Integrand size = 25, antiderivative size = 438 \[ \int \frac {x^3}{\sqrt {c+d x} \left (a-b x^2\right )^{5/2}} \, dx=\frac {a (c-d x) \sqrt {c+d x}}{3 b \left (b c^2-a d^2\right ) \left (a-b x^2\right )^{3/2}}-\frac {\sqrt {c+d x} \left (2 c \left (3 b c^2-a d^2\right )-d \left (7 b c^2-3 a d^2\right ) x\right )}{6 b \left (b c^2-a d^2\right )^2 \sqrt {a-b x^2}}+\frac {\sqrt {a} d \left (7 b c^2-3 a d^2\right ) \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{6 b^{3/2} \left (b c^2-a d^2\right )^2 \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}-\frac {\sqrt {a} c d \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{6 b^{3/2} \left (b c^2-a d^2\right ) \sqrt {c+d x} \sqrt {a-b x^2}} \] Output:
1/3*a*(-d*x+c)*(d*x+c)^(1/2)/b/(-a*d^2+b*c^2)/(-b*x^2+a)^(3/2)-1/6*(d*x+c) ^(1/2)*(2*c*(-a*d^2+3*b*c^2)-d*(-3*a*d^2+7*b*c^2)*x)/b/(-a*d^2+b*c^2)^2/(- b*x^2+a)^(1/2)+1/6*a^(1/2)*d*(-3*a*d^2+7*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))/b^(3/2)/(-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)-1/6*a^(1/2)*c*d*(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) )/b^(3/2)/(-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 = 23.58 (sec) , antiderivative size = 576, normalized size of antiderivative = 1.32 \[ \int \frac {x^3}{\sqrt {c+d x} \left (a-b x^2\right )^{5/2}} \, dx=\frac {\sqrt {a-b x^2} \left (\frac {b (c+d x) \left (-a^2 d^3 x+b^2 c^2 x^2 (6 c-7 d x)+a b \left (-4 c^3+5 c^2 d x-2 c d^2 x^2+3 d^3 x^3\right )\right )}{\left (a-b x^2\right )^2}+\frac {d^2 \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} \left (7 b c^2-3 a d^2\right ) \left (a-b x^2\right )+i \sqrt {b} \left (7 b^{3/2} c^3-7 \sqrt {a} b c^2 d-3 a \sqrt {b} c d^2+3 a^{3/2} d^3\right ) \sqrt {\frac {d \left (\frac {\sqrt {a}}{\sqrt {b}}+x\right )}{c+d x}} \sqrt {-\frac {\frac {\sqrt {a} d}{\sqrt {b}}-d x}{c+d x}} (c+d x)^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}}{\sqrt {c+d x}}\right )|\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )-i \sqrt {b} \left (6 b^{3/2} c^3-7 \sqrt {a} b c^2 d-2 a \sqrt {b} c d^2+3 a^{3/2} d^3\right ) \sqrt {\frac {d \left (\frac {\sqrt {a}}{\sqrt {b}}+x\right )}{c+d x}} \sqrt {-\frac {\frac {\sqrt {a} d}{\sqrt {b}}-d x}{c+d x}} (c+d x)^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}}{\sqrt {c+d x}}\right ),\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )}{\sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} \left (a-b x^2\right )}\right )}{6 b^2 \left (b c^2-a d^2\right )^2 \sqrt {c+d x}} \] Input:
Integrate[x^3/(Sqrt[c + d*x]*(a - b*x^2)^(5/2)),x]
Output:
(Sqrt[a - b*x^2]*((b*(c + d*x)*(-(a^2*d^3*x) + b^2*c^2*x^2*(6*c - 7*d*x) + a*b*(-4*c^3 + 5*c^2*d*x - 2*c*d^2*x^2 + 3*d^3*x^3)))/(a - b*x^2)^2 + (d^2 *Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*(7*b*c^2 - 3*a*d^2)*(a - b*x^2) + I*Sqrt[b ]*(7*b^(3/2)*c^3 - 7*Sqrt[a]*b*c^2*d - 3*a*Sqrt[b]*c*d^2 + 3*a^(3/2)*d^3)* 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)/S qrt[b]]/Sqrt[c + d*x]], (Sqrt[b]*c + Sqrt[a]*d)/(Sqrt[b]*c - Sqrt[a]*d)] - I*Sqrt[b]*(6*b^(3/2)*c^3 - 7*Sqrt[a]*b*c^2*d - 2*a*Sqrt[b]*c*d^2 + 3*a^(3 /2)*d^3)*Sqrt[(d*(Sqrt[a]/Sqrt[b] + x))/(c + d*x)]*Sqrt[-(((Sqrt[a]*d)/Sqr t[b] - d*x)/(c + d*x))]*(c + d*x)^(3/2)*EllipticF[I*ArcSinh[Sqrt[-c + (Sqr t[a]*d)/Sqrt[b]]/Sqrt[c + d*x]], (Sqrt[b]*c + Sqrt[a]*d)/(Sqrt[b]*c - Sqrt [a]*d)])/(Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*(a - b*x^2))))/(6*b^2*(b*c^2 - a* d^2)^2*Sqrt[c + d*x])
Time = 0.67 (sec) , antiderivative size = 441, normalized size of antiderivative = 1.01, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.440, Rules used = {602, 27, 686, 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.
\(\displaystyle \int \frac {x^3}{\left (a-b x^2\right )^{5/2} \sqrt {c+d x}} \, dx\) |
\(\Big \downarrow \) 602 |
\(\displaystyle \frac {\int \frac {a \left (a c d-3 \left (2 b c^2-a d^2\right ) x\right )}{2 b \sqrt {c+d x} \left (a-b x^2\right )^{3/2}}dx}{3 a \left (b c^2-a d^2\right )}+\frac {a \sqrt {c+d x} (c-d x)}{3 b \left (a-b x^2\right )^{3/2} \left (b c^2-a d^2\right )}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int \frac {a c d-3 \left (2 b c^2-a d^2\right ) x}{\sqrt {c+d x} \left (a-b x^2\right )^{3/2}}dx}{6 b \left (b c^2-a d^2\right )}+\frac {a \sqrt {c+d x} (c-d x)}{3 b \left (a-b x^2\right )^{3/2} \left (b c^2-a d^2\right )}\) |
\(\Big \downarrow \) 686 |
\(\displaystyle \frac {-\frac {\int \frac {a b d \left (2 c \left (3 b c^2-a d^2\right )+d \left (7 b c^2-3 a d^2\right ) x\right )}{2 \sqrt {c+d x} \sqrt {a-b x^2}}dx}{a b \left (b c^2-a d^2\right )}-\frac {\sqrt {c+d x} \left (2 c \left (3 b c^2-a d^2\right )-d x \left (7 b c^2-3 a d^2\right )\right )}{\sqrt {a-b x^2} \left (b c^2-a d^2\right )}}{6 b \left (b c^2-a d^2\right )}+\frac {a \sqrt {c+d x} (c-d x)}{3 b \left (a-b x^2\right )^{3/2} \left (b c^2-a d^2\right )}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {-\frac {d \int \frac {2 c \left (3 b c^2-a d^2\right )+d \left (7 b c^2-3 a d^2\right ) x}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{2 \left (b c^2-a d^2\right )}-\frac {\sqrt {c+d x} \left (2 c \left (3 b c^2-a d^2\right )-d x \left (7 b c^2-3 a d^2\right )\right )}{\sqrt {a-b x^2} \left (b c^2-a d^2\right )}}{6 b \left (b c^2-a d^2\right )}+\frac {a \sqrt {c+d x} (c-d x)}{3 b \left (a-b x^2\right )^{3/2} \left (b c^2-a d^2\right )}\) |
\(\Big \downarrow \) 600 |
\(\displaystyle \frac {-\frac {d \left (\left (7 b c^2-3 a d^2\right ) \int \frac {\sqrt {c+d x}}{\sqrt {a-b x^2}}dx-c \left (b c^2-a d^2\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx\right )}{2 \left (b c^2-a d^2\right )}-\frac {\sqrt {c+d x} \left (2 c \left (3 b c^2-a d^2\right )-d x \left (7 b c^2-3 a d^2\right )\right )}{\sqrt {a-b x^2} \left (b c^2-a d^2\right )}}{6 b \left (b c^2-a d^2\right )}+\frac {a \sqrt {c+d x} (c-d x)}{3 b \left (a-b x^2\right )^{3/2} \left (b c^2-a d^2\right )}\) |
\(\Big \downarrow \) 509 |
\(\displaystyle \frac {-\frac {d \left (\frac {\sqrt {1-\frac {b x^2}{a}} \left (7 b c^2-3 a d^2\right ) \int \frac {\sqrt {c+d x}}{\sqrt {1-\frac {b x^2}{a}}}dx}{\sqrt {a-b x^2}}-c \left (b c^2-a d^2\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx\right )}{2 \left (b c^2-a d^2\right )}-\frac {\sqrt {c+d x} \left (2 c \left (3 b c^2-a d^2\right )-d x \left (7 b c^2-3 a d^2\right )\right )}{\sqrt {a-b x^2} \left (b c^2-a d^2\right )}}{6 b \left (b c^2-a d^2\right )}+\frac {a \sqrt {c+d x} (c-d x)}{3 b \left (a-b x^2\right )^{3/2} \left (b c^2-a d^2\right )}\) |
\(\Big \downarrow \) 508 |
\(\displaystyle \frac {-\frac {d \left (-c \left (b c^2-a d^2\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx-\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (7 b c^2-3 a d^2\right ) \int \frac {\sqrt {1-\frac {d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\frac {\sqrt {b} c}{\sqrt {a}}+d}}}{\sqrt {\frac {1}{2} \left (\frac {\sqrt {b} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}}{\sqrt {b} \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )}{2 \left (b c^2-a d^2\right )}-\frac {\sqrt {c+d x} \left (2 c \left (3 b c^2-a d^2\right )-d x \left (7 b c^2-3 a d^2\right )\right )}{\sqrt {a-b x^2} \left (b c^2-a d^2\right )}}{6 b \left (b c^2-a d^2\right )}+\frac {a \sqrt {c+d x} (c-d x)}{3 b \left (a-b x^2\right )^{3/2} \left (b c^2-a d^2\right )}\) |
\(\Big \downarrow \) 327 |
\(\displaystyle \frac {-\frac {d \left (-c \left (b c^2-a d^2\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx-\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (7 b c^2-3 a d^2\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )}{2 \left (b c^2-a d^2\right )}-\frac {\sqrt {c+d x} \left (2 c \left (3 b c^2-a d^2\right )-d x \left (7 b c^2-3 a d^2\right )\right )}{\sqrt {a-b x^2} \left (b c^2-a d^2\right )}}{6 b \left (b c^2-a d^2\right )}+\frac {a \sqrt {c+d x} (c-d x)}{3 b \left (a-b x^2\right )^{3/2} \left (b c^2-a d^2\right )}\) |
\(\Big \downarrow \) 512 |
\(\displaystyle \frac {-\frac {d \left (-\frac {c \sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}}}dx}{\sqrt {a-b x^2}}-\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (7 b c^2-3 a d^2\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )}{2 \left (b c^2-a d^2\right )}-\frac {\sqrt {c+d x} \left (2 c \left (3 b c^2-a d^2\right )-d x \left (7 b c^2-3 a d^2\right )\right )}{\sqrt {a-b x^2} \left (b c^2-a d^2\right )}}{6 b \left (b c^2-a d^2\right )}+\frac {a \sqrt {c+d x} (c-d x)}{3 b \left (a-b x^2\right )^{3/2} \left (b c^2-a d^2\right )}\) |
\(\Big \downarrow \) 511 |
\(\displaystyle \frac {-\frac {d \left (\frac {2 \sqrt {a} c \sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \int \frac {1}{\sqrt {1-\frac {d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\frac {\sqrt {b} c}{\sqrt {a}}+d}} \sqrt {\frac {1}{2} \left (\frac {\sqrt {b} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}}{\sqrt {b} \sqrt {a-b x^2} \sqrt {c+d x}}-\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (7 b c^2-3 a d^2\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )}{2 \left (b c^2-a d^2\right )}-\frac {\sqrt {c+d x} \left (2 c \left (3 b c^2-a d^2\right )-d x \left (7 b c^2-3 a d^2\right )\right )}{\sqrt {a-b x^2} \left (b c^2-a d^2\right )}}{6 b \left (b c^2-a d^2\right )}+\frac {a \sqrt {c+d x} (c-d x)}{3 b \left (a-b x^2\right )^{3/2} \left (b c^2-a d^2\right )}\) |
\(\Big \downarrow \) 321 |
\(\displaystyle \frac {-\frac {d \left (\frac {2 \sqrt {a} c \sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} \sqrt {a-b x^2} \sqrt {c+d x}}-\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (7 b c^2-3 a d^2\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )}{2 \left (b c^2-a d^2\right )}-\frac {\sqrt {c+d x} \left (2 c \left (3 b c^2-a d^2\right )-d x \left (7 b c^2-3 a d^2\right )\right )}{\sqrt {a-b x^2} \left (b c^2-a d^2\right )}}{6 b \left (b c^2-a d^2\right )}+\frac {a \sqrt {c+d x} (c-d x)}{3 b \left (a-b x^2\right )^{3/2} \left (b c^2-a d^2\right )}\) |
Input:
Int[x^3/(Sqrt[c + d*x]*(a - b*x^2)^(5/2)),x]
Output:
(a*(c - d*x)*Sqrt[c + d*x])/(3*b*(b*c^2 - a*d^2)*(a - b*x^2)^(3/2)) + (-(( Sqrt[c + d*x]*(2*c*(3*b*c^2 - a*d^2) - d*(7*b*c^2 - 3*a*d^2)*x))/((b*c^2 - a*d^2)*Sqrt[a - b*x^2])) - (d*((-2*Sqrt[a]*(7*b*c^2 - 3*a*d^2)*Sqrt[c + d *x]*Sqrt[1 - (b*x^2)/a]*EllipticE[ArcSin[Sqrt[1 - (Sqrt[b]*x)/Sqrt[a]]/Sqr t[2]], (2*d)/((Sqrt[b]*c)/Sqrt[a] + d)])/(Sqrt[b]*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]*Elli pticF[ArcSin[Sqrt[1 - (Sqrt[b]*x)/Sqrt[a]]/Sqrt[2]], (2*d)/((Sqrt[b]*c)/Sq rt[a] + d)])/(Sqrt[b]*Sqrt[c + d*x]*Sqrt[a - b*x^2])))/(2*(b*c^2 - a*d^2)) )/(6*b*(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[((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[(x_)^(m_)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo l] :> With[{Qx = PolynomialQuotient[x^m, a + b*x^2, x], e = Coeff[Polynomia lRemainder[x^m, a + b*x^2, x], x, 0], f = Coeff[PolynomialRemainder[x^m, a + b*x^2, x], x, 1]}, Simp[(-(c + d*x)^(n + 1))*(a + b*x^2)^(p + 1)*((a*(d*e - c*f) + (b*c*e + a*d*f)*x)/(2*a*(p + 1)*(b*c^2 + a*d^2))), x] + Simp[1/(2 *a*(p + 1)*(b*c^2 + a*d^2)) Int[(c + d*x)^n*(a + b*x^2)^(p + 1)*ExpandToS um[2*a*(p + 1)*(b*c^2 + a*d^2)*Qx + e*(b*c^2*(2*p + 3) + a*d^2*(n + 2*p + 3 )) - a*c*d*f*n + d*(b*c*e + a*d*f)*(n + 2*p + 4)*x, x], x], x]] /; FreeQ[{a , b, c, d, n}, x] && IGtQ[m, 1] && LtQ[p, -1] && NeQ[b*c^2 + a*d^2, 0]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p _), x_Symbol] :> Simp[(-(d + e*x)^(m + 1))*(f*a*c*e - a*g*c*d + c*(c*d*f + a*e*g)*x)*((a + c*x^2)^(p + 1)/(2*a*c*(p + 1)*(c*d^2 + a*e^2))), x] + Simp[ 1/(2*a*c*(p + 1)*(c*d^2 + a*e^2)) Int[(d + e*x)^m*(a + c*x^2)^(p + 1)*Sim p[f*(c^2*d^2*(2*p + 3) + a*c*e^2*(m + 2*p + 3)) - a*c*d*e*g*m + c*e*(c*d*f + a*e*g)*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g}, x] && LtQ [p, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
Leaf count of result is larger than twice the leaf count of optimal. \(841\) vs. \(2(368)=736\).
Time = 7.48 (sec) , antiderivative size = 842, normalized size of antiderivative = 1.92
method | result | size |
elliptic | \(\frac {\sqrt {\left (d x +c \right ) \left (-b \,x^{2}+a \right )}\, \left (\frac {\left (\frac {d a x}{3 \left (a \,d^{2}-b \,c^{2}\right ) b^{3}}-\frac {c a}{3 \left (a \,d^{2}-b \,c^{2}\right ) b^{3}}\right ) \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}{\left (x^{2}-\frac {a}{b}\right )^{2}}-\frac {2 \left (-b d x -b c \right ) \left (-\frac {d \left (3 a \,d^{2}-7 b \,c^{2}\right ) x}{12 b^{2} \left (a \,d^{2}-b \,c^{2}\right )^{2}}+\frac {c \left (a \,d^{2}-3 b \,c^{2}\right )}{6 b^{2} \left (a \,d^{2}-b \,c^{2}\right )^{2}}\right )}{\sqrt {\left (x^{2}-\frac {a}{b}\right ) \left (-b d x -b c \right )}}+\frac {2 \left (-\frac {d c}{6 \left (a \,d^{2}-b \,c^{2}\right ) b}-\frac {d c \left (a \,d^{2}-3 b \,c^{2}\right )}{6 b \left (a \,d^{2}-b \,c^{2}\right )^{2}}+\frac {c d \left (3 a \,d^{2}-7 b \,c^{2}\right )}{6 b \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 {d^{2} \left (3 a \,d^{2}-7 b \,c^{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}}}\, \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 )}{6 b \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}}\) | \(842\) |
default | \(\text {Expression too large to display}\) | \(2459\) |
Input:
int(x^3/(d*x+c)^(1/2)/(-b*x^2+a)^(5/2),x,method=_RETURNVERBOSE)
Output:
((d*x+c)*(-b*x^2+a))^(1/2)/(d*x+c)^(1/2)/(-b*x^2+a)^(1/2)*((1/3*d/(a*d^2-b *c^2)/b^3*a*x-1/3*c/(a*d^2-b*c^2)/b^3*a)*(-b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2 )/(x^2-a/b)^2-2*(-b*d*x-b*c)*(-1/12*d*(3*a*d^2-7*b*c^2)/b^2/(a*d^2-b*c^2)^ 2*x+1/6*c*(a*d^2-3*b*c^2)/b^2/(a*d^2-b*c^2)^2)/((x^2-a/b)*(-b*d*x-b*c))^(1 /2)+2*(-1/6*d*c/(a*d^2-b*c^2)/b-1/6/b*d*c*(a*d^2-3*b*c^2)/(a*d^2-b*c^2)^2+ 1/6/b*c*d*(3*a*d^2-7*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))+1/6*d^2*(3*a*d^2-7*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)*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))))
Time = 0.11 (sec) , antiderivative size = 508, normalized size of antiderivative = 1.16 \[ \int \frac {x^3}{\sqrt {c+d x} \left (a-b x^2\right )^{5/2}} \, dx=\frac {{\left (11 \, a^{2} b c^{3} - 3 \, a^{3} c d^{2} + {\left (11 \, b^{3} c^{3} - 3 \, a b^{2} c d^{2}\right )} x^{4} - 2 \, {\left (11 \, a b^{2} c^{3} - 3 \, a^{2} b c d^{2}\right )} x^{2}\right )} \sqrt {-b d} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b c^{2} + 3 \, a d^{2}\right )}}{3 \, b d^{2}}, -\frac {8 \, {\left (b c^{3} - 9 \, a c d^{2}\right )}}{27 \, b d^{3}}, \frac {3 \, d x + c}{3 \, d}\right ) - 3 \, {\left (7 \, a^{2} b c^{2} d - 3 \, a^{3} d^{3} + {\left (7 \, b^{3} c^{2} d - 3 \, a b^{2} d^{3}\right )} x^{4} - 2 \, {\left (7 \, a b^{2} c^{2} d - 3 \, a^{2} b d^{3}\right )} x^{2}\right )} \sqrt {-b d} {\rm weierstrassZeta}\left (\frac {4 \, {\left (b c^{2} + 3 \, a d^{2}\right )}}{3 \, b d^{2}}, -\frac {8 \, {\left (b c^{3} - 9 \, a c d^{2}\right )}}{27 \, b d^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b c^{2} + 3 \, a d^{2}\right )}}{3 \, b d^{2}}, -\frac {8 \, {\left (b c^{3} - 9 \, a c d^{2}\right )}}{27 \, b d^{3}}, \frac {3 \, d x + c}{3 \, d}\right )\right ) - 3 \, {\left (4 \, a b^{2} c^{3} + {\left (7 \, b^{3} c^{2} d - 3 \, a b^{2} d^{3}\right )} x^{3} - 2 \, {\left (3 \, b^{3} c^{3} - a b^{2} c d^{2}\right )} x^{2} - {\left (5 \, a b^{2} c^{2} d - a^{2} b d^{3}\right )} x\right )} \sqrt {-b x^{2} + a} \sqrt {d x + c}}{18 \, {\left (a^{2} b^{4} c^{4} - 2 \, a^{3} b^{3} c^{2} d^{2} + a^{4} b^{2} d^{4} + {\left (b^{6} c^{4} - 2 \, a b^{5} c^{2} d^{2} + a^{2} b^{4} d^{4}\right )} x^{4} - 2 \, {\left (a b^{5} c^{4} - 2 \, a^{2} b^{4} c^{2} d^{2} + a^{3} b^{3} d^{4}\right )} x^{2}\right )}} \] Input:
integrate(x^3/(d*x+c)^(1/2)/(-b*x^2+a)^(5/2),x, algorithm="fricas")
Output:
1/18*((11*a^2*b*c^3 - 3*a^3*c*d^2 + (11*b^3*c^3 - 3*a*b^2*c*d^2)*x^4 - 2*( 11*a*b^2*c^3 - 3*a^2*b*c*d^2)*x^2)*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*(7*a^2*b*c^2*d - 3*a^3*d^3 + (7*b^3*c^2*d - 3*a*b^2*d^3)*x^4 - 2*(7 *a*b^2*c^2*d - 3*a^2*b*d^3)*x^2)*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*(4*a*b^2*c^3 + (7*b^3*c^2*d - 3*a*b^2*d^3)*x^3 - 2*(3*b^3*c^ 3 - a*b^2*c*d^2)*x^2 - (5*a*b^2*c^2*d - a^2*b*d^3)*x)*sqrt(-b*x^2 + a)*sqr t(d*x + c))/(a^2*b^4*c^4 - 2*a^3*b^3*c^2*d^2 + a^4*b^2*d^4 + (b^6*c^4 - 2* a*b^5*c^2*d^2 + a^2*b^4*d^4)*x^4 - 2*(a*b^5*c^4 - 2*a^2*b^4*c^2*d^2 + a^3* b^3*d^4)*x^2)
\[ \int \frac {x^3}{\sqrt {c+d x} \left (a-b x^2\right )^{5/2}} \, dx=\int \frac {x^{3}}{\left (a - b x^{2}\right )^{\frac {5}{2}} \sqrt {c + d x}}\, dx \] Input:
integrate(x**3/(d*x+c)**(1/2)/(-b*x**2+a)**(5/2),x)
Output:
Integral(x**3/((a - b*x**2)**(5/2)*sqrt(c + d*x)), x)
\[ \int \frac {x^3}{\sqrt {c+d x} \left (a-b x^2\right )^{5/2}} \, dx=\int { \frac {x^{3}}{{\left (-b x^{2} + a\right )}^{\frac {5}{2}} \sqrt {d x + c}} \,d x } \] Input:
integrate(x^3/(d*x+c)^(1/2)/(-b*x^2+a)^(5/2),x, algorithm="maxima")
Output:
integrate(x^3/((-b*x^2 + a)^(5/2)*sqrt(d*x + c)), x)
\[ \int \frac {x^3}{\sqrt {c+d x} \left (a-b x^2\right )^{5/2}} \, dx=\int { \frac {x^{3}}{{\left (-b x^{2} + a\right )}^{\frac {5}{2}} \sqrt {d x + c}} \,d x } \] Input:
integrate(x^3/(d*x+c)^(1/2)/(-b*x^2+a)^(5/2),x, algorithm="giac")
Output:
integrate(x^3/((-b*x^2 + a)^(5/2)*sqrt(d*x + c)), x)
Timed out. \[ \int \frac {x^3}{\sqrt {c+d x} \left (a-b x^2\right )^{5/2}} \, dx=\int \frac {x^3}{{\left (a-b\,x^2\right )}^{5/2}\,\sqrt {c+d\,x}} \,d x \] Input:
int(x^3/((a - b*x^2)^(5/2)*(c + d*x)^(1/2)),x)
Output:
int(x^3/((a - b*x^2)^(5/2)*(c + d*x)^(1/2)), x)
\[ \int \frac {x^3}{\sqrt {c+d x} \left (a-b x^2\right )^{5/2}} \, dx=\text {too large to display} \] Input:
int(x^3/(d*x+c)^(1/2)/(-b*x^2+a)^(5/2),x)
Output:
(sqrt(a - b*x**2)*int(sqrt(c + d*x)/(sqrt(a - b*x**2)*a**2*c**2 - sqrt(a - b*x**2)*a**2*d**2*x**2 - 2*sqrt(a - b*x**2)*a*b*c**2*x**2 + 2*sqrt(a - b* x**2)*a*b*d**2*x**4 + sqrt(a - b*x**2)*b**2*c**2*x**4 - sqrt(a - b*x**2)*b **2*d**2*x**6),x)*a**4*c*d**3 - 4*sqrt(a - b*x**2)*int(sqrt(c + d*x)/(sqrt (a - b*x**2)*a**2*c**2 - sqrt(a - b*x**2)*a**2*d**2*x**2 - 2*sqrt(a - b*x* *2)*a*b*c**2*x**2 + 2*sqrt(a - b*x**2)*a*b*d**2*x**4 + sqrt(a - b*x**2)*b* *2*c**2*x**4 - sqrt(a - b*x**2)*b**2*d**2*x**6),x)*a**3*b*c**3*d - 2*sqrt( a - b*x**2)*int(sqrt(c + d*x)/(sqrt(a - b*x**2)*a**2*c**2 - sqrt(a - b*x** 2)*a**2*d**2*x**2 - 2*sqrt(a - b*x**2)*a*b*c**2*x**2 + 2*sqrt(a - b*x**2)* a*b*d**2*x**4 + sqrt(a - b*x**2)*b**2*c**2*x**4 - sqrt(a - b*x**2)*b**2*d* *2*x**6),x)*a**3*b*c*d**3*x**2 + 8*sqrt(a - b*x**2)*int(sqrt(c + d*x)/(sqr t(a - b*x**2)*a**2*c**2 - sqrt(a - b*x**2)*a**2*d**2*x**2 - 2*sqrt(a - b*x **2)*a*b*c**2*x**2 + 2*sqrt(a - b*x**2)*a*b*d**2*x**4 + sqrt(a - b*x**2)*b **2*c**2*x**4 - sqrt(a - b*x**2)*b**2*d**2*x**6),x)*a**2*b**2*c**3*d*x**2 + sqrt(a - b*x**2)*int(sqrt(c + d*x)/(sqrt(a - b*x**2)*a**2*c**2 - sqrt(a - b*x**2)*a**2*d**2*x**2 - 2*sqrt(a - b*x**2)*a*b*c**2*x**2 + 2*sqrt(a - b *x**2)*a*b*d**2*x**4 + sqrt(a - b*x**2)*b**2*c**2*x**4 - sqrt(a - b*x**2)* b**2*d**2*x**6),x)*a**2*b**2*c*d**3*x**4 - 4*sqrt(a - b*x**2)*int(sqrt(c + d*x)/(sqrt(a - b*x**2)*a**2*c**2 - sqrt(a - b*x**2)*a**2*d**2*x**2 - 2*sq rt(a - b*x**2)*a*b*c**2*x**2 + 2*sqrt(a - b*x**2)*a*b*d**2*x**4 + sqrt(...