Integrand size = 21, antiderivative size = 144 \[ \int \frac {\left (a+b x^2\right )^{5/2}}{\left (c+d x^2\right )^4} \, dx=\frac {x \left (a+b x^2\right )^{5/2}}{6 c \left (c+d x^2\right )^3}+\frac {5 a x \left (a+b x^2\right )^{3/2}}{24 c^2 \left (c+d x^2\right )^2}+\frac {5 a^2 x \sqrt {a+b x^2}}{16 c^3 \left (c+d x^2\right )}+\frac {5 a^3 \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a+b x^2}}\right )}{16 c^{7/2} \sqrt {b c-a d}} \]
1/6*x*(b*x^2+a)^(5/2)/c/(d*x^2+c)^3+5/24*a*x*(b*x^2+a)^(3/2)/c^2/(d*x^2+c) ^2+5/16*a^3*arctanh(x*(-a*d+b*c)^(1/2)/c^(1/2)/(b*x^2+a)^(1/2))/c^(7/2)/(- a*d+b*c)^(1/2)+5/16*a^2*x*(b*x^2+a)^(1/2)/c^3/(d*x^2+c)
Time = 10.60 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.40 \[ \int \frac {\left (a+b x^2\right )^{5/2}}{\left (c+d x^2\right )^4} \, dx=\frac {x \sqrt {a+b x^2} \left (\frac {\sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \left (8 b^2 c^2 x^4+2 a b c x^2 \left (13 c+5 d x^2\right )+a^2 \left (33 c^2+40 c d x^2+15 d^2 x^4\right )\right )}{\left (c+d x^2\right )^2 \sqrt {1+\frac {d x^2}{c}}}+\frac {15 a^2 \arcsin \left (\frac {\sqrt {\left (-\frac {b}{a}+\frac {d}{c}\right ) x^2}}{\sqrt {1+\frac {d x^2}{c}}}\right )}{\sqrt {\frac {(-b c+a d) x^2}{a c}}}\right )}{48 c^4 \sqrt {1+\frac {b x^2}{a}}} \]
(x*Sqrt[a + b*x^2]*((Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*(8*b^2*c^2*x^4 + 2*a*b*c*x^2*(13*c + 5*d*x^2) + a^2*(33*c^2 + 40*c*d*x^2 + 15*d^2*x^4)))/ ((c + d*x^2)^2*Sqrt[1 + (d*x^2)/c]) + (15*a^2*ArcSin[Sqrt[(-(b/a) + d/c)*x ^2]/Sqrt[1 + (d*x^2)/c]])/Sqrt[((-(b*c) + a*d)*x^2)/(a*c)]))/(48*c^4*Sqrt[ 1 + (b*x^2)/a])
Time = 0.25 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.08, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {292, 292, 292, 291, 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.
| \(\displaystyle \int \frac {\left (a+b x^2\right )^{5/2}}{\left (c+d x^2\right )^4} \, dx\) |
| \(\Big \downarrow \) 292 |
| \(\displaystyle \frac {5 a \int \frac {\left (b x^2+a\right )^{3/2}}{\left (d x^2+c\right )^3}dx}{6 c}+\frac {x \left (a+b x^2\right )^{5/2}}{6 c \left (c+d x^2\right )^3}\) |
| \(\Big \downarrow \) 292 |
| \(\displaystyle \frac {5 a \left (\frac {3 a \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^2}dx}{4 c}+\frac {x \left (a+b x^2\right )^{3/2}}{4 c \left (c+d x^2\right )^2}\right )}{6 c}+\frac {x \left (a+b x^2\right )^{5/2}}{6 c \left (c+d x^2\right )^3}\) |
| \(\Big \downarrow \) 292 |
| \(\displaystyle \frac {5 a \left (\frac {3 a \left (\frac {a \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )}dx}{2 c}+\frac {x \sqrt {a+b x^2}}{2 c \left (c+d x^2\right )}\right )}{4 c}+\frac {x \left (a+b x^2\right )^{3/2}}{4 c \left (c+d x^2\right )^2}\right )}{6 c}+\frac {x \left (a+b x^2\right )^{5/2}}{6 c \left (c+d x^2\right )^3}\) |
| \(\Big \downarrow \) 291 |
| \(\displaystyle \frac {5 a \left (\frac {3 a \left (\frac {a \int \frac {1}{c-\frac {(b c-a d) x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{2 c}+\frac {x \sqrt {a+b x^2}}{2 c \left (c+d x^2\right )}\right )}{4 c}+\frac {x \left (a+b x^2\right )^{3/2}}{4 c \left (c+d x^2\right )^2}\right )}{6 c}+\frac {x \left (a+b x^2\right )^{5/2}}{6 c \left (c+d x^2\right )^3}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {5 a \left (\frac {3 a \left (\frac {a \text {arctanh}\left (\frac {x \sqrt {b c-a d}}{\sqrt {c} \sqrt {a+b x^2}}\right )}{2 c^{3/2} \sqrt {b c-a d}}+\frac {x \sqrt {a+b x^2}}{2 c \left (c+d x^2\right )}\right )}{4 c}+\frac {x \left (a+b x^2\right )^{3/2}}{4 c \left (c+d x^2\right )^2}\right )}{6 c}+\frac {x \left (a+b x^2\right )^{5/2}}{6 c \left (c+d x^2\right )^3}\) |
(x*(a + b*x^2)^(5/2))/(6*c*(c + d*x^2)^3) + (5*a*((x*(a + b*x^2)^(3/2))/(4 *c*(c + d*x^2)^2) + (3*a*((x*Sqrt[a + b*x^2])/(2*c*(c + d*x^2)) + (a*ArcTa nh[(Sqrt[b*c - a*d]*x)/(Sqrt[c]*Sqrt[a + b*x^2])])/(2*c^(3/2)*Sqrt[b*c - a *d])))/(4*c)))/(6*c)
3.1.69.3.1 Defintions of rubi rules used
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[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] :> Si mp[(-x)*(a + b*x^2)^(p + 1)*((c + d*x^2)^q/(2*a*(p + 1))), x] - Simp[c*(q/( a*(p + 1))) Int[(a + b*x^2)^(p + 1)*(c + d*x^2)^(q - 1), x], x] /; FreeQ[ {a, b, c, d, p}, x] && NeQ[b*c - a*d, 0] && EqQ[2*(p + q + 1) + 1, 0] && Gt Q[q, 0] && NeQ[p, -1]
Time = 2.67 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.00
| method | result | size |
| pseudoelliptic | \(-\frac {-33 x \sqrt {b \,x^{2}+a}\, \left (\left (\frac {8}{33} b^{2} x^{4}+\frac {26}{33} a b \,x^{2}+a^{2}\right ) c^{2}+\frac {40 x^{2} \left (\frac {b \,x^{2}}{4}+a \right ) d a c}{33}+\frac {5 a^{2} d^{2} x^{4}}{11}\right ) \sqrt {\left (a d -b c \right ) c}+15 a^{3} \left (d \,x^{2}+c \right )^{3} \arctan \left (\frac {c \sqrt {b \,x^{2}+a}}{x \sqrt {\left (a d -b c \right ) c}}\right )}{48 \sqrt {\left (a d -b c \right ) c}\, c^{3} \left (d \,x^{2}+c \right )^{3}}\) | \(144\) |
| default | \(\text {Expression too large to display}\) | \(19519\) |
-1/48/((a*d-b*c)*c)^(1/2)*(-33*x*(b*x^2+a)^(1/2)*((8/33*b^2*x^4+26/33*a*b* x^2+a^2)*c^2+40/33*x^2*(1/4*b*x^2+a)*d*a*c+5/11*a^2*d^2*x^4)*((a*d-b*c)*c) ^(1/2)+15*a^3*(d*x^2+c)^3*arctan(c*(b*x^2+a)^(1/2)/x/((a*d-b*c)*c)^(1/2))) /c^3/(d*x^2+c)^3
Leaf count of result is larger than twice the leaf count of optimal. 333 vs. \(2 (120) = 240\).
Time = 0.37 (sec) , antiderivative size = 706, normalized size of antiderivative = 4.90 \[ \int \frac {\left (a+b x^2\right )^{5/2}}{\left (c+d x^2\right )^4} \, dx=\left [\frac {15 \, {\left (a^{3} d^{3} x^{6} + 3 \, a^{3} c d^{2} x^{4} + 3 \, a^{3} c^{2} d x^{2} + a^{3} c^{3}\right )} \sqrt {b c^{2} - a c d} \log \left (\frac {{\left (8 \, b^{2} c^{2} - 8 \, a b c d + a^{2} d^{2}\right )} x^{4} + a^{2} c^{2} + 2 \, {\left (4 \, a b c^{2} - 3 \, a^{2} c d\right )} x^{2} + 4 \, {\left ({\left (2 \, b c - a d\right )} x^{3} + a c x\right )} \sqrt {b c^{2} - a c d} \sqrt {b x^{2} + a}}{d^{2} x^{4} + 2 \, c d x^{2} + c^{2}}\right ) + 4 \, {\left ({\left (8 \, b^{3} c^{4} + 2 \, a b^{2} c^{3} d + 5 \, a^{2} b c^{2} d^{2} - 15 \, a^{3} c d^{3}\right )} x^{5} + 2 \, {\left (13 \, a b^{2} c^{4} + 7 \, a^{2} b c^{3} d - 20 \, a^{3} c^{2} d^{2}\right )} x^{3} + 33 \, {\left (a^{2} b c^{4} - a^{3} c^{3} d\right )} x\right )} \sqrt {b x^{2} + a}}{192 \, {\left (b c^{8} - a c^{7} d + {\left (b c^{5} d^{3} - a c^{4} d^{4}\right )} x^{6} + 3 \, {\left (b c^{6} d^{2} - a c^{5} d^{3}\right )} x^{4} + 3 \, {\left (b c^{7} d - a c^{6} d^{2}\right )} x^{2}\right )}}, -\frac {15 \, {\left (a^{3} d^{3} x^{6} + 3 \, a^{3} c d^{2} x^{4} + 3 \, a^{3} c^{2} d x^{2} + a^{3} c^{3}\right )} \sqrt {-b c^{2} + a c d} \arctan \left (\frac {\sqrt {-b c^{2} + a c d} {\left ({\left (2 \, b c - a d\right )} x^{2} + a c\right )} \sqrt {b x^{2} + a}}{2 \, {\left ({\left (b^{2} c^{2} - a b c d\right )} x^{3} + {\left (a b c^{2} - a^{2} c d\right )} x\right )}}\right ) - 2 \, {\left ({\left (8 \, b^{3} c^{4} + 2 \, a b^{2} c^{3} d + 5 \, a^{2} b c^{2} d^{2} - 15 \, a^{3} c d^{3}\right )} x^{5} + 2 \, {\left (13 \, a b^{2} c^{4} + 7 \, a^{2} b c^{3} d - 20 \, a^{3} c^{2} d^{2}\right )} x^{3} + 33 \, {\left (a^{2} b c^{4} - a^{3} c^{3} d\right )} x\right )} \sqrt {b x^{2} + a}}{96 \, {\left (b c^{8} - a c^{7} d + {\left (b c^{5} d^{3} - a c^{4} d^{4}\right )} x^{6} + 3 \, {\left (b c^{6} d^{2} - a c^{5} d^{3}\right )} x^{4} + 3 \, {\left (b c^{7} d - a c^{6} d^{2}\right )} x^{2}\right )}}\right ] \]
[1/192*(15*(a^3*d^3*x^6 + 3*a^3*c*d^2*x^4 + 3*a^3*c^2*d*x^2 + a^3*c^3)*sqr t(b*c^2 - a*c*d)*log(((8*b^2*c^2 - 8*a*b*c*d + a^2*d^2)*x^4 + a^2*c^2 + 2* (4*a*b*c^2 - 3*a^2*c*d)*x^2 + 4*((2*b*c - a*d)*x^3 + a*c*x)*sqrt(b*c^2 - a *c*d)*sqrt(b*x^2 + a))/(d^2*x^4 + 2*c*d*x^2 + c^2)) + 4*((8*b^3*c^4 + 2*a* b^2*c^3*d + 5*a^2*b*c^2*d^2 - 15*a^3*c*d^3)*x^5 + 2*(13*a*b^2*c^4 + 7*a^2* b*c^3*d - 20*a^3*c^2*d^2)*x^3 + 33*(a^2*b*c^4 - a^3*c^3*d)*x)*sqrt(b*x^2 + a))/(b*c^8 - a*c^7*d + (b*c^5*d^3 - a*c^4*d^4)*x^6 + 3*(b*c^6*d^2 - a*c^5 *d^3)*x^4 + 3*(b*c^7*d - a*c^6*d^2)*x^2), -1/96*(15*(a^3*d^3*x^6 + 3*a^3*c *d^2*x^4 + 3*a^3*c^2*d*x^2 + a^3*c^3)*sqrt(-b*c^2 + a*c*d)*arctan(1/2*sqrt (-b*c^2 + a*c*d)*((2*b*c - a*d)*x^2 + a*c)*sqrt(b*x^2 + a)/((b^2*c^2 - a*b *c*d)*x^3 + (a*b*c^2 - a^2*c*d)*x)) - 2*((8*b^3*c^4 + 2*a*b^2*c^3*d + 5*a^ 2*b*c^2*d^2 - 15*a^3*c*d^3)*x^5 + 2*(13*a*b^2*c^4 + 7*a^2*b*c^3*d - 20*a^3 *c^2*d^2)*x^3 + 33*(a^2*b*c^4 - a^3*c^3*d)*x)*sqrt(b*x^2 + a))/(b*c^8 - a* c^7*d + (b*c^5*d^3 - a*c^4*d^4)*x^6 + 3*(b*c^6*d^2 - a*c^5*d^3)*x^4 + 3*(b *c^7*d - a*c^6*d^2)*x^2)]
Timed out. \[ \int \frac {\left (a+b x^2\right )^{5/2}}{\left (c+d x^2\right )^4} \, dx=\text {Timed out} \]
\[ \int \frac {\left (a+b x^2\right )^{5/2}}{\left (c+d x^2\right )^4} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}}}{{\left (d x^{2} + c\right )}^{4}} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 846 vs. \(2 (120) = 240\).
Time = 1.31 (sec) , antiderivative size = 846, normalized size of antiderivative = 5.88 \[ \int \frac {\left (a+b x^2\right )^{5/2}}{\left (c+d x^2\right )^4} \, dx=-\frac {5 \, a^{3} \sqrt {b} \arctan \left (\frac {{\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} d + 2 \, b c - a d}{2 \, \sqrt {-b^{2} c^{2} + a b c d}}\right )}{16 \, \sqrt {-b^{2} c^{2} + a b c d} c^{3}} + \frac {48 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{10} b^{\frac {7}{2}} c^{3} d^{2} - 15 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{10} a^{3} \sqrt {b} d^{5} + 192 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{8} b^{\frac {9}{2}} c^{4} d + 48 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{8} a b^{\frac {7}{2}} c^{3} d^{2} - 150 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{8} a^{3} b^{\frac {3}{2}} c d^{4} + 75 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{8} a^{4} \sqrt {b} d^{5} + 256 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} b^{\frac {11}{2}} c^{5} - 64 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} a b^{\frac {9}{2}} c^{4} d + 288 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} a^{2} b^{\frac {7}{2}} c^{3} d^{2} - 440 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} a^{3} b^{\frac {5}{2}} c^{2} d^{3} + 440 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} a^{4} b^{\frac {3}{2}} c d^{4} - 150 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} a^{5} \sqrt {b} d^{5} + 192 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} a^{2} b^{\frac {9}{2}} c^{4} d + 48 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} a^{3} b^{\frac {7}{2}} c^{3} d^{2} + 360 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} a^{4} b^{\frac {5}{2}} c^{2} d^{3} - 420 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} a^{5} b^{\frac {3}{2}} c d^{4} + 150 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} a^{6} \sqrt {b} d^{5} + 48 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a^{4} b^{\frac {7}{2}} c^{3} d^{2} + 72 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a^{5} b^{\frac {5}{2}} c^{2} d^{3} + 120 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a^{6} b^{\frac {3}{2}} c d^{4} - 75 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a^{7} \sqrt {b} d^{5} + 8 \, a^{6} b^{\frac {5}{2}} c^{2} d^{3} + 10 \, a^{7} b^{\frac {3}{2}} c d^{4} + 15 \, a^{8} \sqrt {b} d^{5}}{24 \, {\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} d + 4 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} b c - 2 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a d + a^{2} d\right )}^{3} c^{3} d^{3}} \]
-5/16*a^3*sqrt(b)*arctan(1/2*((sqrt(b)*x - sqrt(b*x^2 + a))^2*d + 2*b*c - a*d)/sqrt(-b^2*c^2 + a*b*c*d))/(sqrt(-b^2*c^2 + a*b*c*d)*c^3) + 1/24*(48*( sqrt(b)*x - sqrt(b*x^2 + a))^10*b^(7/2)*c^3*d^2 - 15*(sqrt(b)*x - sqrt(b*x ^2 + a))^10*a^3*sqrt(b)*d^5 + 192*(sqrt(b)*x - sqrt(b*x^2 + a))^8*b^(9/2)* c^4*d + 48*(sqrt(b)*x - sqrt(b*x^2 + a))^8*a*b^(7/2)*c^3*d^2 - 150*(sqrt(b )*x - sqrt(b*x^2 + a))^8*a^3*b^(3/2)*c*d^4 + 75*(sqrt(b)*x - sqrt(b*x^2 + a))^8*a^4*sqrt(b)*d^5 + 256*(sqrt(b)*x - sqrt(b*x^2 + a))^6*b^(11/2)*c^5 - 64*(sqrt(b)*x - sqrt(b*x^2 + a))^6*a*b^(9/2)*c^4*d + 288*(sqrt(b)*x - sqr t(b*x^2 + a))^6*a^2*b^(7/2)*c^3*d^2 - 440*(sqrt(b)*x - sqrt(b*x^2 + a))^6* a^3*b^(5/2)*c^2*d^3 + 440*(sqrt(b)*x - sqrt(b*x^2 + a))^6*a^4*b^(3/2)*c*d^ 4 - 150*(sqrt(b)*x - sqrt(b*x^2 + a))^6*a^5*sqrt(b)*d^5 + 192*(sqrt(b)*x - sqrt(b*x^2 + a))^4*a^2*b^(9/2)*c^4*d + 48*(sqrt(b)*x - sqrt(b*x^2 + a))^4 *a^3*b^(7/2)*c^3*d^2 + 360*(sqrt(b)*x - sqrt(b*x^2 + a))^4*a^4*b^(5/2)*c^2 *d^3 - 420*(sqrt(b)*x - sqrt(b*x^2 + a))^4*a^5*b^(3/2)*c*d^4 + 150*(sqrt(b )*x - sqrt(b*x^2 + a))^4*a^6*sqrt(b)*d^5 + 48*(sqrt(b)*x - sqrt(b*x^2 + a) )^2*a^4*b^(7/2)*c^3*d^2 + 72*(sqrt(b)*x - sqrt(b*x^2 + a))^2*a^5*b^(5/2)*c ^2*d^3 + 120*(sqrt(b)*x - sqrt(b*x^2 + a))^2*a^6*b^(3/2)*c*d^4 - 75*(sqrt( b)*x - sqrt(b*x^2 + a))^2*a^7*sqrt(b)*d^5 + 8*a^6*b^(5/2)*c^2*d^3 + 10*a^7 *b^(3/2)*c*d^4 + 15*a^8*sqrt(b)*d^5)/(((sqrt(b)*x - sqrt(b*x^2 + a))^4*d + 4*(sqrt(b)*x - sqrt(b*x^2 + a))^2*b*c - 2*(sqrt(b)*x - sqrt(b*x^2 + a)...
Timed out. \[ \int \frac {\left (a+b x^2\right )^{5/2}}{\left (c+d x^2\right )^4} \, dx=\int \frac {{\left (b\,x^2+a\right )}^{5/2}}{{\left (d\,x^2+c\right )}^4} \,d x \]