Integrand size = 21, antiderivative size = 156 \[ \int \frac {\left (c+d x^2\right )^{5/2}}{a+b x^2} \, dx=\frac {d (7 b c-4 a d) x \sqrt {c+d x^2}}{8 b^2}+\frac {d x \left (c+d x^2\right )^{3/2}}{4 b}+\frac {(b c-a d)^{5/2} \arctan \left (\frac {\sqrt {b c-a d} x}{\sqrt {a} \sqrt {c+d x^2}}\right )}{\sqrt {a} b^3}+\frac {\sqrt {d} \left (15 b^2 c^2-20 a b c d+8 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{8 b^3} \]
1/4*d*x*(d*x^2+c)^(3/2)/b+(-a*d+b*c)^(5/2)*arctan(x*(-a*d+b*c)^(1/2)/a^(1/ 2)/(d*x^2+c)^(1/2))/b^3/a^(1/2)+1/8*(8*a^2*d^2-20*a*b*c*d+15*b^2*c^2)*arct anh(x*d^(1/2)/(d*x^2+c)^(1/2))*d^(1/2)/b^3+1/8*d*(-4*a*d+7*b*c)*x*(d*x^2+c )^(1/2)/b^2
Time = 0.36 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.03 \[ \int \frac {\left (c+d x^2\right )^{5/2}}{a+b x^2} \, dx=\frac {b d x \sqrt {c+d x^2} \left (9 b c-4 a d+2 b d x^2\right )-\frac {8 (b c-a d)^{5/2} \arctan \left (\frac {a \sqrt {d}+b x \left (\sqrt {d} x-\sqrt {c+d x^2}\right )}{\sqrt {a} \sqrt {b c-a d}}\right )}{\sqrt {a}}-\sqrt {d} \left (15 b^2 c^2-20 a b c d+8 a^2 d^2\right ) \log \left (-\sqrt {d} x+\sqrt {c+d x^2}\right )}{8 b^3} \]
(b*d*x*Sqrt[c + d*x^2]*(9*b*c - 4*a*d + 2*b*d*x^2) - (8*(b*c - a*d)^(5/2)* ArcTan[(a*Sqrt[d] + b*x*(Sqrt[d]*x - Sqrt[c + d*x^2]))/(Sqrt[a]*Sqrt[b*c - a*d])])/Sqrt[a] - Sqrt[d]*(15*b^2*c^2 - 20*a*b*c*d + 8*a^2*d^2)*Log[-(Sqr t[d]*x) + Sqrt[c + d*x^2]])/(8*b^3)
Time = 0.34 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.09, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {318, 403, 398, 224, 219, 291, 218}
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.
| \(\displaystyle \int \frac {\left (c+d x^2\right )^{5/2}}{a+b x^2} \, dx\) |
| \(\Big \downarrow \) 318 |
| \(\displaystyle \frac {\int \frac {\sqrt {d x^2+c} \left (d (7 b c-4 a d) x^2+c (4 b c-a d)\right )}{b x^2+a}dx}{4 b}+\frac {d x \left (c+d x^2\right )^{3/2}}{4 b}\) |
| \(\Big \downarrow \) 403 |
| \(\displaystyle \frac {\frac {\int \frac {d \left (15 b^2 c^2-20 a b d c+8 a^2 d^2\right ) x^2+c \left (8 b^2 c^2-9 a b d c+4 a^2 d^2\right )}{\left (b x^2+a\right ) \sqrt {d x^2+c}}dx}{2 b}+\frac {d x \sqrt {c+d x^2} (7 b c-4 a d)}{2 b}}{4 b}+\frac {d x \left (c+d x^2\right )^{3/2}}{4 b}\) |
| \(\Big \downarrow \) 398 |
| \(\displaystyle \frac {\frac {\frac {d \left (8 a^2 d^2-20 a b c d+15 b^2 c^2\right ) \int \frac {1}{\sqrt {d x^2+c}}dx}{b}+\frac {8 (b c-a d)^3 \int \frac {1}{\left (b x^2+a\right ) \sqrt {d x^2+c}}dx}{b}}{2 b}+\frac {d x \sqrt {c+d x^2} (7 b c-4 a d)}{2 b}}{4 b}+\frac {d x \left (c+d x^2\right )^{3/2}}{4 b}\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle \frac {\frac {\frac {d \left (8 a^2 d^2-20 a b c d+15 b^2 c^2\right ) \int \frac {1}{1-\frac {d x^2}{d x^2+c}}d\frac {x}{\sqrt {d x^2+c}}}{b}+\frac {8 (b c-a d)^3 \int \frac {1}{\left (b x^2+a\right ) \sqrt {d x^2+c}}dx}{b}}{2 b}+\frac {d x \sqrt {c+d x^2} (7 b c-4 a d)}{2 b}}{4 b}+\frac {d x \left (c+d x^2\right )^{3/2}}{4 b}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\frac {\frac {8 (b c-a d)^3 \int \frac {1}{\left (b x^2+a\right ) \sqrt {d x^2+c}}dx}{b}+\frac {\sqrt {d} \left (8 a^2 d^2-20 a b c d+15 b^2 c^2\right ) \text {arctanh}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{b}}{2 b}+\frac {d x \sqrt {c+d x^2} (7 b c-4 a d)}{2 b}}{4 b}+\frac {d x \left (c+d x^2\right )^{3/2}}{4 b}\) |
| \(\Big \downarrow \) 291 |
| \(\displaystyle \frac {\frac {\frac {8 (b c-a d)^3 \int \frac {1}{a-\frac {(a d-b c) x^2}{d x^2+c}}d\frac {x}{\sqrt {d x^2+c}}}{b}+\frac {\sqrt {d} \left (8 a^2 d^2-20 a b c d+15 b^2 c^2\right ) \text {arctanh}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{b}}{2 b}+\frac {d x \sqrt {c+d x^2} (7 b c-4 a d)}{2 b}}{4 b}+\frac {d x \left (c+d x^2\right )^{3/2}}{4 b}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {\frac {\frac {\sqrt {d} \left (8 a^2 d^2-20 a b c d+15 b^2 c^2\right ) \text {arctanh}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{b}+\frac {8 (b c-a d)^{5/2} \arctan \left (\frac {x \sqrt {b c-a d}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{\sqrt {a} b}}{2 b}+\frac {d x \sqrt {c+d x^2} (7 b c-4 a d)}{2 b}}{4 b}+\frac {d x \left (c+d x^2\right )^{3/2}}{4 b}\) |
(d*x*(c + d*x^2)^(3/2))/(4*b) + ((d*(7*b*c - 4*a*d)*x*Sqrt[c + d*x^2])/(2* b) + ((8*(b*c - a*d)^(5/2)*ArcTan[(Sqrt[b*c - a*d]*x)/(Sqrt[a]*Sqrt[c + d* x^2])])/(Sqrt[a]*b) + (Sqrt[d]*(15*b^2*c^2 - 20*a*b*c*d + 8*a^2*d^2)*ArcTa nh[(Sqrt[d]*x)/Sqrt[c + d*x^2]])/b)/(2*b))/(4*b)
3.7.98.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
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[1/(Sqrt[(a_) + (b_.)*(x_)^2]*((c_) + (d_.)*(x_)^2)), x_Symbol] :> Subst [Int[1/(c - (b*c - a*d)*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim p[d*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q - 1)/(b*(2*(p + q) + 1))), x] + S imp[1/(b*(2*(p + q) + 1)) Int[(a + b*x^2)^p*(c + d*x^2)^(q - 2)*Simp[c*(b *c*(2*(p + q) + 1) - a*d) + d*(b*c*(2*(p + 2*q - 1) + 1) - a*d*(2*(q - 1) + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, p}, x] && NeQ[b*c - a*d, 0] && G tQ[q, 1] && NeQ[2*(p + q) + 1, 0] && !IGtQ[p, 1] && IntBinomialQ[a, b, c, d, 2, p, q, x]
Int[((e_) + (f_.)*(x_)^2)/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]) , x_Symbol] :> Simp[f/b Int[1/Sqrt[c + d*x^2], x], x] + Simp[(b*e - a*f)/ b Int[1/((a + b*x^2)*Sqrt[c + d*x^2]), x], x] /; FreeQ[{a, b, c, d, e, f} , x]
Int[((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*( x_)^2), x_Symbol] :> Simp[f*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^q/(b*(2*(p + q + 1) + 1))), x] + Simp[1/(b*(2*(p + q + 1) + 1)) Int[(a + b*x^2)^p*(c + d*x^2)^(q - 1)*Simp[c*(b*e - a*f + b*e*2*(p + q + 1)) + (d*(b*e - a*f) + f*2*q*(b*c - a*d) + b*d*e*2*(p + q + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && GtQ[q, 0] && NeQ[2*(p + q + 1) + 1, 0]
Time = 3.05 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.87
| method | result | size |
| pseudoelliptic | \(-\frac {\frac {d \left (b \sqrt {d \,x^{2}+c}\, \left (-2 b d \,x^{2}+4 a d -9 b c \right ) x -\frac {\left (8 a^{2} d^{2}-20 a b c d +15 b^{2} c^{2}\right ) \operatorname {arctanh}\left (\frac {\sqrt {d \,x^{2}+c}}{x \sqrt {d}}\right )}{\sqrt {d}}\right )}{4}+\frac {2 \left (a d -b c \right )^{3} \operatorname {arctanh}\left (\frac {\sqrt {d \,x^{2}+c}\, a}{x \sqrt {\left (a d -b c \right ) a}}\right )}{\sqrt {\left (a d -b c \right ) a}}}{2 b^{3}}\) | \(135\) |
| risch | \(-\frac {x d \left (-2 b d \,x^{2}+4 a d -9 b c \right ) \sqrt {d \,x^{2}+c}}{8 b^{2}}+\frac {\frac {\sqrt {d}\, \left (8 a^{2} d^{2}-20 a b c d +15 b^{2} c^{2}\right ) \ln \left (x \sqrt {d}+\sqrt {d \,x^{2}+c}\right )}{b}-\frac {\left (4 a^{3} d^{3}-12 a^{2} b c \,d^{2}+12 a \,b^{2} c^{2} d -4 b^{3} c^{3}\right ) \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {d \left (x +\frac {\sqrt {-a b}}{b}\right )^{2}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x +\frac {\sqrt {-a b}}{b}}\right )}{\sqrt {-a b}\, b \sqrt {-\frac {a d -b c}{b}}}-\frac {\left (-4 a^{3} d^{3}+12 a^{2} b c \,d^{2}-12 a \,b^{2} c^{2} d +4 b^{3} c^{3}\right ) \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {d \left (x -\frac {\sqrt {-a b}}{b}\right )^{2}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x -\frac {\sqrt {-a b}}{b}}\right )}{\sqrt {-a b}\, b \sqrt {-\frac {a d -b c}{b}}}}{8 b^{2}}\) | \(470\) |
| default | \(\text {Expression too large to display}\) | \(2074\) |
-1/2/b^3*(1/4*d*(b*(d*x^2+c)^(1/2)*(-2*b*d*x^2+4*a*d-9*b*c)*x-(8*a^2*d^2-2 0*a*b*c*d+15*b^2*c^2)/d^(1/2)*arctanh((d*x^2+c)^(1/2)/x/d^(1/2)))+2*(a*d-b *c)^3/((a*d-b*c)*a)^(1/2)*arctanh((d*x^2+c)^(1/2)/x*a/((a*d-b*c)*a)^(1/2)) )
Time = 0.85 (sec) , antiderivative size = 931, normalized size of antiderivative = 5.97 \[ \int \frac {\left (c+d x^2\right )^{5/2}}{a+b x^2} \, dx=\left [\frac {{\left (15 \, b^{2} c^{2} - 20 \, a b c d + 8 \, a^{2} d^{2}\right )} \sqrt {d} \log \left (-2 \, d x^{2} - 2 \, \sqrt {d x^{2} + c} \sqrt {d} x - c\right ) + 4 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt {-\frac {b c - a d}{a}} \log \left (\frac {{\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{4} + a^{2} c^{2} - 2 \, {\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{2} - 4 \, {\left (a^{2} c x - {\left (a b c - 2 \, a^{2} d\right )} x^{3}\right )} \sqrt {d x^{2} + c} \sqrt {-\frac {b c - a d}{a}}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right ) + 2 \, {\left (2 \, b^{2} d^{2} x^{3} + {\left (9 \, b^{2} c d - 4 \, a b d^{2}\right )} x\right )} \sqrt {d x^{2} + c}}{16 \, b^{3}}, -\frac {{\left (15 \, b^{2} c^{2} - 20 \, a b c d + 8 \, a^{2} d^{2}\right )} \sqrt {-d} \arctan \left (\frac {\sqrt {-d} x}{\sqrt {d x^{2} + c}}\right ) - 2 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt {-\frac {b c - a d}{a}} \log \left (\frac {{\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{4} + a^{2} c^{2} - 2 \, {\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{2} - 4 \, {\left (a^{2} c x - {\left (a b c - 2 \, a^{2} d\right )} x^{3}\right )} \sqrt {d x^{2} + c} \sqrt {-\frac {b c - a d}{a}}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right ) - {\left (2 \, b^{2} d^{2} x^{3} + {\left (9 \, b^{2} c d - 4 \, a b d^{2}\right )} x\right )} \sqrt {d x^{2} + c}}{8 \, b^{3}}, \frac {8 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt {\frac {b c - a d}{a}} \arctan \left (\frac {{\left ({\left (b c - 2 \, a d\right )} x^{2} - a c\right )} \sqrt {d x^{2} + c} \sqrt {\frac {b c - a d}{a}}}{2 \, {\left ({\left (b c d - a d^{2}\right )} x^{3} + {\left (b c^{2} - a c d\right )} x\right )}}\right ) + {\left (15 \, b^{2} c^{2} - 20 \, a b c d + 8 \, a^{2} d^{2}\right )} \sqrt {d} \log \left (-2 \, d x^{2} - 2 \, \sqrt {d x^{2} + c} \sqrt {d} x - c\right ) + 2 \, {\left (2 \, b^{2} d^{2} x^{3} + {\left (9 \, b^{2} c d - 4 \, a b d^{2}\right )} x\right )} \sqrt {d x^{2} + c}}{16 \, b^{3}}, -\frac {{\left (15 \, b^{2} c^{2} - 20 \, a b c d + 8 \, a^{2} d^{2}\right )} \sqrt {-d} \arctan \left (\frac {\sqrt {-d} x}{\sqrt {d x^{2} + c}}\right ) - 4 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt {\frac {b c - a d}{a}} \arctan \left (\frac {{\left ({\left (b c - 2 \, a d\right )} x^{2} - a c\right )} \sqrt {d x^{2} + c} \sqrt {\frac {b c - a d}{a}}}{2 \, {\left ({\left (b c d - a d^{2}\right )} x^{3} + {\left (b c^{2} - a c d\right )} x\right )}}\right ) - {\left (2 \, b^{2} d^{2} x^{3} + {\left (9 \, b^{2} c d - 4 \, a b d^{2}\right )} x\right )} \sqrt {d x^{2} + c}}{8 \, b^{3}}\right ] \]
[1/16*((15*b^2*c^2 - 20*a*b*c*d + 8*a^2*d^2)*sqrt(d)*log(-2*d*x^2 - 2*sqrt (d*x^2 + c)*sqrt(d)*x - c) + 4*(b^2*c^2 - 2*a*b*c*d + a^2*d^2)*sqrt(-(b*c - a*d)/a)*log(((b^2*c^2 - 8*a*b*c*d + 8*a^2*d^2)*x^4 + a^2*c^2 - 2*(3*a*b* c^2 - 4*a^2*c*d)*x^2 - 4*(a^2*c*x - (a*b*c - 2*a^2*d)*x^3)*sqrt(d*x^2 + c) *sqrt(-(b*c - a*d)/a))/(b^2*x^4 + 2*a*b*x^2 + a^2)) + 2*(2*b^2*d^2*x^3 + ( 9*b^2*c*d - 4*a*b*d^2)*x)*sqrt(d*x^2 + c))/b^3, -1/8*((15*b^2*c^2 - 20*a*b *c*d + 8*a^2*d^2)*sqrt(-d)*arctan(sqrt(-d)*x/sqrt(d*x^2 + c)) - 2*(b^2*c^2 - 2*a*b*c*d + a^2*d^2)*sqrt(-(b*c - a*d)/a)*log(((b^2*c^2 - 8*a*b*c*d + 8 *a^2*d^2)*x^4 + a^2*c^2 - 2*(3*a*b*c^2 - 4*a^2*c*d)*x^2 - 4*(a^2*c*x - (a* b*c - 2*a^2*d)*x^3)*sqrt(d*x^2 + c)*sqrt(-(b*c - a*d)/a))/(b^2*x^4 + 2*a*b *x^2 + a^2)) - (2*b^2*d^2*x^3 + (9*b^2*c*d - 4*a*b*d^2)*x)*sqrt(d*x^2 + c) )/b^3, 1/16*(8*(b^2*c^2 - 2*a*b*c*d + a^2*d^2)*sqrt((b*c - a*d)/a)*arctan( 1/2*((b*c - 2*a*d)*x^2 - a*c)*sqrt(d*x^2 + c)*sqrt((b*c - a*d)/a)/((b*c*d - a*d^2)*x^3 + (b*c^2 - a*c*d)*x)) + (15*b^2*c^2 - 20*a*b*c*d + 8*a^2*d^2) *sqrt(d)*log(-2*d*x^2 - 2*sqrt(d*x^2 + c)*sqrt(d)*x - c) + 2*(2*b^2*d^2*x^ 3 + (9*b^2*c*d - 4*a*b*d^2)*x)*sqrt(d*x^2 + c))/b^3, -1/8*((15*b^2*c^2 - 2 0*a*b*c*d + 8*a^2*d^2)*sqrt(-d)*arctan(sqrt(-d)*x/sqrt(d*x^2 + c)) - 4*(b^ 2*c^2 - 2*a*b*c*d + a^2*d^2)*sqrt((b*c - a*d)/a)*arctan(1/2*((b*c - 2*a*d) *x^2 - a*c)*sqrt(d*x^2 + c)*sqrt((b*c - a*d)/a)/((b*c*d - a*d^2)*x^3 + (b* c^2 - a*c*d)*x)) - (2*b^2*d^2*x^3 + (9*b^2*c*d - 4*a*b*d^2)*x)*sqrt(d*x...
\[ \int \frac {\left (c+d x^2\right )^{5/2}}{a+b x^2} \, dx=\int \frac {\left (c + d x^{2}\right )^{\frac {5}{2}}}{a + b x^{2}}\, dx \]
\[ \int \frac {\left (c+d x^2\right )^{5/2}}{a+b x^2} \, dx=\int { \frac {{\left (d x^{2} + c\right )}^{\frac {5}{2}}}{b x^{2} + a} \,d x } \]
Exception generated. \[ \int \frac {\left (c+d x^2\right )^{5/2}}{a+b x^2} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:index.cc index_m i_lex_is_greater E rror: Bad Argument Value
Timed out. \[ \int \frac {\left (c+d x^2\right )^{5/2}}{a+b x^2} \, dx=\int \frac {{\left (d\,x^2+c\right )}^{5/2}}{b\,x^2+a} \,d x \]