Integrand size = 26, antiderivative size = 420 \[ \int \frac {(A+B x) \sqrt {a+c x^2}}{(d+e x)^{5/2}} \, dx=-\frac {2 \left (4 B c d^3-A c d^2 e+2 a B d e^2+a A e^3+e \left (5 B c d^2-2 A c d e+3 a B e^2\right ) x\right ) \sqrt {a+c x^2}}{3 e^2 \left (c d^2+a e^2\right ) (d+e x)^{3/2}}-\frac {4 \sqrt {-a} \sqrt {c} \left (4 B c d^2-A c d e+3 a B e^2\right ) \sqrt {d+e x} \sqrt {1+\frac {c x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{3 e^3 \left (c d^2+a e^2\right ) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {a+c x^2}}+\frac {4 \sqrt {-a} \sqrt {c} (4 B d-A e) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {1+\frac {c x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right ),-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{3 e^3 \sqrt {d+e x} \sqrt {a+c x^2}} \]
-2/3*(4*B*c*d^3-A*c*d^2*e+2*B*a*d*e^2+A*a*e^3+e*(-2*A*c*d*e+3*B*a*e^2+5*B* c*d^2)*x)*(c*x^2+a)^(1/2)/e^2/(a*e^2+c*d^2)/(e*x+d)^(3/2)-4/3*(-A*c*d*e+3* B*a*e^2+4*B*c*d^2)*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/(a*e^2+c*d^2)/(c*x^2+a)^(1/2)/((e*x+d)*c^(1/2)/(e*( -a)^(1/2)+d*c^(1/2)))^(1/2)+4/3*(-A*e+4*B*d)*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)*c^(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/(e*x+d)^(1/2)/(c*x^2+a)^(1/2)
Result contains complex when optimal does not.
Time = 24.48 (sec) , antiderivative size = 609, normalized size of antiderivative = 1.45 \[ \int \frac {(A+B x) \sqrt {a+c x^2}}{(d+e x)^{5/2}} \, dx=-\frac {2 \sqrt {a+c x^2} \left (a A e^3-A c d e (d+2 e x)+a B e^2 (2 d+3 e x)+B c d^2 (4 d+5 e x)\right )}{3 e^2 \left (c d^2+a e^2\right ) (d+e x)^{3/2}}+\frac {4 \left (e^2 \sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}} \left (4 B c d^2-A c d e+3 a B e^2\right ) \left (a+c x^2\right )-\sqrt {c} \left (-i \sqrt {c} d+\sqrt {a} e\right ) \left (-4 B c d^2+A c d e-3 a B e^2\right ) \sqrt {\frac {e \left (\frac {i \sqrt {a}}{\sqrt {c}}+x\right )}{d+e x}} \sqrt {-\frac {\frac {i \sqrt {a} e}{\sqrt {c}}-e x}{d+e x}} (d+e x)^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right )|\frac {\sqrt {c} d-i \sqrt {a} e}{\sqrt {c} d+i \sqrt {a} e}\right )+\sqrt {a} \sqrt {c} e \left (\sqrt {c} d+i \sqrt {a} e\right ) \left (-4 B \sqrt {c} d+3 i \sqrt {a} B e+A \sqrt {c} e\right ) \sqrt {\frac {e \left (\frac {i \sqrt {a}}{\sqrt {c}}+x\right )}{d+e x}} \sqrt {-\frac {\frac {i \sqrt {a} e}{\sqrt {c}}-e x}{d+e x}} (d+e x)^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right ),\frac {\sqrt {c} d-i \sqrt {a} e}{\sqrt {c} d+i \sqrt {a} e}\right )\right )}{3 e^4 \sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}} \left (c d^2+a e^2\right ) \sqrt {d+e x} \sqrt {a+c x^2}} \]
(-2*Sqrt[a + c*x^2]*(a*A*e^3 - A*c*d*e*(d + 2*e*x) + a*B*e^2*(2*d + 3*e*x) + B*c*d^2*(4*d + 5*e*x)))/(3*e^2*(c*d^2 + a*e^2)*(d + e*x)^(3/2)) + (4*(e ^2*Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]*(4*B*c*d^2 - A*c*d*e + 3*a*B*e^2)*(a + c*x^2) - Sqrt[c]*((-I)*Sqrt[c]*d + Sqrt[a]*e)*(-4*B*c*d^2 + A*c*d*e - 3*a *B*e^2)*Sqrt[(e*((I*Sqrt[a])/Sqrt[c] + x))/(d + e*x)]*Sqrt[-(((I*Sqrt[a]*e )/Sqrt[c] - e*x)/(d + e*x))]*(d + e*x)^(3/2)*EllipticE[I*ArcSinh[Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]/Sqrt[d + e*x]], (Sqrt[c]*d - I*Sqrt[a]*e)/(Sqrt[c] *d + I*Sqrt[a]*e)] + Sqrt[a]*Sqrt[c]*e*(Sqrt[c]*d + I*Sqrt[a]*e)*(-4*B*Sqr t[c]*d + (3*I)*Sqrt[a]*B*e + 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))]*(d + e*x)^( 3/2)*EllipticF[I*ArcSinh[Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]/Sqrt[d + e*x]], (Sqrt[c]*d - I*Sqrt[a]*e)/(Sqrt[c]*d + I*Sqrt[a]*e)]))/(3*e^4*Sqrt[-d - (I *Sqrt[a]*e)/Sqrt[c]]*(c*d^2 + a*e^2)*Sqrt[d + e*x]*Sqrt[a + c*x^2])
Time = 1.04 (sec) , antiderivative size = 767, normalized size of antiderivative = 1.83, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {680, 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.
| \(\displaystyle \int \frac {\sqrt {a+c x^2} (A+B x)}{(d+e x)^{5/2}} \, dx\) |
| \(\Big \downarrow \) 680 |
| \(\displaystyle -\frac {2 \int \frac {c \left (a e (B d-A e)-\left (4 B c d^2-A c e d+3 a B e^2\right ) x\right )}{\sqrt {d+e x} \sqrt {c x^2+a}}dx}{3 e^2 \left (a e^2+c d^2\right )}-\frac {2 \sqrt {a+c x^2} \left (e x \left (3 a B e^2-2 A c d e+5 B c d^2\right )+a A e^3+2 a B d e^2-A c d^2 e+4 B c d^3\right )}{3 e^2 (d+e x)^{3/2} \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {2 c \int \frac {a e (B d-A e)-\left (4 B c d^2-A c e d+3 a B e^2\right ) x}{\sqrt {d+e x} \sqrt {c x^2+a}}dx}{3 e^2 \left (a e^2+c d^2\right )}-\frac {2 \sqrt {a+c x^2} \left (e x \left (3 a B e^2-2 A c d e+5 B c d^2\right )+a A e^3+2 a B d e^2-A c d^2 e+4 B c d^3\right )}{3 e^2 (d+e x)^{3/2} \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 599 |
| \(\displaystyle \frac {4 c \int -\frac {(4 B d-A e) \left (c d^2+a e^2\right )-\left (4 B c d^2-A c e d+3 a B e^2\right ) (d+e x)}{\sqrt {\frac {c d^2}{e^2}-\frac {2 c (d+e x) d}{e^2}+\frac {c (d+e x)^2}{e^2}+a}}d\sqrt {d+e x}}{3 e^4 \left (a e^2+c d^2\right )}-\frac {2 \sqrt {a+c x^2} \left (e x \left (3 a B e^2-2 A c d e+5 B c d^2\right )+a A e^3+2 a B d e^2-A c d^2 e+4 B c d^3\right )}{3 e^2 (d+e x)^{3/2} \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {4 c \int \frac {(4 B d-A e) \left (c d^2+a e^2\right )-\left (4 B c d^2-A c e d+3 a B e^2\right ) (d+e x)}{\sqrt {\frac {c d^2}{e^2}-\frac {2 c (d+e x) d}{e^2}+\frac {c (d+e x)^2}{e^2}+a}}d\sqrt {d+e x}}{3 e^4 \left (a e^2+c d^2\right )}-\frac {2 \sqrt {a+c x^2} \left (e x \left (3 a B e^2-2 A c d e+5 B c d^2\right )+a A e^3+2 a B d e^2-A c d^2 e+4 B c d^3\right )}{3 e^2 (d+e x)^{3/2} \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 1511 |
| \(\displaystyle \frac {4 c \left (\frac {\sqrt {a e^2+c d^2} \left (-\sqrt {c} \sqrt {a e^2+c d^2} (4 B d-A e)+3 a B e^2-A c d e+4 B c d^2\right ) \int \frac {1}{\sqrt {\frac {c d^2}{e^2}-\frac {2 c (d+e x) d}{e^2}+\frac {c (d+e x)^2}{e^2}+a}}d\sqrt {d+e x}}{\sqrt {c}}-\frac {\sqrt {a e^2+c d^2} \left (3 a B e^2-A c d e+4 B c d^2\right ) \int \frac {1-\frac {\sqrt {c} (d+e x)}{\sqrt {c d^2+a e^2}}}{\sqrt {\frac {c d^2}{e^2}-\frac {2 c (d+e x) d}{e^2}+\frac {c (d+e x)^2}{e^2}+a}}d\sqrt {d+e x}}{\sqrt {c}}\right )}{3 e^4 \left (a e^2+c d^2\right )}-\frac {2 \sqrt {a+c x^2} \left (e x \left (3 a B e^2-2 A c d e+5 B c d^2\right )+a A e^3+2 a B d e^2-A c d^2 e+4 B c d^3\right )}{3 e^2 (d+e x)^{3/2} \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 1416 |
| \(\displaystyle \frac {4 c \left (\frac {\left (a e^2+c d^2\right )^{3/4} \left (\frac {\sqrt {c} (d+e x)}{\sqrt {a e^2+c d^2}}+1\right ) \sqrt {\frac {a+\frac {c d^2}{e^2}-\frac {2 c d (d+e x)}{e^2}+\frac {c (d+e x)^2}{e^2}}{\left (a+\frac {c d^2}{e^2}\right ) \left (\frac {\sqrt {c} (d+e x)}{\sqrt {a e^2+c d^2}}+1\right )^2}} \left (-\sqrt {c} \sqrt {a e^2+c d^2} (4 B d-A e)+3 a B e^2-A c d e+4 B c d^2\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt [4]{c d^2+a e^2}}\right ),\frac {1}{2} \left (\frac {\sqrt {c} d}{\sqrt {c d^2+a e^2}}+1\right )\right )}{2 c^{3/4} \sqrt {a+\frac {c d^2}{e^2}-\frac {2 c d (d+e x)}{e^2}+\frac {c (d+e x)^2}{e^2}}}-\frac {\sqrt {a e^2+c d^2} \left (3 a B e^2-A c d e+4 B c d^2\right ) \int \frac {1-\frac {\sqrt {c} (d+e x)}{\sqrt {c d^2+a e^2}}}{\sqrt {\frac {c d^2}{e^2}-\frac {2 c (d+e x) d}{e^2}+\frac {c (d+e x)^2}{e^2}+a}}d\sqrt {d+e x}}{\sqrt {c}}\right )}{3 e^4 \left (a e^2+c d^2\right )}-\frac {2 \sqrt {a+c x^2} \left (e x \left (3 a B e^2-2 A c d e+5 B c d^2\right )+a A e^3+2 a B d e^2-A c d^2 e+4 B c d^3\right )}{3 e^2 (d+e x)^{3/2} \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 1509 |
| \(\displaystyle \frac {4 c \left (\frac {\left (a e^2+c d^2\right )^{3/4} \left (\frac {\sqrt {c} (d+e x)}{\sqrt {a e^2+c d^2}}+1\right ) \sqrt {\frac {a+\frac {c d^2}{e^2}-\frac {2 c d (d+e x)}{e^2}+\frac {c (d+e x)^2}{e^2}}{\left (a+\frac {c d^2}{e^2}\right ) \left (\frac {\sqrt {c} (d+e x)}{\sqrt {a e^2+c d^2}}+1\right )^2}} \left (-\sqrt {c} \sqrt {a e^2+c d^2} (4 B d-A e)+3 a B e^2-A c d e+4 B c d^2\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt [4]{c d^2+a e^2}}\right ),\frac {1}{2} \left (\frac {\sqrt {c} d}{\sqrt {c d^2+a e^2}}+1\right )\right )}{2 c^{3/4} \sqrt {a+\frac {c d^2}{e^2}-\frac {2 c d (d+e x)}{e^2}+\frac {c (d+e x)^2}{e^2}}}-\frac {\sqrt {a e^2+c d^2} \left (3 a B e^2-A c d e+4 B c d^2\right ) \left (\frac {\sqrt [4]{a e^2+c d^2} \left (\frac {\sqrt {c} (d+e x)}{\sqrt {a e^2+c d^2}}+1\right ) \sqrt {\frac {a+\frac {c d^2}{e^2}-\frac {2 c d (d+e x)}{e^2}+\frac {c (d+e x)^2}{e^2}}{\left (a+\frac {c d^2}{e^2}\right ) \left (\frac {\sqrt {c} (d+e x)}{\sqrt {a e^2+c d^2}}+1\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt [4]{c d^2+a e^2}}\right )|\frac {1}{2} \left (\frac {\sqrt {c} d}{\sqrt {c d^2+a e^2}}+1\right )\right )}{\sqrt [4]{c} \sqrt {a+\frac {c d^2}{e^2}-\frac {2 c d (d+e x)}{e^2}+\frac {c (d+e x)^2}{e^2}}}-\frac {\sqrt {d+e x} \sqrt {a+\frac {c d^2}{e^2}-\frac {2 c d (d+e x)}{e^2}+\frac {c (d+e x)^2}{e^2}}}{\left (a+\frac {c d^2}{e^2}\right ) \left (\frac {\sqrt {c} (d+e x)}{\sqrt {a e^2+c d^2}}+1\right )}\right )}{\sqrt {c}}\right )}{3 e^4 \left (a e^2+c d^2\right )}-\frac {2 \sqrt {a+c x^2} \left (e x \left (3 a B e^2-2 A c d e+5 B c d^2\right )+a A e^3+2 a B d e^2-A c d^2 e+4 B c d^3\right )}{3 e^2 (d+e x)^{3/2} \left (a e^2+c d^2\right )}\) |
(-2*(4*B*c*d^3 - A*c*d^2*e + 2*a*B*d*e^2 + a*A*e^3 + e*(5*B*c*d^2 - 2*A*c* d*e + 3*a*B*e^2)*x)*Sqrt[a + c*x^2])/(3*e^2*(c*d^2 + a*e^2)*(d + e*x)^(3/2 )) + (4*c*(-((Sqrt[c*d^2 + a*e^2]*(4*B*c*d^2 - A*c*d*e + 3*a*B*e^2)*(-((Sq rt[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])*Sqr t[(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*A rcTan[(c^(1/4)*Sqrt[d + e*x])/(c*d^2 + a*e^2)^(1/4)], (1 + (Sqrt[c]*d)/Sqr t[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])))/Sqrt[c]) + ((c*d^2 + a*e^2)^(3/4)*(4*B*c*d^2 - A*c*d*e + 3*a*B*e^2 - Sqrt[c]*(4*B*d - A*e)*Sqrt[c*d^2 + a*e^2])*(1 + (Sq rt[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)]*EllipticF[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^(3/4)* 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*(c*d^2 + a*e^2))
3.15.75.3.1 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[((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))*((a + c*x^2)^p/(e^2*(m + 1)*(m + 2)*(c*d^2 + a*e^2)))*((d*g - e*f*(m + 2))*(c*d^2 + a*e^2) - 2*c*d^2*p*(e* f - d*g) - e*(g*(m + 1)*(c*d^2 + a*e^2) + 2*c*d*p*(e*f - d*g))*x), x] - Sim p[p/(e^2*(m + 1)*(m + 2)*(c*d^2 + a*e^2)) Int[(d + e*x)^(m + 2)*(a + c*x^ 2)^(p - 1)*Simp[2*a*c*e*(e*f - d*g)*(m + 2) - c*(2*c*d*(d*g*(2*p + 1) - e*f *(m + 2*p + 2)) - 2*a*e^2*g*(m + 1))*x, x], x], x] /; FreeQ[{a, c, d, e, f, g}, x] && GtQ[p, 0] && LtQ[m, -2] && LtQ[m + 2*p, 0] && !ILtQ[m + 2*p + 3 , 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]
Leaf count of result is larger than twice the leaf count of optimal. \(782\) vs. \(2(354)=708\).
Time = 2.07 (sec) , antiderivative size = 783, normalized size of antiderivative = 1.86
| method | result | size |
| elliptic | \(\frac {\sqrt {\left (e x +d \right ) \left (c \,x^{2}+a \right )}\, \left (-\frac {2 \left (A e -B d \right ) \sqrt {c e \,x^{3}+c d \,x^{2}+a e x +a d}}{3 e^{4} \left (x +\frac {d}{e}\right )^{2}}+\frac {2 \left (c e \,x^{2}+a e \right ) \left (2 A c d e -3 B a \,e^{2}-5 B c \,d^{2}\right )}{3 \left (e^{2} a +c \,d^{2}\right ) e^{3} \sqrt {\left (x +\frac {d}{e}\right ) \left (c e \,x^{2}+a e \right )}}+\frac {2 \left (\frac {\left (A e -2 B d \right ) c}{e^{3}}-\frac {\left (A e -B d \right ) c}{3 e^{3}}-\frac {c d \left (2 A c d e -3 B a \,e^{2}-5 B c \,d^{2}\right )}{3 e^{3} \left (e^{2} a +c \,d^{2}\right )}\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 {B c}{e^{2}}-\frac {c \left (2 A c d e -3 B a \,e^{2}-5 B c \,d^{2}\right )}{3 e^{2} \left (e^{2} a +c \,d^{2}\right )}\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}}\) | \(783\) |
| default | \(\text {Expression too large to display}\) | \(3552\) |
((e*x+d)*(c*x^2+a))^(1/2)/(e*x+d)^(1/2)/(c*x^2+a)^(1/2)*(-2/3*(A*e-B*d)/e^ 4*(c*e*x^3+c*d*x^2+a*e*x+a*d)^(1/2)/(x+d/e)^2+2/3*(c*e*x^2+a*e)/(a*e^2+c*d ^2)/e^3*(2*A*c*d*e-3*B*a*e^2-5*B*c*d^2)/((x+d/e)*(c*e*x^2+a*e))^(1/2)+2*(( A*e-2*B*d)*c/e^3-1/3*(A*e-B*d)/e^3*c-1/3*c/e^3*d*(2*A*c*d*e-3*B*a*e^2-5*B* c*d^2)/(a*e^2+c*d^2))*(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*(B*c/e^2-1/3*c/e^2*(2*A*c*d*e-3*B*a*e^2-5*B*c*d^2)/(a*e^ 2+c*d^2))*(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 = 0.11 (sec) , antiderivative size = 508, normalized size of antiderivative = 1.21 \[ \int \frac {(A+B x) \sqrt {a+c x^2}}{(d+e x)^{5/2}} \, dx=-\frac {2 \, {\left (2 \, {\left (4 \, B c d^{5} - A c d^{4} e + 6 \, B a d^{3} e^{2} - 3 \, A a d^{2} e^{3} + {\left (4 \, B c d^{3} e^{2} - A c d^{2} e^{3} + 6 \, B a d e^{4} - 3 \, A a e^{5}\right )} x^{2} + 2 \, {\left (4 \, B c d^{4} e - A c d^{3} e^{2} + 6 \, B a d^{2} e^{3} - 3 \, A a d e^{4}\right )} x\right )} \sqrt {c e} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, \frac {3 \, e x + d}{3 \, e}\right ) + 6 \, {\left (4 \, B c d^{4} e - A c d^{3} e^{2} + 3 \, B a d^{2} e^{3} + {\left (4 \, B c d^{2} e^{3} - A c d e^{4} + 3 \, B a e^{5}\right )} x^{2} + 2 \, {\left (4 \, B c d^{3} e^{2} - A c d^{2} e^{3} + 3 \, B a d e^{4}\right )} x\right )} \sqrt {c e} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, \frac {3 \, e x + d}{3 \, e}\right )\right ) + 3 \, {\left (4 \, B c d^{3} e^{2} - A c d^{2} e^{3} + 2 \, B a d e^{4} + A a e^{5} + {\left (5 \, B c d^{2} e^{3} - 2 \, A c d e^{4} + 3 \, B a e^{5}\right )} x\right )} \sqrt {c x^{2} + a} \sqrt {e x + d}\right )}}{9 \, {\left (c d^{4} e^{4} + a d^{2} e^{6} + {\left (c d^{2} e^{6} + a e^{8}\right )} x^{2} + 2 \, {\left (c d^{3} e^{5} + a d e^{7}\right )} x\right )}} \]
-2/9*(2*(4*B*c*d^5 - A*c*d^4*e + 6*B*a*d^3*e^2 - 3*A*a*d^2*e^3 + (4*B*c*d^ 3*e^2 - A*c*d^2*e^3 + 6*B*a*d*e^4 - 3*A*a*e^5)*x^2 + 2*(4*B*c*d^4*e - A*c* d^3*e^2 + 6*B*a*d^2*e^3 - 3*A*a*d*e^4)*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^4*e - A*c*d^3*e^2 + 3*B*a*d^2*e^3 + (4*B*c*d^2*e^3 - A*c*d*e^4 + 3*B*a*e^5)*x^2 + 2*(4*B*c*d^3*e^2 - A*c*d^2*e^3 + 3*B*a*d*e^4 )*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), 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)) + 3*(4*B*c*d^3*e^2 - A*c*d^2*e^3 + 2*B*a*d*e^4 + A*a*e^5 + (5*B*c*d^2*e^3 - 2*A*c*d*e^4 + 3*B *a*e^5)*x)*sqrt(c*x^2 + a)*sqrt(e*x + d))/(c*d^4*e^4 + a*d^2*e^6 + (c*d^2* e^6 + a*e^8)*x^2 + 2*(c*d^3*e^5 + a*d*e^7)*x)
\[ \int \frac {(A+B x) \sqrt {a+c x^2}}{(d+e x)^{5/2}} \, dx=\int \frac {\left (A + B x\right ) \sqrt {a + c x^{2}}}{\left (d + e x\right )^{\frac {5}{2}}}\, dx \]
\[ \int \frac {(A+B x) \sqrt {a+c x^2}}{(d+e x)^{5/2}} \, dx=\int { \frac {\sqrt {c x^{2} + a} {\left (B x + A\right )}}{{\left (e x + d\right )}^{\frac {5}{2}}} \,d x } \]
\[ \int \frac {(A+B x) \sqrt {a+c x^2}}{(d+e x)^{5/2}} \, dx=\int { \frac {\sqrt {c x^{2} + a} {\left (B x + A\right )}}{{\left (e x + d\right )}^{\frac {5}{2}}} \,d x } \]
Timed out. \[ \int \frac {(A+B x) \sqrt {a+c x^2}}{(d+e x)^{5/2}} \, dx=\int \frac {\sqrt {c\,x^2+a}\,\left (A+B\,x\right )}{{\left (d+e\,x\right )}^{5/2}} \,d x \]