Integrand size = 23, antiderivative size = 325 \[ \int \sqrt {a+b x^2} \left (c+d x^2\right )^{3/2} \, dx=\frac {\left (7 a c+\frac {3 b c^2}{d}-\frac {2 a^2 d}{b}\right ) x \sqrt {c+d x^2}}{15 \sqrt {a+b x^2}}+\frac {(3 b c+a d) x \sqrt {a+b x^2} \sqrt {c+d x^2}}{15 b}+\frac {1}{5} x \sqrt {a+b x^2} \left (c+d x^2\right )^{3/2}-\frac {\sqrt {a} \left (3 b^2 c^2+7 a b c d-2 a^2 d^2\right ) \sqrt {c+d x^2} E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{15 b^{3/2} d \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}+\frac {a^{3/2} (9 b c-a d) \sqrt {c+d x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{15 b^{3/2} \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}} \] Output:
1/15*(7*a*c+3*b*c^2/d-2*a^2*d/b)*x*(d*x^2+c)^(1/2)/(b*x^2+a)^(1/2)+1/15*(a *d+3*b*c)*x*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/b+1/5*x*(b*x^2+a)^(1/2)*(d*x^2 +c)^(3/2)-1/15*a^(1/2)*(-2*a^2*d^2+7*a*b*c*d+3*b^2*c^2)*(d*x^2+c)^(1/2)*El lipticE(b^(1/2)*x/a^(1/2)/(1+b*x^2/a)^(1/2),(1-a*d/b/c)^(1/2))/b^(3/2)/d/( b*x^2+a)^(1/2)/(a*(d*x^2+c)/c/(b*x^2+a))^(1/2)+1/15*a^(3/2)*(-a*d+9*b*c)*( d*x^2+c)^(1/2)*InverseJacobiAM(arctan(b^(1/2)*x/a^(1/2)),(1-a*d/b/c)^(1/2) )/b^(3/2)/(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 = 2.19 (sec) , antiderivative size = 246, normalized size of antiderivative = 0.76 \[ \int \sqrt {a+b x^2} \left (c+d x^2\right )^{3/2} \, dx=\frac {\sqrt {\frac {b}{a}} d x \left (a+b x^2\right ) \left (c+d x^2\right ) \left (6 b c+a d+3 b d x^2\right )+i c \left (-3 b^2 c^2-7 a b c d+2 a^2 d^2\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )-i c \left (-3 b^2 c^2+2 a b c d+a^2 d^2\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )}{15 b \sqrt {\frac {b}{a}} d \sqrt {a+b x^2} \sqrt {c+d x^2}} \] Input:
Integrate[Sqrt[a + b*x^2]*(c + d*x^2)^(3/2),x]
Output:
(Sqrt[b/a]*d*x*(a + b*x^2)*(c + d*x^2)*(6*b*c + a*d + 3*b*d*x^2) + I*c*(-3 *b^2*c^2 - 7*a*b*c*d + 2*a^2*d^2)*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]* EllipticE[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] - I*c*(-3*b^2*c^2 + 2*a*b*c *d + a^2*d^2)*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticF[I*ArcSinh[ Sqrt[b/a]*x], (a*d)/(b*c)])/(15*b*Sqrt[b/a]*d*Sqrt[a + b*x^2]*Sqrt[c + d*x ^2])
Time = 0.46 (sec) , antiderivative size = 314, normalized size of antiderivative = 0.97, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {318, 403, 27, 406, 320, 388, 313}
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.
| \(\displaystyle \int \sqrt {a+b x^2} \left (c+d x^2\right )^{3/2} \, dx\) |
| \(\Big \downarrow \) 318 |
| \(\displaystyle \frac {\int \frac {\sqrt {b x^2+a} \left (2 d (3 b c-a d) x^2+c (5 b c-a d)\right )}{\sqrt {d x^2+c}}dx}{5 b}+\frac {d x \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{5 b}\) |
| \(\Big \downarrow \) 403 |
| \(\displaystyle \frac {\frac {\int \frac {d \left (\left (3 b^2 c^2+7 a b d c-2 a^2 d^2\right ) x^2+a c (9 b c-a d)\right )}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{3 d}+\frac {2}{3} x \sqrt {a+b x^2} \sqrt {c+d x^2} (3 b c-a d)}{5 b}+\frac {d x \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{5 b}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{3} \int \frac {\left (3 b^2 c^2+7 a b d c-2 a^2 d^2\right ) x^2+a c (9 b c-a d)}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx+\frac {2}{3} x \sqrt {a+b x^2} \sqrt {c+d x^2} (3 b c-a d)}{5 b}+\frac {d x \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{5 b}\) |
| \(\Big \downarrow \) 406 |
| \(\displaystyle \frac {\frac {1}{3} \left (\left (-2 a^2 d^2+7 a b c d+3 b^2 c^2\right ) \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx+a c (9 b c-a d) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx\right )+\frac {2}{3} x \sqrt {a+b x^2} \sqrt {c+d x^2} (3 b c-a d)}{5 b}+\frac {d x \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{5 b}\) |
| \(\Big \downarrow \) 320 |
| \(\displaystyle \frac {\frac {1}{3} \left (\left (-2 a^2 d^2+7 a b c d+3 b^2 c^2\right ) \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx+\frac {c^{3/2} \sqrt {a+b x^2} (9 b c-a d) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )+\frac {2}{3} x \sqrt {a+b x^2} \sqrt {c+d x^2} (3 b c-a d)}{5 b}+\frac {d x \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{5 b}\) |
| \(\Big \downarrow \) 388 |
| \(\displaystyle \frac {\frac {1}{3} \left (\left (-2 a^2 d^2+7 a b c d+3 b^2 c^2\right ) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {c \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{3/2}}dx}{b}\right )+\frac {c^{3/2} \sqrt {a+b x^2} (9 b c-a d) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )+\frac {2}{3} x \sqrt {a+b x^2} \sqrt {c+d x^2} (3 b c-a d)}{5 b}+\frac {d x \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{5 b}\) |
| \(\Big \downarrow \) 313 |
| \(\displaystyle \frac {\frac {1}{3} \left (\left (-2 a^2 d^2+7 a b c d+3 b^2 c^2\right ) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )+\frac {c^{3/2} \sqrt {a+b x^2} (9 b c-a d) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )+\frac {2}{3} x \sqrt {a+b x^2} \sqrt {c+d x^2} (3 b c-a d)}{5 b}+\frac {d x \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{5 b}\) |
Input:
Int[Sqrt[a + b*x^2]*(c + d*x^2)^(3/2),x]
Output:
(d*x*(a + b*x^2)^(3/2)*Sqrt[c + d*x^2])/(5*b) + ((2*(3*b*c - a*d)*x*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/3 + ((3*b^2*c^2 + 7*a*b*c*d - 2*a^2*d^2)*((x*Sq rt[a + b*x^2])/(b*Sqrt[c + d*x^2]) - (Sqrt[c]*Sqrt[a + b*x^2]*EllipticE[Ar cTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(b*Sqrt[d]*Sqrt[(c*(a + b*x^2 ))/(a*(c + d*x^2))]*Sqrt[c + d*x^2])) + (c^(3/2)*(9*b*c - a*d)*Sqrt[a + b* x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(Sqrt[d]*Sqr t[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]))/3)/(5*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[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[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[(x_)^2/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[x*(Sqrt[a + b*x^2]/(b*Sqrt[c + d*x^2])), x] - Simp[c/b Int[Sqrt[ a + b*x^2]/(c + d*x^2)^(3/2), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && PosQ[b/a] && PosQ[d/c] && !SimplerSqrtQ[b/a, d/c]
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]
Int[((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*( x_)^2), x_Symbol] :> Simp[e Int[(a + b*x^2)^p*(c + d*x^2)^q, x], x] + Sim p[f Int[x^2*(a + b*x^2)^p*(c + d*x^2)^q, x], x] /; FreeQ[{a, b, c, d, e, f, p, q}, x]
Time = 3.46 (sec) , antiderivative size = 411, normalized size of antiderivative = 1.26
| method | result | size |
| risch | \(\frac {x \left (3 b d \,x^{2}+a d +6 b c \right ) \sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}{15 b}-\frac {\left (-\frac {\left (2 a^{2} d^{2}-7 a b c d -3 b^{2} c^{2}\right ) c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}\, d}+\frac {a^{2} c d \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}-\frac {9 b \,c^{2} a \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}\right ) \sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}}{15 b \sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\) | \(411\) |
| elliptic | \(\frac {\sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}\, \left (\frac {d \,x^{3} \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}{5}+\frac {\left (a \,d^{2}+2 b c d -\frac {d \left (4 a d +4 b c \right )}{5}\right ) x \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}{3 b d}+\frac {\left (a \,c^{2}-\frac {\left (a \,d^{2}+2 b c d -\frac {d \left (4 a d +4 b c \right )}{5}\right ) a c}{3 b d}\right ) \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}-\frac {\left (\frac {7 a c d}{5}+b \,c^{2}-\frac {\left (a \,d^{2}+2 b c d -\frac {d \left (4 a d +4 b c \right )}{5}\right ) \left (2 a d +2 b c \right )}{3 b d}\right ) c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}\, d}\right )}{\sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\) | \(423\) |
| default | \(\frac {\sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}\, \left (3 \sqrt {-\frac {b}{a}}\, b^{2} d^{3} x^{7}+4 \sqrt {-\frac {b}{a}}\, a b \,d^{3} x^{5}+9 \sqrt {-\frac {b}{a}}\, b^{2} c \,d^{2} x^{5}+\sqrt {-\frac {b}{a}}\, a^{2} d^{3} x^{3}+10 \sqrt {-\frac {b}{a}}\, a b c \,d^{2} x^{3}+6 \sqrt {-\frac {b}{a}}\, b^{2} c^{2} d \,x^{3}+\sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a^{2} c \,d^{2}+2 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a b \,c^{2} d -3 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b^{2} c^{3}-2 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a^{2} c \,d^{2}+7 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a b \,c^{2} d +3 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b^{2} c^{3}+\sqrt {-\frac {b}{a}}\, a^{2} c \,d^{2} x +6 \sqrt {-\frac {b}{a}}\, a b \,c^{2} d x \right )}{15 d \left (b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c \right ) b \sqrt {-\frac {b}{a}}}\) | \(545\) |
Input:
int((b*x^2+a)^(1/2)*(d*x^2+c)^(3/2),x,method=_RETURNVERBOSE)
Output:
1/15*x*(3*b*d*x^2+a*d+6*b*c)*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/b-1/15/b*(-(2 *a^2*d^2-7*a*b*c*d-3*b^2*c^2)*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)/d*(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))) +a^2*c*d/(-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))-9*b *c^2*a/(-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)))*((b* x^2+a)*(d*x^2+c))^(1/2)/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)
Time = 0.10 (sec) , antiderivative size = 233, normalized size of antiderivative = 0.72 \[ \int \sqrt {a+b x^2} \left (c+d x^2\right )^{3/2} \, dx=-\frac {{\left (3 \, b^{2} c^{3} + 7 \, a b c^{2} d - 2 \, a^{2} c d^{2}\right )} \sqrt {b d} x \sqrt {-\frac {c}{d}} E(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a d}{b c}) - {\left (3 \, b^{2} c^{3} + 7 \, a b c^{2} d - a^{2} d^{3} - {\left (2 \, a^{2} - 9 \, a b\right )} c d^{2}\right )} \sqrt {b d} x \sqrt {-\frac {c}{d}} F(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a d}{b c}) - {\left (3 \, b^{2} d^{3} x^{4} + 3 \, b^{2} c^{2} d + 7 \, a b c d^{2} - 2 \, a^{2} d^{3} + {\left (6 \, b^{2} c d^{2} + a b d^{3}\right )} x^{2}\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c}}{15 \, b^{2} d^{2} x} \] Input:
integrate((b*x^2+a)^(1/2)*(d*x^2+c)^(3/2),x, algorithm="fricas")
Output:
-1/15*((3*b^2*c^3 + 7*a*b*c^2*d - 2*a^2*c*d^2)*sqrt(b*d)*x*sqrt(-c/d)*elli ptic_e(arcsin(sqrt(-c/d)/x), a*d/(b*c)) - (3*b^2*c^3 + 7*a*b*c^2*d - a^2*d ^3 - (2*a^2 - 9*a*b)*c*d^2)*sqrt(b*d)*x*sqrt(-c/d)*elliptic_f(arcsin(sqrt( -c/d)/x), a*d/(b*c)) - (3*b^2*d^3*x^4 + 3*b^2*c^2*d + 7*a*b*c*d^2 - 2*a^2* d^3 + (6*b^2*c*d^2 + a*b*d^3)*x^2)*sqrt(b*x^2 + a)*sqrt(d*x^2 + c))/(b^2*d ^2*x)
\[ \int \sqrt {a+b x^2} \left (c+d x^2\right )^{3/2} \, dx=\int \sqrt {a + b x^{2}} \left (c + d x^{2}\right )^{\frac {3}{2}}\, dx \] Input:
integrate((b*x**2+a)**(1/2)*(d*x**2+c)**(3/2),x)
Output:
Integral(sqrt(a + b*x**2)*(c + d*x**2)**(3/2), x)
\[ \int \sqrt {a+b x^2} \left (c+d x^2\right )^{3/2} \, dx=\int { \sqrt {b x^{2} + a} {\left (d x^{2} + c\right )}^{\frac {3}{2}} \,d x } \] Input:
integrate((b*x^2+a)^(1/2)*(d*x^2+c)^(3/2),x, algorithm="maxima")
Output:
integrate(sqrt(b*x^2 + a)*(d*x^2 + c)^(3/2), x)
\[ \int \sqrt {a+b x^2} \left (c+d x^2\right )^{3/2} \, dx=\int { \sqrt {b x^{2} + a} {\left (d x^{2} + c\right )}^{\frac {3}{2}} \,d x } \] Input:
integrate((b*x^2+a)^(1/2)*(d*x^2+c)^(3/2),x, algorithm="giac")
Output:
integrate(sqrt(b*x^2 + a)*(d*x^2 + c)^(3/2), x)
Timed out. \[ \int \sqrt {a+b x^2} \left (c+d x^2\right )^{3/2} \, dx=\int \sqrt {b\,x^2+a}\,{\left (d\,x^2+c\right )}^{3/2} \,d x \] Input:
int((a + b*x^2)^(1/2)*(c + d*x^2)^(3/2),x)
Output:
int((a + b*x^2)^(1/2)*(c + d*x^2)^(3/2), x)
\[ \int \sqrt {a+b x^2} \left (c+d x^2\right )^{3/2} \, dx=\frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, a d x +6 \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, b c x +3 \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, b d \,x^{3}-2 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x^{2}}{b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}d x \right ) a^{2} d^{2}+7 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x^{2}}{b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}d x \right ) a b c d +3 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x^{2}}{b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}d x \right ) b^{2} c^{2}-\left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}}{b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}d x \right ) a^{2} c d +9 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}}{b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}d x \right ) a b \,c^{2}}{15 b} \] Input:
int((b*x^2+a)^(1/2)*(d*x^2+c)^(3/2),x)
Output:
(sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*d*x + 6*sqrt(c + d*x**2)*sqrt(a + b*x **2)*b*c*x + 3*sqrt(c + d*x**2)*sqrt(a + b*x**2)*b*d*x**3 - 2*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**2)/(a*c + a*d*x**2 + b*c*x**2 + b*d*x**4),x) *a**2*d**2 + 7*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**2)/(a*c + a*d*x** 2 + b*c*x**2 + b*d*x**4),x)*a*b*c*d + 3*int((sqrt(c + d*x**2)*sqrt(a + b*x **2)*x**2)/(a*c + a*d*x**2 + b*c*x**2 + b*d*x**4),x)*b**2*c**2 - int((sqrt (c + d*x**2)*sqrt(a + b*x**2))/(a*c + a*d*x**2 + b*c*x**2 + b*d*x**4),x)*a **2*c*d + 9*int((sqrt(c + d*x**2)*sqrt(a + b*x**2))/(a*c + a*d*x**2 + b*c* x**2 + b*d*x**4),x)*a*b*c**2)/(15*b)