Integrand size = 26, antiderivative size = 352 \[ \int \frac {(A+B x) \sqrt {a+c x^2}}{(d+e x)^{3/2}} \, dx=\frac {2 (4 B d-3 A e+B e x) \sqrt {a+c x^2}}{3 e^2 \sqrt {d+e x}}+\frac {4 \sqrt {-a} \sqrt {c} (4 B d-3 A e) \sqrt {d+e x} \sqrt {1+\frac {c x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{3 e^3 \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {a+c x^2}}-\frac {4 \sqrt {-a} \left (4 B c d^2-3 A c d e+a B e^2\right ) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {1+\frac {c x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right ),-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{3 \sqrt {c} e^3 \sqrt {d+e x} \sqrt {a+c x^2}} \]
2/3*(B*e*x-3*A*e+4*B*d)*(c*x^2+a)^(1/2)/e^2/(e*x+d)^(1/2)+4/3*(-3*A*e+4*B* d)*EllipticE(1/2*(1-x*c^(1/2)/(-a)^(1/2))^(1/2)*2^(1/2),(-2*a*e/(-a*e+d*(- a)^(1/2)*c^(1/2)))^(1/2))*(-a)^(1/2)*c^(1/2)*(e*x+d)^(1/2)*(1+c*x^2/a)^(1/ 2)/e^3/(c*x^2+a)^(1/2)/((e*x+d)*c^(1/2)/(e*(-a)^(1/2)+d*c^(1/2)))^(1/2)-4/ 3*(-3*A*c*d*e+B*a*e^2+4*B*c*d^2)*EllipticF(1/2*(1-x*c^(1/2)/(-a)^(1/2))^(1 /2)*2^(1/2),(-2*a*e/(-a*e+d*(-a)^(1/2)*c^(1/2)))^(1/2))*(-a)^(1/2)*(1+c*x^ 2/a)^(1/2)*((e*x+d)*c^(1/2)/(e*(-a)^(1/2)+d*c^(1/2)))^(1/2)/e^3/c^(1/2)/(e *x+d)^(1/2)/(c*x^2+a)^(1/2)
Result contains complex when optimal does not.
Time = 23.46 (sec) , antiderivative size = 512, normalized size of antiderivative = 1.45 \[ \int \frac {(A+B x) \sqrt {a+c x^2}}{(d+e x)^{3/2}} \, dx=\frac {\sqrt {d+e x} \left (\frac {2 (4 B d-3 A e+B e x) \left (a+c x^2\right )}{e^2 (d+e x)}+\frac {2 (d+e x) \left (\frac {2 e^2 (-4 B d+3 A e) \sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}} \left (a+c x^2\right )}{(d+e x)^2}+\frac {2 \sqrt {c} \left (-i \sqrt {c} d+\sqrt {a} e\right ) (-4 B d+3 A e) \sqrt {\frac {e \left (\frac {i \sqrt {a}}{\sqrt {c}}+x\right )}{d+e x}} \sqrt {-\frac {\frac {i \sqrt {a} e}{\sqrt {c}}-e x}{d+e x}} E\left (i \text {arcsinh}\left (\frac {\sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right )|\frac {\sqrt {c} d-i \sqrt {a} e}{\sqrt {c} d+i \sqrt {a} e}\right )}{\sqrt {d+e x}}-\frac {2 \sqrt {a} e \left (-4 B \sqrt {c} d-i \sqrt {a} B e+3 A \sqrt {c} e\right ) \sqrt {\frac {e \left (\frac {i \sqrt {a}}{\sqrt {c}}+x\right )}{d+e x}} \sqrt {-\frac {\frac {i \sqrt {a} e}{\sqrt {c}}-e x}{d+e x}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right ),\frac {\sqrt {c} d-i \sqrt {a} e}{\sqrt {c} d+i \sqrt {a} e}\right )}{\sqrt {d+e x}}\right )}{e^4 \sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}\right )}{3 \sqrt {a+c x^2}} \]
(Sqrt[d + e*x]*((2*(4*B*d - 3*A*e + B*e*x)*(a + c*x^2))/(e^2*(d + e*x)) + (2*(d + e*x)*((2*e^2*(-4*B*d + 3*A*e)*Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]*(a + c*x^2))/(d + e*x)^2 + (2*Sqrt[c]*((-I)*Sqrt[c]*d + Sqrt[a]*e)*(-4*B*d + 3*A*e)*Sqrt[(e*((I*Sqrt[a])/Sqrt[c] + x))/(d + e*x)]*Sqrt[-(((I*Sqrt[a]*e) /Sqrt[c] - e*x)/(d + e*x))]*EllipticE[I*ArcSinh[Sqrt[-d - (I*Sqrt[a]*e)/Sq rt[c]]/Sqrt[d + e*x]], (Sqrt[c]*d - I*Sqrt[a]*e)/(Sqrt[c]*d + I*Sqrt[a]*e) ])/Sqrt[d + e*x] - (2*Sqrt[a]*e*(-4*B*Sqrt[c]*d - I*Sqrt[a]*B*e + 3*A*Sqrt [c]*e)*Sqrt[(e*((I*Sqrt[a])/Sqrt[c] + x))/(d + e*x)]*Sqrt[-(((I*Sqrt[a]*e) /Sqrt[c] - e*x)/(d + e*x))]*EllipticF[I*ArcSinh[Sqrt[-d - (I*Sqrt[a]*e)/Sq rt[c]]/Sqrt[d + e*x]], (Sqrt[c]*d - I*Sqrt[a]*e)/(Sqrt[c]*d + I*Sqrt[a]*e) ])/Sqrt[d + e*x]))/(e^4*Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]])))/(3*Sqrt[a + c* x^2])
Time = 0.78 (sec) , antiderivative size = 686, normalized size of antiderivative = 1.95, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {681, 25, 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.
| \(\displaystyle \int \frac {\sqrt {a+c x^2} (A+B x)}{(d+e x)^{3/2}} \, dx\) |
| \(\Big \downarrow \) 681 |
| \(\displaystyle \frac {2 \sqrt {a+c x^2} (-3 A e+4 B d+B e x)}{3 e^2 \sqrt {d+e x}}-\frac {2 \int -\frac {a B e-c (4 B d-3 A e) x}{\sqrt {d+e x} \sqrt {c x^2+a}}dx}{3 e^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 \int \frac {a B e-c (4 B d-3 A e) x}{\sqrt {d+e x} \sqrt {c x^2+a}}dx}{3 e^2}+\frac {2 \sqrt {a+c x^2} (-3 A e+4 B d+B e x)}{3 e^2 \sqrt {d+e x}}\) |
| \(\Big \downarrow \) 599 |
| \(\displaystyle \frac {2 \sqrt {a+c x^2} (-3 A e+4 B d+B e x)}{3 e^2 \sqrt {d+e x}}-\frac {4 \int -\frac {4 B c d^2-3 A c e d+a B e^2-c (4 B d-3 A e) (d+e x)}{\sqrt {\frac {c d^2}{e^2}-\frac {2 c (d+e x) d}{e^2}+\frac {c (d+e x)^2}{e^2}+a}}d\sqrt {d+e x}}{3 e^4}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {4 \int \frac {4 B c d^2-3 A c e d+a B e^2-c (4 B d-3 A e) (d+e x)}{\sqrt {\frac {c d^2}{e^2}-\frac {2 c (d+e x) d}{e^2}+\frac {c (d+e x)^2}{e^2}+a}}d\sqrt {d+e x}}{3 e^4}+\frac {2 \sqrt {a+c x^2} (-3 A e+4 B d+B e x)}{3 e^2 \sqrt {d+e x}}\) |
| \(\Big \downarrow \) 1511 |
| \(\displaystyle \frac {2 \sqrt {a+c x^2} (-3 A e+4 B d+B e x)}{3 e^2 \sqrt {d+e x}}-\frac {4 \left (-\left (\left (-\sqrt {c} \sqrt {a e^2+c d^2} (4 B d-3 A e)+a B e^2-3 A c d e+4 B c d^2\right ) \int \frac {1}{\sqrt {\frac {c d^2}{e^2}-\frac {2 c (d+e x) d}{e^2}+\frac {c (d+e x)^2}{e^2}+a}}d\sqrt {d+e x}\right )-\sqrt {c} \sqrt {a e^2+c d^2} (4 B d-3 A e) \int \frac {1-\frac {\sqrt {c} (d+e x)}{\sqrt {c d^2+a e^2}}}{\sqrt {\frac {c d^2}{e^2}-\frac {2 c (d+e x) d}{e^2}+\frac {c (d+e x)^2}{e^2}+a}}d\sqrt {d+e x}\right )}{3 e^4}\) |
| \(\Big \downarrow \) 1416 |
| \(\displaystyle \frac {2 \sqrt {a+c x^2} (-3 A e+4 B d+B e x)}{3 e^2 \sqrt {d+e x}}-\frac {4 \left (-\sqrt {c} \sqrt {a e^2+c d^2} (4 B d-3 A e) \int \frac {1-\frac {\sqrt {c} (d+e x)}{\sqrt {c d^2+a e^2}}}{\sqrt {\frac {c d^2}{e^2}-\frac {2 c (d+e x) d}{e^2}+\frac {c (d+e x)^2}{e^2}+a}}d\sqrt {d+e x}-\frac {\sqrt [4]{a e^2+c d^2} \left (\frac {\sqrt {c} (d+e x)}{\sqrt {a e^2+c d^2}}+1\right ) \sqrt {\frac {a+\frac {c d^2}{e^2}-\frac {2 c d (d+e x)}{e^2}+\frac {c (d+e x)^2}{e^2}}{\left (a+\frac {c d^2}{e^2}\right ) \left (\frac {\sqrt {c} (d+e x)}{\sqrt {a e^2+c d^2}}+1\right )^2}} \left (-\sqrt {c} \sqrt {a e^2+c d^2} (4 B d-3 A e)+a B e^2-3 A c d e+4 B c d^2\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt [4]{c d^2+a e^2}}\right ),\frac {1}{2} \left (\frac {\sqrt {c} d}{\sqrt {c d^2+a e^2}}+1\right )\right )}{2 \sqrt [4]{c} \sqrt {a+\frac {c d^2}{e^2}-\frac {2 c d (d+e x)}{e^2}+\frac {c (d+e x)^2}{e^2}}}\right )}{3 e^4}\) |
| \(\Big \downarrow \) 1509 |
| \(\displaystyle \frac {2 \sqrt {a+c x^2} (-3 A e+4 B d+B e x)}{3 e^2 \sqrt {d+e x}}-\frac {4 \left (-\frac {\sqrt [4]{a e^2+c d^2} \left (\frac {\sqrt {c} (d+e x)}{\sqrt {a e^2+c d^2}}+1\right ) \sqrt {\frac {a+\frac {c d^2}{e^2}-\frac {2 c d (d+e x)}{e^2}+\frac {c (d+e x)^2}{e^2}}{\left (a+\frac {c d^2}{e^2}\right ) \left (\frac {\sqrt {c} (d+e x)}{\sqrt {a e^2+c d^2}}+1\right )^2}} \left (-\sqrt {c} \sqrt {a e^2+c d^2} (4 B d-3 A e)+a B e^2-3 A c d e+4 B c d^2\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt [4]{c d^2+a e^2}}\right ),\frac {1}{2} \left (\frac {\sqrt {c} d}{\sqrt {c d^2+a e^2}}+1\right )\right )}{2 \sqrt [4]{c} \sqrt {a+\frac {c d^2}{e^2}-\frac {2 c d (d+e x)}{e^2}+\frac {c (d+e x)^2}{e^2}}}-\sqrt {c} \sqrt {a e^2+c d^2} (4 B d-3 A e) \left (\frac {\sqrt [4]{a e^2+c d^2} \left (\frac {\sqrt {c} (d+e x)}{\sqrt {a e^2+c d^2}}+1\right ) \sqrt {\frac {a+\frac {c d^2}{e^2}-\frac {2 c d (d+e x)}{e^2}+\frac {c (d+e x)^2}{e^2}}{\left (a+\frac {c d^2}{e^2}\right ) \left (\frac {\sqrt {c} (d+e x)}{\sqrt {a e^2+c d^2}}+1\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt [4]{c d^2+a e^2}}\right )|\frac {1}{2} \left (\frac {\sqrt {c} d}{\sqrt {c d^2+a e^2}}+1\right )\right )}{\sqrt [4]{c} \sqrt {a+\frac {c d^2}{e^2}-\frac {2 c d (d+e x)}{e^2}+\frac {c (d+e x)^2}{e^2}}}-\frac {\sqrt {d+e x} \sqrt {a+\frac {c d^2}{e^2}-\frac {2 c d (d+e x)}{e^2}+\frac {c (d+e x)^2}{e^2}}}{\left (a+\frac {c d^2}{e^2}\right ) \left (\frac {\sqrt {c} (d+e x)}{\sqrt {a e^2+c d^2}}+1\right )}\right )\right )}{3 e^4}\) |
(2*(4*B*d - 3*A*e + B*e*x)*Sqrt[a + c*x^2])/(3*e^2*Sqrt[d + e*x]) - (4*(-( Sqrt[c]*(4*B*d - 3*A*e)*Sqrt[c*d^2 + a*e^2]*(-((Sqrt[d + e*x]*Sqrt[a + (c* d^2)/e^2 - (2*c*d*(d + e*x))/e^2 + (c*(d + e*x)^2)/e^2])/((a + (c*d^2)/e^2 )*(1 + (Sqrt[c]*(d + e*x))/Sqrt[c*d^2 + a*e^2]))) + ((c*d^2 + a*e^2)^(1/4) *(1 + (Sqrt[c]*(d + e*x))/Sqrt[c*d^2 + a*e^2])*Sqrt[(a + (c*d^2)/e^2 - (2* c*d*(d + e*x))/e^2 + (c*(d + e*x)^2)/e^2)/((a + (c*d^2)/e^2)*(1 + (Sqrt[c] *(d + e*x))/Sqrt[c*d^2 + a*e^2])^2)]*EllipticE[2*ArcTan[(c^(1/4)*Sqrt[d + e*x])/(c*d^2 + a*e^2)^(1/4)], (1 + (Sqrt[c]*d)/Sqrt[c*d^2 + a*e^2])/2])/(c ^(1/4)*Sqrt[a + (c*d^2)/e^2 - (2*c*d*(d + e*x))/e^2 + (c*(d + e*x)^2)/e^2] ))) - ((c*d^2 + a*e^2)^(1/4)*(4*B*c*d^2 - 3*A*c*d*e + a*B*e^2 - Sqrt[c]*(4 *B*d - 3*A*e)*Sqrt[c*d^2 + a*e^2])*(1 + (Sqrt[c]*(d + e*x))/Sqrt[c*d^2 + a *e^2])*Sqrt[(a + (c*d^2)/e^2 - (2*c*d*(d + e*x))/e^2 + (c*(d + e*x)^2)/e^2 )/((a + (c*d^2)/e^2)*(1 + (Sqrt[c]*(d + e*x))/Sqrt[c*d^2 + a*e^2])^2)]*Ell ipticF[2*ArcTan[(c^(1/4)*Sqrt[d + e*x])/(c*d^2 + a*e^2)^(1/4)], (1 + (Sqrt [c]*d)/Sqrt[c*d^2 + a*e^2])/2])/(2*c^(1/4)*Sqrt[a + (c*d^2)/e^2 - (2*c*d*( d + e*x))/e^2 + (c*(d + e*x)^2)/e^2])))/(3*e^4)
3.15.74.3.1 Defintions of rubi rules used
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[(d + e*x)^(m + 1)*(e*f*(m + 2*p + 2) - d*g*(2*p + 1) + e*g*(m + 1)*x)*((a + c*x^2)^p/(e^2*(m + 1)*(m + 2*p + 2))), x] + Simp[p/ (e^2*(m + 1)*(m + 2*p + 2)) Int[(d + e*x)^(m + 1)*(a + c*x^2)^(p - 1)*Sim p[g*(2*a*e + 2*a*e*m) + (g*(2*c*d + 4*c*d*p) - 2*c*e*f*(m + 2*p + 2))*x, x] , x], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && GtQ[p, 0] && (LtQ[m, -1] || EqQ[p, 1] || (IntegerQ[p] && !RationalQ[m])) && NeQ[m, -1] && !ILtQ[m + 2 *p + 1, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
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]
Leaf count of result is larger than twice the leaf count of optimal. \(694\) vs. \(2(286)=572\).
Time = 3.78 (sec) , antiderivative size = 695, normalized size of antiderivative = 1.97
| method | result | size |
| elliptic | \(\frac {\sqrt {\left (e x +d \right ) \left (c \,x^{2}+a \right )}\, \left (-\frac {2 \left (c e \,x^{2}+a e \right ) \left (A e -B d \right )}{e^{3} \sqrt {\left (x +\frac {d}{e}\right ) \left (c e \,x^{2}+a e \right )}}+\frac {2 B \sqrt {c e \,x^{3}+c d \,x^{2}+a e x +a d}}{3 e^{2}}+\frac {2 \left (-\frac {A c d e -B a \,e^{2}-B c \,d^{2}}{e^{3}}+\frac {\left (A e -B d \right ) c d}{e^{3}}-\frac {B a}{3 e}\right ) \left (\frac {d}{e}-\frac {\sqrt {-a c}}{c}\right ) \sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x -\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x +\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}+\frac {\sqrt {-a c}}{c}}}\, F\left (\sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}, \sqrt {\frac {-\frac {d}{e}+\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\right )}{\sqrt {c e \,x^{3}+c d \,x^{2}+a e x +a d}}+\frac {2 \left (\frac {2 \left (A e -B d \right ) c}{e^{2}}-\frac {2 B c d}{3 e^{2}}\right ) \left (\frac {d}{e}-\frac {\sqrt {-a c}}{c}\right ) \sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x -\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x +\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}+\frac {\sqrt {-a c}}{c}}}\, \left (\left (-\frac {d}{e}-\frac {\sqrt {-a c}}{c}\right ) E\left (\sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}, \sqrt {\frac {-\frac {d}{e}+\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\right )+\frac {\sqrt {-a c}\, F\left (\sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}, \sqrt {\frac {-\frac {d}{e}+\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\right )}{c}\right )}{\sqrt {c e \,x^{3}+c d \,x^{2}+a e x +a d}}\right )}{\sqrt {e x +d}\, \sqrt {c \,x^{2}+a}}\) | \(695\) |
| risch | \(\text {Expression too large to display}\) | \(1246\) |
| default | \(\text {Expression too large to display}\) | \(1463\) |
((e*x+d)*(c*x^2+a))^(1/2)/(e*x+d)^(1/2)/(c*x^2+a)^(1/2)*(-2*(c*e*x^2+a*e)* (A*e-B*d)/e^3/((x+d/e)*(c*e*x^2+a*e))^(1/2)+2/3*B/e^2*(c*e*x^3+c*d*x^2+a*e *x+a*d)^(1/2)+2*(-(A*c*d*e-B*a*e^2-B*c*d^2)/e^3+(A*e-B*d)/e^3*c*d-1/3*B*a/ e)*(d/e-(-a*c)^(1/2)/c)*((x+d/e)/(d/e-(-a*c)^(1/2)/c))^(1/2)*((x-(-a*c)^(1 /2)/c)/(-d/e-(-a*c)^(1/2)/c))^(1/2)*((x+(-a*c)^(1/2)/c)/(-d/e+(-a*c)^(1/2) /c))^(1/2)/(c*e*x^3+c*d*x^2+a*e*x+a*d)^(1/2)*EllipticF(((x+d/e)/(d/e-(-a*c )^(1/2)/c))^(1/2),((-d/e+(-a*c)^(1/2)/c)/(-d/e-(-a*c)^(1/2)/c))^(1/2))+2*( 2*(A*e-B*d)/e^2*c-2/3*B/e^2*c*d)*(d/e-(-a*c)^(1/2)/c)*((x+d/e)/(d/e-(-a*c) ^(1/2)/c))^(1/2)*((x-(-a*c)^(1/2)/c)/(-d/e-(-a*c)^(1/2)/c))^(1/2)*((x+(-a* c)^(1/2)/c)/(-d/e+(-a*c)^(1/2)/c))^(1/2)/(c*e*x^3+c*d*x^2+a*e*x+a*d)^(1/2) *((-d/e-(-a*c)^(1/2)/c)*EllipticE(((x+d/e)/(d/e-(-a*c)^(1/2)/c))^(1/2),((- d/e+(-a*c)^(1/2)/c)/(-d/e-(-a*c)^(1/2)/c))^(1/2))+(-a*c)^(1/2)/c*EllipticF (((x+d/e)/(d/e-(-a*c)^(1/2)/c))^(1/2),((-d/e+(-a*c)^(1/2)/c)/(-d/e-(-a*c)^ (1/2)/c))^(1/2))))
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.13 (sec) , antiderivative size = 306, normalized size of antiderivative = 0.87 \[ \int \frac {(A+B x) \sqrt {a+c x^2}}{(d+e x)^{3/2}} \, dx=\frac {2 \, {\left (2 \, {\left (4 \, B c d^{3} - 3 \, A c d^{2} e + 3 \, B a d e^{2} + {\left (4 \, B c d^{2} e - 3 \, A c d e^{2} + 3 \, B a e^{3}\right )} x\right )} \sqrt {c e} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, \frac {3 \, e x + d}{3 \, e}\right ) + 6 \, {\left (4 \, B c d^{2} e - 3 \, A c d e^{2} + {\left (4 \, B c d e^{2} - 3 \, A c e^{3}\right )} x\right )} \sqrt {c e} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, \frac {3 \, e x + d}{3 \, e}\right )\right ) + 3 \, {\left (B c e^{3} x + 4 \, B c d e^{2} - 3 \, A c e^{3}\right )} \sqrt {c x^{2} + a} \sqrt {e x + d}\right )}}{9 \, {\left (c e^{5} x + c d e^{4}\right )}} \]
2/9*(2*(4*B*c*d^3 - 3*A*c*d^2*e + 3*B*a*d*e^2 + (4*B*c*d^2*e - 3*A*c*d*e^2 + 3*B*a*e^3)*x)*sqrt(c*e)*weierstrassPInverse(4/3*(c*d^2 - 3*a*e^2)/(c*e^ 2), -8/27*(c*d^3 + 9*a*d*e^2)/(c*e^3), 1/3*(3*e*x + d)/e) + 6*(4*B*c*d^2*e - 3*A*c*d*e^2 + (4*B*c*d*e^2 - 3*A*c*e^3)*x)*sqrt(c*e)*weierstrassZeta(4/ 3*(c*d^2 - 3*a*e^2)/(c*e^2), -8/27*(c*d^3 + 9*a*d*e^2)/(c*e^3), weierstras sPInverse(4/3*(c*d^2 - 3*a*e^2)/(c*e^2), -8/27*(c*d^3 + 9*a*d*e^2)/(c*e^3) , 1/3*(3*e*x + d)/e)) + 3*(B*c*e^3*x + 4*B*c*d*e^2 - 3*A*c*e^3)*sqrt(c*x^2 + a)*sqrt(e*x + d))/(c*e^5*x + c*d*e^4)
\[ \int \frac {(A+B x) \sqrt {a+c x^2}}{(d+e x)^{3/2}} \, dx=\int \frac {\left (A + B x\right ) \sqrt {a + c x^{2}}}{\left (d + e x\right )^{\frac {3}{2}}}\, dx \]
\[ \int \frac {(A+B x) \sqrt {a+c x^2}}{(d+e x)^{3/2}} \, dx=\int { \frac {\sqrt {c x^{2} + a} {\left (B x + A\right )}}{{\left (e x + d\right )}^{\frac {3}{2}}} \,d x } \]
\[ \int \frac {(A+B x) \sqrt {a+c x^2}}{(d+e x)^{3/2}} \, dx=\int { \frac {\sqrt {c x^{2} + a} {\left (B x + A\right )}}{{\left (e x + d\right )}^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {(A+B x) \sqrt {a+c x^2}}{(d+e x)^{3/2}} \, dx=\int \frac {\sqrt {c\,x^2+a}\,\left (A+B\,x\right )}{{\left (d+e\,x\right )}^{3/2}} \,d x \]