Integrand size = 26, antiderivative size = 388 \[ \int \frac {(A+B x) (d+e x)^{3/2}}{\sqrt {a+c x^2}} \, dx=\frac {2 (3 B d+5 A e) \sqrt {d+e x} \sqrt {a+c x^2}}{15 c}+\frac {2 B (d+e x)^{3/2} \sqrt {a+c x^2}}{5 c}-\frac {2 \sqrt {-a} \left (3 B c d^2+20 A c d e-9 a B e^2\right ) \sqrt {d+e x} \sqrt {1+\frac {c x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{15 c^{3/2} e \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {a+c x^2}}+\frac {2 \sqrt {-a} (3 B d+5 A e) \left (c d^2+a e^2\right ) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {1+\frac {c x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right ),-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{15 c^{3/2} e \sqrt {d+e x} \sqrt {a+c x^2}} \]
2/5*B*(e*x+d)^(3/2)*(c*x^2+a)^(1/2)/c+2/15*(5*A*e+3*B*d)*(e*x+d)^(1/2)*(c* x^2+a)^(1/2)/c-2/15*(20*A*c*d*e-9*B*a*e^2+3*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)*(e*x+d)^(1/2)*(1+c*x^2/a)^(1/2)/c^(3/2)/e/(c*x^2+a)^(1/2)/((e *x+d)*c^(1/2)/(e*(-a)^(1/2)+d*c^(1/2)))^(1/2)+2/15*(5*A*e+3*B*d)*(a*e^2+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)/c^(3/2)/e/(e*x+d)^(1/2)/(c*x^2+a)^(1/2)
Result contains complex when optimal does not.
Time = 24.34 (sec) , antiderivative size = 550, normalized size of antiderivative = 1.42 \[ \int \frac {(A+B x) (d+e x)^{3/2}}{\sqrt {a+c x^2}} \, dx=\frac {\sqrt {d+e x} \left (\frac {2 (6 B d+5 A e+3 B e x) \left (a+c x^2\right )}{c}+\frac {2 \left (e^2 \sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}} \left (3 B c d^2+20 A c d e-9 a B e^2\right ) \left (a+c x^2\right )+\sqrt {c} \left (-i \sqrt {c} d+\sqrt {a} e\right ) \left (3 B c d^2+20 A c d e-9 a B e^2\right ) \sqrt {\frac {e \left (\frac {i \sqrt {a}}{\sqrt {c}}+x\right )}{d+e x}} \sqrt {-\frac {\frac {i \sqrt {a} e}{\sqrt {c}}-e x}{d+e x}} (d+e x)^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right )|\frac {\sqrt {c} d-i \sqrt {a} e}{\sqrt {c} d+i \sqrt {a} e}\right )+i \sqrt {c} e \left (\sqrt {c} d+i \sqrt {a} e\right ) \left (15 A c d-9 a B e+i \sqrt {a} \sqrt {c} (3 B d+5 A e)\right ) \sqrt {\frac {e \left (\frac {i \sqrt {a}}{\sqrt {c}}+x\right )}{d+e x}} \sqrt {-\frac {\frac {i \sqrt {a} e}{\sqrt {c}}-e x}{d+e x}} (d+e x)^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right ),\frac {\sqrt {c} d-i \sqrt {a} e}{\sqrt {c} d+i \sqrt {a} e}\right )\right )}{c^2 e^2 \sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}} (d+e x)}\right )}{15 \sqrt {a+c x^2}} \]
(Sqrt[d + e*x]*((2*(6*B*d + 5*A*e + 3*B*e*x)*(a + c*x^2))/c + (2*(e^2*Sqrt [-d - (I*Sqrt[a]*e)/Sqrt[c]]*(3*B*c*d^2 + 20*A*c*d*e - 9*a*B*e^2)*(a + c*x ^2) + Sqrt[c]*((-I)*Sqrt[c]*d + Sqrt[a]*e)*(3*B*c*d^2 + 20*A*c*d*e - 9*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)] + I*Sqrt[c]*e*(Sqrt[c]*d + I*Sqrt[a]*e)*(15*A*c*d - 9*a*B *e + I*Sqrt[a]*Sqrt[c]*(3*B*d + 5*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))]*(d + e*x)^(3/ 2)*EllipticF[I*ArcSinh[Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]/Sqrt[d + e*x]], (S qrt[c]*d - I*Sqrt[a]*e)/(Sqrt[c]*d + I*Sqrt[a]*e)]))/(c^2*e^2*Sqrt[-d - (I *Sqrt[a]*e)/Sqrt[c]]*(d + e*x))))/(15*Sqrt[a + c*x^2])
Time = 1.00 (sec) , antiderivative size = 727, normalized size of antiderivative = 1.87, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {687, 27, 687, 27, 599, 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.
| \(\displaystyle \int \frac {(A+B x) (d+e x)^{3/2}}{\sqrt {a+c x^2}} \, dx\) |
| \(\Big \downarrow \) 687 |
| \(\displaystyle \frac {2 \int \frac {\sqrt {d+e x} (5 A c d-3 a B e+c (3 B d+5 A e) x)}{2 \sqrt {c x^2+a}}dx}{5 c}+\frac {2 B \sqrt {a+c x^2} (d+e x)^{3/2}}{5 c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\sqrt {d+e x} (5 A c d-3 a B e+c (3 B d+5 A e) x)}{\sqrt {c x^2+a}}dx}{5 c}+\frac {2 B \sqrt {a+c x^2} (d+e x)^{3/2}}{5 c}\) |
| \(\Big \downarrow \) 687 |
| \(\displaystyle \frac {\frac {2 \int \frac {c \left (15 A c d^2-12 a B e d-5 a A e^2+\left (3 B c d^2+20 A c e d-9 a B e^2\right ) x\right )}{2 \sqrt {d+e x} \sqrt {c x^2+a}}dx}{3 c}+\frac {2}{3} \sqrt {a+c x^2} \sqrt {d+e x} (5 A e+3 B d)}{5 c}+\frac {2 B \sqrt {a+c x^2} (d+e x)^{3/2}}{5 c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{3} \int \frac {15 A c d^2-12 a B e d-5 a A e^2+\left (3 B c d^2+20 A c e d-9 a B e^2\right ) x}{\sqrt {d+e x} \sqrt {c x^2+a}}dx+\frac {2}{3} \sqrt {a+c x^2} \sqrt {d+e x} (5 A e+3 B d)}{5 c}+\frac {2 B \sqrt {a+c x^2} (d+e x)^{3/2}}{5 c}\) |
| \(\Big \downarrow \) 599 |
| \(\displaystyle \frac {\frac {2}{3} \sqrt {a+c x^2} \sqrt {d+e x} (5 A e+3 B d)-\frac {2 \int \frac {(3 B d+5 A e) \left (c d^2+a e^2\right )-\left (3 B c d^2+20 A c e d-9 a B e^2\right ) (d+e x)}{\sqrt {\frac {c d^2}{e^2}-\frac {2 c (d+e x) d}{e^2}+\frac {c (d+e x)^2}{e^2}+a}}d\sqrt {d+e x}}{3 e^2}}{5 c}+\frac {2 B \sqrt {a+c x^2} (d+e x)^{3/2}}{5 c}\) |
| \(\Big \downarrow \) 1511 |
| \(\displaystyle \frac {\frac {2}{3} \sqrt {a+c x^2} \sqrt {d+e x} (5 A e+3 B d)-\frac {2 \left (\frac {\sqrt {a e^2+c d^2} \left (-9 a B e^2+20 A c d e+3 B c d^2\right ) \int \frac {1-\frac {\sqrt {c} (d+e x)}{\sqrt {c d^2+a e^2}}}{\sqrt {\frac {c d^2}{e^2}-\frac {2 c (d+e x) d}{e^2}+\frac {c (d+e x)^2}{e^2}+a}}d\sqrt {d+e x}}{\sqrt {c}}-\frac {\sqrt {a e^2+c d^2} \left (-\sqrt {c} \sqrt {a e^2+c d^2} (5 A e+3 B d)-9 a B e^2+20 A c d e+3 B c d^2\right ) \int \frac {1}{\sqrt {\frac {c d^2}{e^2}-\frac {2 c (d+e x) d}{e^2}+\frac {c (d+e x)^2}{e^2}+a}}d\sqrt {d+e x}}{\sqrt {c}}\right )}{3 e^2}}{5 c}+\frac {2 B \sqrt {a+c x^2} (d+e x)^{3/2}}{5 c}\) |
| \(\Big \downarrow \) 1416 |
| \(\displaystyle \frac {\frac {2}{3} \sqrt {a+c x^2} \sqrt {d+e x} (5 A e+3 B d)-\frac {2 \left (\frac {\sqrt {a e^2+c d^2} \left (-9 a B e^2+20 A c d e+3 B c d^2\right ) \int \frac {1-\frac {\sqrt {c} (d+e x)}{\sqrt {c d^2+a e^2}}}{\sqrt {\frac {c d^2}{e^2}-\frac {2 c (d+e x) d}{e^2}+\frac {c (d+e x)^2}{e^2}+a}}d\sqrt {d+e x}}{\sqrt {c}}-\frac {\left (a e^2+c d^2\right )^{3/4} \left (\frac {\sqrt {c} (d+e x)}{\sqrt {a e^2+c d^2}}+1\right ) \sqrt {\frac {a+\frac {c d^2}{e^2}-\frac {2 c d (d+e x)}{e^2}+\frac {c (d+e x)^2}{e^2}}{\left (a+\frac {c d^2}{e^2}\right ) \left (\frac {\sqrt {c} (d+e x)}{\sqrt {a e^2+c d^2}}+1\right )^2}} \left (-\sqrt {c} \sqrt {a e^2+c d^2} (5 A e+3 B d)-9 a B e^2+20 A c d e+3 B c d^2\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt [4]{c d^2+a e^2}}\right ),\frac {1}{2} \left (\frac {\sqrt {c} d}{\sqrt {c d^2+a e^2}}+1\right )\right )}{2 c^{3/4} \sqrt {a+\frac {c d^2}{e^2}-\frac {2 c d (d+e x)}{e^2}+\frac {c (d+e x)^2}{e^2}}}\right )}{3 e^2}}{5 c}+\frac {2 B \sqrt {a+c x^2} (d+e x)^{3/2}}{5 c}\) |
| \(\Big \downarrow \) 1509 |
| \(\displaystyle \frac {\frac {2}{3} \sqrt {a+c x^2} \sqrt {d+e x} (5 A e+3 B d)-\frac {2 \left (\frac {\sqrt {a e^2+c d^2} \left (-9 a B e^2+20 A c d e+3 B c d^2\right ) \left (\frac {\sqrt [4]{a e^2+c d^2} \left (\frac {\sqrt {c} (d+e x)}{\sqrt {a e^2+c d^2}}+1\right ) \sqrt {\frac {a+\frac {c d^2}{e^2}-\frac {2 c d (d+e x)}{e^2}+\frac {c (d+e x)^2}{e^2}}{\left (a+\frac {c d^2}{e^2}\right ) \left (\frac {\sqrt {c} (d+e x)}{\sqrt {a e^2+c d^2}}+1\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt [4]{c d^2+a e^2}}\right )|\frac {1}{2} \left (\frac {\sqrt {c} d}{\sqrt {c d^2+a e^2}}+1\right )\right )}{\sqrt [4]{c} \sqrt {a+\frac {c d^2}{e^2}-\frac {2 c d (d+e x)}{e^2}+\frac {c (d+e x)^2}{e^2}}}-\frac {\sqrt {d+e x} \sqrt {a+\frac {c d^2}{e^2}-\frac {2 c d (d+e x)}{e^2}+\frac {c (d+e x)^2}{e^2}}}{\left (a+\frac {c d^2}{e^2}\right ) \left (\frac {\sqrt {c} (d+e x)}{\sqrt {a e^2+c d^2}}+1\right )}\right )}{\sqrt {c}}-\frac {\left (a e^2+c d^2\right )^{3/4} \left (\frac {\sqrt {c} (d+e x)}{\sqrt {a e^2+c d^2}}+1\right ) \sqrt {\frac {a+\frac {c d^2}{e^2}-\frac {2 c d (d+e x)}{e^2}+\frac {c (d+e x)^2}{e^2}}{\left (a+\frac {c d^2}{e^2}\right ) \left (\frac {\sqrt {c} (d+e x)}{\sqrt {a e^2+c d^2}}+1\right )^2}} \left (-\sqrt {c} \sqrt {a e^2+c d^2} (5 A e+3 B d)-9 a B e^2+20 A c d e+3 B c d^2\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt [4]{c d^2+a e^2}}\right ),\frac {1}{2} \left (\frac {\sqrt {c} d}{\sqrt {c d^2+a e^2}}+1\right )\right )}{2 c^{3/4} \sqrt {a+\frac {c d^2}{e^2}-\frac {2 c d (d+e x)}{e^2}+\frac {c (d+e x)^2}{e^2}}}\right )}{3 e^2}}{5 c}+\frac {2 B \sqrt {a+c x^2} (d+e x)^{3/2}}{5 c}\) |
(2*B*(d + e*x)^(3/2)*Sqrt[a + c*x^2])/(5*c) + ((2*(3*B*d + 5*A*e)*Sqrt[d + e*x]*Sqrt[a + c*x^2])/3 - (2*((Sqrt[c*d^2 + a*e^2]*(3*B*c*d^2 + 20*A*c*d* e - 9*a*B*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))/Sq rt[c*d^2 + a*e^2]))) + ((c*d^2 + a*e^2)^(1/4)*(1 + (Sqrt[c]*(d + e*x))/Sqr t[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])))/Sqrt[c] - ((c*d^2 + a*e^2 )^(3/4)*(3*B*c*d^2 + 20*A*c*d*e - 9*a*B*e^2 - Sqrt[c]*(3*B*d + 5*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)]*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^2))/(5*c)
3.15.80.3.1 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[g*(d + e*x)^m*((a + c*x^2)^(p + 1)/(c*(m + 2*p + 2)) ), x] + Simp[1/(c*(m + 2*p + 2)) Int[(d + e*x)^(m - 1)*(a + c*x^2)^p*Simp [c*d*f*(m + 2*p + 2) - a*e*g*m + c*(e*f*(m + 2*p + 2) + d*g*m)*x, x], x], x ] /; FreeQ[{a, c, d, e, f, g, p}, x] && GtQ[m, 0] && NeQ[m + 2*p + 2, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p]) && !(IGtQ[m, 0] && Eq Q[f, 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. \(694\) vs. \(2(316)=632\).
Time = 2.10 (sec) , antiderivative size = 695, normalized size of antiderivative = 1.79
| method | result | size |
| elliptic | \(\frac {\sqrt {\left (e x +d \right ) \left (c \,x^{2}+a \right )}\, \left (\frac {2 B e x \sqrt {c e \,x^{3}+c d \,x^{2}+a e x +a d}}{5 c}+\frac {2 \left (A \,e^{2}+\frac {6}{5} B d e \right ) \sqrt {c e \,x^{3}+c d \,x^{2}+a e x +a d}}{3 c e}+\frac {2 \left (A \,d^{2}-\frac {2 B e a d}{5 c}-\frac {\left (A \,e^{2}+\frac {6}{5} B d e \right ) a}{3 c}\right ) \left (\frac {d}{e}-\frac {\sqrt {-a c}}{c}\right ) \sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x -\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x +\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}+\frac {\sqrt {-a c}}{c}}}\, F\left (\sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}, \sqrt {\frac {-\frac {d}{e}+\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\right )}{\sqrt {c e \,x^{3}+c d \,x^{2}+a e x +a d}}+\frac {2 \left (2 A d e +B \,d^{2}-\frac {3 B \,e^{2} a}{5 c}-\frac {2 \left (A \,e^{2}+\frac {6}{5} B d e \right ) d}{3 e}\right ) \left (\frac {d}{e}-\frac {\sqrt {-a c}}{c}\right ) \sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x -\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x +\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}+\frac {\sqrt {-a c}}{c}}}\, \left (\left (-\frac {d}{e}-\frac {\sqrt {-a c}}{c}\right ) E\left (\sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}, \sqrt {\frac {-\frac {d}{e}+\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\right )+\frac {\sqrt {-a c}\, F\left (\sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}, \sqrt {\frac {-\frac {d}{e}+\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\right )}{c}\right )}{\sqrt {c e \,x^{3}+c d \,x^{2}+a e x +a d}}\right )}{\sqrt {e x +d}\, \sqrt {c \,x^{2}+a}}\) | \(695\) |
| risch | \(\text {Expression too large to display}\) | \(1049\) |
| default | \(\text {Expression too large to display}\) | \(2127\) |
((e*x+d)*(c*x^2+a))^(1/2)/(e*x+d)^(1/2)/(c*x^2+a)^(1/2)*(2/5*B*e/c*x*(c*e* x^3+c*d*x^2+a*e*x+a*d)^(1/2)+2/3*(A*e^2+6/5*B*d*e)/c/e*(c*e*x^3+c*d*x^2+a* e*x+a*d)^(1/2)+2*(A*d^2-2/5*B*e/c*a*d-1/3*(A*e^2+6/5*B*d*e)/c*a)*(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*d*e+B*d^ 2-3/5*B*e^2/c*a-2/3*(A*e^2+6/5*B*d*e)/e*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 = 0.09 (sec) , antiderivative size = 267, normalized size of antiderivative = 0.69 \[ \int \frac {(A+B x) (d+e x)^{3/2}}{\sqrt {a+c x^2}} \, dx=-\frac {2 \, {\left ({\left (3 \, B c d^{3} - 25 \, A c d^{2} e + 27 \, B a d e^{2} + 15 \, A a e^{3}\right )} \sqrt {c e} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, \frac {3 \, e x + d}{3 \, e}\right ) + 3 \, {\left (3 \, B c d^{2} e + 20 \, A c d e^{2} - 9 \, B a e^{3}\right )} \sqrt {c e} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, \frac {3 \, e x + d}{3 \, e}\right )\right ) - 3 \, {\left (3 \, B c e^{3} x + 6 \, B c d e^{2} + 5 \, A c e^{3}\right )} \sqrt {c x^{2} + a} \sqrt {e x + d}\right )}}{45 \, c^{2} e^{2}} \]
-2/45*((3*B*c*d^3 - 25*A*c*d^2*e + 27*B*a*d*e^2 + 15*A*a*e^3)*sqrt(c*e)*we ierstrassPInverse(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*(3*B*c*d^2*e + 20*A*c*d*e^2 - 9*B*a*e^3)* 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*(3*B*c*e^3*x + 6*B*c *d*e^2 + 5*A*c*e^3)*sqrt(c*x^2 + a)*sqrt(e*x + d))/(c^2*e^2)
\[ \int \frac {(A+B x) (d+e x)^{3/2}}{\sqrt {a+c x^2}} \, dx=\int \frac {\left (A + B x\right ) \left (d + e x\right )^{\frac {3}{2}}}{\sqrt {a + c x^{2}}}\, dx \]
\[ \int \frac {(A+B x) (d+e x)^{3/2}}{\sqrt {a+c x^2}} \, dx=\int { \frac {{\left (B x + A\right )} {\left (e x + d\right )}^{\frac {3}{2}}}{\sqrt {c x^{2} + a}} \,d x } \]
\[ \int \frac {(A+B x) (d+e x)^{3/2}}{\sqrt {a+c x^2}} \, dx=\int { \frac {{\left (B x + A\right )} {\left (e x + d\right )}^{\frac {3}{2}}}{\sqrt {c x^{2} + a}} \,d x } \]
Timed out. \[ \int \frac {(A+B x) (d+e x)^{3/2}}{\sqrt {a+c x^2}} \, dx=\int \frac {\left (A+B\,x\right )\,{\left (d+e\,x\right )}^{3/2}}{\sqrt {c\,x^2+a}} \,d x \]