Integrand size = 23, antiderivative size = 407 \[ \int \frac {x}{(c+d x)^{5/2} \sqrt {a-b x^2}} \, dx=-\frac {2 c \sqrt {a-b x^2}}{3 \left (b c^2-a d^2\right ) (c+d x)^{3/2}}-\frac {2 \left (b c^2+3 a d^2\right ) \sqrt {a-b x^2}}{3 \left (b c^2-a d^2\right )^2 \sqrt {c+d x}}+\frac {2 \sqrt {a} \sqrt {b} \left (b c^2+3 a d^2\right ) \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{3 d \left (b c^2-a d^2\right )^2 \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}-\frac {2 \sqrt {a} \sqrt {b} c \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{3 d \left (b c^2-a d^2\right ) \sqrt {c+d x} \sqrt {a-b x^2}} \] Output:
-2/3*c*(-b*x^2+a)^(1/2)/(-a*d^2+b*c^2)/(d*x+c)^(3/2)-2/3*(3*a*d^2+b*c^2)*( -b*x^2+a)^(1/2)/(-a*d^2+b*c^2)^2/(d*x+c)^(1/2)+2/3*a^(1/2)*b^(1/2)*(3*a*d^ 2+b*c^2)*(d*x+c)^(1/2)*(1-b*x^2/a)^(1/2)*EllipticE(1/2*(1-b^(1/2)*x/a^(1/2 ))^(1/2)*2^(1/2),2^(1/2)*(a^(1/2)*d/(b^(1/2)*c+a^(1/2)*d))^(1/2))/d/(-a*d^ 2+b*c^2)^2/(b^(1/2)*(d*x+c)/(b^(1/2)*c+a^(1/2)*d))^(1/2)/(-b*x^2+a)^(1/2)- 2/3*a^(1/2)*b^(1/2)*c*(b^(1/2)*(d*x+c)/(b^(1/2)*c+a^(1/2)*d))^(1/2)*(1-b*x ^2/a)^(1/2)*EllipticF(1/2*(1-b^(1/2)*x/a^(1/2))^(1/2)*2^(1/2),2^(1/2)*(a^( 1/2)*d/(b^(1/2)*c+a^(1/2)*d))^(1/2))/d/(-a*d^2+b*c^2)/(d*x+c)^(1/2)/(-b*x^ 2+a)^(1/2)
Result contains complex when optimal does not.
Time = 12.13 (sec) , antiderivative size = 456, normalized size of antiderivative = 1.12 \[ \int \frac {x}{(c+d x)^{5/2} \sqrt {a-b x^2}} \, dx=\frac {2 \sqrt {a-b x^2} \left (\frac {-b c^3+a c d^2}{c+d x}+\frac {i b \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} \left (b c^2+3 a d^2\right ) \sqrt {\frac {d \left (\frac {\sqrt {a}}{\sqrt {b}}+x\right )}{c+d x}} \sqrt {-\frac {\frac {\sqrt {a} d}{\sqrt {b}}-d x}{c+d x}} (c+d x)^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}}{\sqrt {c+d x}}\right )|\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )}{d^2 \left (-a+b x^2\right )}-\frac {i \sqrt {a} \sqrt {b} \left (b c^2-4 \sqrt {a} \sqrt {b} c d+3 a d^2\right ) \sqrt {\frac {d \left (\frac {\sqrt {a}}{\sqrt {b}}+x\right )}{c+d x}} \sqrt {-\frac {\frac {\sqrt {a} d}{\sqrt {b}}-d x}{c+d x}} (c+d x)^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}}{\sqrt {c+d x}}\right ),\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )}{d \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} \left (-a+b x^2\right )}\right )}{3 \left (b c^2-a d^2\right )^2 \sqrt {c+d x}} \] Input:
Integrate[x/((c + d*x)^(5/2)*Sqrt[a - b*x^2]),x]
Output:
(2*Sqrt[a - b*x^2]*((-(b*c^3) + a*c*d^2)/(c + d*x) + (I*b*Sqrt[-c + (Sqrt[ a]*d)/Sqrt[b]]*(b*c^2 + 3*a*d^2)*Sqrt[(d*(Sqrt[a]/Sqrt[b] + x))/(c + d*x)] *Sqrt[-(((Sqrt[a]*d)/Sqrt[b] - d*x)/(c + d*x))]*(c + d*x)^(3/2)*EllipticE[ I*ArcSinh[Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]/Sqrt[c + d*x]], (Sqrt[b]*c + Sqrt [a]*d)/(Sqrt[b]*c - Sqrt[a]*d)])/(d^2*(-a + b*x^2)) - (I*Sqrt[a]*Sqrt[b]*( b*c^2 - 4*Sqrt[a]*Sqrt[b]*c*d + 3*a*d^2)*Sqrt[(d*(Sqrt[a]/Sqrt[b] + x))/(c + d*x)]*Sqrt[-(((Sqrt[a]*d)/Sqrt[b] - d*x)/(c + d*x))]*(c + d*x)^(3/2)*El lipticF[I*ArcSinh[Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]/Sqrt[c + d*x]], (Sqrt[b]* c + Sqrt[a]*d)/(Sqrt[b]*c - Sqrt[a]*d)])/(d*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]] *(-a + b*x^2))))/(3*(b*c^2 - a*d^2)^2*Sqrt[c + d*x])
Time = 0.59 (sec) , antiderivative size = 409, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.478, Rules used = {594, 27, 688, 27, 600, 509, 508, 327, 512, 511, 321}
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.
| \(\displaystyle \int \frac {x}{\sqrt {a-b x^2} (c+d x)^{5/2}} \, dx\) |
| \(\Big \downarrow \) 594 |
| \(\displaystyle \frac {2 \int -\frac {3 a d-b c x}{2 (c+d x)^{3/2} \sqrt {a-b x^2}}dx}{3 \left (b c^2-a d^2\right )}-\frac {2 c \sqrt {a-b x^2}}{3 (c+d x)^{3/2} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\int \frac {3 a d-b c x}{(c+d x)^{3/2} \sqrt {a-b x^2}}dx}{3 \left (b c^2-a d^2\right )}-\frac {2 c \sqrt {a-b x^2}}{3 (c+d x)^{3/2} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 688 |
| \(\displaystyle -\frac {\frac {2 \int \frac {b \left (4 a c d+\left (b c^2+3 a d^2\right ) x\right )}{2 \sqrt {c+d x} \sqrt {a-b x^2}}dx}{b c^2-a d^2}+\frac {2 \sqrt {a-b x^2} \left (3 a d^2+b c^2\right )}{\sqrt {c+d x} \left (b c^2-a d^2\right )}}{3 \left (b c^2-a d^2\right )}-\frac {2 c \sqrt {a-b x^2}}{3 (c+d x)^{3/2} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {b \int \frac {4 a c d+\left (b c^2+3 a d^2\right ) x}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{b c^2-a d^2}+\frac {2 \sqrt {a-b x^2} \left (3 a d^2+b c^2\right )}{\sqrt {c+d x} \left (b c^2-a d^2\right )}}{3 \left (b c^2-a d^2\right )}-\frac {2 c \sqrt {a-b x^2}}{3 (c+d x)^{3/2} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 600 |
| \(\displaystyle -\frac {\frac {b \left (\frac {\left (3 a d^2+b c^2\right ) \int \frac {\sqrt {c+d x}}{\sqrt {a-b x^2}}dx}{d}-\frac {c \left (b c^2-a d^2\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}\right )}{b c^2-a d^2}+\frac {2 \sqrt {a-b x^2} \left (3 a d^2+b c^2\right )}{\sqrt {c+d x} \left (b c^2-a d^2\right )}}{3 \left (b c^2-a d^2\right )}-\frac {2 c \sqrt {a-b x^2}}{3 (c+d x)^{3/2} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 509 |
| \(\displaystyle -\frac {\frac {b \left (\frac {\sqrt {1-\frac {b x^2}{a}} \left (3 a d^2+b c^2\right ) \int \frac {\sqrt {c+d x}}{\sqrt {1-\frac {b x^2}{a}}}dx}{d \sqrt {a-b x^2}}-\frac {c \left (b c^2-a d^2\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}\right )}{b c^2-a d^2}+\frac {2 \sqrt {a-b x^2} \left (3 a d^2+b c^2\right )}{\sqrt {c+d x} \left (b c^2-a d^2\right )}}{3 \left (b c^2-a d^2\right )}-\frac {2 c \sqrt {a-b x^2}}{3 (c+d x)^{3/2} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 508 |
| \(\displaystyle -\frac {\frac {b \left (-\frac {c \left (b c^2-a d^2\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}-\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (3 a d^2+b c^2\right ) \int \frac {\sqrt {1-\frac {d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\frac {\sqrt {b} c}{\sqrt {a}}+d}}}{\sqrt {\frac {1}{2} \left (\frac {\sqrt {b} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )}{b c^2-a d^2}+\frac {2 \sqrt {a-b x^2} \left (3 a d^2+b c^2\right )}{\sqrt {c+d x} \left (b c^2-a d^2\right )}}{3 \left (b c^2-a d^2\right )}-\frac {2 c \sqrt {a-b x^2}}{3 (c+d x)^{3/2} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle -\frac {\frac {b \left (-\frac {c \left (b c^2-a d^2\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}-\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (3 a d^2+b c^2\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )}{b c^2-a d^2}+\frac {2 \sqrt {a-b x^2} \left (3 a d^2+b c^2\right )}{\sqrt {c+d x} \left (b c^2-a d^2\right )}}{3 \left (b c^2-a d^2\right )}-\frac {2 c \sqrt {a-b x^2}}{3 (c+d x)^{3/2} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 512 |
| \(\displaystyle -\frac {\frac {b \left (-\frac {c \sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}}}dx}{d \sqrt {a-b x^2}}-\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (3 a d^2+b c^2\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )}{b c^2-a d^2}+\frac {2 \sqrt {a-b x^2} \left (3 a d^2+b c^2\right )}{\sqrt {c+d x} \left (b c^2-a d^2\right )}}{3 \left (b c^2-a d^2\right )}-\frac {2 c \sqrt {a-b x^2}}{3 (c+d x)^{3/2} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 511 |
| \(\displaystyle -\frac {\frac {b \left (\frac {2 \sqrt {a} c \sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \int \frac {1}{\sqrt {1-\frac {d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\frac {\sqrt {b} c}{\sqrt {a}}+d}} \sqrt {\frac {1}{2} \left (\frac {\sqrt {b} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {c+d x}}-\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (3 a d^2+b c^2\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )}{b c^2-a d^2}+\frac {2 \sqrt {a-b x^2} \left (3 a d^2+b c^2\right )}{\sqrt {c+d x} \left (b c^2-a d^2\right )}}{3 \left (b c^2-a d^2\right )}-\frac {2 c \sqrt {a-b x^2}}{3 (c+d x)^{3/2} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle -\frac {\frac {b \left (\frac {2 \sqrt {a} c \sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {c+d x}}-\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (3 a d^2+b c^2\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )}{b c^2-a d^2}+\frac {2 \sqrt {a-b x^2} \left (3 a d^2+b c^2\right )}{\sqrt {c+d x} \left (b c^2-a d^2\right )}}{3 \left (b c^2-a d^2\right )}-\frac {2 c \sqrt {a-b x^2}}{3 (c+d x)^{3/2} \left (b c^2-a d^2\right )}\) |
Input:
Int[x/((c + d*x)^(5/2)*Sqrt[a - b*x^2]),x]
Output:
(-2*c*Sqrt[a - b*x^2])/(3*(b*c^2 - a*d^2)*(c + d*x)^(3/2)) - ((2*(b*c^2 + 3*a*d^2)*Sqrt[a - b*x^2])/((b*c^2 - a*d^2)*Sqrt[c + d*x]) + (b*((-2*Sqrt[a ]*(b*c^2 + 3*a*d^2)*Sqrt[c + d*x]*Sqrt[1 - (b*x^2)/a]*EllipticE[ArcSin[Sqr t[1 - (Sqrt[b]*x)/Sqrt[a]]/Sqrt[2]], (2*d)/((Sqrt[b]*c)/Sqrt[a] + d)])/(Sq rt[b]*d*Sqrt[(Sqrt[b]*(c + d*x))/(Sqrt[b]*c + Sqrt[a]*d)]*Sqrt[a - b*x^2]) + (2*Sqrt[a]*c*(b*c^2 - a*d^2)*Sqrt[(Sqrt[b]*(c + d*x))/(Sqrt[b]*c + Sqrt [a]*d)]*Sqrt[1 - (b*x^2)/a]*EllipticF[ArcSin[Sqrt[1 - (Sqrt[b]*x)/Sqrt[a]] /Sqrt[2]], (2*d)/((Sqrt[b]*c)/Sqrt[a] + d)])/(Sqrt[b]*d*Sqrt[c + d*x]*Sqrt [a - b*x^2])))/(b*c^2 - a*d^2))/(3*(b*c^2 - a*d^2))
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S imp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]))*EllipticF[ArcSin[Rt[-d/c, 2]*x], b*(c /(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0] && !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])
Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ (Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) )], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
Int[Sqrt[(c_) + (d_.)*(x_)]/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> With[{q = Rt[-b/a, 2]}, Simp[-2*(Sqrt[c + d*x]/(Sqrt[a]*q*Sqrt[q*((c + d*x)/(d + c *q))])) Subst[Int[Sqrt[1 - 2*d*(x^2/(d + c*q))]/Sqrt[1 - x^2], x], x, Sqr t[(1 - q*x)/2]], x]] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] && GtQ[a, 0]
Int[Sqrt[(c_) + (d_.)*(x_)]/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[Sq rt[1 + b*(x^2/a)]/Sqrt[a + b*x^2] Int[Sqrt[c + d*x]/Sqrt[1 + b*(x^2/a)], x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] && !GtQ[a, 0]
Int[1/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> Wit h[{q = Rt[-b/a, 2]}, Simp[-2*(Sqrt[q*((c + d*x)/(d + c*q))]/(Sqrt[a]*q*Sqrt [c + d*x])) Subst[Int[1/(Sqrt[1 - 2*d*(x^2/(d + c*q))]*Sqrt[1 - x^2]), x] , x, Sqrt[(1 - q*x)/2]], x]] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] && GtQ[ a, 0]
Int[1/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> Sim p[Sqrt[1 + b*(x^2/a)]/Sqrt[a + b*x^2] Int[1/(Sqrt[c + d*x]*Sqrt[1 + b*(x^ 2/a)]), x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] && !GtQ[a, 0]
Int[(x_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-c)*(c + d*x)^(n + 1)*((a + b*x^2)^(p + 1)/((n + 1)*(b*c^2 + a*d^2))) , x] + Simp[1/((n + 1)*(b*c^2 + a*d^2)) Int[(c + d*x)^(n + 1)*(a + b*x^2) ^p*(a*d*(n + 1) + b*c*(n + 2*p + 3)*x), x], x] /; FreeQ[{a, b, c, d, p}, x] && LtQ[n, -1] && NeQ[b*c^2 + a*d^2, 0]
Int[((A_.) + (B_.)*(x_))/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2] ), x_Symbol] :> Simp[B/d Int[Sqrt[c + d*x]/Sqrt[a + b*x^2], x], x] - Simp [(B*c - A*d)/d Int[1/(Sqrt[c + d*x]*Sqrt[a + b*x^2]), x], x] /; FreeQ[{a, b, c, d, A, B}, x] && NegQ[b/a]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p _.), x_Symbol] :> Simp[(e*f - d*g)*(d + e*x)^(m + 1)*((a + c*x^2)^(p + 1)/( (m + 1)*(c*d^2 + a*e^2))), x] + Simp[1/((m + 1)*(c*d^2 + a*e^2)) Int[(d + e*x)^(m + 1)*(a + c*x^2)^p*Simp[(c*d*f + a*e*g)*(m + 1) - c*(e*f - d*g)*(m + 2*p + 3)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g, p}, x] && LtQ[m, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
Leaf count of result is larger than twice the leaf count of optimal. \(734\) vs. \(2(337)=674\).
Time = 6.07 (sec) , antiderivative size = 735, normalized size of antiderivative = 1.81
| method | result | size |
| elliptic | \(\frac {\sqrt {\left (d x +c \right ) \left (-b \,x^{2}+a \right )}\, \left (\frac {2 c \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}{3 d^{2} \left (a \,d^{2}-b \,c^{2}\right ) \left (x +\frac {c}{d}\right )^{2}}-\frac {2 \left (-b d \,x^{2}+a d \right ) \left (3 a \,d^{2}+b \,c^{2}\right )}{3 d \left (a \,d^{2}-b \,c^{2}\right )^{2} \sqrt {\left (x +\frac {c}{d}\right ) \left (-b d \,x^{2}+a d \right )}}+\frac {2 \left (-\frac {b c}{3 d \left (a \,d^{2}-b \,c^{2}\right )}-\frac {b c \left (3 a \,d^{2}+b \,c^{2}\right )}{3 d \left (a \,d^{2}-b \,c^{2}\right )^{2}}\right ) \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right ) \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x -\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x +\frac {\sqrt {a b}}{b}}{-\frac {c}{d}+\frac {\sqrt {a b}}{b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}, \sqrt {\frac {-\frac {c}{d}+\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\right )}{\sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}-\frac {2 \left (3 a \,d^{2}+b \,c^{2}\right ) b \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right ) \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x -\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x +\frac {\sqrt {a b}}{b}}{-\frac {c}{d}+\frac {\sqrt {a b}}{b}}}\, \left (\left (-\frac {c}{d}-\frac {\sqrt {a b}}{b}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}, \sqrt {\frac {-\frac {c}{d}+\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\right )+\frac {\sqrt {a b}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}, \sqrt {\frac {-\frac {c}{d}+\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\right )}{b}\right )}{3 \left (a \,d^{2}-b \,c^{2}\right )^{2} \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}\right )}{\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}\) | \(735\) |
| default | \(\text {Expression too large to display}\) | \(2119\) |
Input:
int(x/(d*x+c)^(5/2)/(-b*x^2+a)^(1/2),x,method=_RETURNVERBOSE)
Output:
((d*x+c)*(-b*x^2+a))^(1/2)/(d*x+c)^(1/2)/(-b*x^2+a)^(1/2)*(2/3/d^2/(a*d^2- b*c^2)*c*(-b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)/(x+c/d)^2-2/3*(-b*d*x^2+a*d)/d /(a*d^2-b*c^2)^2*(3*a*d^2+b*c^2)/((x+c/d)*(-b*d*x^2+a*d))^(1/2)+2*(-1/3*b* c/d/(a*d^2-b*c^2)-1/3*b*c/d*(3*a*d^2+b*c^2)/(a*d^2-b*c^2)^2)*(c/d-1/b*(a*b )^(1/2))*((x+c/d)/(c/d-1/b*(a*b)^(1/2)))^(1/2)*((x-1/b*(a*b)^(1/2))/(-c/d- 1/b*(a*b)^(1/2)))^(1/2)*((x+1/b*(a*b)^(1/2))/(-c/d+1/b*(a*b)^(1/2)))^(1/2) /(-b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)*EllipticF(((x+c/d)/(c/d-1/b*(a*b)^(1/2 )))^(1/2),((-c/d+1/b*(a*b)^(1/2))/(-c/d-1/b*(a*b)^(1/2)))^(1/2))-2/3*(3*a* d^2+b*c^2)*b/(a*d^2-b*c^2)^2*(c/d-1/b*(a*b)^(1/2))*((x+c/d)/(c/d-1/b*(a*b) ^(1/2)))^(1/2)*((x-1/b*(a*b)^(1/2))/(-c/d-1/b*(a*b)^(1/2)))^(1/2)*((x+1/b* (a*b)^(1/2))/(-c/d+1/b*(a*b)^(1/2)))^(1/2)/(-b*d*x^3-b*c*x^2+a*d*x+a*c)^(1 /2)*((-c/d-1/b*(a*b)^(1/2))*EllipticE(((x+c/d)/(c/d-1/b*(a*b)^(1/2)))^(1/2 ),((-c/d+1/b*(a*b)^(1/2))/(-c/d-1/b*(a*b)^(1/2)))^(1/2))+1/b*(a*b)^(1/2)*E llipticF(((x+c/d)/(c/d-1/b*(a*b)^(1/2)))^(1/2),((-c/d+1/b*(a*b)^(1/2))/(-c /d-1/b*(a*b)^(1/2)))^(1/2))))
Time = 0.12 (sec) , antiderivative size = 424, normalized size of antiderivative = 1.04 \[ \int \frac {x}{(c+d x)^{5/2} \sqrt {a-b x^2}} \, dx=-\frac {2 \, {\left ({\left (b c^{5} - 9 \, a c^{3} d^{2} + {\left (b c^{3} d^{2} - 9 \, a c d^{4}\right )} x^{2} + 2 \, {\left (b c^{4} d - 9 \, a c^{2} d^{3}\right )} x\right )} \sqrt {-b d} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b c^{2} + 3 \, a d^{2}\right )}}{3 \, b d^{2}}, -\frac {8 \, {\left (b c^{3} - 9 \, a c d^{2}\right )}}{27 \, b d^{3}}, \frac {3 \, d x + c}{3 \, d}\right ) + 3 \, {\left (b c^{4} d + 3 \, a c^{2} d^{3} + {\left (b c^{2} d^{3} + 3 \, a d^{5}\right )} x^{2} + 2 \, {\left (b c^{3} d^{2} + 3 \, a c d^{4}\right )} x\right )} \sqrt {-b d} {\rm weierstrassZeta}\left (\frac {4 \, {\left (b c^{2} + 3 \, a d^{2}\right )}}{3 \, b d^{2}}, -\frac {8 \, {\left (b c^{3} - 9 \, a c d^{2}\right )}}{27 \, b d^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b c^{2} + 3 \, a d^{2}\right )}}{3 \, b d^{2}}, -\frac {8 \, {\left (b c^{3} - 9 \, a c d^{2}\right )}}{27 \, b d^{3}}, \frac {3 \, d x + c}{3 \, d}\right )\right ) + 3 \, {\left (2 \, b c^{3} d^{2} + 2 \, a c d^{4} + {\left (b c^{2} d^{3} + 3 \, a d^{5}\right )} x\right )} \sqrt {-b x^{2} + a} \sqrt {d x + c}\right )}}{9 \, {\left (b^{2} c^{6} d^{2} - 2 \, a b c^{4} d^{4} + a^{2} c^{2} d^{6} + {\left (b^{2} c^{4} d^{4} - 2 \, a b c^{2} d^{6} + a^{2} d^{8}\right )} x^{2} + 2 \, {\left (b^{2} c^{5} d^{3} - 2 \, a b c^{3} d^{5} + a^{2} c d^{7}\right )} x\right )}} \] Input:
integrate(x/(d*x+c)^(5/2)/(-b*x^2+a)^(1/2),x, algorithm="fricas")
Output:
-2/9*((b*c^5 - 9*a*c^3*d^2 + (b*c^3*d^2 - 9*a*c*d^4)*x^2 + 2*(b*c^4*d - 9* a*c^2*d^3)*x)*sqrt(-b*d)*weierstrassPInverse(4/3*(b*c^2 + 3*a*d^2)/(b*d^2) , -8/27*(b*c^3 - 9*a*c*d^2)/(b*d^3), 1/3*(3*d*x + c)/d) + 3*(b*c^4*d + 3*a *c^2*d^3 + (b*c^2*d^3 + 3*a*d^5)*x^2 + 2*(b*c^3*d^2 + 3*a*c*d^4)*x)*sqrt(- b*d)*weierstrassZeta(4/3*(b*c^2 + 3*a*d^2)/(b*d^2), -8/27*(b*c^3 - 9*a*c*d ^2)/(b*d^3), weierstrassPInverse(4/3*(b*c^2 + 3*a*d^2)/(b*d^2), -8/27*(b*c ^3 - 9*a*c*d^2)/(b*d^3), 1/3*(3*d*x + c)/d)) + 3*(2*b*c^3*d^2 + 2*a*c*d^4 + (b*c^2*d^3 + 3*a*d^5)*x)*sqrt(-b*x^2 + a)*sqrt(d*x + c))/(b^2*c^6*d^2 - 2*a*b*c^4*d^4 + a^2*c^2*d^6 + (b^2*c^4*d^4 - 2*a*b*c^2*d^6 + a^2*d^8)*x^2 + 2*(b^2*c^5*d^3 - 2*a*b*c^3*d^5 + a^2*c*d^7)*x)
\[ \int \frac {x}{(c+d x)^{5/2} \sqrt {a-b x^2}} \, dx=\int \frac {x}{\sqrt {a - b x^{2}} \left (c + d x\right )^{\frac {5}{2}}}\, dx \] Input:
integrate(x/(d*x+c)**(5/2)/(-b*x**2+a)**(1/2),x)
Output:
Integral(x/(sqrt(a - b*x**2)*(c + d*x)**(5/2)), x)
\[ \int \frac {x}{(c+d x)^{5/2} \sqrt {a-b x^2}} \, dx=\int { \frac {x}{\sqrt {-b x^{2} + a} {\left (d x + c\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate(x/(d*x+c)^(5/2)/(-b*x^2+a)^(1/2),x, algorithm="maxima")
Output:
integrate(x/(sqrt(-b*x^2 + a)*(d*x + c)^(5/2)), x)
\[ \int \frac {x}{(c+d x)^{5/2} \sqrt {a-b x^2}} \, dx=\int { \frac {x}{\sqrt {-b x^{2} + a} {\left (d x + c\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate(x/(d*x+c)^(5/2)/(-b*x^2+a)^(1/2),x, algorithm="giac")
Output:
integrate(x/(sqrt(-b*x^2 + a)*(d*x + c)^(5/2)), x)
Timed out. \[ \int \frac {x}{(c+d x)^{5/2} \sqrt {a-b x^2}} \, dx=\int \frac {x}{\sqrt {a-b\,x^2}\,{\left (c+d\,x\right )}^{5/2}} \,d x \] Input:
int(x/((a - b*x^2)^(1/2)*(c + d*x)^(5/2)),x)
Output:
int(x/((a - b*x^2)^(1/2)*(c + d*x)^(5/2)), x)
\[ \int \frac {x}{(c+d x)^{5/2} \sqrt {a-b x^2}} \, dx=\int \frac {\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}\, x}{-b \,d^{3} x^{5}-3 b c \,d^{2} x^{4}+a \,d^{3} x^{3}-3 b \,c^{2} d \,x^{3}+3 a c \,d^{2} x^{2}-b \,c^{3} x^{2}+3 a \,c^{2} d x +a \,c^{3}}d x \] Input:
int(x/(d*x+c)^(5/2)/(-b*x^2+a)^(1/2),x)
Output:
int((sqrt(c + d*x)*sqrt(a - b*x**2)*x)/(a*c**3 + 3*a*c**2*d*x + 3*a*c*d**2 *x**2 + a*d**3*x**3 - b*c**3*x**2 - 3*b*c**2*d*x**3 - 3*b*c*d**2*x**4 - b* d**3*x**5),x)