Integrand size = 26, antiderivative size = 187 \[ \int \frac {x^3 \left (a+b x^2\right )^{3/2}}{\sqrt {c+d x^2}} \, dx=\frac {(b c-a d) (5 b c+a d) \sqrt {a+b x^2} \sqrt {c+d x^2}}{16 b d^3}-\frac {(5 b c+a d) \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{24 b d^2}+\frac {\left (a+b x^2\right )^{5/2} \sqrt {c+d x^2}}{6 b d}-\frac {(b c-a d)^2 (5 b c+a d) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x^2}}{\sqrt {b} \sqrt {c+d x^2}}\right )}{16 b^{3/2} d^{7/2}} \]
-1/16*(-a*d+b*c)^2*(a*d+5*b*c)*arctanh(d^(1/2)*(b*x^2+a)^(1/2)/b^(1/2)/(d* x^2+c)^(1/2))/b^(3/2)/d^(7/2)-1/24*(a*d+5*b*c)*(b*x^2+a)^(3/2)*(d*x^2+c)^( 1/2)/b/d^2+1/6*(b*x^2+a)^(5/2)*(d*x^2+c)^(1/2)/b/d+1/16*(-a*d+b*c)*(a*d+5* b*c)*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/b/d^3
Time = 1.58 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.79 \[ \int \frac {x^3 \left (a+b x^2\right )^{3/2}}{\sqrt {c+d x^2}} \, dx=\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \left (3 a^2 d^2+2 a b d \left (-11 c+7 d x^2\right )+b^2 \left (15 c^2-10 c d x^2+8 d^2 x^4\right )\right )}{48 b d^3}-\frac {(b c-a d)^2 (5 b c+a d) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {d} \sqrt {a+b x^2}}\right )}{16 b^{3/2} d^{7/2}} \]
(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]*(3*a^2*d^2 + 2*a*b*d*(-11*c + 7*d*x^2) + b^2*(15*c^2 - 10*c*d*x^2 + 8*d^2*x^4)))/(48*b*d^3) - ((b*c - a*d)^2*(5*b*c + a*d)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^2])/(Sqrt[d]*Sqrt[a + b*x^2])])/(16* b^(3/2)*d^(7/2))
Time = 0.26 (sec) , antiderivative size = 181, normalized size of antiderivative = 0.97, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {354, 90, 60, 60, 66, 221}
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.
\(\displaystyle \int \frac {x^3 \left (a+b x^2\right )^{3/2}}{\sqrt {c+d x^2}} \, dx\) |
\(\Big \downarrow \) 354 |
\(\displaystyle \frac {1}{2} \int \frac {x^2 \left (b x^2+a\right )^{3/2}}{\sqrt {d x^2+c}}dx^2\) |
\(\Big \downarrow \) 90 |
\(\displaystyle \frac {1}{2} \left (\frac {\left (a+b x^2\right )^{5/2} \sqrt {c+d x^2}}{3 b d}-\frac {(a d+5 b c) \int \frac {\left (b x^2+a\right )^{3/2}}{\sqrt {d x^2+c}}dx^2}{6 b d}\right )\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {1}{2} \left (\frac {\left (a+b x^2\right )^{5/2} \sqrt {c+d x^2}}{3 b d}-\frac {(a d+5 b c) \left (\frac {\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{2 d}-\frac {3 (b c-a d) \int \frac {\sqrt {b x^2+a}}{\sqrt {d x^2+c}}dx^2}{4 d}\right )}{6 b d}\right )\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {1}{2} \left (\frac {\left (a+b x^2\right )^{5/2} \sqrt {c+d x^2}}{3 b d}-\frac {(a d+5 b c) \left (\frac {\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{2 d}-\frac {3 (b c-a d) \left (\frac {\sqrt {a+b x^2} \sqrt {c+d x^2}}{d}-\frac {(b c-a d) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx^2}{2 d}\right )}{4 d}\right )}{6 b d}\right )\) |
\(\Big \downarrow \) 66 |
\(\displaystyle \frac {1}{2} \left (\frac {\left (a+b x^2\right )^{5/2} \sqrt {c+d x^2}}{3 b d}-\frac {(a d+5 b c) \left (\frac {\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{2 d}-\frac {3 (b c-a d) \left (\frac {\sqrt {a+b x^2} \sqrt {c+d x^2}}{d}-\frac {(b c-a d) \int \frac {1}{b-d x^4}d\frac {\sqrt {b x^2+a}}{\sqrt {d x^2+c}}}{d}\right )}{4 d}\right )}{6 b d}\right )\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {1}{2} \left (\frac {\left (a+b x^2\right )^{5/2} \sqrt {c+d x^2}}{3 b d}-\frac {(a d+5 b c) \left (\frac {\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{2 d}-\frac {3 (b c-a d) \left (\frac {\sqrt {a+b x^2} \sqrt {c+d x^2}}{d}-\frac {(b c-a d) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x^2}}{\sqrt {b} \sqrt {c+d x^2}}\right )}{\sqrt {b} d^{3/2}}\right )}{4 d}\right )}{6 b d}\right )\) |
(((a + b*x^2)^(5/2)*Sqrt[c + d*x^2])/(3*b*d) - ((5*b*c + a*d)*(((a + b*x^2 )^(3/2)*Sqrt[c + d*x^2])/(2*d) - (3*(b*c - a*d)*((Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/d - ((b*c - a*d)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x^2])/(Sqrt[b]*Sqrt[ c + d*x^2])])/(Sqrt[b]*d^(3/2))))/(4*d)))/(6*b*d))/2
3.10.45.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( b*(m + n + 1))) Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] && !(IGtQ[m, 0] && ( !Integer Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) && !ILtQ[m + n + 2, 0] && IntLinear Q[a, b, c, d, m, n, x]
Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[ 2 Subst[Int[1/(b - d*x^2), x], x, Sqrt[a + b*x]/Sqrt[c + d*x]], x] /; Fre eQ[{a, b, c, d}, x] && !GtQ[c - a*(d/b), 0]
Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[b*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), x] + Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(d*f*(n + p + 2)) Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
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]
Time = 3.20 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.13
method | result | size |
risch | \(\frac {\left (8 b^{2} d^{2} x^{4}+14 x^{2} a b \,d^{2}-10 x^{2} b^{2} c d +3 a^{2} d^{2}-22 a b c d +15 b^{2} c^{2}\right ) \sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}}{48 b \,d^{3}}-\frac {\left (a^{3} d^{3}+3 a^{2} b c \,d^{2}-9 a \,b^{2} c^{2} d +5 b^{3} c^{3}\right ) \ln \left (\frac {\frac {1}{2} a d +\frac {1}{2} b c +b d \,x^{2}}{\sqrt {b d}}+\sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}\right ) \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}}{32 b \,d^{3} \sqrt {b d}\, \sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}}\) | \(212\) |
default | \(-\frac {\sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}\, \left (-16 b^{2} d^{2} x^{4} \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \sqrt {b d}-28 \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \sqrt {b d}\, a b \,d^{2} x^{2}+20 \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \sqrt {b d}\, b^{2} c d \,x^{2}+3 \ln \left (\frac {2 b d \,x^{2}+2 \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a^{3} d^{3}+9 \ln \left (\frac {2 b d \,x^{2}+2 \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a^{2} b c \,d^{2}-27 \ln \left (\frac {2 b d \,x^{2}+2 \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a \,b^{2} c^{2} d +15 \ln \left (\frac {2 b d \,x^{2}+2 \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) b^{3} c^{3}-6 \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \sqrt {b d}\, a^{2} d^{2}+44 \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \sqrt {b d}\, a b c d -30 \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \sqrt {b d}\, b^{2} c^{2}\right )}{96 b \,d^{3} \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \sqrt {b d}}\) | \(455\) |
elliptic | \(\frac {\sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \left (\frac {b \,x^{4} \sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}}{6 d}+\frac {7 \sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}\, x^{2} a}{24 d}-\frac {5 b \sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}\, x^{2} c}{24 d^{2}}+\frac {\sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}\, a^{2}}{16 b d}-\frac {11 \sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}\, a c}{24 d^{2}}+\frac {5 b \sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}\, c^{2}}{16 d^{3}}-\frac {\ln \left (\frac {\frac {1}{2} a d +\frac {1}{2} b c +b d \,x^{2}}{\sqrt {b d}}+\sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}\right ) a^{3}}{32 b \sqrt {b d}}-\frac {3 \ln \left (\frac {\frac {1}{2} a d +\frac {1}{2} b c +b d \,x^{2}}{\sqrt {b d}}+\sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}\right ) a^{2} c}{32 d \sqrt {b d}}+\frac {9 b \ln \left (\frac {\frac {1}{2} a d +\frac {1}{2} b c +b d \,x^{2}}{\sqrt {b d}}+\sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}\right ) a \,c^{2}}{32 d^{2} \sqrt {b d}}-\frac {5 b^{2} \ln \left (\frac {\frac {1}{2} a d +\frac {1}{2} b c +b d \,x^{2}}{\sqrt {b d}}+\sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}\right ) c^{3}}{32 d^{3} \sqrt {b d}}\right )}{\sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}}\) | \(473\) |
1/48/b*(8*b^2*d^2*x^4+14*a*b*d^2*x^2-10*b^2*c*d*x^2+3*a^2*d^2-22*a*b*c*d+1 5*b^2*c^2)*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/d^3-1/32/b*(a^3*d^3+3*a^2*b*c*d ^2-9*a*b^2*c^2*d+5*b^3*c^3)/d^3*ln((1/2*a*d+1/2*b*c+b*d*x^2)/(b*d)^(1/2)+( b*d*x^4+(a*d+b*c)*x^2+a*c)^(1/2))/(b*d)^(1/2)*((b*x^2+a)*(d*x^2+c))^(1/2)/ (b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)
Time = 0.27 (sec) , antiderivative size = 440, normalized size of antiderivative = 2.35 \[ \int \frac {x^3 \left (a+b x^2\right )^{3/2}}{\sqrt {c+d x^2}} \, dx=\left [\frac {3 \, {\left (5 \, b^{3} c^{3} - 9 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} + a^{3} d^{3}\right )} \sqrt {b d} \log \left (8 \, b^{2} d^{2} x^{4} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x^{2} - 4 \, {\left (2 \, b d x^{2} + b c + a d\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c} \sqrt {b d}\right ) + 4 \, {\left (8 \, b^{3} d^{3} x^{4} + 15 \, b^{3} c^{2} d - 22 \, a b^{2} c d^{2} + 3 \, a^{2} b d^{3} - 2 \, {\left (5 \, b^{3} c d^{2} - 7 \, a b^{2} d^{3}\right )} x^{2}\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c}}{192 \, b^{2} d^{4}}, \frac {3 \, {\left (5 \, b^{3} c^{3} - 9 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} + a^{3} d^{3}\right )} \sqrt {-b d} \arctan \left (\frac {{\left (2 \, b d x^{2} + b c + a d\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c} \sqrt {-b d}}{2 \, {\left (b^{2} d^{2} x^{4} + a b c d + {\left (b^{2} c d + a b d^{2}\right )} x^{2}\right )}}\right ) + 2 \, {\left (8 \, b^{3} d^{3} x^{4} + 15 \, b^{3} c^{2} d - 22 \, a b^{2} c d^{2} + 3 \, a^{2} b d^{3} - 2 \, {\left (5 \, b^{3} c d^{2} - 7 \, a b^{2} d^{3}\right )} x^{2}\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c}}{96 \, b^{2} d^{4}}\right ] \]
[1/192*(3*(5*b^3*c^3 - 9*a*b^2*c^2*d + 3*a^2*b*c*d^2 + a^3*d^3)*sqrt(b*d)* log(8*b^2*d^2*x^4 + b^2*c^2 + 6*a*b*c*d + a^2*d^2 + 8*(b^2*c*d + a*b*d^2)* x^2 - 4*(2*b*d*x^2 + b*c + a*d)*sqrt(b*x^2 + a)*sqrt(d*x^2 + c)*sqrt(b*d)) + 4*(8*b^3*d^3*x^4 + 15*b^3*c^2*d - 22*a*b^2*c*d^2 + 3*a^2*b*d^3 - 2*(5*b ^3*c*d^2 - 7*a*b^2*d^3)*x^2)*sqrt(b*x^2 + a)*sqrt(d*x^2 + c))/(b^2*d^4), 1 /96*(3*(5*b^3*c^3 - 9*a*b^2*c^2*d + 3*a^2*b*c*d^2 + a^3*d^3)*sqrt(-b*d)*ar ctan(1/2*(2*b*d*x^2 + b*c + a*d)*sqrt(b*x^2 + a)*sqrt(d*x^2 + c)*sqrt(-b*d )/(b^2*d^2*x^4 + a*b*c*d + (b^2*c*d + a*b*d^2)*x^2)) + 2*(8*b^3*d^3*x^4 + 15*b^3*c^2*d - 22*a*b^2*c*d^2 + 3*a^2*b*d^3 - 2*(5*b^3*c*d^2 - 7*a*b^2*d^3 )*x^2)*sqrt(b*x^2 + a)*sqrt(d*x^2 + c))/(b^2*d^4)]
\[ \int \frac {x^3 \left (a+b x^2\right )^{3/2}}{\sqrt {c+d x^2}} \, dx=\int \frac {x^{3} \left (a + b x^{2}\right )^{\frac {3}{2}}}{\sqrt {c + d x^{2}}}\, dx \]
Exception generated. \[ \int \frac {x^3 \left (a+b x^2\right )^{3/2}}{\sqrt {c+d x^2}} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(a*d-b*c>0)', see `assume?` for m ore detail
Time = 0.33 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.20 \[ \int \frac {x^3 \left (a+b x^2\right )^{3/2}}{\sqrt {c+d x^2}} \, dx=\frac {{\left (\sqrt {b^{2} c + {\left (b x^{2} + a\right )} b d - a b d} \sqrt {b x^{2} + a} {\left (2 \, {\left (b x^{2} + a\right )} {\left (\frac {4 \, {\left (b x^{2} + a\right )}}{b^{2} d} - \frac {5 \, b^{3} c d^{3} + a b^{2} d^{4}}{b^{4} d^{5}}\right )} + \frac {3 \, {\left (5 \, b^{4} c^{2} d^{2} - 4 \, a b^{3} c d^{3} - a^{2} b^{2} d^{4}\right )}}{b^{4} d^{5}}\right )} + \frac {3 \, {\left (5 \, b^{3} c^{3} - 9 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} + a^{3} d^{3}\right )} \log \left ({\left | -\sqrt {b x^{2} + a} \sqrt {b d} + \sqrt {b^{2} c + {\left (b x^{2} + a\right )} b d - a b d} \right |}\right )}{\sqrt {b d} b d^{3}}\right )} b}{48 \, {\left | b \right |}} \]
1/48*(sqrt(b^2*c + (b*x^2 + a)*b*d - a*b*d)*sqrt(b*x^2 + a)*(2*(b*x^2 + a) *(4*(b*x^2 + a)/(b^2*d) - (5*b^3*c*d^3 + a*b^2*d^4)/(b^4*d^5)) + 3*(5*b^4* c^2*d^2 - 4*a*b^3*c*d^3 - a^2*b^2*d^4)/(b^4*d^5)) + 3*(5*b^3*c^3 - 9*a*b^2 *c^2*d + 3*a^2*b*c*d^2 + a^3*d^3)*log(abs(-sqrt(b*x^2 + a)*sqrt(b*d) + sqr t(b^2*c + (b*x^2 + a)*b*d - a*b*d)))/(sqrt(b*d)*b*d^3))*b/abs(b)
Timed out. \[ \int \frac {x^3 \left (a+b x^2\right )^{3/2}}{\sqrt {c+d x^2}} \, dx=\int \frac {x^3\,{\left (b\,x^2+a\right )}^{3/2}}{\sqrt {d\,x^2+c}} \,d x \]