Integrand size = 23, antiderivative size = 340 \[ \int \frac {x (c+d x)^{5/2}}{\left (a-b x^2\right )^{3/2}} \, dx=\frac {(c+d x)^{5/2}}{b \sqrt {a-b x^2}}+\frac {5 d^2 \sqrt {c+d x} \sqrt {a-b x^2}}{3 b^2}+\frac {20 \sqrt {a} c d \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{3 b^{3/2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}-\frac {5 \sqrt {a} d \left (b c^2-a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{3 b^{5/2} \sqrt {c+d x} \sqrt {a-b x^2}} \] Output:
(d*x+c)^(5/2)/b/(-b*x^2+a)^(1/2)+5/3*d^2*(d*x+c)^(1/2)*(-b*x^2+a)^(1/2)/b^ 2+20/3*a^(1/2)*c*d*(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)/(b^(1/2)*(d*x+c)/(b^(1/2)*c+a^(1/2)*d))^(1/2)/(-b*x^2+a)^(1/2)-5 /3*a^(1/2)*d*(-a*d^2+b*c^2)*(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^(5/2)/(d*x+c)^(1/2)/(-b*x^2+a )^(1/2)
Result contains complex when optimal does not.
Time = 11.69 (sec) , antiderivative size = 472, normalized size of antiderivative = 1.39 \[ \int \frac {x (c+d x)^{5/2}}{\left (a-b x^2\right )^{3/2}} \, dx=\frac {\sqrt {a-b x^2} \left (-\frac {(c+d x) \left (5 a d^2+b \left (3 c^2+6 c d x-2 d^2 x^2\right )\right )}{b^2 \left (-a+b x^2\right )}-\frac {5 \left (4 c d^2 \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} \left (a-b x^2\right )+4 i \sqrt {b} c \left (\sqrt {b} c-\sqrt {a} d\right ) \sqrt {\frac {d \left (\frac {\sqrt {a}}{\sqrt {b}}+x\right )}{c+d x}} \sqrt {-\frac {\frac {\sqrt {a} d}{\sqrt {b}}-d x}{c+d x}} (c+d x)^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}}{\sqrt {c+d x}}\right )|\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )-i \left (3 b c^2-4 \sqrt {a} \sqrt {b} c d+a d^2\right ) \sqrt {\frac {d \left (\frac {\sqrt {a}}{\sqrt {b}}+x\right )}{c+d x}} \sqrt {-\frac {\frac {\sqrt {a} d}{\sqrt {b}}-d x}{c+d x}} (c+d x)^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}}{\sqrt {c+d x}}\right ),\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )\right )}{b^2 \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} \left (-a+b x^2\right )}\right )}{3 \sqrt {c+d x}} \] Input:
Integrate[(x*(c + d*x)^(5/2))/(a - b*x^2)^(3/2),x]
Output:
(Sqrt[a - b*x^2]*(-(((c + d*x)*(5*a*d^2 + b*(3*c^2 + 6*c*d*x - 2*d^2*x^2)) )/(b^2*(-a + b*x^2))) - (5*(4*c*d^2*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*(a - b* x^2) + (4*I)*Sqrt[b]*c*(Sqrt[b]*c - Sqrt[a]*d)*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]], (Sq rt[b]*c + Sqrt[a]*d)/(Sqrt[b]*c - Sqrt[a]*d)] - I*(3*b*c^2 - 4*Sqrt[a]*Sqr t[b]*c*d + 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)*EllipticF[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)]))/(b^2*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*(-a + b*x^2))))/(3* Sqrt[c + d*x])
Time = 0.50 (sec) , antiderivative size = 337, normalized size of antiderivative = 0.99, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.435, Rules used = {592, 497, 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.
| \(\displaystyle \int \frac {x (c+d x)^{5/2}}{\left (a-b x^2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 592 |
| \(\displaystyle \frac {(c+d x)^{5/2}}{b \sqrt {a-b x^2}}-\frac {5 d \int \frac {(c+d x)^{3/2}}{\sqrt {a-b x^2}}dx}{2 b}\) |
| \(\Big \downarrow \) 497 |
| \(\displaystyle \frac {(c+d x)^{5/2}}{b \sqrt {a-b x^2}}-\frac {5 d \left (-\frac {2 \int -\frac {3 b c^2+4 b d x c+a d^2}{2 \sqrt {c+d x} \sqrt {a-b x^2}}dx}{3 b}-\frac {2 d \sqrt {a-b x^2} \sqrt {c+d x}}{3 b}\right )}{2 b}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {(c+d x)^{5/2}}{b \sqrt {a-b x^2}}-\frac {5 d \left (\frac {\int \frac {3 b c^2+4 b d x c+a d^2}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{3 b}-\frac {2 d \sqrt {a-b x^2} \sqrt {c+d x}}{3 b}\right )}{2 b}\) |
| \(\Big \downarrow \) 600 |
| \(\displaystyle \frac {(c+d x)^{5/2}}{b \sqrt {a-b x^2}}-\frac {5 d \left (\frac {4 b c \int \frac {\sqrt {c+d x}}{\sqrt {a-b x^2}}dx-\left (b c^2-a d^2\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{3 b}-\frac {2 d \sqrt {a-b x^2} \sqrt {c+d x}}{3 b}\right )}{2 b}\) |
| \(\Big \downarrow \) 509 |
| \(\displaystyle \frac {(c+d x)^{5/2}}{b \sqrt {a-b x^2}}-\frac {5 d \left (\frac {\frac {4 b c \sqrt {1-\frac {b x^2}{a}} \int \frac {\sqrt {c+d x}}{\sqrt {1-\frac {b x^2}{a}}}dx}{\sqrt {a-b x^2}}-\left (b c^2-a d^2\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{3 b}-\frac {2 d \sqrt {a-b x^2} \sqrt {c+d x}}{3 b}\right )}{2 b}\) |
| \(\Big \downarrow \) 508 |
| \(\displaystyle \frac {(c+d x)^{5/2}}{b \sqrt {a-b x^2}}-\frac {5 d \left (\frac {-\left (\left (b c^2-a d^2\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx\right )-\frac {8 \sqrt {a} \sqrt {b} c \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \int \frac {\sqrt {1-\frac {d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\frac {\sqrt {b} c}{\sqrt {a}}+d}}}{\sqrt {\frac {1}{2} \left (\frac {\sqrt {b} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}}{\sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}}{3 b}-\frac {2 d \sqrt {a-b x^2} \sqrt {c+d x}}{3 b}\right )}{2 b}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {(c+d x)^{5/2}}{b \sqrt {a-b x^2}}-\frac {5 d \left (\frac {-\left (b c^2-a d^2\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx-\frac {8 \sqrt {a} \sqrt {b} c \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}}{3 b}-\frac {2 d \sqrt {a-b x^2} \sqrt {c+d x}}{3 b}\right )}{2 b}\) |
| \(\Big \downarrow \) 512 |
| \(\displaystyle \frac {(c+d x)^{5/2}}{b \sqrt {a-b x^2}}-\frac {5 d \left (\frac {-\frac {\sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}}}dx}{\sqrt {a-b x^2}}-\frac {8 \sqrt {a} \sqrt {b} c \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}}{3 b}-\frac {2 d \sqrt {a-b x^2} \sqrt {c+d x}}{3 b}\right )}{2 b}\) |
| \(\Big \downarrow \) 511 |
| \(\displaystyle \frac {(c+d x)^{5/2}}{b \sqrt {a-b x^2}}-\frac {5 d \left (\frac {\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \int \frac {1}{\sqrt {1-\frac {d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\frac {\sqrt {b} c}{\sqrt {a}}+d}} \sqrt {\frac {1}{2} \left (\frac {\sqrt {b} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}}{\sqrt {b} \sqrt {a-b x^2} \sqrt {c+d x}}-\frac {8 \sqrt {a} \sqrt {b} c \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}}{3 b}-\frac {2 d \sqrt {a-b x^2} \sqrt {c+d x}}{3 b}\right )}{2 b}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle \frac {(c+d x)^{5/2}}{b \sqrt {a-b x^2}}-\frac {5 d \left (\frac {\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} \sqrt {a-b x^2} \sqrt {c+d x}}-\frac {8 \sqrt {a} \sqrt {b} c \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}}{3 b}-\frac {2 d \sqrt {a-b x^2} \sqrt {c+d x}}{3 b}\right )}{2 b}\) |
Input:
Int[(x*(c + d*x)^(5/2))/(a - b*x^2)^(3/2),x]
Output:
(c + d*x)^(5/2)/(b*Sqrt[a - b*x^2]) - (5*d*((-2*d*Sqrt[c + d*x]*Sqrt[a - b *x^2])/(3*b) + ((-8*Sqrt[a]*Sqrt[b]*c*Sqrt[c + d*x]*Sqrt[1 - (b*x^2)/a]*El lipticE[ArcSin[Sqrt[1 - (Sqrt[b]*x)/Sqrt[a]]/Sqrt[2]], (2*d)/((Sqrt[b]*c)/ Sqrt[a] + d)])/(Sqrt[(Sqrt[b]*(c + d*x))/(Sqrt[b]*c + Sqrt[a]*d)]*Sqrt[a - b*x^2]) + (2*Sqrt[a]*(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)/Sq rt[a]]/Sqrt[2]], (2*d)/((Sqrt[b]*c)/Sqrt[a] + d)])/(Sqrt[b]*Sqrt[c + d*x]* Sqrt[a - b*x^2]))/(3*b)))/(2*b)
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[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ d*(c + d*x)^(n - 1)*((a + b*x^2)^(p + 1)/(b*(n + 2*p + 1))), x] + Simp[1/(b *(n + 2*p + 1)) Int[(c + d*x)^(n - 2)*(a + b*x^2)^p*Simp[b*c^2*(n + 2*p + 1) - a*d^2*(n - 1) + 2*b*c*d*(n + p)*x, x], x], x] /; FreeQ[{a, b, c, d, n , p}, x] && If[RationalQ[n], GtQ[n, 1], SumSimplerQ[n, -2]] && NeQ[n + 2*p + 1, 0] && IntQuadraticQ[a, 0, b, c, d, n, p, x]
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 + d*x)^n*((a + b*x^2)^(p + 1)/(2*b*(p + 1))), x] - Simp[d*(n/(2*b* (p + 1))) Int[(c + d*x)^(n - 1)*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d}, x] && LtQ[p, -1] && GtQ[n, 0] && (IntegerQ[n] || IntegerQ[p] || I ntegersQ[2*n, 2*p])
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]
Leaf count of result is larger than twice the leaf count of optimal. \(670\) vs. \(2(272)=544\).
Time = 6.47 (sec) , antiderivative size = 671, normalized size of antiderivative = 1.97
| method | result | size |
| elliptic | \(\frac {\sqrt {\left (d x +c \right ) \left (-b \,x^{2}+a \right )}\, \left (-\frac {2 \left (-b d x -b c \right ) \left (\frac {c d x}{b^{2}}+\frac {a \,d^{2}+b \,c^{2}}{2 b^{3}}\right )}{\sqrt {\left (x^{2}-\frac {a}{b}\right ) \left (-b d x -b c \right )}}+\frac {2 d^{2} \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}{3 b^{2}}+\frac {2 \left (-\frac {d \left (a \,d^{2}+b \,c^{2}\right )}{2 b^{2}}-\frac {2 c^{2} d}{b}-\frac {d^{3} a}{3 b^{2}}\right ) \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right ) \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x -\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x +\frac {\sqrt {a b}}{b}}{-\frac {c}{d}+\frac {\sqrt {a b}}{b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}, \sqrt {\frac {-\frac {c}{d}+\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\right )}{\sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}-\frac {20 c \,d^{2} \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right ) \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x -\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x +\frac {\sqrt {a b}}{b}}{-\frac {c}{d}+\frac {\sqrt {a b}}{b}}}\, \left (\left (-\frac {c}{d}-\frac {\sqrt {a b}}{b}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}, \sqrt {\frac {-\frac {c}{d}+\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\right )+\frac {\sqrt {a b}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}, \sqrt {\frac {-\frac {c}{d}+\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\right )}{b}\right )}{3 b \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}\right )}{\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}\) | \(671\) |
| risch | \(\frac {2 d^{2} \sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}{3 b^{2}}-\frac {\left (-\frac {6 \left (-b d x -b c \right ) \left (-c d x -\frac {a \,d^{2}+b \,c^{2}}{2 b}\right )}{\sqrt {\left (x^{2}-\frac {a}{b}\right ) \left (-b d x -b c \right )}}+\frac {3 \left (-d \left (a \,d^{2}+3 b \,c^{2}\right )+\frac {d \left (a \,d^{2}+b \,c^{2}\right )}{2}+2 b \,c^{2} d \right ) \sqrt {a b}\, \sqrt {2}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \operatorname {EllipticF}\left (\frac {\sqrt {2}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}}{2}, \sqrt {-\frac {2 \sqrt {a b}}{b \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right )}}\right )}{b \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}+\frac {10 c \,d^{2} \sqrt {a b}\, \sqrt {2}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \left (\left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right ) \operatorname {EllipticE}\left (\frac {\sqrt {2}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}}{2}, \sqrt {-\frac {2 \sqrt {a b}}{b \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right )}}\right )-\frac {c \operatorname {EllipticF}\left (\frac {\sqrt {2}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}}{2}, \sqrt {-\frac {2 \sqrt {a b}}{b \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right )}}\right )}{d}\right )}{\sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}+\frac {4 a \,d^{3} \sqrt {a b}\, \sqrt {2}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \operatorname {EllipticF}\left (\frac {\sqrt {2}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}}{2}, \sqrt {-\frac {2 \sqrt {a b}}{b \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right )}}\right )}{b \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}+\frac {9 c^{2} d \sqrt {a b}\, \sqrt {2}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \operatorname {EllipticF}\left (\frac {\sqrt {2}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}}{2}, \sqrt {-\frac {2 \sqrt {a b}}{b \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right )}}\right )}{\sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}\right ) \sqrt {\left (d x +c \right ) \left (-b \,x^{2}+a \right )}}{3 b^{2} \sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}\) | \(898\) |
| default | \(\frac {\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}\, \left (5 \sqrt {-\frac {\left (d x +c \right ) b}{d \sqrt {a b}-b c}}\, \sqrt {\frac {\left (-b x +\sqrt {a b}\right ) d}{d \sqrt {a b}+b c}}\, \sqrt {\frac {\left (b x +\sqrt {a b}\right ) d}{d \sqrt {a b}-b c}}\, \operatorname {EllipticF}\left (\sqrt {-\frac {\left (d x +c \right ) b}{d \sqrt {a b}-b c}}, \sqrt {-\frac {d \sqrt {a b}-b c}{d \sqrt {a b}+b c}}\right ) \sqrt {a b}\, a \,d^{3}-5 \sqrt {-\frac {\left (d x +c \right ) b}{d \sqrt {a b}-b c}}\, \sqrt {\frac {\left (-b x +\sqrt {a b}\right ) d}{d \sqrt {a b}+b c}}\, \sqrt {\frac {\left (b x +\sqrt {a b}\right ) d}{d \sqrt {a b}-b c}}\, \operatorname {EllipticF}\left (\sqrt {-\frac {\left (d x +c \right ) b}{d \sqrt {a b}-b c}}, \sqrt {-\frac {d \sqrt {a b}-b c}{d \sqrt {a b}+b c}}\right ) \sqrt {a b}\, b \,c^{2} d +15 \sqrt {-\frac {\left (d x +c \right ) b}{d \sqrt {a b}-b c}}\, \sqrt {\frac {\left (-b x +\sqrt {a b}\right ) d}{d \sqrt {a b}+b c}}\, \sqrt {\frac {\left (b x +\sqrt {a b}\right ) d}{d \sqrt {a b}-b c}}\, \operatorname {EllipticF}\left (\sqrt {-\frac {\left (d x +c \right ) b}{d \sqrt {a b}-b c}}, \sqrt {-\frac {d \sqrt {a b}-b c}{d \sqrt {a b}+b c}}\right ) a b c \,d^{2}-15 \sqrt {-\frac {\left (d x +c \right ) b}{d \sqrt {a b}-b c}}\, \sqrt {\frac {\left (-b x +\sqrt {a b}\right ) d}{d \sqrt {a b}+b c}}\, \sqrt {\frac {\left (b x +\sqrt {a b}\right ) d}{d \sqrt {a b}-b c}}\, \operatorname {EllipticF}\left (\sqrt {-\frac {\left (d x +c \right ) b}{d \sqrt {a b}-b c}}, \sqrt {-\frac {d \sqrt {a b}-b c}{d \sqrt {a b}+b c}}\right ) b^{2} c^{3}-20 \sqrt {-\frac {\left (d x +c \right ) b}{d \sqrt {a b}-b c}}\, \sqrt {\frac {\left (-b x +\sqrt {a b}\right ) d}{d \sqrt {a b}+b c}}\, \sqrt {\frac {\left (b x +\sqrt {a b}\right ) d}{d \sqrt {a b}-b c}}\, \operatorname {EllipticE}\left (\sqrt {-\frac {\left (d x +c \right ) b}{d \sqrt {a b}-b c}}, \sqrt {-\frac {d \sqrt {a b}-b c}{d \sqrt {a b}+b c}}\right ) a b c \,d^{2}+20 \sqrt {-\frac {\left (d x +c \right ) b}{d \sqrt {a b}-b c}}\, \sqrt {\frac {\left (-b x +\sqrt {a b}\right ) d}{d \sqrt {a b}+b c}}\, \sqrt {\frac {\left (b x +\sqrt {a b}\right ) d}{d \sqrt {a b}-b c}}\, \operatorname {EllipticE}\left (\sqrt {-\frac {\left (d x +c \right ) b}{d \sqrt {a b}-b c}}, \sqrt {-\frac {d \sqrt {a b}-b c}{d \sqrt {a b}+b c}}\right ) b^{2} c^{3}-2 b^{2} d^{3} x^{3}+4 b^{2} c \,d^{2} x^{2}+5 a b \,d^{3} x +9 b^{2} c^{2} d x +5 a b c \,d^{2}+3 c^{3} b^{2}\right )}{3 \left (-b d \,x^{3}-b c \,x^{2}+a d x +a c \right ) b^{3}}\) | \(952\) |
Input:
int(x*(d*x+c)^(5/2)/(-b*x^2+a)^(3/2),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*(-b*d*x-b*c) *(c*d*x/b^2+1/2*(a*d^2+b*c^2)/b^3)/((x^2-a/b)*(-b*d*x-b*c))^(1/2)+2/3*d^2/ b^2*(-b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)+2*(-1/2*d*(a*d^2+b*c^2)/b^2-2*c^2*d /b-1/3*d^3/b^2*a)*(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)*Ellipti cF(((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))-20/3*c*d^2/b*(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 = 0.10 (sec) , antiderivative size = 290, normalized size of antiderivative = 0.85 \[ \int \frac {x (c+d x)^{5/2}}{\left (a-b x^2\right )^{3/2}} \, dx=-\frac {5 \, {\left (5 \, a b c^{2} + 3 \, a^{2} d^{2} - {\left (5 \, b^{2} c^{2} + 3 \, a b d^{2}\right )} x^{2}\right )} \sqrt {-b d} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b c^{2} + 3 \, a d^{2}\right )}}{3 \, b d^{2}}, -\frac {8 \, {\left (b c^{3} - 9 \, a c d^{2}\right )}}{27 \, b d^{3}}, \frac {3 \, d x + c}{3 \, d}\right ) + 60 \, {\left (b^{2} c d x^{2} - a b c d\right )} \sqrt {-b d} {\rm weierstrassZeta}\left (\frac {4 \, {\left (b c^{2} + 3 \, a d^{2}\right )}}{3 \, b d^{2}}, -\frac {8 \, {\left (b c^{3} - 9 \, a c d^{2}\right )}}{27 \, b d^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b c^{2} + 3 \, a d^{2}\right )}}{3 \, b d^{2}}, -\frac {8 \, {\left (b c^{3} - 9 \, a c d^{2}\right )}}{27 \, b d^{3}}, \frac {3 \, d x + c}{3 \, d}\right )\right ) - 3 \, {\left (2 \, b^{2} d^{2} x^{2} - 6 \, b^{2} c d x - 3 \, b^{2} c^{2} - 5 \, a b d^{2}\right )} \sqrt {-b x^{2} + a} \sqrt {d x + c}}{9 \, {\left (b^{4} x^{2} - a b^{3}\right )}} \] Input:
integrate(x*(d*x+c)^(5/2)/(-b*x^2+a)^(3/2),x, algorithm="fricas")
Output:
-1/9*(5*(5*a*b*c^2 + 3*a^2*d^2 - (5*b^2*c^2 + 3*a*b*d^2)*x^2)*sqrt(-b*d)*w eierstrassPInverse(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) + 60*(b^2*c*d*x^2 - a*b*c*d)*sqrt(-b*d)*weie rstrassZeta(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^2*d^2*x^2 - 6*b^2*c*d*x - 3*b ^2*c^2 - 5*a*b*d^2)*sqrt(-b*x^2 + a)*sqrt(d*x + c))/(b^4*x^2 - a*b^3)
\[ \int \frac {x (c+d x)^{5/2}}{\left (a-b x^2\right )^{3/2}} \, dx=\int \frac {x \left (c + d x\right )^{\frac {5}{2}}}{\left (a - b x^{2}\right )^{\frac {3}{2}}}\, dx \] Input:
integrate(x*(d*x+c)**(5/2)/(-b*x**2+a)**(3/2),x)
Output:
Integral(x*(c + d*x)**(5/2)/(a - b*x**2)**(3/2), x)
\[ \int \frac {x (c+d x)^{5/2}}{\left (a-b x^2\right )^{3/2}} \, dx=\int { \frac {{\left (d x + c\right )}^{\frac {5}{2}} x}{{\left (-b x^{2} + a\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate(x*(d*x+c)^(5/2)/(-b*x^2+a)^(3/2),x, algorithm="maxima")
Output:
integrate((d*x + c)^(5/2)*x/(-b*x^2 + a)^(3/2), x)
\[ \int \frac {x (c+d x)^{5/2}}{\left (a-b x^2\right )^{3/2}} \, dx=\int { \frac {{\left (d x + c\right )}^{\frac {5}{2}} x}{{\left (-b x^{2} + a\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate(x*(d*x+c)^(5/2)/(-b*x^2+a)^(3/2),x, algorithm="giac")
Output:
integrate((d*x + c)^(5/2)*x/(-b*x^2 + a)^(3/2), x)
Timed out. \[ \int \frac {x (c+d x)^{5/2}}{\left (a-b x^2\right )^{3/2}} \, dx=\int \frac {x\,{\left (c+d\,x\right )}^{5/2}}{{\left (a-b\,x^2\right )}^{3/2}} \,d x \] Input:
int((x*(c + d*x)^(5/2))/(a - b*x^2)^(3/2),x)
Output:
int((x*(c + d*x)^(5/2))/(a - b*x^2)^(3/2), x)
\[ \int \frac {x (c+d x)^{5/2}}{\left (a-b x^2\right )^{3/2}} \, dx=\text {too large to display} \] Input:
int(x*(d*x+c)^(5/2)/(-b*x^2+a)^(3/2),x)
Output:
( - 20*sqrt(a - b*x**2)*int(sqrt(c + d*x)/(sqrt(a - b*x**2)*a*c**2 - sqrt( a - b*x**2)*a*d**2*x**2 - sqrt(a - b*x**2)*b*c**2*x**2 + sqrt(a - b*x**2)* b*d**2*x**4),x)*a**4*c*d**4 + 20*sqrt(a - b*x**2)*int(sqrt(c + d*x)/(sqrt( a - b*x**2)*a*c**2 - sqrt(a - b*x**2)*a*d**2*x**2 - sqrt(a - b*x**2)*b*c** 2*x**2 + sqrt(a - b*x**2)*b*d**2*x**4),x)*a**3*b*c**3*d**2 + 20*sqrt(a - b *x**2)*int(sqrt(c + d*x)/(sqrt(a - b*x**2)*a*c**2 - sqrt(a - b*x**2)*a*d** 2*x**2 - sqrt(a - b*x**2)*b*c**2*x**2 + sqrt(a - b*x**2)*b*d**2*x**4),x)*a **3*b*c*d**4*x**2 - 20*sqrt(a - b*x**2)*int(sqrt(c + d*x)/(sqrt(a - b*x**2 )*a*c**2 - sqrt(a - b*x**2)*a*d**2*x**2 - sqrt(a - b*x**2)*b*c**2*x**2 + s qrt(a - b*x**2)*b*d**2*x**4),x)*a**2*b**2*c**3*d**2*x**2 - 5*sqrt(a - b*x* *2)*int((sqrt(c + d*x)*sqrt(a - b*x**2)*x**3)/(a**2*c + a**2*d*x - 2*a*b*c *x**2 - 2*a*b*d*x**3 + b**2*c*x**4 + b**2*d*x**5),x)*a**2*b**2*c*d**3 + 5* sqrt(a - b*x**2)*int((sqrt(c + d*x)*sqrt(a - b*x**2)*x**3)/(a**2*c + a**2* d*x - 2*a*b*c*x**2 - 2*a*b*d*x**3 + b**2*c*x**4 + b**2*d*x**5),x)*a*b**3*c **3*d + 5*sqrt(a - b*x**2)*int((sqrt(c + d*x)*sqrt(a - b*x**2)*x**3)/(a**2 *c + a**2*d*x - 2*a*b*c*x**2 - 2*a*b*d*x**3 + b**2*c*x**4 + b**2*d*x**5),x )*a*b**3*c*d**3*x**2 - 5*sqrt(a - b*x**2)*int((sqrt(c + d*x)*sqrt(a - b*x* *2)*x**3)/(a**2*c + a**2*d*x - 2*a*b*c*x**2 - 2*a*b*d*x**3 + b**2*c*x**4 + b**2*d*x**5),x)*b**4*c**3*d*x**2 - 15*sqrt(a - b*x**2)*int((sqrt(c + d*x) *sqrt(a - b*x**2)*x**2)/(a**2*c + a**2*d*x - 2*a*b*c*x**2 - 2*a*b*d*x**...