Integrand size = 21, antiderivative size = 406 \[ \int \frac {(c+d x)^{3/2}}{\sqrt {a+\frac {b}{x^2}}} \, dx=\frac {2 c \sqrt {c+d x} \left (b+a x^2\right )}{5 a \sqrt {a+\frac {b}{x^2}} x}+\frac {2 (c+d x)^{3/2} \left (b+a x^2\right )}{5 a \sqrt {a+\frac {b}{x^2}} x}+\frac {2 \sqrt {b} \left (a c^2-3 b d^2\right ) \sqrt {c+d x} \sqrt {1+\frac {a x^2}{b}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {-a} x}{\sqrt {b}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {-a} \sqrt {b} d}{a c-\sqrt {-a} \sqrt {b} d}\right )}{5 (-a)^{3/2} d \sqrt {a+\frac {b}{x^2}} x \sqrt {\frac {a (c+d x)}{a c-\sqrt {-a} \sqrt {b} d}}}-\frac {2 \sqrt {b} c \left (a c^2+b d^2\right ) \sqrt {\frac {a (c+d x)}{a c-\sqrt {-a} \sqrt {b} d}} \sqrt {1+\frac {a x^2}{b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {-a} x}{\sqrt {b}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {-a} \sqrt {b} d}{a c-\sqrt {-a} \sqrt {b} d}\right )}{5 (-a)^{3/2} d \sqrt {a+\frac {b}{x^2}} x \sqrt {c+d x}} \]
2/5*(d*x+c)^(3/2)*(a*x^2+b)/a/x/(a+b/x^2)^(1/2)+2/5*c*(a*x^2+b)*(d*x+c)^(1 /2)/a/x/(a+b/x^2)^(1/2)+2/5*(a*c^2-3*b*d^2)*EllipticE(1/2*(1-x*(-a)^(1/2)/ b^(1/2))^(1/2)*2^(1/2),(-2*d*(-a)^(1/2)*b^(1/2)/(a*c-d*(-a)^(1/2)*b^(1/2)) )^(1/2))*b^(1/2)*(d*x+c)^(1/2)*(1+a*x^2/b)^(1/2)/(-a)^(3/2)/d/x/(a+b/x^2)^ (1/2)/(a*(d*x+c)/(a*c-d*(-a)^(1/2)*b^(1/2)))^(1/2)-2/5*c*(a*c^2+b*d^2)*Ell ipticF(1/2*(1-x*(-a)^(1/2)/b^(1/2))^(1/2)*2^(1/2),(-2*d*(-a)^(1/2)*b^(1/2) /(a*c-d*(-a)^(1/2)*b^(1/2)))^(1/2))*b^(1/2)*(1+a*x^2/b)^(1/2)*(a*(d*x+c)/( a*c-d*(-a)^(1/2)*b^(1/2)))^(1/2)/(-a)^(3/2)/d/x/(a+b/x^2)^(1/2)/(d*x+c)^(1 /2)
Result contains complex when optimal does not.
Time = 24.58 (sec) , antiderivative size = 525, normalized size of antiderivative = 1.29 \[ \int \frac {(c+d x)^{3/2}}{\sqrt {a+\frac {b}{x^2}}} \, dx=\frac {\sqrt {c+d x} \left (\frac {2 (2 c+d x) \left (b+a x^2\right )}{a}+\frac {2 \left (d^2 \sqrt {-c-\frac {i \sqrt {b} d}{\sqrt {a}}} \left (a c^2-3 b d^2\right ) \left (b+a x^2\right )+\sqrt {a} \left (-i a^{3/2} c^3+a \sqrt {b} c^2 d+3 i \sqrt {a} b c d^2-3 b^{3/2} d^3\right ) \sqrt {\frac {d \left (\frac {i \sqrt {b}}{\sqrt {a}}+x\right )}{c+d x}} \sqrt {-\frac {\frac {i \sqrt {b} d}{\sqrt {a}}-d x}{c+d x}} (c+d x)^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {-c-\frac {i \sqrt {b} d}{\sqrt {a}}}}{\sqrt {c+d x}}\right )|\frac {\sqrt {a} c-i \sqrt {b} d}{\sqrt {a} c+i \sqrt {b} d}\right )-\sqrt {a} \sqrt {b} d \left (a c^2+4 i \sqrt {a} \sqrt {b} c d-3 b d^2\right ) \sqrt {\frac {d \left (\frac {i \sqrt {b}}{\sqrt {a}}+x\right )}{c+d x}} \sqrt {-\frac {\frac {i \sqrt {b} d}{\sqrt {a}}-d x}{c+d x}} (c+d x)^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-c-\frac {i \sqrt {b} d}{\sqrt {a}}}}{\sqrt {c+d x}}\right ),\frac {\sqrt {a} c-i \sqrt {b} d}{\sqrt {a} c+i \sqrt {b} d}\right )\right )}{a^2 d^2 \sqrt {-c-\frac {i \sqrt {b} d}{\sqrt {a}}} (c+d x)}\right )}{5 \sqrt {a+\frac {b}{x^2}} x} \]
(Sqrt[c + d*x]*((2*(2*c + d*x)*(b + a*x^2))/a + (2*(d^2*Sqrt[-c - (I*Sqrt[ b]*d)/Sqrt[a]]*(a*c^2 - 3*b*d^2)*(b + a*x^2) + Sqrt[a]*((-I)*a^(3/2)*c^3 + a*Sqrt[b]*c^2*d + (3*I)*Sqrt[a]*b*c*d^2 - 3*b^(3/2)*d^3)*Sqrt[(d*((I*Sqrt [b])/Sqrt[a] + x))/(c + d*x)]*Sqrt[-(((I*Sqrt[b]*d)/Sqrt[a] - d*x)/(c + d* x))]*(c + d*x)^(3/2)*EllipticE[I*ArcSinh[Sqrt[-c - (I*Sqrt[b]*d)/Sqrt[a]]/ Sqrt[c + d*x]], (Sqrt[a]*c - I*Sqrt[b]*d)/(Sqrt[a]*c + I*Sqrt[b]*d)] - Sqr t[a]*Sqrt[b]*d*(a*c^2 + (4*I)*Sqrt[a]*Sqrt[b]*c*d - 3*b*d^2)*Sqrt[(d*((I*S qrt[b])/Sqrt[a] + x))/(c + d*x)]*Sqrt[-(((I*Sqrt[b]*d)/Sqrt[a] - d*x)/(c + d*x))]*(c + d*x)^(3/2)*EllipticF[I*ArcSinh[Sqrt[-c - (I*Sqrt[b]*d)/Sqrt[a ]]/Sqrt[c + d*x]], (Sqrt[a]*c - I*Sqrt[b]*d)/(Sqrt[a]*c + I*Sqrt[b]*d)]))/ (a^2*d^2*Sqrt[-c - (I*Sqrt[b]*d)/Sqrt[a]]*(c + d*x))))/(5*Sqrt[a + b/x^2]* x)
Time = 0.82 (sec) , antiderivative size = 719, normalized size of antiderivative = 1.77, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {1780, 596, 687, 27, 599, 25, 1511, 1416, 1509}
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.
| \(\displaystyle \int \frac {(c+d x)^{3/2}}{\sqrt {a+\frac {b}{x^2}}} \, dx\) |
| \(\Big \downarrow \) 1780 |
| \(\displaystyle \frac {\sqrt {a x^2+b} \int \frac {x (c+d x)^{3/2}}{\sqrt {a x^2+b}}dx}{x \sqrt {a+\frac {b}{x^2}}}\) |
| \(\Big \downarrow \) 596 |
| \(\displaystyle \frac {\sqrt {a x^2+b} \left (\frac {2 \sqrt {a x^2+b} (c+d x)^{3/2}}{5 a}-\frac {3 \int \frac {(b d-a c x) \sqrt {c+d x}}{\sqrt {a x^2+b}}dx}{5 a}\right )}{x \sqrt {a+\frac {b}{x^2}}}\) |
| \(\Big \downarrow \) 687 |
| \(\displaystyle \frac {\sqrt {a x^2+b} \left (\frac {2 \sqrt {a x^2+b} (c+d x)^{3/2}}{5 a}-\frac {3 \left (\frac {2 \int \frac {a \left (4 b c d-\left (a c^2-3 b d^2\right ) x\right )}{2 \sqrt {c+d x} \sqrt {a x^2+b}}dx}{3 a}-\frac {2}{3} c \sqrt {a x^2+b} \sqrt {c+d x}\right )}{5 a}\right )}{x \sqrt {a+\frac {b}{x^2}}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {a x^2+b} \left (\frac {2 \sqrt {a x^2+b} (c+d x)^{3/2}}{5 a}-\frac {3 \left (\frac {1}{3} \int \frac {4 b c d-\left (a c^2-3 b d^2\right ) x}{\sqrt {c+d x} \sqrt {a x^2+b}}dx-\frac {2}{3} c \sqrt {a x^2+b} \sqrt {c+d x}\right )}{5 a}\right )}{x \sqrt {a+\frac {b}{x^2}}}\) |
| \(\Big \downarrow \) 599 |
| \(\displaystyle \frac {\sqrt {a x^2+b} \left (\frac {2 \sqrt {a x^2+b} (c+d x)^{3/2}}{5 a}-\frac {3 \left (-\frac {2 \int -\frac {c \left (a c^2+b d^2\right )-\left (a c^2-3 b d^2\right ) (c+d x)}{\sqrt {\frac {a c^2}{d^2}-\frac {2 a (c+d x) c}{d^2}+\frac {a (c+d x)^2}{d^2}+b}}d\sqrt {c+d x}}{3 d^2}-\frac {2}{3} c \sqrt {a x^2+b} \sqrt {c+d x}\right )}{5 a}\right )}{x \sqrt {a+\frac {b}{x^2}}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\sqrt {a x^2+b} \left (\frac {2 \sqrt {a x^2+b} (c+d x)^{3/2}}{5 a}-\frac {3 \left (\frac {2 \int \frac {c \left (a c^2+b d^2\right )-\left (a c^2-3 b d^2\right ) (c+d x)}{\sqrt {\frac {a c^2}{d^2}-\frac {2 a (c+d x) c}{d^2}+\frac {a (c+d x)^2}{d^2}+b}}d\sqrt {c+d x}}{3 d^2}-\frac {2}{3} c \sqrt {a x^2+b} \sqrt {c+d x}\right )}{5 a}\right )}{x \sqrt {a+\frac {b}{x^2}}}\) |
| \(\Big \downarrow \) 1511 |
| \(\displaystyle \frac {\sqrt {a x^2+b} \left (\frac {2 \sqrt {a x^2+b} (c+d x)^{3/2}}{5 a}-\frac {3 \left (-\frac {2 \left (\frac {\sqrt {a c^2+b d^2} \left (-\sqrt {a} c \sqrt {a c^2+b d^2}+a c^2-3 b d^2\right ) \int \frac {1}{\sqrt {\frac {a c^2}{d^2}-\frac {2 a (c+d x) c}{d^2}+\frac {a (c+d x)^2}{d^2}+b}}d\sqrt {c+d x}}{\sqrt {a}}-\frac {\left (a c^2-3 b d^2\right ) \sqrt {a c^2+b d^2} \int \frac {1-\frac {\sqrt {a} (c+d x)}{\sqrt {a c^2+b d^2}}}{\sqrt {\frac {a c^2}{d^2}-\frac {2 a (c+d x) c}{d^2}+\frac {a (c+d x)^2}{d^2}+b}}d\sqrt {c+d x}}{\sqrt {a}}\right )}{3 d^2}-\frac {2}{3} c \sqrt {a x^2+b} \sqrt {c+d x}\right )}{5 a}\right )}{x \sqrt {a+\frac {b}{x^2}}}\) |
| \(\Big \downarrow \) 1416 |
| \(\displaystyle \frac {\sqrt {a x^2+b} \left (\frac {2 \sqrt {a x^2+b} (c+d x)^{3/2}}{5 a}-\frac {3 \left (-\frac {2 \left (\frac {\left (a c^2+b d^2\right )^{3/4} \left (-\sqrt {a} c \sqrt {a c^2+b d^2}+a c^2-3 b d^2\right ) \left (\frac {\sqrt {a} (c+d x)}{\sqrt {a c^2+b d^2}}+1\right ) \sqrt {\frac {\frac {a c^2}{d^2}-\frac {2 a c (c+d x)}{d^2}+\frac {a (c+d x)^2}{d^2}+b}{\left (\frac {a c^2}{d^2}+b\right ) \left (\frac {\sqrt {a} (c+d x)}{\sqrt {a c^2+b d^2}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{a} \sqrt {c+d x}}{\sqrt [4]{a c^2+b d^2}}\right ),\frac {1}{2} \left (\frac {\sqrt {a} c}{\sqrt {a c^2+b d^2}}+1\right )\right )}{2 a^{3/4} \sqrt {\frac {a c^2}{d^2}-\frac {2 a c (c+d x)}{d^2}+\frac {a (c+d x)^2}{d^2}+b}}-\frac {\left (a c^2-3 b d^2\right ) \sqrt {a c^2+b d^2} \int \frac {1-\frac {\sqrt {a} (c+d x)}{\sqrt {a c^2+b d^2}}}{\sqrt {\frac {a c^2}{d^2}-\frac {2 a (c+d x) c}{d^2}+\frac {a (c+d x)^2}{d^2}+b}}d\sqrt {c+d x}}{\sqrt {a}}\right )}{3 d^2}-\frac {2}{3} c \sqrt {a x^2+b} \sqrt {c+d x}\right )}{5 a}\right )}{x \sqrt {a+\frac {b}{x^2}}}\) |
| \(\Big \downarrow \) 1509 |
| \(\displaystyle \frac {\sqrt {a x^2+b} \left (\frac {2 \sqrt {a x^2+b} (c+d x)^{3/2}}{5 a}-\frac {3 \left (-\frac {2 \left (\frac {\left (a c^2+b d^2\right )^{3/4} \left (-\sqrt {a} c \sqrt {a c^2+b d^2}+a c^2-3 b d^2\right ) \left (\frac {\sqrt {a} (c+d x)}{\sqrt {a c^2+b d^2}}+1\right ) \sqrt {\frac {\frac {a c^2}{d^2}-\frac {2 a c (c+d x)}{d^2}+\frac {a (c+d x)^2}{d^2}+b}{\left (\frac {a c^2}{d^2}+b\right ) \left (\frac {\sqrt {a} (c+d x)}{\sqrt {a c^2+b d^2}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{a} \sqrt {c+d x}}{\sqrt [4]{a c^2+b d^2}}\right ),\frac {1}{2} \left (\frac {\sqrt {a} c}{\sqrt {a c^2+b d^2}}+1\right )\right )}{2 a^{3/4} \sqrt {\frac {a c^2}{d^2}-\frac {2 a c (c+d x)}{d^2}+\frac {a (c+d x)^2}{d^2}+b}}-\frac {\left (a c^2-3 b d^2\right ) \sqrt {a c^2+b d^2} \left (\frac {\sqrt [4]{a c^2+b d^2} \left (\frac {\sqrt {a} (c+d x)}{\sqrt {a c^2+b d^2}}+1\right ) \sqrt {\frac {\frac {a c^2}{d^2}-\frac {2 a c (c+d x)}{d^2}+\frac {a (c+d x)^2}{d^2}+b}{\left (\frac {a c^2}{d^2}+b\right ) \left (\frac {\sqrt {a} (c+d x)}{\sqrt {a c^2+b d^2}}+1\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{a} \sqrt {c+d x}}{\sqrt [4]{a c^2+b d^2}}\right )|\frac {1}{2} \left (\frac {\sqrt {a} c}{\sqrt {a c^2+b d^2}}+1\right )\right )}{\sqrt [4]{a} \sqrt {\frac {a c^2}{d^2}-\frac {2 a c (c+d x)}{d^2}+\frac {a (c+d x)^2}{d^2}+b}}-\frac {\sqrt {c+d x} \sqrt {\frac {a c^2}{d^2}-\frac {2 a c (c+d x)}{d^2}+\frac {a (c+d x)^2}{d^2}+b}}{\left (\frac {a c^2}{d^2}+b\right ) \left (\frac {\sqrt {a} (c+d x)}{\sqrt {a c^2+b d^2}}+1\right )}\right )}{\sqrt {a}}\right )}{3 d^2}-\frac {2}{3} c \sqrt {a x^2+b} \sqrt {c+d x}\right )}{5 a}\right )}{x \sqrt {a+\frac {b}{x^2}}}\) |
(Sqrt[b + a*x^2]*((2*(c + d*x)^(3/2)*Sqrt[b + a*x^2])/(5*a) - (3*((-2*c*Sq rt[c + d*x]*Sqrt[b + a*x^2])/3 - (2*(-(((a*c^2 - 3*b*d^2)*Sqrt[a*c^2 + b*d ^2]*(-((Sqrt[c + d*x]*Sqrt[b + (a*c^2)/d^2 - (2*a*c*(c + d*x))/d^2 + (a*(c + d*x)^2)/d^2])/((b + (a*c^2)/d^2)*(1 + (Sqrt[a]*(c + d*x))/Sqrt[a*c^2 + b*d^2]))) + ((a*c^2 + b*d^2)^(1/4)*(1 + (Sqrt[a]*(c + d*x))/Sqrt[a*c^2 + b *d^2])*Sqrt[(b + (a*c^2)/d^2 - (2*a*c*(c + d*x))/d^2 + (a*(c + d*x)^2)/d^2 )/((b + (a*c^2)/d^2)*(1 + (Sqrt[a]*(c + d*x))/Sqrt[a*c^2 + b*d^2])^2)]*Ell ipticE[2*ArcTan[(a^(1/4)*Sqrt[c + d*x])/(a*c^2 + b*d^2)^(1/4)], (1 + (Sqrt [a]*c)/Sqrt[a*c^2 + b*d^2])/2])/(a^(1/4)*Sqrt[b + (a*c^2)/d^2 - (2*a*c*(c + d*x))/d^2 + (a*(c + d*x)^2)/d^2])))/Sqrt[a]) + ((a*c^2 + b*d^2)^(3/4)*(a *c^2 - 3*b*d^2 - Sqrt[a]*c*Sqrt[a*c^2 + b*d^2])*(1 + (Sqrt[a]*(c + d*x))/S qrt[a*c^2 + b*d^2])*Sqrt[(b + (a*c^2)/d^2 - (2*a*c*(c + d*x))/d^2 + (a*(c + d*x)^2)/d^2)/((b + (a*c^2)/d^2)*(1 + (Sqrt[a]*(c + d*x))/Sqrt[a*c^2 + b* d^2])^2)]*EllipticF[2*ArcTan[(a^(1/4)*Sqrt[c + d*x])/(a*c^2 + b*d^2)^(1/4) ], (1 + (Sqrt[a]*c)/Sqrt[a*c^2 + b*d^2])/2])/(2*a^(3/4)*Sqrt[b + (a*c^2)/d ^2 - (2*a*c*(c + d*x))/d^2 + (a*(c + d*x)^2)/d^2])))/(3*d^2)))/(5*a)))/(Sq rt[a + b/x^2]*x)
3.7.1.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(x_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c + d*x)^n*((a + b*x^2)^(p + 1)/(b*(n + 2*p + 2))), x] - Simp[n/(b*(n + 2*p + 2)) Int[(c + d*x)^(n - 1)*(a + b*x^2)^p*(a*d - b*c*x), x], x] /; FreeQ[{a, b, c, d, p}, x] && GtQ[n, 0] && NeQ[n + 2*p + 2, 0]
Int[((A_.) + (B_.)*(x_))/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2] ), x_Symbol] :> Simp[-2/d^2 Subst[Int[(B*c - A*d - B*x^2)/Sqrt[(b*c^2 + a *d^2)/d^2 - 2*b*c*(x^2/d^2) + b*(x^4/d^2)], x], x, Sqrt[c + d*x]], x] /; Fr eeQ[{a, b, c, d, A, B}, x] && PosQ[b/a]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p _.), x_Symbol] :> Simp[g*(d + e*x)^m*((a + c*x^2)^(p + 1)/(c*(m + 2*p + 2)) ), x] + Simp[1/(c*(m + 2*p + 2)) Int[(d + e*x)^(m - 1)*(a + c*x^2)^p*Simp [c*d*f*(m + 2*p + 2) - a*e*g*m + c*(e*f*(m + 2*p + 2) + d*g*m)*x, x], x], x ] /; FreeQ[{a, c, d, e, f, g, p}, x] && GtQ[m, 0] && NeQ[m + 2*p + 2, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p]) && !(IGtQ[m, 0] && Eq Q[f, 0])
Int[1/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c /a, 4]}, Simp[(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2*x^2)^2)]/ (2*q*Sqrt[a + b*x^2 + c*x^4]))*EllipticF[2*ArcTan[q*x], 1/2 - b*(q^2/(4*c)) ], x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbo l] :> With[{q = Rt[c/a, 4]}, Simp[(-d)*x*(Sqrt[a + b*x^2 + c*x^4]/(a*(1 + q ^2*x^2))), x] + Simp[d*(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2* x^2)^2)]/(q*Sqrt[a + b*x^2 + c*x^4]))*EllipticE[2*ArcTan[q*x], 1/2 - b*(q^2 /(4*c))], x] /; EqQ[e + d*q^2, 0]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbo l] :> With[{q = Rt[c/a, 2]}, Simp[(e + d*q)/q Int[1/Sqrt[a + b*x^2 + c*x^ 4], x], x] - Simp[e/q Int[(1 - q*x^2)/Sqrt[a + b*x^2 + c*x^4], x], x] /; NeQ[e + d*q, 0]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && Pos Q[c/a]
Int[((a_.) + (c_.)*(x_)^(mn2_.))^(p_)*((d_) + (e_.)*(x_)^(n_.))^(q_.), x_Sy mbol] :> Simp[x^(2*n*FracPart[p])*((a + c/x^(2*n))^FracPart[p]/(c + a*x^(2* n))^FracPart[p]) Int[((d + e*x^n)^q*(c + a*x^(2*n))^p)/x^(2*n*p), x], x] /; FreeQ[{a, c, d, e, n, p, q}, x] && EqQ[mn2, -2*n] && !IntegerQ[p] && ! IntegerQ[q] && PosQ[n]
Time = 2.42 (sec) , antiderivative size = 623, normalized size of antiderivative = 1.53
| method | result | size |
| risch | \(\frac {2 \left (d x +2 c \right ) \left (a \,x^{2}+b \right ) \sqrt {d x +c}}{5 a \sqrt {\frac {a \,x^{2}+b}{x^{2}}}\, x}+\frac {\left (-\frac {8 b c d \left (\frac {c}{d}-\frac {\sqrt {-a b}}{a}\right ) \sqrt {\frac {\frac {c}{d}+x}{\frac {c}{d}-\frac {\sqrt {-a b}}{a}}}\, \sqrt {\frac {x -\frac {\sqrt {-a b}}{a}}{-\frac {c}{d}-\frac {\sqrt {-a b}}{a}}}\, \sqrt {\frac {x +\frac {\sqrt {-a b}}{a}}{-\frac {c}{d}+\frac {\sqrt {-a b}}{a}}}\, F\left (\sqrt {\frac {\frac {c}{d}+x}{\frac {c}{d}-\frac {\sqrt {-a b}}{a}}}, \sqrt {\frac {-\frac {c}{d}+\frac {\sqrt {-a b}}{a}}{-\frac {c}{d}-\frac {\sqrt {-a b}}{a}}}\right )}{\sqrt {a d \,x^{3}+a c \,x^{2}+b d x +b c}}+\frac {2 \left (a \,c^{2}-3 b \,d^{2}\right ) \left (\frac {c}{d}-\frac {\sqrt {-a b}}{a}\right ) \sqrt {\frac {\frac {c}{d}+x}{\frac {c}{d}-\frac {\sqrt {-a b}}{a}}}\, \sqrt {\frac {x -\frac {\sqrt {-a b}}{a}}{-\frac {c}{d}-\frac {\sqrt {-a b}}{a}}}\, \sqrt {\frac {x +\frac {\sqrt {-a b}}{a}}{-\frac {c}{d}+\frac {\sqrt {-a b}}{a}}}\, \left (\left (-\frac {c}{d}-\frac {\sqrt {-a b}}{a}\right ) E\left (\sqrt {\frac {\frac {c}{d}+x}{\frac {c}{d}-\frac {\sqrt {-a b}}{a}}}, \sqrt {\frac {-\frac {c}{d}+\frac {\sqrt {-a b}}{a}}{-\frac {c}{d}-\frac {\sqrt {-a b}}{a}}}\right )+\frac {\sqrt {-a b}\, F\left (\sqrt {\frac {\frac {c}{d}+x}{\frac {c}{d}-\frac {\sqrt {-a b}}{a}}}, \sqrt {\frac {-\frac {c}{d}+\frac {\sqrt {-a b}}{a}}{-\frac {c}{d}-\frac {\sqrt {-a b}}{a}}}\right )}{a}\right )}{\sqrt {a d \,x^{3}+a c \,x^{2}+b d x +b c}}\right ) \sqrt {\left (a \,x^{2}+b \right ) \left (d x +c \right )}}{5 a \sqrt {\frac {a \,x^{2}+b}{x^{2}}}\, x \sqrt {d x +c}}\) | \(623\) |
| default | \(\text {Expression too large to display}\) | \(1145\) |
2/5*(d*x+2*c)*(a*x^2+b)*(d*x+c)^(1/2)/a/((a*x^2+b)/x^2)^(1/2)/x+1/5/a*(-8* b*c*d*(c/d-1/a*(-a*b)^(1/2))*((c/d+x)/(c/d-1/a*(-a*b)^(1/2)))^(1/2)*((x-1/ a*(-a*b)^(1/2))/(-c/d-1/a*(-a*b)^(1/2)))^(1/2)*((x+1/a*(-a*b)^(1/2))/(-c/d +1/a*(-a*b)^(1/2)))^(1/2)/(a*d*x^3+a*c*x^2+b*d*x+b*c)^(1/2)*EllipticF(((c/ d+x)/(c/d-1/a*(-a*b)^(1/2)))^(1/2),((-c/d+1/a*(-a*b)^(1/2))/(-c/d-1/a*(-a* b)^(1/2)))^(1/2))+2*(a*c^2-3*b*d^2)*(c/d-1/a*(-a*b)^(1/2))*((c/d+x)/(c/d-1 /a*(-a*b)^(1/2)))^(1/2)*((x-1/a*(-a*b)^(1/2))/(-c/d-1/a*(-a*b)^(1/2)))^(1/ 2)*((x+1/a*(-a*b)^(1/2))/(-c/d+1/a*(-a*b)^(1/2)))^(1/2)/(a*d*x^3+a*c*x^2+b *d*x+b*c)^(1/2)*((-c/d-1/a*(-a*b)^(1/2))*EllipticE(((c/d+x)/(c/d-1/a*(-a*b )^(1/2)))^(1/2),((-c/d+1/a*(-a*b)^(1/2))/(-c/d-1/a*(-a*b)^(1/2)))^(1/2))+1 /a*(-a*b)^(1/2)*EllipticF(((c/d+x)/(c/d-1/a*(-a*b)^(1/2)))^(1/2),((-c/d+1/ a*(-a*b)^(1/2))/(-c/d-1/a*(-a*b)^(1/2)))^(1/2))))/((a*x^2+b)/x^2)^(1/2)/x* ((a*x^2+b)*(d*x+c))^(1/2)/(d*x+c)^(1/2)
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.09 (sec) , antiderivative size = 235, normalized size of antiderivative = 0.58 \[ \int \frac {(c+d x)^{3/2}}{\sqrt {a+\frac {b}{x^2}}} \, dx=-\frac {2 \, {\left ({\left (a c^{3} + 9 \, b c d^{2}\right )} \sqrt {a d} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (a c^{2} - 3 \, b d^{2}\right )}}{3 \, a d^{2}}, -\frac {8 \, {\left (a c^{3} + 9 \, b c d^{2}\right )}}{27 \, a d^{3}}, \frac {3 \, d x + c}{3 \, d}\right ) + 3 \, {\left (a c^{2} d - 3 \, b d^{3}\right )} \sqrt {a d} {\rm weierstrassZeta}\left (\frac {4 \, {\left (a c^{2} - 3 \, b d^{2}\right )}}{3 \, a d^{2}}, -\frac {8 \, {\left (a c^{3} + 9 \, b c d^{2}\right )}}{27 \, a d^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (a c^{2} - 3 \, b d^{2}\right )}}{3 \, a d^{2}}, -\frac {8 \, {\left (a c^{3} + 9 \, b c d^{2}\right )}}{27 \, a d^{3}}, \frac {3 \, d x + c}{3 \, d}\right )\right ) - 3 \, {\left (a d^{3} x^{2} + 2 \, a c d^{2} x\right )} \sqrt {d x + c} \sqrt {\frac {a x^{2} + b}{x^{2}}}\right )}}{15 \, a^{2} d^{2}} \]
-2/15*((a*c^3 + 9*b*c*d^2)*sqrt(a*d)*weierstrassPInverse(4/3*(a*c^2 - 3*b* d^2)/(a*d^2), -8/27*(a*c^3 + 9*b*c*d^2)/(a*d^3), 1/3*(3*d*x + c)/d) + 3*(a *c^2*d - 3*b*d^3)*sqrt(a*d)*weierstrassZeta(4/3*(a*c^2 - 3*b*d^2)/(a*d^2), -8/27*(a*c^3 + 9*b*c*d^2)/(a*d^3), weierstrassPInverse(4/3*(a*c^2 - 3*b*d ^2)/(a*d^2), -8/27*(a*c^3 + 9*b*c*d^2)/(a*d^3), 1/3*(3*d*x + c)/d)) - 3*(a *d^3*x^2 + 2*a*c*d^2*x)*sqrt(d*x + c)*sqrt((a*x^2 + b)/x^2))/(a^2*d^2)
\[ \int \frac {(c+d x)^{3/2}}{\sqrt {a+\frac {b}{x^2}}} \, dx=\int \frac {\left (c + d x\right )^{\frac {3}{2}}}{\sqrt {a + \frac {b}{x^{2}}}}\, dx \]
\[ \int \frac {(c+d x)^{3/2}}{\sqrt {a+\frac {b}{x^2}}} \, dx=\int { \frac {{\left (d x + c\right )}^{\frac {3}{2}}}{\sqrt {a + \frac {b}{x^{2}}}} \,d x } \]
\[ \int \frac {(c+d x)^{3/2}}{\sqrt {a+\frac {b}{x^2}}} \, dx=\int { \frac {{\left (d x + c\right )}^{\frac {3}{2}}}{\sqrt {a + \frac {b}{x^{2}}}} \,d x } \]
Timed out. \[ \int \frac {(c+d x)^{3/2}}{\sqrt {a+\frac {b}{x^2}}} \, dx=\int \frac {{\left (c+d\,x\right )}^{3/2}}{\sqrt {a+\frac {b}{x^2}}} \,d x \]