Integrand size = 21, antiderivative size = 172 \[ \int \frac {\left (c+d x^2\right )^3}{\left (a+b x^2\right )^{5/2}} \, dx=-\frac {d \left (4 b^2 c^2+8 a b c d-15 a^2 d^2\right ) x \sqrt {a+b x^2}}{6 a^2 b^3}+\frac {(b c-a d) (2 b c+5 a d) x \left (c+d x^2\right )}{3 a^2 b^2 \sqrt {a+b x^2}}+\frac {(b c-a d) x \left (c+d x^2\right )^2}{3 a b \left (a+b x^2\right )^{3/2}}+\frac {d^2 (6 b c-5 a d) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 b^{7/2}} \]
1/3*(-a*d+b*c)*x*(d*x^2+c)^2/a/b/(b*x^2+a)^(3/2)+1/2*d^2*(-5*a*d+6*b*c)*ar ctanh(x*b^(1/2)/(b*x^2+a)^(1/2))/b^(7/2)+1/3*(-a*d+b*c)*(5*a*d+2*b*c)*x*(d *x^2+c)/a^2/b^2/(b*x^2+a)^(1/2)-1/6*d*(-15*a^2*d^2+8*a*b*c*d+4*b^2*c^2)*x* (b*x^2+a)^(1/2)/a^2/b^3
Time = 0.27 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.83 \[ \int \frac {\left (c+d x^2\right )^3}{\left (a+b x^2\right )^{5/2}} \, dx=\frac {x \left (15 a^4 d^3+4 b^4 c^3 x^2+3 a^2 b^2 d^2 x^2 \left (-8 c+d x^2\right )+6 a b^3 c^2 \left (c+d x^2\right )+2 a^3 b d^2 \left (-9 c+10 d x^2\right )\right )}{6 a^2 b^3 \left (a+b x^2\right )^{3/2}}+\frac {d^2 (-6 b c+5 a d) \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{2 b^{7/2}} \]
(x*(15*a^4*d^3 + 4*b^4*c^3*x^2 + 3*a^2*b^2*d^2*x^2*(-8*c + d*x^2) + 6*a*b^ 3*c^2*(c + d*x^2) + 2*a^3*b*d^2*(-9*c + 10*d*x^2)))/(6*a^2*b^3*(a + b*x^2) ^(3/2)) + (d^2*(-6*b*c + 5*a*d)*Log[-(Sqrt[b]*x) + Sqrt[a + b*x^2]])/(2*b^ (7/2))
Time = 0.34 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.09, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {315, 401, 27, 299, 224, 219}
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.
\(\displaystyle \int \frac {\left (c+d x^2\right )^3}{\left (a+b x^2\right )^{5/2}} \, dx\) |
\(\Big \downarrow \) 315 |
\(\displaystyle \frac {\int \frac {\left (d x^2+c\right ) \left (c (2 b c+a d)-d (2 b c-5 a d) x^2\right )}{\left (b x^2+a\right )^{3/2}}dx}{3 a b}+\frac {x \left (c+d x^2\right )^2 (b c-a d)}{3 a b \left (a+b x^2\right )^{3/2}}\) |
\(\Big \downarrow \) 401 |
\(\displaystyle \frac {\frac {x \left (c+d x^2\right ) \left (\frac {2 b c^2}{a}-\frac {5 a d^2}{b}+3 c d\right )}{\sqrt {a+b x^2}}-\frac {\int \frac {d \left (\left (4 b^2 c^2+8 a b d c-15 a^2 d^2\right ) x^2+a c (2 b c-5 a d)\right )}{\sqrt {b x^2+a}}dx}{a b}}{3 a b}+\frac {x \left (c+d x^2\right )^2 (b c-a d)}{3 a b \left (a+b x^2\right )^{3/2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\frac {x \left (c+d x^2\right ) \left (\frac {2 b c^2}{a}-\frac {5 a d^2}{b}+3 c d\right )}{\sqrt {a+b x^2}}-\frac {d \int \frac {\left (4 b^2 c^2+8 a b d c-15 a^2 d^2\right ) x^2+a c (2 b c-5 a d)}{\sqrt {b x^2+a}}dx}{a b}}{3 a b}+\frac {x \left (c+d x^2\right )^2 (b c-a d)}{3 a b \left (a+b x^2\right )^{3/2}}\) |
\(\Big \downarrow \) 299 |
\(\displaystyle \frac {\frac {x \left (c+d x^2\right ) \left (\frac {2 b c^2}{a}-\frac {5 a d^2}{b}+3 c d\right )}{\sqrt {a+b x^2}}-\frac {d \left (\frac {x \sqrt {a+b x^2} \left (-15 a^2 d^2+8 a b c d+4 b^2 c^2\right )}{2 b}-\frac {3 a^2 d (6 b c-5 a d) \int \frac {1}{\sqrt {b x^2+a}}dx}{2 b}\right )}{a b}}{3 a b}+\frac {x \left (c+d x^2\right )^2 (b c-a d)}{3 a b \left (a+b x^2\right )^{3/2}}\) |
\(\Big \downarrow \) 224 |
\(\displaystyle \frac {\frac {x \left (c+d x^2\right ) \left (\frac {2 b c^2}{a}-\frac {5 a d^2}{b}+3 c d\right )}{\sqrt {a+b x^2}}-\frac {d \left (\frac {x \sqrt {a+b x^2} \left (-15 a^2 d^2+8 a b c d+4 b^2 c^2\right )}{2 b}-\frac {3 a^2 d (6 b c-5 a d) \int \frac {1}{1-\frac {b x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{2 b}\right )}{a b}}{3 a b}+\frac {x \left (c+d x^2\right )^2 (b c-a d)}{3 a b \left (a+b x^2\right )^{3/2}}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {\frac {x \left (c+d x^2\right ) \left (\frac {2 b c^2}{a}-\frac {5 a d^2}{b}+3 c d\right )}{\sqrt {a+b x^2}}-\frac {d \left (\frac {x \sqrt {a+b x^2} \left (-15 a^2 d^2+8 a b c d+4 b^2 c^2\right )}{2 b}-\frac {3 a^2 d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (6 b c-5 a d)}{2 b^{3/2}}\right )}{a b}}{3 a b}+\frac {x \left (c+d x^2\right )^2 (b c-a d)}{3 a b \left (a+b x^2\right )^{3/2}}\) |
((b*c - a*d)*x*(c + d*x^2)^2)/(3*a*b*(a + b*x^2)^(3/2)) + ((((2*b*c^2)/a + 3*c*d - (5*a*d^2)/b)*x*(c + d*x^2))/Sqrt[a + b*x^2] - (d*(((4*b^2*c^2 + 8 *a*b*c*d - 15*a^2*d^2)*x*Sqrt[a + b*x^2])/(2*b) - (3*a^2*d*(6*b*c - 5*a*d) *ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(2*b^(3/2))))/(a*b))/(3*a*b)
3.1.90.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] && !GtQ[a, 0]
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[d*x *((a + b*x^2)^(p + 1)/(b*(2*p + 3))), x] - Simp[(a*d - b*c*(2*p + 3))/(b*(2 *p + 3)) Int[(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && NeQ[2*p + 3, 0]
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim p[(a*d - c*b)*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q - 1)/(2*a*b*(p + 1))), x] - Simp[1/(2*a*b*(p + 1)) Int[(a + b*x^2)^(p + 1)*(c + d*x^2)^(q - 2)*S imp[c*(a*d - c*b*(2*p + 3)) + d*(a*d*(2*(q - 1) + 1) - b*c*(2*(p + q) + 1)) *x^2, x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && LtQ[p, - 1] && GtQ[q, 1] && IntBinomialQ[a, b, c, d, 2, p, q, x]
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*(x _)^2), x_Symbol] :> Simp[(-(b*e - a*f))*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^ q/(a*b*2*(p + 1))), x] + Simp[1/(a*b*2*(p + 1)) Int[(a + b*x^2)^(p + 1)*( c + d*x^2)^(q - 1)*Simp[c*(b*e*2*(p + 1) + b*e - a*f) + d*(b*e*2*(p + 1) + (b*e - a*f)*(2*q + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && L tQ[p, -1] && GtQ[q, 0]
Time = 2.48 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.83
method | result | size |
pseudoelliptic | \(-\frac {5 \left (\left (b \,x^{2}+a \right )^{\frac {3}{2}} d^{2} \left (a d -\frac {6 b c}{5}\right ) a^{2} \operatorname {arctanh}\left (\frac {\sqrt {b \,x^{2}+a}}{x \sqrt {b}}\right )-x \left (-\frac {6 \left (-\frac {10 d \,x^{2}}{9}+c \right ) d^{2} a^{3} b^{\frac {3}{2}}}{5}-\frac {8 x^{2} \left (-\frac {d \,x^{2}}{8}+c \right ) d^{2} a^{2} b^{\frac {5}{2}}}{5}+\frac {2 a \,c^{2} \left (d \,x^{2}+c \right ) b^{\frac {7}{2}}}{5}+\sqrt {b}\, a^{4} d^{3}+\frac {4 b^{\frac {9}{2}} c^{3} x^{2}}{15}\right )\right )}{2 b^{\frac {7}{2}} \left (b \,x^{2}+a \right )^{\frac {3}{2}} a^{2}}\) | \(142\) |
default | \(c^{3} \left (\frac {x}{3 a \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {2 x}{3 a^{2} \sqrt {b \,x^{2}+a}}\right )+d^{3} \left (\frac {x^{5}}{2 b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}-\frac {5 a \left (-\frac {x^{3}}{3 b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {-\frac {x}{b \sqrt {b \,x^{2}+a}}+\frac {\ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{b^{\frac {3}{2}}}}{b}\right )}{2 b}\right )+3 c \,d^{2} \left (-\frac {x^{3}}{3 b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {-\frac {x}{b \sqrt {b \,x^{2}+a}}+\frac {\ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{b^{\frac {3}{2}}}}{b}\right )+3 c^{2} d \left (-\frac {x}{2 b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {a \left (\frac {x}{3 a \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {2 x}{3 a^{2} \sqrt {b \,x^{2}+a}}\right )}{2 b}\right )\) | \(246\) |
risch | \(\frac {d^{3} x \sqrt {b \,x^{2}+a}}{2 b^{3}}-\frac {\frac {d^{2} \left (5 a d -6 b c \right ) \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{\sqrt {b}}-\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \left (-\frac {\sqrt {\left (x -\frac {\sqrt {-a b}}{b}\right )^{2} b +2 \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}}{3 \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )^{2}}-\frac {\sqrt {\left (x -\frac {\sqrt {-a b}}{b}\right )^{2} b +2 \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}}{3 a \left (x -\frac {\sqrt {-a b}}{b}\right )}\right )}{2 b a}-\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \left (\frac {\sqrt {\left (x +\frac {\sqrt {-a b}}{b}\right )^{2} b -2 \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}}{3 \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )^{2}}-\frac {\sqrt {\left (x +\frac {\sqrt {-a b}}{b}\right )^{2} b -2 \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}}{3 a \left (x +\frac {\sqrt {-a b}}{b}\right )}\right )}{2 b a}-\frac {\left (5 a^{3} d^{3}-9 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d +b^{3} c^{3}\right ) \sqrt {\left (x +\frac {\sqrt {-a b}}{b}\right )^{2} b -2 \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}}{2 a^{2} b \left (x +\frac {\sqrt {-a b}}{b}\right )}-\frac {\left (5 a^{3} d^{3}-9 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d +b^{3} c^{3}\right ) \sqrt {\left (x -\frac {\sqrt {-a b}}{b}\right )^{2} b +2 \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}}{2 a^{2} b \left (x -\frac {\sqrt {-a b}}{b}\right )}}{2 b^{3}}\) | \(589\) |
-5/2*((b*x^2+a)^(3/2)*d^2*(a*d-6/5*b*c)*a^2*arctanh((b*x^2+a)^(1/2)/x/b^(1 /2))-x*(-6/5*(-10/9*d*x^2+c)*d^2*a^3*b^(3/2)-8/5*x^2*(-1/8*d*x^2+c)*d^2*a^ 2*b^(5/2)+2/5*a*c^2*(d*x^2+c)*b^(7/2)+b^(1/2)*a^4*d^3+4/15*b^(9/2)*c^3*x^2 ))/b^(7/2)/(b*x^2+a)^(3/2)/a^2
Time = 0.30 (sec) , antiderivative size = 486, normalized size of antiderivative = 2.83 \[ \int \frac {\left (c+d x^2\right )^3}{\left (a+b x^2\right )^{5/2}} \, dx=\left [-\frac {3 \, {\left (6 \, a^{4} b c d^{2} - 5 \, a^{5} d^{3} + {\left (6 \, a^{2} b^{3} c d^{2} - 5 \, a^{3} b^{2} d^{3}\right )} x^{4} + 2 \, {\left (6 \, a^{3} b^{2} c d^{2} - 5 \, a^{4} b d^{3}\right )} x^{2}\right )} \sqrt {b} \log \left (-2 \, b x^{2} + 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) - 2 \, {\left (3 \, a^{2} b^{3} d^{3} x^{5} + 2 \, {\left (2 \, b^{5} c^{3} + 3 \, a b^{4} c^{2} d - 12 \, a^{2} b^{3} c d^{2} + 10 \, a^{3} b^{2} d^{3}\right )} x^{3} + 3 \, {\left (2 \, a b^{4} c^{3} - 6 \, a^{3} b^{2} c d^{2} + 5 \, a^{4} b d^{3}\right )} x\right )} \sqrt {b x^{2} + a}}{12 \, {\left (a^{2} b^{6} x^{4} + 2 \, a^{3} b^{5} x^{2} + a^{4} b^{4}\right )}}, -\frac {3 \, {\left (6 \, a^{4} b c d^{2} - 5 \, a^{5} d^{3} + {\left (6 \, a^{2} b^{3} c d^{2} - 5 \, a^{3} b^{2} d^{3}\right )} x^{4} + 2 \, {\left (6 \, a^{3} b^{2} c d^{2} - 5 \, a^{4} b d^{3}\right )} x^{2}\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) - {\left (3 \, a^{2} b^{3} d^{3} x^{5} + 2 \, {\left (2 \, b^{5} c^{3} + 3 \, a b^{4} c^{2} d - 12 \, a^{2} b^{3} c d^{2} + 10 \, a^{3} b^{2} d^{3}\right )} x^{3} + 3 \, {\left (2 \, a b^{4} c^{3} - 6 \, a^{3} b^{2} c d^{2} + 5 \, a^{4} b d^{3}\right )} x\right )} \sqrt {b x^{2} + a}}{6 \, {\left (a^{2} b^{6} x^{4} + 2 \, a^{3} b^{5} x^{2} + a^{4} b^{4}\right )}}\right ] \]
[-1/12*(3*(6*a^4*b*c*d^2 - 5*a^5*d^3 + (6*a^2*b^3*c*d^2 - 5*a^3*b^2*d^3)*x ^4 + 2*(6*a^3*b^2*c*d^2 - 5*a^4*b*d^3)*x^2)*sqrt(b)*log(-2*b*x^2 + 2*sqrt( b*x^2 + a)*sqrt(b)*x - a) - 2*(3*a^2*b^3*d^3*x^5 + 2*(2*b^5*c^3 + 3*a*b^4* c^2*d - 12*a^2*b^3*c*d^2 + 10*a^3*b^2*d^3)*x^3 + 3*(2*a*b^4*c^3 - 6*a^3*b^ 2*c*d^2 + 5*a^4*b*d^3)*x)*sqrt(b*x^2 + a))/(a^2*b^6*x^4 + 2*a^3*b^5*x^2 + a^4*b^4), -1/6*(3*(6*a^4*b*c*d^2 - 5*a^5*d^3 + (6*a^2*b^3*c*d^2 - 5*a^3*b^ 2*d^3)*x^4 + 2*(6*a^3*b^2*c*d^2 - 5*a^4*b*d^3)*x^2)*sqrt(-b)*arctan(sqrt(- b)*x/sqrt(b*x^2 + a)) - (3*a^2*b^3*d^3*x^5 + 2*(2*b^5*c^3 + 3*a*b^4*c^2*d - 12*a^2*b^3*c*d^2 + 10*a^3*b^2*d^3)*x^3 + 3*(2*a*b^4*c^3 - 6*a^3*b^2*c*d^ 2 + 5*a^4*b*d^3)*x)*sqrt(b*x^2 + a))/(a^2*b^6*x^4 + 2*a^3*b^5*x^2 + a^4*b^ 4)]
\[ \int \frac {\left (c+d x^2\right )^3}{\left (a+b x^2\right )^{5/2}} \, dx=\int \frac {\left (c + d x^{2}\right )^{3}}{\left (a + b x^{2}\right )^{\frac {5}{2}}}\, dx \]
Time = 0.20 (sec) , antiderivative size = 254, normalized size of antiderivative = 1.48 \[ \int \frac {\left (c+d x^2\right )^3}{\left (a+b x^2\right )^{5/2}} \, dx=\frac {d^{3} x^{5}}{2 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} b} - c d^{2} x {\left (\frac {3 \, x^{2}}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} b} + \frac {2 \, a}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} b^{2}}\right )} + \frac {5 \, a d^{3} x {\left (\frac {3 \, x^{2}}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} b} + \frac {2 \, a}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} b^{2}}\right )}}{6 \, b} + \frac {2 \, c^{3} x}{3 \, \sqrt {b x^{2} + a} a^{2}} + \frac {c^{3} x}{3 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a} - \frac {c^{2} d x}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} b} + \frac {c^{2} d x}{\sqrt {b x^{2} + a} a b} - \frac {c d^{2} x}{\sqrt {b x^{2} + a} b^{2}} + \frac {5 \, a d^{3} x}{6 \, \sqrt {b x^{2} + a} b^{3}} + \frac {3 \, c d^{2} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{b^{\frac {5}{2}}} - \frac {5 \, a d^{3} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{2 \, b^{\frac {7}{2}}} \]
1/2*d^3*x^5/((b*x^2 + a)^(3/2)*b) - c*d^2*x*(3*x^2/((b*x^2 + a)^(3/2)*b) + 2*a/((b*x^2 + a)^(3/2)*b^2)) + 5/6*a*d^3*x*(3*x^2/((b*x^2 + a)^(3/2)*b) + 2*a/((b*x^2 + a)^(3/2)*b^2))/b + 2/3*c^3*x/(sqrt(b*x^2 + a)*a^2) + 1/3*c^ 3*x/((b*x^2 + a)^(3/2)*a) - c^2*d*x/((b*x^2 + a)^(3/2)*b) + c^2*d*x/(sqrt( b*x^2 + a)*a*b) - c*d^2*x/(sqrt(b*x^2 + a)*b^2) + 5/6*a*d^3*x/(sqrt(b*x^2 + a)*b^3) + 3*c*d^2*arcsinh(b*x/sqrt(a*b))/b^(5/2) - 5/2*a*d^3*arcsinh(b*x /sqrt(a*b))/b^(7/2)
Time = 0.29 (sec) , antiderivative size = 158, normalized size of antiderivative = 0.92 \[ \int \frac {\left (c+d x^2\right )^3}{\left (a+b x^2\right )^{5/2}} \, dx=\frac {{\left ({\left (\frac {3 \, d^{3} x^{2}}{b} + \frac {2 \, {\left (2 \, b^{6} c^{3} + 3 \, a b^{5} c^{2} d - 12 \, a^{2} b^{4} c d^{2} + 10 \, a^{3} b^{3} d^{3}\right )}}{a^{2} b^{5}}\right )} x^{2} + \frac {3 \, {\left (2 \, a b^{5} c^{3} - 6 \, a^{3} b^{3} c d^{2} + 5 \, a^{4} b^{2} d^{3}\right )}}{a^{2} b^{5}}\right )} x}{6 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}}} - \frac {{\left (6 \, b c d^{2} - 5 \, a d^{3}\right )} \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{2 \, b^{\frac {7}{2}}} \]
1/6*((3*d^3*x^2/b + 2*(2*b^6*c^3 + 3*a*b^5*c^2*d - 12*a^2*b^4*c*d^2 + 10*a ^3*b^3*d^3)/(a^2*b^5))*x^2 + 3*(2*a*b^5*c^3 - 6*a^3*b^3*c*d^2 + 5*a^4*b^2* d^3)/(a^2*b^5))*x/(b*x^2 + a)^(3/2) - 1/2*(6*b*c*d^2 - 5*a*d^3)*log(abs(-s qrt(b)*x + sqrt(b*x^2 + a)))/b^(7/2)
Timed out. \[ \int \frac {\left (c+d x^2\right )^3}{\left (a+b x^2\right )^{5/2}} \, dx=\int \frac {{\left (d\,x^2+c\right )}^3}{{\left (b\,x^2+a\right )}^{5/2}} \,d x \]