Integrand size = 23, antiderivative size = 510 \[ \int \frac {\left (c+d x^2\right )^{5/2}}{\left (a+b x^2\right )^{11/2}} \, dx=\frac {4 (b c-a d) (2 b c+a d) x \sqrt {c+d x^2}}{63 a^2 b^2 \left (a+b x^2\right )^{7/2}}+\frac {\left (48 b^2 c^2+19 a b c d+8 a^2 d^2\right ) x \sqrt {c+d x^2}}{315 a^3 b^2 \left (a+b x^2\right )^{5/2}}+\frac {\left (64 b^3 c^3-36 a b^2 c^2 d-15 a^2 b c d^2-8 a^3 d^3\right ) x \sqrt {c+d x^2}}{315 a^4 b^2 (b c-a d) \left (a+b x^2\right )^{3/2}}+\frac {(b c-a d) x \left (c+d x^2\right )^{3/2}}{9 a b \left (a+b x^2\right )^{9/2}}+\frac {\left (128 b^4 c^4-184 a b^3 c^3 d+27 a^2 b^2 c^2 d^2+11 a^3 b c d^3+8 a^4 d^4\right ) \sqrt {c+d x^2} E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{315 a^{9/2} b^{5/2} (b c-a d)^2 \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}-\frac {2 d \left (32 b^3 c^3-42 a b^2 c^2 d+3 a^2 b c d^2+2 a^3 d^3\right ) \sqrt {c+d x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{315 a^{7/2} b^{5/2} (b c-a d)^2 \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}} \] Output:
4/63*(-a*d+b*c)*(a*d+2*b*c)*x*(d*x^2+c)^(1/2)/a^2/b^2/(b*x^2+a)^(7/2)+1/31 5*(8*a^2*d^2+19*a*b*c*d+48*b^2*c^2)*x*(d*x^2+c)^(1/2)/a^3/b^2/(b*x^2+a)^(5 /2)+1/315*(-8*a^3*d^3-15*a^2*b*c*d^2-36*a*b^2*c^2*d+64*b^3*c^3)*x*(d*x^2+c )^(1/2)/a^4/b^2/(-a*d+b*c)/(b*x^2+a)^(3/2)+1/9*(-a*d+b*c)*x*(d*x^2+c)^(3/2 )/a/b/(b*x^2+a)^(9/2)+1/315*(8*a^4*d^4+11*a^3*b*c*d^3+27*a^2*b^2*c^2*d^2-1 84*a*b^3*c^3*d+128*b^4*c^4)*(d*x^2+c)^(1/2)*EllipticE(b^(1/2)*x/a^(1/2)/(1 +b*x^2/a)^(1/2),(1-a*d/b/c)^(1/2))/a^(9/2)/b^(5/2)/(-a*d+b*c)^2/(b*x^2+a)^ (1/2)/(a*(d*x^2+c)/c/(b*x^2+a))^(1/2)-2/315*d*(2*a^3*d^3+3*a^2*b*c*d^2-42* a*b^2*c^2*d+32*b^3*c^3)*(d*x^2+c)^(1/2)*InverseJacobiAM(arctan(b^(1/2)*x/a ^(1/2)),(1-a*d/b/c)^(1/2))/a^(7/2)/b^(5/2)/(-a*d+b*c)^2/(b*x^2+a)^(1/2)/(a *(d*x^2+c)/c/(b*x^2+a))^(1/2)
Result contains complex when optimal does not.
Time = 5.72 (sec) , antiderivative size = 480, normalized size of antiderivative = 0.94 \[ \int \frac {\left (c+d x^2\right )^{5/2}}{\left (a+b x^2\right )^{11/2}} \, dx=\frac {\sqrt {\frac {b}{a}} \left (\sqrt {\frac {b}{a}} x \left (c+d x^2\right ) \left (35 a^4 (b c-a d)^4+5 a^3 (b c-a d)^3 (8 b c+11 a d) \left (a+b x^2\right )+a^2 (b c-a d)^2 \left (48 b^2 c^2+19 a b c d+8 a^2 d^2\right ) \left (a+b x^2\right )^2+a (-b c+a d) \left (-64 b^3 c^3+36 a b^2 c^2 d+15 a^2 b c d^2+8 a^3 d^3\right ) \left (a+b x^2\right )^3+\left (128 b^4 c^4-184 a b^3 c^3 d+27 a^2 b^2 c^2 d^2+11 a^3 b c d^3+8 a^4 d^4\right ) \left (a+b x^2\right )^4\right )+i c \left (a+b x^2\right )^4 \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \left (\left (128 b^4 c^4-184 a b^3 c^3 d+27 a^2 b^2 c^2 d^2+11 a^3 b c d^3+8 a^4 d^4\right ) E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )-\left (128 b^4 c^4-248 a b^3 c^3 d+111 a^2 b^2 c^2 d^2+5 a^3 b c d^3+4 a^4 d^4\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )\right )\right )}{315 a^4 b^3 (b c-a d)^2 \left (a+b x^2\right )^{9/2} \sqrt {c+d x^2}} \] Input:
Integrate[(c + d*x^2)^(5/2)/(a + b*x^2)^(11/2),x]
Output:
(Sqrt[b/a]*(Sqrt[b/a]*x*(c + d*x^2)*(35*a^4*(b*c - a*d)^4 + 5*a^3*(b*c - a *d)^3*(8*b*c + 11*a*d)*(a + b*x^2) + a^2*(b*c - a*d)^2*(48*b^2*c^2 + 19*a* b*c*d + 8*a^2*d^2)*(a + b*x^2)^2 + a*(-(b*c) + a*d)*(-64*b^3*c^3 + 36*a*b^ 2*c^2*d + 15*a^2*b*c*d^2 + 8*a^3*d^3)*(a + b*x^2)^3 + (128*b^4*c^4 - 184*a *b^3*c^3*d + 27*a^2*b^2*c^2*d^2 + 11*a^3*b*c*d^3 + 8*a^4*d^4)*(a + b*x^2)^ 4) + I*c*(a + b*x^2)^4*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*((128*b^4*c ^4 - 184*a*b^3*c^3*d + 27*a^2*b^2*c^2*d^2 + 11*a^3*b*c*d^3 + 8*a^4*d^4)*El lipticE[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] - (128*b^4*c^4 - 248*a*b^3*c^ 3*d + 111*a^2*b^2*c^2*d^2 + 5*a^3*b*c*d^3 + 4*a^4*d^4)*EllipticF[I*ArcSinh [Sqrt[b/a]*x], (a*d)/(b*c)])))/(315*a^4*b^3*(b*c - a*d)^2*(a + b*x^2)^(9/2 )*Sqrt[c + d*x^2])
Time = 0.70 (sec) , antiderivative size = 549, normalized size of antiderivative = 1.08, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.435, Rules used = {315, 401, 25, 402, 27, 402, 25, 400, 313, 320}
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.
| \(\displaystyle \int \frac {\left (c+d x^2\right )^{5/2}}{\left (a+b x^2\right )^{11/2}} \, dx\) |
| \(\Big \downarrow \) 315 |
| \(\displaystyle \frac {\int \frac {\sqrt {d x^2+c} \left (d (5 b c+4 a d) x^2+c (8 b c+a d)\right )}{\left (b x^2+a\right )^{9/2}}dx}{9 a b}+\frac {x \left (c+d x^2\right )^{3/2} (b c-a d)}{9 a b \left (a+b x^2\right )^{9/2}}\) |
| \(\Big \downarrow \) 401 |
| \(\displaystyle \frac {\frac {4 x \sqrt {c+d x^2} (b c-a d) (a d+2 b c)}{7 a b \left (a+b x^2\right )^{7/2}}-\frac {\int -\frac {d \left (40 b^2 c^2+15 a b d c+8 a^2 d^2\right ) x^2+c \left (48 b^2 c^2+11 a b d c+4 a^2 d^2\right )}{\left (b x^2+a\right )^{7/2} \sqrt {d x^2+c}}dx}{7 a b}}{9 a b}+\frac {x \left (c+d x^2\right )^{3/2} (b c-a d)}{9 a b \left (a+b x^2\right )^{9/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int \frac {d \left (40 b^2 c^2+15 a b d c+8 a^2 d^2\right ) x^2+c \left (48 b^2 c^2+11 a b d c+4 a^2 d^2\right )}{\left (b x^2+a\right )^{7/2} \sqrt {d x^2+c}}dx}{7 a b}+\frac {4 x \sqrt {c+d x^2} (b c-a d) (a d+2 b c)}{7 a b \left (a+b x^2\right )^{7/2}}}{9 a b}+\frac {x \left (c+d x^2\right )^{3/2} (b c-a d)}{9 a b \left (a+b x^2\right )^{9/2}}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {\frac {\frac {x \sqrt {c+d x^2} \left (8 a^2 d^2+19 a b c d+48 b^2 c^2\right )}{5 a \left (a+b x^2\right )^{5/2}}-\frac {\int -\frac {3 (b c-a d) \left (d \left (48 b^2 c^2+19 a b d c+8 a^2 d^2\right ) x^2+4 c \left (16 b^2 c^2+3 a b d c+a^2 d^2\right )\right )}{\left (b x^2+a\right )^{5/2} \sqrt {d x^2+c}}dx}{5 a (b c-a d)}}{7 a b}+\frac {4 x \sqrt {c+d x^2} (b c-a d) (a d+2 b c)}{7 a b \left (a+b x^2\right )^{7/2}}}{9 a b}+\frac {x \left (c+d x^2\right )^{3/2} (b c-a d)}{9 a b \left (a+b x^2\right )^{9/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {3 \int \frac {d \left (48 b^2 c^2+19 a b d c+8 a^2 d^2\right ) x^2+4 c \left (16 b^2 c^2+3 a b d c+a^2 d^2\right )}{\left (b x^2+a\right )^{5/2} \sqrt {d x^2+c}}dx}{5 a}+\frac {x \sqrt {c+d x^2} \left (8 a^2 d^2+19 a b c d+48 b^2 c^2\right )}{5 a \left (a+b x^2\right )^{5/2}}}{7 a b}+\frac {4 x \sqrt {c+d x^2} (b c-a d) (a d+2 b c)}{7 a b \left (a+b x^2\right )^{7/2}}}{9 a b}+\frac {x \left (c+d x^2\right )^{3/2} (b c-a d)}{9 a b \left (a+b x^2\right )^{9/2}}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {\frac {\frac {3 \left (\frac {x \sqrt {c+d x^2} \left (-8 a^3 d^3-15 a^2 b c d^2-36 a b^2 c^2 d+64 b^3 c^3\right )}{3 a \left (a+b x^2\right )^{3/2} (b c-a d)}-\frac {\int -\frac {d \left (64 b^3 c^3-36 a b^2 d c^2-15 a^2 b d^2 c-8 a^3 d^3\right ) x^2+c \left (128 b^3 c^3-120 a b^2 d c^2-9 a^2 b d^2 c-4 a^3 d^3\right )}{\left (b x^2+a\right )^{3/2} \sqrt {d x^2+c}}dx}{3 a (b c-a d)}\right )}{5 a}+\frac {x \sqrt {c+d x^2} \left (8 a^2 d^2+19 a b c d+48 b^2 c^2\right )}{5 a \left (a+b x^2\right )^{5/2}}}{7 a b}+\frac {4 x \sqrt {c+d x^2} (b c-a d) (a d+2 b c)}{7 a b \left (a+b x^2\right )^{7/2}}}{9 a b}+\frac {x \left (c+d x^2\right )^{3/2} (b c-a d)}{9 a b \left (a+b x^2\right )^{9/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\frac {3 \left (\frac {\int \frac {d \left (64 b^3 c^3-36 a b^2 d c^2-15 a^2 b d^2 c-8 a^3 d^3\right ) x^2+c \left (128 b^3 c^3-120 a b^2 d c^2-9 a^2 b d^2 c-4 a^3 d^3\right )}{\left (b x^2+a\right )^{3/2} \sqrt {d x^2+c}}dx}{3 a (b c-a d)}+\frac {x \sqrt {c+d x^2} \left (-8 a^3 d^3-15 a^2 b c d^2-36 a b^2 c^2 d+64 b^3 c^3\right )}{3 a \left (a+b x^2\right )^{3/2} (b c-a d)}\right )}{5 a}+\frac {x \sqrt {c+d x^2} \left (8 a^2 d^2+19 a b c d+48 b^2 c^2\right )}{5 a \left (a+b x^2\right )^{5/2}}}{7 a b}+\frac {4 x \sqrt {c+d x^2} (b c-a d) (a d+2 b c)}{7 a b \left (a+b x^2\right )^{7/2}}}{9 a b}+\frac {x \left (c+d x^2\right )^{3/2} (b c-a d)}{9 a b \left (a+b x^2\right )^{9/2}}\) |
| \(\Big \downarrow \) 400 |
| \(\displaystyle \frac {\frac {\frac {3 \left (\frac {\frac {\left (8 a^4 d^4+11 a^3 b c d^3+27 a^2 b^2 c^2 d^2-184 a b^3 c^3 d+128 b^4 c^4\right ) \int \frac {\sqrt {d x^2+c}}{\left (b x^2+a\right )^{3/2}}dx}{b c-a d}-\frac {2 c d \left (2 a^3 d^3+3 a^2 b c d^2-42 a b^2 c^2 d+32 b^3 c^3\right ) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{b c-a d}}{3 a (b c-a d)}+\frac {x \sqrt {c+d x^2} \left (-8 a^3 d^3-15 a^2 b c d^2-36 a b^2 c^2 d+64 b^3 c^3\right )}{3 a \left (a+b x^2\right )^{3/2} (b c-a d)}\right )}{5 a}+\frac {x \sqrt {c+d x^2} \left (8 a^2 d^2+19 a b c d+48 b^2 c^2\right )}{5 a \left (a+b x^2\right )^{5/2}}}{7 a b}+\frac {4 x \sqrt {c+d x^2} (b c-a d) (a d+2 b c)}{7 a b \left (a+b x^2\right )^{7/2}}}{9 a b}+\frac {x \left (c+d x^2\right )^{3/2} (b c-a d)}{9 a b \left (a+b x^2\right )^{9/2}}\) |
| \(\Big \downarrow \) 313 |
| \(\displaystyle \frac {\frac {\frac {3 \left (\frac {\frac {\sqrt {c+d x^2} \left (8 a^4 d^4+11 a^3 b c d^3+27 a^2 b^2 c^2 d^2-184 a b^3 c^3 d+128 b^4 c^4\right ) E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{\sqrt {a} \sqrt {b} \sqrt {a+b x^2} (b c-a d) \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}-\frac {2 c d \left (2 a^3 d^3+3 a^2 b c d^2-42 a b^2 c^2 d+32 b^3 c^3\right ) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{b c-a d}}{3 a (b c-a d)}+\frac {x \sqrt {c+d x^2} \left (-8 a^3 d^3-15 a^2 b c d^2-36 a b^2 c^2 d+64 b^3 c^3\right )}{3 a \left (a+b x^2\right )^{3/2} (b c-a d)}\right )}{5 a}+\frac {x \sqrt {c+d x^2} \left (8 a^2 d^2+19 a b c d+48 b^2 c^2\right )}{5 a \left (a+b x^2\right )^{5/2}}}{7 a b}+\frac {4 x \sqrt {c+d x^2} (b c-a d) (a d+2 b c)}{7 a b \left (a+b x^2\right )^{7/2}}}{9 a b}+\frac {x \left (c+d x^2\right )^{3/2} (b c-a d)}{9 a b \left (a+b x^2\right )^{9/2}}\) |
| \(\Big \downarrow \) 320 |
| \(\displaystyle \frac {\frac {\frac {x \sqrt {c+d x^2} \left (8 a^2 d^2+19 a b c d+48 b^2 c^2\right )}{5 a \left (a+b x^2\right )^{5/2}}+\frac {3 \left (\frac {x \sqrt {c+d x^2} \left (-8 a^3 d^3-15 a^2 b c d^2-36 a b^2 c^2 d+64 b^3 c^3\right )}{3 a \left (a+b x^2\right )^{3/2} (b c-a d)}+\frac {\frac {\sqrt {c+d x^2} \left (8 a^4 d^4+11 a^3 b c d^3+27 a^2 b^2 c^2 d^2-184 a b^3 c^3 d+128 b^4 c^4\right ) E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{\sqrt {a} \sqrt {b} \sqrt {a+b x^2} (b c-a d) \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}-\frac {2 c^{3/2} \sqrt {d} \sqrt {a+b x^2} \left (2 a^3 d^3+3 a^2 b c d^2-42 a b^2 c^2 d+32 b^3 c^3\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}}{3 a (b c-a d)}\right )}{5 a}}{7 a b}+\frac {4 x \sqrt {c+d x^2} (b c-a d) (a d+2 b c)}{7 a b \left (a+b x^2\right )^{7/2}}}{9 a b}+\frac {x \left (c+d x^2\right )^{3/2} (b c-a d)}{9 a b \left (a+b x^2\right )^{9/2}}\) |
Input:
Int[(c + d*x^2)^(5/2)/(a + b*x^2)^(11/2),x]
Output:
((b*c - a*d)*x*(c + d*x^2)^(3/2))/(9*a*b*(a + b*x^2)^(9/2)) + ((4*(b*c - a *d)*(2*b*c + a*d)*x*Sqrt[c + d*x^2])/(7*a*b*(a + b*x^2)^(7/2)) + (((48*b^2 *c^2 + 19*a*b*c*d + 8*a^2*d^2)*x*Sqrt[c + d*x^2])/(5*a*(a + b*x^2)^(5/2)) + (3*(((64*b^3*c^3 - 36*a*b^2*c^2*d - 15*a^2*b*c*d^2 - 8*a^3*d^3)*x*Sqrt[c + d*x^2])/(3*a*(b*c - a*d)*(a + b*x^2)^(3/2)) + (((128*b^4*c^4 - 184*a*b^ 3*c^3*d + 27*a^2*b^2*c^2*d^2 + 11*a^3*b*c*d^3 + 8*a^4*d^4)*Sqrt[c + d*x^2] *EllipticE[ArcTan[(Sqrt[b]*x)/Sqrt[a]], 1 - (a*d)/(b*c)])/(Sqrt[a]*Sqrt[b] *(b*c - a*d)*Sqrt[a + b*x^2]*Sqrt[(a*(c + d*x^2))/(c*(a + b*x^2))]) - (2*c ^(3/2)*Sqrt[d]*(32*b^3*c^3 - 42*a*b^2*c^2*d + 3*a^2*b*c*d^2 + 2*a^3*d^3)*S qrt[a + b*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(a *(b*c - a*d)*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]))/(3*a* (b*c - a*d))))/(5*a))/(7*a*b))/(9*a*b)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[Sqrt[(a_) + (b_.)*(x_)^2]/((c_) + (d_.)*(x_)^2)^(3/2), x_Symbol] :> Sim p[(Sqrt[a + b*x^2]/(c*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sqrt[c*((a + b*x^2)/(a*(c + d*x^2)))]))*EllipticE[ArcTan[Rt[d/c, 2]*x], 1 - b*(c/(a*d))], x] /; FreeQ [{a, b, c, d}, x] && PosQ[b/a] && PosQ[d/c]
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[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S imp[(Sqrt[a + b*x^2]/(a*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sqrt[c*((a + b*x^2)/(a*( c + d*x^2)))]))*EllipticF[ArcTan[Rt[d/c, 2]*x], 1 - b*(c/(a*d))], x] /; Fre eQ[{a, b, c, d}, x] && PosQ[d/c] && PosQ[b/a] && !SimplerSqrtQ[b/a, d/c]
Int[((e_) + (f_.)*(x_)^2)/(Sqrt[(a_) + (b_.)*(x_)^2]*((c_) + (d_.)*(x_)^2)^ (3/2)), x_Symbol] :> Simp[(b*e - a*f)/(b*c - a*d) Int[1/(Sqrt[a + b*x^2]* Sqrt[c + d*x^2]), x], x] - Simp[(d*e - c*f)/(b*c - a*d) Int[Sqrt[a + b*x^ 2]/(c + d*x^2)^(3/2), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && PosQ[b/a] & & PosQ[d/c]
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]
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 + 1)/(a*2*(b*c - a*d)*(p + 1))), x] + Simp[1/(a*2*(b*c - a*d)*(p + 1)) Int[(a + b*x^2)^(p + 1)*(c + d*x^2)^q*Simp[c*(b*e - a*f) + e*2*(b*c - a*d) *(p + 1) + d*(b*e - a*f)*(2*(p + q + 2) + 1)*x^2, x], x], x] /; FreeQ[{a, b , c, d, e, f, q}, x] && LtQ[p, -1]
Time = 8.00 (sec) , antiderivative size = 899, normalized size of antiderivative = 1.76
| method | result | size |
| elliptic | \(\frac {\sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}\, \left (\frac {\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) x \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}{9 a \,b^{7} \left (x^{2}+\frac {a}{b}\right )^{5}}-\frac {\left (11 a^{2} d^{2}-3 a b c d -8 b^{2} c^{2}\right ) x \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}{63 a^{2} b^{6} \left (x^{2}+\frac {a}{b}\right )^{4}}+\frac {\left (8 a^{2} d^{2}+19 a b c d +48 b^{2} c^{2}\right ) x \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}{315 a^{3} b^{5} \left (x^{2}+\frac {a}{b}\right )^{3}}+\frac {\left (8 a^{3} d^{3}+15 a^{2} b c \,d^{2}+36 a \,b^{2} c^{2} d -64 b^{3} c^{3}\right ) x \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}{315 \left (a d -b c \right ) a^{4} b^{4} \left (x^{2}+\frac {a}{b}\right )^{2}}+\frac {\left (b d \,x^{2}+b c \right ) x \left (8 d^{4} a^{4}+11 a^{3} b c \,d^{3}+27 a^{2} b^{2} c^{2} d^{2}-184 a \,b^{3} c^{3} d +128 c^{4} b^{4}\right )}{315 b^{3} a^{5} \left (a d -b c \right )^{2} \sqrt {\left (x^{2}+\frac {a}{b}\right ) \left (b d \,x^{2}+b c \right )}}+\frac {\left (\frac {d \left (8 a^{3} d^{3}+15 a^{2} b c \,d^{2}+36 a \,b^{2} c^{2} d -64 b^{3} c^{3}\right )}{315 a^{4} b^{3} \left (a d -b c \right )}-\frac {8 d^{4} a^{4}+11 a^{3} b c \,d^{3}+27 a^{2} b^{2} c^{2} d^{2}-184 a \,b^{3} c^{3} d +128 c^{4} b^{4}}{315 b^{3} \left (a d -b c \right ) a^{5}}-\frac {c \left (8 d^{4} a^{4}+11 a^{3} b c \,d^{3}+27 a^{2} b^{2} c^{2} d^{2}-184 a \,b^{3} c^{3} d +128 c^{4} b^{4}\right )}{315 b^{2} a^{5} \left (a d -b c \right )^{2}}\right ) \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}+\frac {\left (8 d^{4} a^{4}+11 a^{3} b c \,d^{3}+27 a^{2} b^{2} c^{2} d^{2}-184 a \,b^{3} c^{3} d +128 c^{4} b^{4}\right ) c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{315 b^{2} \left (a d -b c \right )^{2} a^{5} \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}\right )}{\sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\) | \(899\) |
| default | \(\text {Expression too large to display}\) | \(3930\) |
Input:
int((d*x^2+c)^(5/2)/(b*x^2+a)^(11/2),x,method=_RETURNVERBOSE)
Output:
((b*x^2+a)*(d*x^2+c))^(1/2)/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)*(1/9*(a^2*d^2- 2*a*b*c*d+b^2*c^2)/a/b^7*x*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)/(x^2+a/b)^5 -1/63*(11*a^2*d^2-3*a*b*c*d-8*b^2*c^2)/a^2/b^6*x*(b*d*x^4+a*d*x^2+b*c*x^2+ a*c)^(1/2)/(x^2+a/b)^4+1/315*(8*a^2*d^2+19*a*b*c*d+48*b^2*c^2)/a^3/b^5*x*( b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)/(x^2+a/b)^3+1/315*(8*a^3*d^3+15*a^2*b*c *d^2+36*a*b^2*c^2*d-64*b^3*c^3)/(a*d-b*c)/a^4/b^4*x*(b*d*x^4+a*d*x^2+b*c*x ^2+a*c)^(1/2)/(x^2+a/b)^2+1/315*(b*d*x^2+b*c)/b^3/a^5/(a*d-b*c)^2*x*(8*a^4 *d^4+11*a^3*b*c*d^3+27*a^2*b^2*c^2*d^2-184*a*b^3*c^3*d+128*b^4*c^4)/((x^2+ a/b)*(b*d*x^2+b*c))^(1/2)+(1/315*d*(8*a^3*d^3+15*a^2*b*c*d^2+36*a*b^2*c^2* d-64*b^3*c^3)/a^4/b^3/(a*d-b*c)-1/315/b^3/(a*d-b*c)*(8*a^4*d^4+11*a^3*b*c* d^3+27*a^2*b^2*c^2*d^2-184*a*b^3*c^3*d+128*b^4*c^4)/a^5-1/315/b^2*c/a^5/(a *d-b*c)^2*(8*a^4*d^4+11*a^3*b*c*d^3+27*a^2*b^2*c^2*d^2-184*a*b^3*c^3*d+128 *b^4*c^4))/(-b/a)^(1/2)*(1+b*x^2/a)^(1/2)*(1+d*x^2/c)^(1/2)/(b*d*x^4+a*d*x ^2+b*c*x^2+a*c)^(1/2)*EllipticF(x*(-b/a)^(1/2),(-1+(a*d+b*c)/c/b)^(1/2))+1 /315/b^2*(8*a^4*d^4+11*a^3*b*c*d^3+27*a^2*b^2*c^2*d^2-184*a*b^3*c^3*d+128* b^4*c^4)/(a*d-b*c)^2/a^5*c/(-b/a)^(1/2)*(1+b*x^2/a)^(1/2)*(1+d*x^2/c)^(1/2 )/(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)*(EllipticF(x*(-b/a)^(1/2),(-1+(a*d+b *c)/c/b)^(1/2))-EllipticE(x*(-b/a)^(1/2),(-1+(a*d+b*c)/c/b)^(1/2))))
Leaf count of result is larger than twice the leaf count of optimal. 1632 vs. \(2 (469) = 938\).
Time = 0.18 (sec) , antiderivative size = 1632, normalized size of antiderivative = 3.20 \[ \int \frac {\left (c+d x^2\right )^{5/2}}{\left (a+b x^2\right )^{11/2}} \, dx=\text {Too large to display} \] Input:
integrate((d*x^2+c)^(5/2)/(b*x^2+a)^(11/2),x, algorithm="fricas")
Output:
-1/315*((128*a^5*b^5*c^4 - 184*a^6*b^4*c^3*d + 27*a^7*b^3*c^2*d^2 + 11*a^8 *b^2*c*d^3 + 8*a^9*b*d^4 + (128*b^10*c^4 - 184*a*b^9*c^3*d + 27*a^2*b^8*c^ 2*d^2 + 11*a^3*b^7*c*d^3 + 8*a^4*b^6*d^4)*x^10 + 5*(128*a*b^9*c^4 - 184*a^ 2*b^8*c^3*d + 27*a^3*b^7*c^2*d^2 + 11*a^4*b^6*c*d^3 + 8*a^5*b^5*d^4)*x^8 + 10*(128*a^2*b^8*c^4 - 184*a^3*b^7*c^3*d + 27*a^4*b^6*c^2*d^2 + 11*a^5*b^5 *c*d^3 + 8*a^6*b^4*d^4)*x^6 + 10*(128*a^3*b^7*c^4 - 184*a^4*b^6*c^3*d + 27 *a^5*b^5*c^2*d^2 + 11*a^6*b^4*c*d^3 + 8*a^7*b^3*d^4)*x^4 + 5*(128*a^4*b^6* c^4 - 184*a^5*b^5*c^3*d + 27*a^6*b^4*c^2*d^2 + 11*a^7*b^3*c*d^3 + 8*a^8*b^ 2*d^4)*x^2)*sqrt(a*c)*sqrt(-b/a)*elliptic_e(arcsin(x*sqrt(-b/a)), a*d/(b*c )) - (128*a^5*b^5*c^4 + (128*b^10*c^4 + 8*(8*a^2*b^8 - 23*a*b^9)*c^3*d - 3 *(28*a^3*b^7 - 9*a^2*b^8)*c^2*d^2 + (6*a^4*b^6 + 11*a^3*b^7)*c*d^3 + 4*(a^ 5*b^5 + 2*a^4*b^6)*d^4)*x^10 + 5*(128*a*b^9*c^4 + 8*(8*a^3*b^7 - 23*a^2*b^ 8)*c^3*d - 3*(28*a^4*b^6 - 9*a^3*b^7)*c^2*d^2 + (6*a^5*b^5 + 11*a^4*b^6)*c *d^3 + 4*(a^6*b^4 + 2*a^5*b^5)*d^4)*x^8 + 10*(128*a^2*b^8*c^4 + 8*(8*a^4*b ^6 - 23*a^3*b^7)*c^3*d - 3*(28*a^5*b^5 - 9*a^4*b^6)*c^2*d^2 + (6*a^6*b^4 + 11*a^5*b^5)*c*d^3 + 4*(a^7*b^3 + 2*a^6*b^4)*d^4)*x^6 + 8*(8*a^7*b^3 - 23* a^6*b^4)*c^3*d - 3*(28*a^8*b^2 - 9*a^7*b^3)*c^2*d^2 + (6*a^9*b + 11*a^8*b^ 2)*c*d^3 + 4*(a^10 + 2*a^9*b)*d^4 + 10*(128*a^3*b^7*c^4 + 8*(8*a^5*b^5 - 2 3*a^4*b^6)*c^3*d - 3*(28*a^6*b^4 - 9*a^5*b^5)*c^2*d^2 + (6*a^7*b^3 + 11*a^ 6*b^4)*c*d^3 + 4*(a^8*b^2 + 2*a^7*b^3)*d^4)*x^4 + 5*(128*a^4*b^6*c^4 + ...
Timed out. \[ \int \frac {\left (c+d x^2\right )^{5/2}}{\left (a+b x^2\right )^{11/2}} \, dx=\text {Timed out} \] Input:
integrate((d*x**2+c)**(5/2)/(b*x**2+a)**(11/2),x)
Output:
Timed out
\[ \int \frac {\left (c+d x^2\right )^{5/2}}{\left (a+b x^2\right )^{11/2}} \, dx=\int { \frac {{\left (d x^{2} + c\right )}^{\frac {5}{2}}}{{\left (b x^{2} + a\right )}^{\frac {11}{2}}} \,d x } \] Input:
integrate((d*x^2+c)^(5/2)/(b*x^2+a)^(11/2),x, algorithm="maxima")
Output:
integrate((d*x^2 + c)^(5/2)/(b*x^2 + a)^(11/2), x)
\[ \int \frac {\left (c+d x^2\right )^{5/2}}{\left (a+b x^2\right )^{11/2}} \, dx=\int { \frac {{\left (d x^{2} + c\right )}^{\frac {5}{2}}}{{\left (b x^{2} + a\right )}^{\frac {11}{2}}} \,d x } \] Input:
integrate((d*x^2+c)^(5/2)/(b*x^2+a)^(11/2),x, algorithm="giac")
Output:
integrate((d*x^2 + c)^(5/2)/(b*x^2 + a)^(11/2), x)
Timed out. \[ \int \frac {\left (c+d x^2\right )^{5/2}}{\left (a+b x^2\right )^{11/2}} \, dx=\int \frac {{\left (d\,x^2+c\right )}^{5/2}}{{\left (b\,x^2+a\right )}^{11/2}} \,d x \] Input:
int((c + d*x^2)^(5/2)/(a + b*x^2)^(11/2),x)
Output:
int((c + d*x^2)^(5/2)/(a + b*x^2)^(11/2), x)
\[ \int \frac {\left (c+d x^2\right )^{5/2}}{\left (a+b x^2\right )^{11/2}} \, dx=\text {too large to display} \] Input:
int((d*x^2+c)^(5/2)/(b*x^2+a)^(11/2),x)
Output:
(3*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*c*d**2*x - 2*sqrt(c + d*x**2)*sqrt( a + b*x**2)*a*d**3*x**3 + 15*sqrt(c + d*x**2)*sqrt(a + b*x**2)*b*c**2*d*x + 8*sqrt(c + d*x**2)*sqrt(a + b*x**2)*b*c*d**2*x**3 + 8*int((sqrt(c + d*x* *2)*sqrt(a + b*x**2)*x**4)/(a**7*c*d + a**7*d**2*x**2 - 4*a**6*b*c**2 + 2* a**6*b*c*d*x**2 + 6*a**6*b*d**2*x**4 - 24*a**5*b**2*c**2*x**2 - 9*a**5*b** 2*c*d*x**4 + 15*a**5*b**2*d**2*x**6 - 60*a**4*b**3*c**2*x**4 - 40*a**4*b** 3*c*d*x**6 + 20*a**4*b**3*d**2*x**8 - 80*a**3*b**4*c**2*x**6 - 65*a**3*b** 4*c*d*x**8 + 15*a**3*b**4*d**2*x**10 - 60*a**2*b**5*c**2*x**8 - 54*a**2*b* *5*c*d*x**10 + 6*a**2*b**5*d**2*x**12 - 24*a*b**6*c**2*x**10 - 23*a*b**6*c *d*x**12 + a*b**6*d**2*x**14 - 4*b**7*c**2*x**12 - 4*b**7*c*d*x**14),x)*a* *8*d**5 - 25*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**4)/(a**7*c*d + a**7 *d**2*x**2 - 4*a**6*b*c**2 + 2*a**6*b*c*d*x**2 + 6*a**6*b*d**2*x**4 - 24*a **5*b**2*c**2*x**2 - 9*a**5*b**2*c*d*x**4 + 15*a**5*b**2*d**2*x**6 - 60*a* *4*b**3*c**2*x**4 - 40*a**4*b**3*c*d*x**6 + 20*a**4*b**3*d**2*x**8 - 80*a* *3*b**4*c**2*x**6 - 65*a**3*b**4*c*d*x**8 + 15*a**3*b**4*d**2*x**10 - 60*a **2*b**5*c**2*x**8 - 54*a**2*b**5*c*d*x**10 + 6*a**2*b**5*d**2*x**12 - 24* a*b**6*c**2*x**10 - 23*a*b**6*c*d*x**12 + a*b**6*d**2*x**14 - 4*b**7*c**2* x**12 - 4*b**7*c*d*x**14),x)*a**7*b*c*d**4 + 40*int((sqrt(c + d*x**2)*sqrt (a + b*x**2)*x**4)/(a**7*c*d + a**7*d**2*x**2 - 4*a**6*b*c**2 + 2*a**6*b*c *d*x**2 + 6*a**6*b*d**2*x**4 - 24*a**5*b**2*c**2*x**2 - 9*a**5*b**2*c*d...