a*d^2+b*c^2))/d^3/(a*d^2+b*c^2)/(p+1)
\[ \int \frac {x^4 \left (a+b x^2\right )^p}{c+d x} \, dx=\int \frac {x^4 \left (a+b x^2\right )^p}{c+d x} \, dx \] Input:
Integrate[(x^4*(a + b*x^2)^p)/(c + d*x),x]
Output:
Integrate[(x^4*(a + b*x^2)^p)/(c + d*x), x]
Time = 0.63 (sec) , antiderivative size = 198, normalized size of antiderivative = 0.99, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {621, 354, 99, 395, 394, 2009}
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.
\(\displaystyle \int \frac {x^4 \left (a+b x^2\right )^p}{c+d x} \, dx\) |
\(\Big \downarrow \) 621 |
\(\displaystyle c \int \frac {x^4 \left (b x^2+a\right )^p}{c^2-d^2 x^2}dx-d \int \frac {x^5 \left (b x^2+a\right )^p}{c^2-d^2 x^2}dx\) |
\(\Big \downarrow \) 354 |
\(\displaystyle c \int \frac {x^4 \left (b x^2+a\right )^p}{c^2-d^2 x^2}dx-\frac {1}{2} d \int \frac {x^4 \left (b x^2+a\right )^p}{c^2-d^2 x^2}dx^2\) |
\(\Big \downarrow \) 99 |
\(\displaystyle c \int \frac {x^4 \left (b x^2+a\right )^p}{c^2-d^2 x^2}dx-\frac {1}{2} d \int \left (\frac {\left (a d^2-b c^2\right ) \left (b x^2+a\right )^p}{b d^4}+\frac {c^4 \left (b x^2+a\right )^p}{d^4 \left (c^2-d^2 x^2\right )}-\frac {\left (b x^2+a\right )^{p+1}}{b d^2}\right )dx^2\) |
\(\Big \downarrow \) 395 |
\(\displaystyle c \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \int \frac {x^4 \left (\frac {b x^2}{a}+1\right )^p}{c^2-d^2 x^2}dx-\frac {1}{2} d \int \left (\frac {\left (a d^2-b c^2\right ) \left (b x^2+a\right )^p}{b d^4}+\frac {c^4 \left (b x^2+a\right )^p}{d^4 \left (c^2-d^2 x^2\right )}-\frac {\left (b x^2+a\right )^{p+1}}{b d^2}\right )dx^2\) |
\(\Big \downarrow \) 394 |
\(\displaystyle \frac {x^5 \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \operatorname {AppellF1}\left (\frac {5}{2},-p,1,\frac {7}{2},-\frac {b x^2}{a},\frac {d^2 x^2}{c^2}\right )}{5 c}-\frac {1}{2} d \int \left (\frac {\left (a d^2-b c^2\right ) \left (b x^2+a\right )^p}{b d^4}+\frac {c^4 \left (b x^2+a\right )^p}{d^4 \left (c^2-d^2 x^2\right )}-\frac {\left (b x^2+a\right )^{p+1}}{b d^2}\right )dx^2\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {x^5 \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \operatorname {AppellF1}\left (\frac {5}{2},-p,1,\frac {7}{2},-\frac {b x^2}{a},\frac {d^2 x^2}{c^2}\right )}{5 c}-\frac {1}{2} d \left (-\frac {\left (b c^2-a d^2\right ) \left (a+b x^2\right )^{p+1}}{b^2 d^4 (p+1)}-\frac {\left (a+b x^2\right )^{p+2}}{b^2 d^2 (p+2)}+\frac {c^4 \left (a+b x^2\right )^{p+1} \operatorname {Hypergeometric2F1}\left (1,p+1,p+2,\frac {d^2 \left (b x^2+a\right )}{b c^2+a d^2}\right )}{d^4 (p+1) \left (a d^2+b c^2\right )}\right )\) |
Input:
Int[(x^4*(a + b*x^2)^p)/(c + d*x),x]
Output:
(x^5*(a + b*x^2)^p*AppellF1[5/2, -p, 1, 7/2, -((b*x^2)/a), (d^2*x^2)/c^2]) /(5*c*(1 + (b*x^2)/a)^p) - (d*(-(((b*c^2 - a*d^2)*(a + b*x^2)^(1 + p))/(b^ 2*d^4*(1 + p))) - (a + b*x^2)^(2 + p)/(b^2*d^2*(2 + p)) + (c^4*(a + b*x^2) ^(1 + p)*Hypergeometric2F1[1, 1 + p, 2 + p, (d^2*(a + b*x^2))/(b*c^2 + a*d ^2)])/(d^4*(b*c^2 + a*d^2)*(1 + p))))/2
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] && (IntegerQ[p] | | (GtQ[m, 0] && GeQ[n, -1]))
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_S ymbol] :> Simp[1/2 Subst[Int[x^((m - 1)/2)*(a + b*x)^p*(c + d*x)^q, x], x , x^2], x] /; FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && IntegerQ [(m - 1)/2]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_ ), x_Symbol] :> Simp[a^p*c^q*((e*x)^(m + 1)/(e*(m + 1)))*AppellF1[(m + 1)/2 , -p, -q, 1 + (m + 1)/2, (-b)*(x^2/a), (-d)*(x^2/c)], x] /; FreeQ[{a, b, c, d, e, m, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, 1] && (Int egerQ[p] || GtQ[a, 0]) && (IntegerQ[q] || GtQ[c, 0])
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_ ), x_Symbol] :> Simp[a^IntPart[p]*((a + b*x^2)^FracPart[p]/(1 + b*(x^2/a))^ FracPart[p]) Int[(e*x)^m*(1 + b*(x^2/a))^p*(c + d*x^2)^q, x], x] /; FreeQ [{a, b, c, d, e, m, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, 1] && !(IntegerQ[p] || GtQ[a, 0])
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]
\[\int \frac {x^{4} \left (b \,x^{2}+a \right )^{p}}{d x +c}d x\]
Input:
int(x^4*(b*x^2+a)^p/(d*x+c),x)
Output:
int(x^4*(b*x^2+a)^p/(d*x+c),x)
\[ \int \frac {x^4 \left (a+b x^2\right )^p}{c+d x} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{p} x^{4}}{d x + c} \,d x } \] Input:
integrate(x^4*(b*x^2+a)^p/(d*x+c),x, algorithm="fricas")
Output:
integral((b*x^2 + a)^p*x^4/(d*x + c), x)
Timed out. \[ \int \frac {x^4 \left (a+b x^2\right )^p}{c+d x} \, dx=\text {Timed out} \] Input:
integrate(x**4*(b*x**2+a)**p/(d*x+c),x)
Output:
Timed out
\[ \int \frac {x^4 \left (a+b x^2\right )^p}{c+d x} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{p} x^{4}}{d x + c} \,d x } \] Input:
integrate(x^4*(b*x^2+a)^p/(d*x+c),x, algorithm="maxima")
Output:
integrate((b*x^2 + a)^p*x^4/(d*x + c), x)
\[ \int \frac {x^4 \left (a+b x^2\right )^p}{c+d x} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{p} x^{4}}{d x + c} \,d x } \] Input:
integrate(x^4*(b*x^2+a)^p/(d*x+c),x, algorithm="giac")
Output:
integrate((b*x^2 + a)^p*x^4/(d*x + c), x)
Timed out. \[ \int \frac {x^4 \left (a+b x^2\right )^p}{c+d x} \, dx=\int \frac {x^4\,{\left (b\,x^2+a\right )}^p}{c+d\,x} \,d x \] Input:
int((x^4*(a + b*x^2)^p)/(c + d*x),x)
Output:
int((x^4*(a + b*x^2)^p)/(c + d*x), x)
\[ \int \frac {x^4 \left (a+b x^2\right )^p}{c+d x} \, dx=\text {too large to display} \] Input:
int(x^4*(b*x^2+a)^p/(d*x+c),x)
Output:
( - 2*(a + b*x**2)**p*a**2*d**3*p**2 - 2*(a + b*x**2)**p*a**2*d**3*p + (a + b*x**2)**p*a**2*d**3 - 2*(a + b*x**2)**p*a*b*c**2*d*p**2 - 7*(a + b*x**2 )**p*a*b*c**2*d*p - 6*(a + b*x**2)**p*a*b*c**2*d - 4*(a + b*x**2)**p*a*b*c *d**2*p**3*x - 12*(a + b*x**2)**p*a*b*c*d**2*p**2*x - 8*(a + b*x**2)**p*a* b*c*d**2*p*x + 4*(a + b*x**2)**p*a*b*d**3*p**3*x**2 + 8*(a + b*x**2)**p*a* b*d**3*p**2*x**2 + 3*(a + b*x**2)**p*a*b*d**3*p*x**2 - 4*(a + b*x**2)**p*b **2*c**3*p**3*x - 18*(a + b*x**2)**p*b**2*c**3*p**2*x - 26*(a + b*x**2)**p *b**2*c**3*p*x - 12*(a + b*x**2)**p*b**2*c**3*x + 4*(a + b*x**2)**p*b**2*c **2*d*p**3*x**2 + 16*(a + b*x**2)**p*b**2*c**2*d*p**2*x**2 + 19*(a + b*x** 2)**p*b**2*c**2*d*p*x**2 + 6*(a + b*x**2)**p*b**2*c**2*d*x**2 - 4*(a + b*x **2)**p*b**2*c*d**2*p**3*x**3 - 14*(a + b*x**2)**p*b**2*c*d**2*p**2*x**3 - 14*(a + b*x**2)**p*b**2*c*d**2*p*x**3 - 4*(a + b*x**2)**p*b**2*c*d**2*x** 3 + 4*(a + b*x**2)**p*b**2*d**3*p**3*x**4 + 12*(a + b*x**2)**p*b**2*d**3*p **2*x**4 + 11*(a + b*x**2)**p*b**2*d**3*p*x**4 + 3*(a + b*x**2)**p*b**2*d* *3*x**4 + 16*int((a + b*x**2)**p/(4*a*c*p**2 + 8*a*c*p + 3*a*c + 4*a*d*p** 2*x + 8*a*d*p*x + 3*a*d*x + 4*b*c*p**2*x**2 + 8*b*c*p*x**2 + 3*b*c*x**2 + 4*b*d*p**2*x**3 + 8*b*d*p*x**3 + 3*b*d*x**3),x)*a**2*b*c**2*d**2*p**5 + 80 *int((a + b*x**2)**p/(4*a*c*p**2 + 8*a*c*p + 3*a*c + 4*a*d*p**2*x + 8*a*d* p*x + 3*a*d*x + 4*b*c*p**2*x**2 + 8*b*c*p*x**2 + 3*b*c*x**2 + 4*b*d*p**2*x **3 + 8*b*d*p*x**3 + 3*b*d*x**3),x)*a**2*b*c**2*d**2*p**4 + 140*int((a ...