Integrand size = 18, antiderivative size = 118 \[ \int \frac {x^6 (c+d x)}{\left (a+b x^2\right )^3} \, dx=\frac {c x}{b^3}+\frac {d x^2}{2 b^3}+\frac {a^2 (a d-b c x)}{4 b^4 \left (a+b x^2\right )^2}-\frac {3 a (4 a d-3 b c x)}{8 b^4 \left (a+b x^2\right )}-\frac {15 \sqrt {a} c \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{8 b^{7/2}}-\frac {3 a d \log \left (a+b x^2\right )}{2 b^4} \] Output:
c*x/b^3+1/2*d*x^2/b^3+1/4*a^2*(-b*c*x+a*d)/b^4/(b*x^2+a)^2-3/8*a*(-3*b*c*x +4*a*d)/b^4/(b*x^2+a)-15/8*a^(1/2)*c*arctan(b^(1/2)*x/a^(1/2))/b^(7/2)-3/2 *a*d*ln(b*x^2+a)/b^4
Time = 0.08 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.87 \[ \int \frac {x^6 (c+d x)}{\left (a+b x^2\right )^3} \, dx=\frac {8 b c x+4 b d x^2+\frac {2 a^2 (a d-b c x)}{\left (a+b x^2\right )^2}+\frac {3 a (-4 a d+3 b c x)}{a+b x^2}-15 \sqrt {a} \sqrt {b} c \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )-12 a d \log \left (a+b x^2\right )}{8 b^4} \] Input:
Integrate[(x^6*(c + d*x))/(a + b*x^2)^3,x]
Output:
(8*b*c*x + 4*b*d*x^2 + (2*a^2*(a*d - b*c*x))/(a + b*x^2)^2 + (3*a*(-4*a*d + 3*b*c*x))/(a + b*x^2) - 15*Sqrt[a]*Sqrt[b]*c*ArcTan[(Sqrt[b]*x)/Sqrt[a]] - 12*a*d*Log[a + b*x^2])/(8*b^4)
Time = 0.85 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.18, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {530, 25, 2345, 2341, 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.
\(\displaystyle \int \frac {x^6 (c+d x)}{\left (a+b x^2\right )^3} \, dx\) |
\(\Big \downarrow \) 530 |
\(\displaystyle \frac {a^2 (a d-b c x)}{4 b^4 \left (a+b x^2\right )^2}-\frac {\int -\frac {\frac {4 a d x^5}{b}+\frac {4 a c x^4}{b}-\frac {4 a^2 d x^3}{b^2}-\frac {4 a^2 c x^2}{b^2}+\frac {4 a^3 d x}{b^3}+\frac {a^3 c}{b^3}}{\left (b x^2+a\right )^2}dx}{4 a}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\int \frac {\frac {4 a d x^5}{b}+\frac {4 a c x^4}{b}-\frac {4 a^2 d x^3}{b^2}-\frac {4 a^2 c x^2}{b^2}+\frac {4 a^3 d x}{b^3}+\frac {a^3 c}{b^3}}{\left (b x^2+a\right )^2}dx}{4 a}+\frac {a^2 (a d-b c x)}{4 b^4 \left (a+b x^2\right )^2}\) |
\(\Big \downarrow \) 2345 |
\(\displaystyle \frac {-\frac {\int \frac {\frac {7 c a^3}{b^3}+\frac {16 d x a^3}{b^3}-\frac {8 d x^3 a^2}{b^2}-\frac {8 c x^2 a^2}{b^2}}{b x^2+a}dx}{2 a}-\frac {3 a^2 (4 a d-3 b c x)}{2 b^4 \left (a+b x^2\right )}}{4 a}+\frac {a^2 (a d-b c x)}{4 b^4 \left (a+b x^2\right )^2}\) |
\(\Big \downarrow \) 2341 |
\(\displaystyle \frac {-\frac {\int \left (-\frac {8 c a^2}{b^3}-\frac {8 d x a^2}{b^3}+\frac {3 \left (5 c a^3+8 d x a^3\right )}{b^3 \left (b x^2+a\right )}\right )dx}{2 a}-\frac {3 a^2 (4 a d-3 b c x)}{2 b^4 \left (a+b x^2\right )}}{4 a}+\frac {a^2 (a d-b c x)}{4 b^4 \left (a+b x^2\right )^2}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {a^2 (a d-b c x)}{4 b^4 \left (a+b x^2\right )^2}+\frac {-\frac {3 a^2 (4 a d-3 b c x)}{2 b^4 \left (a+b x^2\right )}-\frac {\frac {15 a^{5/2} c \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{b^{7/2}}+\frac {12 a^3 d \log \left (a+b x^2\right )}{b^4}-\frac {8 a^2 c x}{b^3}-\frac {4 a^2 d x^2}{b^3}}{2 a}}{4 a}\) |
Input:
Int[(x^6*(c + d*x))/(a + b*x^2)^3,x]
Output:
(a^2*(a*d - b*c*x))/(4*b^4*(a + b*x^2)^2) + ((-3*a^2*(4*a*d - 3*b*c*x))/(2 *b^4*(a + b*x^2)) - ((-8*a^2*c*x)/b^3 - (4*a^2*d*x^2)/b^3 + (15*a^(5/2)*c* ArcTan[(Sqrt[b]*x)/Sqrt[a]])/b^(7/2) + (12*a^3*d*Log[a + b*x^2])/b^4)/(2*a ))/(4*a)
Int[(x_)^(m_.)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symb ol] :> With[{Qx = PolynomialQuotient[x^m*(c + d*x)^n, a + b*x^2, x], e = Co eff[PolynomialRemainder[x^m*(c + d*x)^n, a + b*x^2, x], x, 0], f = Coeff[Po lynomialRemainder[x^m*(c + d*x)^n, a + b*x^2, x], x, 1]}, Simp[(a*f - b*e*x )*((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] + Simp[1/(2*a*(p + 1)) Int[(a + b*x^2)^(p + 1)*ExpandToSum[2*a*(p + 1)*Qx + e*(2*p + 3), x], x], x]] /; FreeQ[{a, b, c, d}, x] && IGtQ[n, 0] && IGtQ[m, 0] && LtQ[p, -1] && EqQ[n, 1] && IntegerQ[2*p]
Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[Pq* (a + b*x^2)^p, x], x] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && IGtQ[p, -2]
Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuot ient[Pq, a + b*x^2, x], f = Coeff[PolynomialRemainder[Pq, a + b*x^2, x], x, 0], g = Coeff[PolynomialRemainder[Pq, a + b*x^2, x], x, 1]}, Simp[(a*g - b *f*x)*((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] + Simp[1/(2*a*(p + 1)) In t[(a + b*x^2)^(p + 1)*ExpandToSum[2*a*(p + 1)*Q + f*(2*p + 3), x], x], x]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && LtQ[p, -1]
Time = 0.22 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.79
method | result | size |
default | \(\frac {\frac {1}{2} d \,x^{2}+c x}{b^{3}}-\frac {a \left (\frac {-\frac {9 b c \,x^{3}}{8}+\frac {3 a d \,x^{2}}{2}-\frac {7 a c x}{8}+\frac {5 a^{2} d}{4 b}}{\left (b \,x^{2}+a \right )^{2}}+\frac {3 d \ln \left (b \,x^{2}+a \right )}{2 b}+\frac {15 c \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \sqrt {a b}}\right )}{b^{3}}\) | \(93\) |
risch | \(\frac {d \,x^{2}}{2 b^{3}}+\frac {c x}{b^{3}}+\frac {\frac {9 a b c \,x^{3}}{8}-\frac {3 a^{2} d \,x^{2}}{2}+\frac {7 a^{2} c x}{8}-\frac {5 a^{3} d}{4 b}}{b^{3} \left (b \,x^{2}+a \right )^{2}}+\frac {15 \ln \left (-\sqrt {-a b}\, x -a \right ) c \sqrt {-a b}}{16 b^{4}}-\frac {3 \ln \left (-\sqrt {-a b}\, x -a \right ) a d}{2 b^{4}}-\frac {15 \ln \left (\sqrt {-a b}\, x -a \right ) c \sqrt {-a b}}{16 b^{4}}-\frac {3 \ln \left (\sqrt {-a b}\, x -a \right ) a d}{2 b^{4}}\) | \(156\) |
Input:
int(x^6*(d*x+c)/(b*x^2+a)^3,x,method=_RETURNVERBOSE)
Output:
1/b^3*(1/2*d*x^2+c*x)-a/b^3*((-9/8*b*c*x^3+3/2*a*d*x^2-7/8*a*c*x+5/4*a^2/b *d)/(b*x^2+a)^2+3/2*d*ln(b*x^2+a)/b+15/8*c/(a*b)^(1/2)*arctan(b*x/(a*b)^(1 /2)))
Time = 0.15 (sec) , antiderivative size = 368, normalized size of antiderivative = 3.12 \[ \int \frac {x^6 (c+d x)}{\left (a+b x^2\right )^3} \, dx=\left [\frac {8 \, b^{3} d x^{6} + 16 \, b^{3} c x^{5} + 16 \, a b^{2} d x^{4} + 50 \, a b^{2} c x^{3} - 16 \, a^{2} b d x^{2} + 30 \, a^{2} b c x - 20 \, a^{3} d + 15 \, {\left (b^{3} c x^{4} + 2 \, a b^{2} c x^{2} + a^{2} b c\right )} \sqrt {-\frac {a}{b}} \log \left (\frac {b x^{2} - 2 \, b x \sqrt {-\frac {a}{b}} - a}{b x^{2} + a}\right ) - 24 \, {\left (a b^{2} d x^{4} + 2 \, a^{2} b d x^{2} + a^{3} d\right )} \log \left (b x^{2} + a\right )}{16 \, {\left (b^{6} x^{4} + 2 \, a b^{5} x^{2} + a^{2} b^{4}\right )}}, \frac {4 \, b^{3} d x^{6} + 8 \, b^{3} c x^{5} + 8 \, a b^{2} d x^{4} + 25 \, a b^{2} c x^{3} - 8 \, a^{2} b d x^{2} + 15 \, a^{2} b c x - 10 \, a^{3} d - 15 \, {\left (b^{3} c x^{4} + 2 \, a b^{2} c x^{2} + a^{2} b c\right )} \sqrt {\frac {a}{b}} \arctan \left (\frac {b x \sqrt {\frac {a}{b}}}{a}\right ) - 12 \, {\left (a b^{2} d x^{4} + 2 \, a^{2} b d x^{2} + a^{3} d\right )} \log \left (b x^{2} + a\right )}{8 \, {\left (b^{6} x^{4} + 2 \, a b^{5} x^{2} + a^{2} b^{4}\right )}}\right ] \] Input:
integrate(x^6*(d*x+c)/(b*x^2+a)^3,x, algorithm="fricas")
Output:
[1/16*(8*b^3*d*x^6 + 16*b^3*c*x^5 + 16*a*b^2*d*x^4 + 50*a*b^2*c*x^3 - 16*a ^2*b*d*x^2 + 30*a^2*b*c*x - 20*a^3*d + 15*(b^3*c*x^4 + 2*a*b^2*c*x^2 + a^2 *b*c)*sqrt(-a/b)*log((b*x^2 - 2*b*x*sqrt(-a/b) - a)/(b*x^2 + a)) - 24*(a*b ^2*d*x^4 + 2*a^2*b*d*x^2 + a^3*d)*log(b*x^2 + a))/(b^6*x^4 + 2*a*b^5*x^2 + a^2*b^4), 1/8*(4*b^3*d*x^6 + 8*b^3*c*x^5 + 8*a*b^2*d*x^4 + 25*a*b^2*c*x^3 - 8*a^2*b*d*x^2 + 15*a^2*b*c*x - 10*a^3*d - 15*(b^3*c*x^4 + 2*a*b^2*c*x^2 + a^2*b*c)*sqrt(a/b)*arctan(b*x*sqrt(a/b)/a) - 12*(a*b^2*d*x^4 + 2*a^2*b* d*x^2 + a^3*d)*log(b*x^2 + a))/(b^6*x^4 + 2*a*b^5*x^2 + a^2*b^4)]
Leaf count of result is larger than twice the leaf count of optimal. 243 vs. \(2 (116) = 232\).
Time = 0.85 (sec) , antiderivative size = 243, normalized size of antiderivative = 2.06 \[ \int \frac {x^6 (c+d x)}{\left (a+b x^2\right )^3} \, dx=\left (- \frac {3 a d}{2 b^{4}} - \frac {15 c \sqrt {- a b^{9}}}{16 b^{8}}\right ) \log {\left (x + \frac {- 24 a d - 16 b^{4} \left (- \frac {3 a d}{2 b^{4}} - \frac {15 c \sqrt {- a b^{9}}}{16 b^{8}}\right )}{15 b c} \right )} + \left (- \frac {3 a d}{2 b^{4}} + \frac {15 c \sqrt {- a b^{9}}}{16 b^{8}}\right ) \log {\left (x + \frac {- 24 a d - 16 b^{4} \left (- \frac {3 a d}{2 b^{4}} + \frac {15 c \sqrt {- a b^{9}}}{16 b^{8}}\right )}{15 b c} \right )} + \frac {- 10 a^{3} d + 7 a^{2} b c x - 12 a^{2} b d x^{2} + 9 a b^{2} c x^{3}}{8 a^{2} b^{4} + 16 a b^{5} x^{2} + 8 b^{6} x^{4}} + \frac {c x}{b^{3}} + \frac {d x^{2}}{2 b^{3}} \] Input:
integrate(x**6*(d*x+c)/(b*x**2+a)**3,x)
Output:
(-3*a*d/(2*b**4) - 15*c*sqrt(-a*b**9)/(16*b**8))*log(x + (-24*a*d - 16*b** 4*(-3*a*d/(2*b**4) - 15*c*sqrt(-a*b**9)/(16*b**8)))/(15*b*c)) + (-3*a*d/(2 *b**4) + 15*c*sqrt(-a*b**9)/(16*b**8))*log(x + (-24*a*d - 16*b**4*(-3*a*d/ (2*b**4) + 15*c*sqrt(-a*b**9)/(16*b**8)))/(15*b*c)) + (-10*a**3*d + 7*a**2 *b*c*x - 12*a**2*b*d*x**2 + 9*a*b**2*c*x**3)/(8*a**2*b**4 + 16*a*b**5*x**2 + 8*b**6*x**4) + c*x/b**3 + d*x**2/(2*b**3)
Time = 0.11 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.97 \[ \int \frac {x^6 (c+d x)}{\left (a+b x^2\right )^3} \, dx=\frac {9 \, a b^{2} c x^{3} - 12 \, a^{2} b d x^{2} + 7 \, a^{2} b c x - 10 \, a^{3} d}{8 \, {\left (b^{6} x^{4} + 2 \, a b^{5} x^{2} + a^{2} b^{4}\right )}} - \frac {15 \, a c \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \, \sqrt {a b} b^{3}} - \frac {3 \, a d \log \left (b x^{2} + a\right )}{2 \, b^{4}} + \frac {d x^{2} + 2 \, c x}{2 \, b^{3}} \] Input:
integrate(x^6*(d*x+c)/(b*x^2+a)^3,x, algorithm="maxima")
Output:
1/8*(9*a*b^2*c*x^3 - 12*a^2*b*d*x^2 + 7*a^2*b*c*x - 10*a^3*d)/(b^6*x^4 + 2 *a*b^5*x^2 + a^2*b^4) - 15/8*a*c*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*b^3) - 3 /2*a*d*log(b*x^2 + a)/b^4 + 1/2*(d*x^2 + 2*c*x)/b^3
Time = 0.13 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.91 \[ \int \frac {x^6 (c+d x)}{\left (a+b x^2\right )^3} \, dx=-\frac {15 \, a c \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \, \sqrt {a b} b^{3}} - \frac {3 \, a d \log \left (b x^{2} + a\right )}{2 \, b^{4}} + \frac {b^{3} d x^{2} + 2 \, b^{3} c x}{2 \, b^{6}} + \frac {9 \, a b^{2} c x^{3} - 12 \, a^{2} b d x^{2} + 7 \, a^{2} b c x - 10 \, a^{3} d}{8 \, {\left (b x^{2} + a\right )}^{2} b^{4}} \] Input:
integrate(x^6*(d*x+c)/(b*x^2+a)^3,x, algorithm="giac")
Output:
-15/8*a*c*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*b^3) - 3/2*a*d*log(b*x^2 + a)/b ^4 + 1/2*(b^3*d*x^2 + 2*b^3*c*x)/b^6 + 1/8*(9*a*b^2*c*x^3 - 12*a^2*b*d*x^2 + 7*a^2*b*c*x - 10*a^3*d)/((b*x^2 + a)^2*b^4)
Time = 0.08 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.94 \[ \int \frac {x^6 (c+d x)}{\left (a+b x^2\right )^3} \, dx=\frac {d\,x^2}{2\,b^3}-\frac {\frac {5\,a^3\,d}{4\,b}+\frac {3\,a^2\,d\,x^2}{2}-\frac {7\,a^2\,c\,x}{8}-\frac {9\,a\,b\,c\,x^3}{8}}{a^2\,b^3+2\,a\,b^4\,x^2+b^5\,x^4}+\frac {c\,x}{b^3}-\frac {3\,a\,d\,\ln \left (b\,x^2+a\right )}{2\,b^4}-\frac {15\,\sqrt {a}\,c\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )}{8\,b^{7/2}} \] Input:
int((x^6*(c + d*x))/(a + b*x^2)^3,x)
Output:
(d*x^2)/(2*b^3) - ((5*a^3*d)/(4*b) + (3*a^2*d*x^2)/2 - (7*a^2*c*x)/8 - (9* a*b*c*x^3)/8)/(a^2*b^3 + b^5*x^4 + 2*a*b^4*x^2) + (c*x)/b^3 - (3*a*d*log(a + b*x^2))/(2*b^4) - (15*a^(1/2)*c*atan((b^(1/2)*x)/a^(1/2)))/(8*b^(7/2))
Time = 0.21 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.69 \[ \int \frac {x^6 (c+d x)}{\left (a+b x^2\right )^3} \, dx=\frac {-15 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{2} c -30 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a b c \,x^{2}-15 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) b^{2} c \,x^{4}-12 \,\mathrm {log}\left (b \,x^{2}+a \right ) a^{3} d -24 \,\mathrm {log}\left (b \,x^{2}+a \right ) a^{2} b d \,x^{2}-12 \,\mathrm {log}\left (b \,x^{2}+a \right ) a \,b^{2} d \,x^{4}-6 a^{3} d +15 a^{2} b c x +25 a \,b^{2} c \,x^{3}+12 a \,b^{2} d \,x^{4}+8 b^{3} c \,x^{5}+4 b^{3} d \,x^{6}}{8 b^{4} \left (b^{2} x^{4}+2 a b \,x^{2}+a^{2}\right )} \] Input:
int(x^6*(d*x+c)/(b*x^2+a)^3,x)
Output:
( - 15*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**2*c - 30*sqrt(b)*s qrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a*b*c*x**2 - 15*sqrt(b)*sqrt(a)*atan( (b*x)/(sqrt(b)*sqrt(a)))*b**2*c*x**4 - 12*log(a + b*x**2)*a**3*d - 24*log( a + b*x**2)*a**2*b*d*x**2 - 12*log(a + b*x**2)*a*b**2*d*x**4 - 6*a**3*d + 15*a**2*b*c*x + 25*a*b**2*c*x**3 + 12*a*b**2*d*x**4 + 8*b**3*c*x**5 + 4*b* *3*d*x**6)/(8*b**4*(a**2 + 2*a*b*x**2 + b**2*x**4))