Integrand size = 26, antiderivative size = 438 \[ \int (A+B x) \sqrt {d+e x} \sqrt {a+c x^2} \, dx=-\frac {2 \sqrt {d+e x} \left (4 B c d^2-7 A c d e+5 a B e^2-3 c e (B d+7 A e) x\right ) \sqrt {a+c x^2}}{105 c e^2}+\frac {2 B \sqrt {d+e x} \left (a+c x^2\right )^{3/2}}{7 c}-\frac {4 \sqrt {-a} \left (4 B c d^3-7 A c d^2 e+8 a B d e^2+21 a A e^3\right ) \sqrt {d+e x} \sqrt {1+\frac {c x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{105 \sqrt {c} e^3 \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {a+c x^2}}+\frac {4 \sqrt {-a} \left (c d^2+a e^2\right ) \left (4 B c d^2-7 A c d e+5 a B e^2\right ) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {1+\frac {c x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right ),-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{105 c^{3/2} e^3 \sqrt {d+e x} \sqrt {a+c x^2}} \]
2/7*B*(c*x^2+a)^(3/2)*(e*x+d)^(1/2)/c-2/105*(4*B*c*d^2-7*A*c*d*e+5*B*a*e^2 -3*c*e*(7*A*e+B*d)*x)*(e*x+d)^(1/2)*(c*x^2+a)^(1/2)/c/e^2-4/105*(21*A*a*e^ 3-7*A*c*d^2*e+8*B*a*d*e^2+4*B*c*d^3)*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)/e^3/c^(1/2)/(c*x^2+a)^(1/2)/((e*x+d)*c^(1/2)/ (e*(-a)^(1/2)+d*c^(1/2)))^(1/2)+4/105*(a*e^2+c*d^2)*(-7*A*c*d*e+5*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)/c^(3/2)/e^3/(e*x+d)^(1/2)/(c*x^2+a)^ (1/2)
Result contains complex when optimal does not.
Time = 24.55 (sec) , antiderivative size = 622, normalized size of antiderivative = 1.42 \[ \int (A+B x) \sqrt {d+e x} \sqrt {a+c x^2} \, dx=\frac {\sqrt {d+e x} \left (\frac {2 \left (a+c x^2\right ) \left (10 a B e^2+7 A c e (d+3 e x)+B c \left (-4 d^2+3 d e x+15 e^2 x^2\right )\right )}{c e^2}+\frac {4 \left (e^2 \sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}} \left (4 B c d^3-7 A c d^2 e+8 a B d e^2+21 a A e^3\right ) \left (a+c x^2\right )-\sqrt {c} \left (i \sqrt {c} d-\sqrt {a} e\right ) \left (4 B c d^3-7 A c d^2 e+8 a B d e^2+21 a A e^3\right ) \sqrt {\frac {e \left (\frac {i \sqrt {a}}{\sqrt {c}}+x\right )}{d+e x}} \sqrt {-\frac {\frac {i \sqrt {a} e}{\sqrt {c}}-e x}{d+e x}} (d+e x)^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right )|\frac {\sqrt {c} d-i \sqrt {a} e}{\sqrt {c} d+i \sqrt {a} e}\right )+\sqrt {a} e \left (\sqrt {c} d+i \sqrt {a} e\right ) \left (B \left (-4 c d^2+3 i \sqrt {a} \sqrt {c} d e-5 a e^2\right )+7 A \left (c d e+3 i \sqrt {a} \sqrt {c} e^2\right )\right ) \sqrt {\frac {e \left (\frac {i \sqrt {a}}{\sqrt {c}}+x\right )}{d+e x}} \sqrt {-\frac {\frac {i \sqrt {a} e}{\sqrt {c}}-e x}{d+e x}} (d+e x)^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right ),\frac {\sqrt {c} d-i \sqrt {a} e}{\sqrt {c} d+i \sqrt {a} e}\right )\right )}{c e^4 \sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}} (d+e x)}\right )}{105 \sqrt {a+c x^2}} \]
(Sqrt[d + e*x]*((2*(a + c*x^2)*(10*a*B*e^2 + 7*A*c*e*(d + 3*e*x) + B*c*(-4 *d^2 + 3*d*e*x + 15*e^2*x^2)))/(c*e^2) + (4*(e^2*Sqrt[-d - (I*Sqrt[a]*e)/S qrt[c]]*(4*B*c*d^3 - 7*A*c*d^2*e + 8*a*B*d*e^2 + 21*a*A*e^3)*(a + c*x^2) - Sqrt[c]*(I*Sqrt[c]*d - Sqrt[a]*e)*(4*B*c*d^3 - 7*A*c*d^2*e + 8*a*B*d*e^2 + 21*a*A*e^3)*Sqrt[(e*((I*Sqrt[a])/Sqrt[c] + x))/(d + e*x)]*Sqrt[-(((I*Sqr t[a]*e)/Sqrt[c] - e*x)/(d + e*x))]*(d + e*x)^(3/2)*EllipticE[I*ArcSinh[Sqr t[-d - (I*Sqrt[a]*e)/Sqrt[c]]/Sqrt[d + e*x]], (Sqrt[c]*d - I*Sqrt[a]*e)/(S qrt[c]*d + I*Sqrt[a]*e)] + Sqrt[a]*e*(Sqrt[c]*d + I*Sqrt[a]*e)*(B*(-4*c*d^ 2 + (3*I)*Sqrt[a]*Sqrt[c]*d*e - 5*a*e^2) + 7*A*(c*d*e + (3*I)*Sqrt[a]*Sqrt [c]*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)*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)]))/(c*e^4*Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]*(d + e*x))) )/(105*Sqrt[a + c*x^2])
Time = 1.06 (sec) , antiderivative size = 790, normalized size of antiderivative = 1.80, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.346, Rules used = {687, 27, 682, 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.
| \(\displaystyle \int \sqrt {a+c x^2} (A+B x) \sqrt {d+e x} \, dx\) |
| \(\Big \downarrow \) 687 |
| \(\displaystyle \frac {2 \int \frac {(7 A c d-a B e+c (B d+7 A e) x) \sqrt {c x^2+a}}{2 \sqrt {d+e x}}dx}{7 c}+\frac {2 B \left (a+c x^2\right )^{3/2} \sqrt {d+e x}}{7 c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {(7 A c d-a B e+c (B d+7 A e) x) \sqrt {c x^2+a}}{\sqrt {d+e x}}dx}{7 c}+\frac {2 B \left (a+c x^2\right )^{3/2} \sqrt {d+e x}}{7 c}\) |
| \(\Big \downarrow \) 682 |
| \(\displaystyle \frac {\frac {4 \int -\frac {c \left (a e \left (B c d^2-28 A c e d+5 a B e^2\right )-c \left (4 B c d^3-7 A c e d^2+8 a B e^2 d+21 a A e^3\right ) x\right )}{2 \sqrt {d+e x} \sqrt {c x^2+a}}dx}{15 c e^2}-\frac {2 \sqrt {a+c x^2} \sqrt {d+e x} \left (5 a B e^2-3 c e x (7 A e+B d)-7 A c d e+4 B c d^2\right )}{15 e^2}}{7 c}+\frac {2 B \left (a+c x^2\right )^{3/2} \sqrt {d+e x}}{7 c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {2 \int \frac {a e \left (B c d^2-28 A c e d+5 a B e^2\right )-c \left (4 B c d^3-7 A c e d^2+8 a B e^2 d+21 a A e^3\right ) x}{\sqrt {d+e x} \sqrt {c x^2+a}}dx}{15 e^2}-\frac {2 \sqrt {a+c x^2} \sqrt {d+e x} \left (5 a B e^2-3 c e x (7 A e+B d)-7 A c d e+4 B c d^2\right )}{15 e^2}}{7 c}+\frac {2 B \left (a+c x^2\right )^{3/2} \sqrt {d+e x}}{7 c}\) |
| \(\Big \downarrow \) 599 |
| \(\displaystyle \frac {\frac {4 \int -\frac {\left (c d^2+a e^2\right ) \left (4 B c d^2-7 A c e d+5 a B e^2\right )-c \left (4 B c d^3-7 A c e d^2+8 a B e^2 d+21 a A e^3\right ) (d+e x)}{\sqrt {\frac {c d^2}{e^2}-\frac {2 c (d+e x) d}{e^2}+\frac {c (d+e x)^2}{e^2}+a}}d\sqrt {d+e x}}{15 e^4}-\frac {2 \sqrt {a+c x^2} \sqrt {d+e x} \left (5 a B e^2-3 c e x (7 A e+B d)-7 A c d e+4 B c d^2\right )}{15 e^2}}{7 c}+\frac {2 B \left (a+c x^2\right )^{3/2} \sqrt {d+e x}}{7 c}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-\frac {4 \int \frac {\left (c d^2+a e^2\right ) \left (4 B c d^2-7 A c e d+5 a B e^2\right )-c \left (4 B c d^3-7 A c e d^2+8 a B e^2 d+21 a A e^3\right ) (d+e x)}{\sqrt {\frac {c d^2}{e^2}-\frac {2 c (d+e x) d}{e^2}+\frac {c (d+e x)^2}{e^2}+a}}d\sqrt {d+e x}}{15 e^4}-\frac {2 \sqrt {a+c x^2} \sqrt {d+e x} \left (5 a B e^2-3 c e x (7 A e+B d)-7 A c d e+4 B c d^2\right )}{15 e^2}}{7 c}+\frac {2 B \left (a+c x^2\right )^{3/2} \sqrt {d+e x}}{7 c}\) |
| \(\Big \downarrow \) 1511 |
| \(\displaystyle \frac {\frac {4 \left (-\sqrt {a e^2+c d^2} \left (\sqrt {a e^2+c d^2} \left (5 a B e^2-7 A c d e+4 B c d^2\right )-\sqrt {c} \left (21 a A e^3+8 a B d e^2-7 A c d^2 e+4 B c d^3\right )\right ) \int \frac {1}{\sqrt {\frac {c d^2}{e^2}-\frac {2 c (d+e x) d}{e^2}+\frac {c (d+e x)^2}{e^2}+a}}d\sqrt {d+e x}-\sqrt {c} \sqrt {a e^2+c d^2} \left (21 a A e^3+8 a B d e^2-7 A c d^2 e+4 B c d^3\right ) \int \frac {1-\frac {\sqrt {c} (d+e x)}{\sqrt {c d^2+a e^2}}}{\sqrt {\frac {c d^2}{e^2}-\frac {2 c (d+e x) d}{e^2}+\frac {c (d+e x)^2}{e^2}+a}}d\sqrt {d+e x}\right )}{15 e^4}-\frac {2 \sqrt {a+c x^2} \sqrt {d+e x} \left (5 a B e^2-3 c e x (7 A e+B d)-7 A c d e+4 B c d^2\right )}{15 e^2}}{7 c}+\frac {2 B \left (a+c x^2\right )^{3/2} \sqrt {d+e x}}{7 c}\) |
| \(\Big \downarrow \) 1416 |
| \(\displaystyle \frac {\frac {4 \left (-\sqrt {c} \sqrt {a e^2+c d^2} \left (21 a A e^3+8 a B d e^2-7 A c d^2 e+4 B c d^3\right ) \int \frac {1-\frac {\sqrt {c} (d+e x)}{\sqrt {c d^2+a e^2}}}{\sqrt {\frac {c d^2}{e^2}-\frac {2 c (d+e x) d}{e^2}+\frac {c (d+e x)^2}{e^2}+a}}d\sqrt {d+e x}-\frac {\left (a e^2+c d^2\right )^{3/4} \left (\frac {\sqrt {c} (d+e x)}{\sqrt {a e^2+c d^2}}+1\right ) \sqrt {\frac {a+\frac {c d^2}{e^2}-\frac {2 c d (d+e x)}{e^2}+\frac {c (d+e x)^2}{e^2}}{\left (a+\frac {c d^2}{e^2}\right ) \left (\frac {\sqrt {c} (d+e x)}{\sqrt {a e^2+c d^2}}+1\right )^2}} \left (\sqrt {a e^2+c d^2} \left (5 a B e^2-7 A c d e+4 B c d^2\right )-\sqrt {c} \left (21 a A e^3+8 a B d e^2-7 A c d^2 e+4 B c d^3\right )\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt [4]{c d^2+a e^2}}\right ),\frac {1}{2} \left (\frac {\sqrt {c} d}{\sqrt {c d^2+a e^2}}+1\right )\right )}{2 \sqrt [4]{c} \sqrt {a+\frac {c d^2}{e^2}-\frac {2 c d (d+e x)}{e^2}+\frac {c (d+e x)^2}{e^2}}}\right )}{15 e^4}-\frac {2 \sqrt {a+c x^2} \sqrt {d+e x} \left (5 a B e^2-3 c e x (7 A e+B d)-7 A c d e+4 B c d^2\right )}{15 e^2}}{7 c}+\frac {2 B \left (a+c x^2\right )^{3/2} \sqrt {d+e x}}{7 c}\) |
| \(\Big \downarrow \) 1509 |
| \(\displaystyle \frac {\frac {4 \left (-\frac {\left (a e^2+c d^2\right )^{3/4} \left (\frac {\sqrt {c} (d+e x)}{\sqrt {a e^2+c d^2}}+1\right ) \sqrt {\frac {a+\frac {c d^2}{e^2}-\frac {2 c d (d+e x)}{e^2}+\frac {c (d+e x)^2}{e^2}}{\left (a+\frac {c d^2}{e^2}\right ) \left (\frac {\sqrt {c} (d+e x)}{\sqrt {a e^2+c d^2}}+1\right )^2}} \left (\sqrt {a e^2+c d^2} \left (5 a B e^2-7 A c d e+4 B c d^2\right )-\sqrt {c} \left (21 a A e^3+8 a B d e^2-7 A c d^2 e+4 B c d^3\right )\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt [4]{c d^2+a e^2}}\right ),\frac {1}{2} \left (\frac {\sqrt {c} d}{\sqrt {c d^2+a e^2}}+1\right )\right )}{2 \sqrt [4]{c} \sqrt {a+\frac {c d^2}{e^2}-\frac {2 c d (d+e x)}{e^2}+\frac {c (d+e x)^2}{e^2}}}-\sqrt {c} \sqrt {a e^2+c d^2} \left (21 a A e^3+8 a B d e^2-7 A c d^2 e+4 B c d^3\right ) \left (\frac {\sqrt [4]{a e^2+c d^2} \left (\frac {\sqrt {c} (d+e x)}{\sqrt {a e^2+c d^2}}+1\right ) \sqrt {\frac {a+\frac {c d^2}{e^2}-\frac {2 c d (d+e x)}{e^2}+\frac {c (d+e x)^2}{e^2}}{\left (a+\frac {c d^2}{e^2}\right ) \left (\frac {\sqrt {c} (d+e x)}{\sqrt {a e^2+c d^2}}+1\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt [4]{c d^2+a e^2}}\right )|\frac {1}{2} \left (\frac {\sqrt {c} d}{\sqrt {c d^2+a e^2}}+1\right )\right )}{\sqrt [4]{c} \sqrt {a+\frac {c d^2}{e^2}-\frac {2 c d (d+e x)}{e^2}+\frac {c (d+e x)^2}{e^2}}}-\frac {\sqrt {d+e x} \sqrt {a+\frac {c d^2}{e^2}-\frac {2 c d (d+e x)}{e^2}+\frac {c (d+e x)^2}{e^2}}}{\left (a+\frac {c d^2}{e^2}\right ) \left (\frac {\sqrt {c} (d+e x)}{\sqrt {a e^2+c d^2}}+1\right )}\right )\right )}{15 e^4}-\frac {2 \sqrt {a+c x^2} \sqrt {d+e x} \left (5 a B e^2-3 c e x (7 A e+B d)-7 A c d e+4 B c d^2\right )}{15 e^2}}{7 c}+\frac {2 B \left (a+c x^2\right )^{3/2} \sqrt {d+e x}}{7 c}\) |
(2*B*Sqrt[d + e*x]*(a + c*x^2)^(3/2))/(7*c) + ((-2*Sqrt[d + e*x]*(4*B*c*d^ 2 - 7*A*c*d*e + 5*a*B*e^2 - 3*c*e*(B*d + 7*A*e)*x)*Sqrt[a + c*x^2])/(15*e^ 2) + (4*(-(Sqrt[c]*Sqrt[c*d^2 + a*e^2]*(4*B*c*d^3 - 7*A*c*d^2*e + 8*a*B*d* e^2 + 21*a*A*e^3)*(-((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)^(3/ 4)*(Sqrt[c*d^2 + a*e^2]*(4*B*c*d^2 - 7*A*c*d*e + 5*a*B*e^2) - Sqrt[c]*(4*B *c*d^3 - 7*A*c*d^2*e + 8*a*B*d*e^2 + 21*a*A*e^3))*(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^(1/4)*Sqrt[a + (c*d^2) /e^2 - (2*c*d*(d + e*x))/e^2 + (c*(d + e*x)^2)/e^2])))/(15*e^4))/(7*c)
3.15.72.3.1 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)*(c*e*f*(m + 2*p + 2) - g*c*d*(2*p + 1) + g*c*e*(m + 2*p + 1)*x)*((a + c*x^2)^p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2))), x] + Simp[2*(p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2))) Int[(d + e*x) ^m*(a + c*x^2)^(p - 1)*Simp[f*a*c*e^2*(m + 2*p + 2) + a*c*d*e*g*m - (c^2*f* d*e*(m + 2*p + 2) - g*(c^2*d^2*(2*p + 1) + a*c*e^2*(m + 2*p + 1)))*x, x], x ], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && GtQ[p, 0] && (IntegerQ[p] || ! RationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])) && !ILtQ[m + 2*p, 0] && (Intege rQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
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. \(793\) vs. \(2(366)=732\).
Time = 2.36 (sec) , antiderivative size = 794, normalized size of antiderivative = 1.81
| method | result | size |
| elliptic | \(\frac {\sqrt {\left (e x +d \right ) \left (c \,x^{2}+a \right )}\, \left (\frac {2 B \,x^{2} \sqrt {c e \,x^{3}+c d \,x^{2}+a e x +a d}}{7}+\frac {2 \left (A c e +\frac {1}{7} B c d \right ) x \sqrt {c e \,x^{3}+c d \,x^{2}+a e x +a d}}{5 c e}+\frac {2 \left (A c d +\frac {2 B a e}{7}-\frac {4 \left (A c e +\frac {1}{7} B c d \right ) d}{5 e}\right ) \sqrt {c e \,x^{3}+c d \,x^{2}+a e x +a d}}{3 c e}+\frac {2 \left (d A a -\frac {2 \left (A c e +\frac {1}{7} B c d \right ) a d}{5 c e}-\frac {\left (A c d +\frac {2 B a e}{7}-\frac {4 \left (A c e +\frac {1}{7} B c d \right ) d}{5 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 (a A e +\frac {3 B a d}{7}-\frac {3 \left (A c e +\frac {1}{7} B c d \right ) a}{5 c}-\frac {2 \left (A c d +\frac {2 B a e}{7}-\frac {4 \left (A c e +\frac {1}{7} B c d \right ) d}{5 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}}\) | \(794\) |
| risch | \(\text {Expression too large to display}\) | \(1110\) |
| default | \(\text {Expression too large to display}\) | \(2551\) |
((e*x+d)*(c*x^2+a))^(1/2)/(e*x+d)^(1/2)/(c*x^2+a)^(1/2)*(2/7*B*x^2*(c*e*x^ 3+c*d*x^2+a*e*x+a*d)^(1/2)+2/5*(A*c*e+1/7*B*c*d)/c/e*x*(c*e*x^3+c*d*x^2+a* e*x+a*d)^(1/2)+2/3*(A*c*d+2/7*B*a*e-4/5*(A*c*e+1/7*B*c*d)/e*d)/c/e*(c*e*x^ 3+c*d*x^2+a*e*x+a*d)^(1/2)+2*(d*A*a-2/5*(A*c*e+1/7*B*c*d)/c/e*a*d-1/3*(A*c *d+2/7*B*a*e-4/5*(A*c*e+1/7*B*c*d)/e*d)/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*(a*A*e+3/7*B*a*d-3/5*(A*c*e+1/7*B *c*d)/c*a-2/3*(A*c*d+2/7*B*a*e-4/5*(A*c*e+1/7*B*c*d)/e*d)/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 = 0.14 (sec) , antiderivative size = 343, normalized size of antiderivative = 0.78 \[ \int (A+B x) \sqrt {d+e x} \sqrt {a+c x^2} \, dx=-\frac {2 \, {\left (2 \, {\left (4 \, B c^{2} d^{4} - 7 \, A c^{2} d^{3} e + 11 \, B a c d^{2} e^{2} - 63 \, A a c d e^{3} + 15 \, B a^{2} e^{4}\right )} \sqrt {c e} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, \frac {3 \, e x + d}{3 \, e}\right ) + 6 \, {\left (4 \, B c^{2} d^{3} e - 7 \, A c^{2} d^{2} e^{2} + 8 \, B a c d e^{3} + 21 \, A a c e^{4}\right )} \sqrt {c e} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, \frac {3 \, e x + d}{3 \, e}\right )\right ) - 3 \, {\left (15 \, B c^{2} e^{4} x^{2} - 4 \, B c^{2} d^{2} e^{2} + 7 \, A c^{2} d e^{3} + 10 \, B a c e^{4} + 3 \, {\left (B c^{2} d e^{3} + 7 \, A c^{2} e^{4}\right )} x\right )} \sqrt {c x^{2} + a} \sqrt {e x + d}\right )}}{315 \, c^{2} e^{4}} \]
-2/315*(2*(4*B*c^2*d^4 - 7*A*c^2*d^3*e + 11*B*a*c*d^2*e^2 - 63*A*a*c*d*e^3 + 15*B*a^2*e^4)*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^2*d^3 *e - 7*A*c^2*d^2*e^2 + 8*B*a*c*d*e^3 + 21*A*a*c*e^4)*sqrt(c*e)*weierstrass Zeta(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), wei erstrassPInverse(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*(15*B*c^2*e^4*x^2 - 4*B*c^2*d^2*e^2 + 7*A *c^2*d*e^3 + 10*B*a*c*e^4 + 3*(B*c^2*d*e^3 + 7*A*c^2*e^4)*x)*sqrt(c*x^2 + a)*sqrt(e*x + d))/(c^2*e^4)
\[ \int (A+B x) \sqrt {d+e x} \sqrt {a+c x^2} \, dx=\int \left (A + B x\right ) \sqrt {a + c x^{2}} \sqrt {d + e x}\, dx \]
\[ \int (A+B x) \sqrt {d+e x} \sqrt {a+c x^2} \, dx=\int { \sqrt {c x^{2} + a} {\left (B x + A\right )} \sqrt {e x + d} \,d x } \]
\[ \int (A+B x) \sqrt {d+e x} \sqrt {a+c x^2} \, dx=\int { \sqrt {c x^{2} + a} {\left (B x + A\right )} \sqrt {e x + d} \,d x } \]
Timed out. \[ \int (A+B x) \sqrt {d+e x} \sqrt {a+c x^2} \, dx=\int \sqrt {c\,x^2+a}\,\left (A+B\,x\right )\,\sqrt {d+e\,x} \,d x \]