Integrand size = 26, antiderivative size = 437 \[ \int \frac {(A+B x) \left (a+c x^2\right )^{3/2}}{(d+e x)^{5/2}} \, dx=-\frac {4 \left (9 a B e^2+4 c d (8 B d-5 A e)+c e (8 B d-5 A e) x\right ) \sqrt {a+c x^2}}{15 e^4 \sqrt {d+e x}}+\frac {2 (8 B d-5 A e+3 B e x) \left (a+c x^2\right )^{3/2}}{15 e^2 (d+e x)^{3/2}}-\frac {8 \sqrt {-a} \sqrt {c} \left (9 a B e^2+4 c d (8 B d-5 A e)\right ) \sqrt {d+e x} \sqrt {1+\frac {c x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{15 e^5 \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {a+c x^2}}+\frac {8 \sqrt {-a} \sqrt {c} \left (32 B c d^3-20 A c d^2 e+17 a B d e^2-5 a A e^3\right ) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {1+\frac {c x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right ),-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{15 e^5 \sqrt {d+e x} \sqrt {a+c x^2}} \]
2/15*(3*B*e*x-5*A*e+8*B*d)*(c*x^2+a)^(3/2)/e^2/(e*x+d)^(3/2)-4/15*(9*B*a*e ^2+4*c*d*(-5*A*e+8*B*d)+c*e*(-5*A*e+8*B*d)*x)*(c*x^2+a)^(1/2)/e^4/(e*x+d)^ (1/2)-8/15*(9*B*a*e^2+4*c*d*(-5*A*e+8*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^5/(c*x^2+a)^(1/2)/((e*x+d)*c ^(1/2)/(e*(-a)^(1/2)+d*c^(1/2)))^(1/2)+8/15*(-5*A*a*e^3-20*A*c*d^2*e+17*B* a*d*e^2+32*B*c*d^3)*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^5/(e*x+d)^(1/2)/( c*x^2+a)^(1/2)
Result contains complex when optimal does not.
Time = 24.18 (sec) , antiderivative size = 628, normalized size of antiderivative = 1.44 \[ \int \frac {(A+B x) \left (a+c x^2\right )^{3/2}}{(d+e x)^{5/2}} \, dx=\frac {\sqrt {d+e x} \left (-\frac {2 \left (a+c x^2\right ) \left (5 a A e^3+5 a B e^2 (2 d+3 e x)-5 A c e \left (8 d^2+10 d e x+e^2 x^2\right )+B c \left (64 d^3+80 d^2 e x+8 d e^2 x^2-3 e^3 x^3\right )\right )}{e^4 (d+e x)^2}-\frac {8 \left (-e^2 \sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}} \left (32 B c d^2-20 A c d e+9 a B e^2\right ) \left (a+c x^2\right )+\sqrt {c} \left (-i \sqrt {c} d+\sqrt {a} e\right ) \left (-32 B c d^2+20 A c d e-9 a B e^2\right ) \sqrt {\frac {e \left (\frac {i \sqrt {a}}{\sqrt {c}}+x\right )}{d+e x}} \sqrt {-\frac {\frac {i \sqrt {a} e}{\sqrt {c}}-e x}{d+e x}} (d+e x)^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right )|\frac {\sqrt {c} d-i \sqrt {a} e}{\sqrt {c} d+i \sqrt {a} e}\right )+\sqrt {a} \sqrt {c} e \left (B \left (32 c d^2+8 i \sqrt {a} \sqrt {c} d e+9 a e^2\right )-5 A \left (4 c d e+i \sqrt {a} \sqrt {c} e^2\right )\right ) \sqrt {\frac {e \left (\frac {i \sqrt {a}}{\sqrt {c}}+x\right )}{d+e x}} \sqrt {-\frac {\frac {i \sqrt {a} e}{\sqrt {c}}-e x}{d+e x}} (d+e x)^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right ),\frac {\sqrt {c} d-i \sqrt {a} e}{\sqrt {c} d+i \sqrt {a} e}\right )\right )}{e^6 \sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}} (d+e x)}\right )}{15 \sqrt {a+c x^2}} \]
(Sqrt[d + e*x]*((-2*(a + c*x^2)*(5*a*A*e^3 + 5*a*B*e^2*(2*d + 3*e*x) - 5*A *c*e*(8*d^2 + 10*d*e*x + e^2*x^2) + B*c*(64*d^3 + 80*d^2*e*x + 8*d*e^2*x^2 - 3*e^3*x^3)))/(e^4*(d + e*x)^2) - (8*(-(e^2*Sqrt[-d - (I*Sqrt[a]*e)/Sqrt [c]]*(32*B*c*d^2 - 20*A*c*d*e + 9*a*B*e^2)*(a + c*x^2)) + Sqrt[c]*((-I)*Sq rt[c]*d + Sqrt[a]*e)*(-32*B*c*d^2 + 20*A*c*d*e - 9*a*B*e^2)*Sqrt[(e*((I*Sq rt[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)] + S qrt[a]*Sqrt[c]*e*(B*(32*c*d^2 + (8*I)*Sqrt[a]*Sqrt[c]*d*e + 9*a*e^2) - 5*A *(4*c*d*e + 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)*E llipticF[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)]))/(e^6*Sqrt[-d - (I*Sqrt[a] *e)/Sqrt[c]]*(d + e*x))))/(15*Sqrt[a + c*x^2])
Time = 1.27 (sec) , antiderivative size = 794, normalized size of antiderivative = 1.82, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.346, Rules used = {681, 25, 681, 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.
| \(\displaystyle \int \frac {\left (a+c x^2\right )^{3/2} (A+B x)}{(d+e x)^{5/2}} \, dx\) |
| \(\Big \downarrow \) 681 |
| \(\displaystyle \frac {2 \left (a+c x^2\right )^{3/2} (-5 A e+8 B d+3 B e x)}{15 e^2 (d+e x)^{3/2}}-\frac {2 \int -\frac {(3 a B e-c (8 B d-5 A e) x) \sqrt {c x^2+a}}{(d+e x)^{3/2}}dx}{5 e^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 \int \frac {(3 a B e-c (8 B d-5 A e) x) \sqrt {c x^2+a}}{(d+e x)^{3/2}}dx}{5 e^2}+\frac {2 \left (a+c x^2\right )^{3/2} (-5 A e+8 B d+3 B e x)}{15 e^2 (d+e x)^{3/2}}\) |
| \(\Big \downarrow \) 681 |
| \(\displaystyle \frac {2 \left (-\frac {2 \int \frac {c \left (a e (8 B d-5 A e)-\left (9 a B e^2+4 c d (8 B d-5 A e)\right ) x\right )}{\sqrt {d+e x} \sqrt {c x^2+a}}dx}{3 e^2}-\frac {2 \sqrt {a+c x^2} \left (9 a B e^2+c e x (8 B d-5 A e)+4 c d (8 B d-5 A e)\right )}{3 e^2 \sqrt {d+e x}}\right )}{5 e^2}+\frac {2 \left (a+c x^2\right )^{3/2} (-5 A e+8 B d+3 B e x)}{15 e^2 (d+e x)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 \left (-\frac {2 c \int \frac {a e (8 B d-5 A e)-\left (9 a B e^2+4 c d (8 B d-5 A e)\right ) x}{\sqrt {d+e x} \sqrt {c x^2+a}}dx}{3 e^2}-\frac {2 \sqrt {a+c x^2} \left (9 a B e^2+c e x (8 B d-5 A e)+4 c d (8 B d-5 A e)\right )}{3 e^2 \sqrt {d+e x}}\right )}{5 e^2}+\frac {2 \left (a+c x^2\right )^{3/2} (-5 A e+8 B d+3 B e x)}{15 e^2 (d+e x)^{3/2}}\) |
| \(\Big \downarrow \) 599 |
| \(\displaystyle \frac {2 \left (\frac {4 c \int -\frac {32 B c d^3-20 A c e d^2+17 a B e^2 d-5 a A e^3-\left (9 a B e^2+4 c d (8 B d-5 A e)\right ) (d+e x)}{\sqrt {\frac {c d^2}{e^2}-\frac {2 c (d+e x) d}{e^2}+\frac {c (d+e x)^2}{e^2}+a}}d\sqrt {d+e x}}{3 e^4}-\frac {2 \sqrt {a+c x^2} \left (9 a B e^2+c e x (8 B d-5 A e)+4 c d (8 B d-5 A e)\right )}{3 e^2 \sqrt {d+e x}}\right )}{5 e^2}+\frac {2 \left (a+c x^2\right )^{3/2} (-5 A e+8 B d+3 B e x)}{15 e^2 (d+e x)^{3/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 \left (-\frac {4 c \int \frac {32 B c d^3-20 A c e d^2+17 a B e^2 d-5 a A e^3-\left (9 a B e^2+4 c d (8 B d-5 A e)\right ) (d+e x)}{\sqrt {\frac {c d^2}{e^2}-\frac {2 c (d+e x) d}{e^2}+\frac {c (d+e x)^2}{e^2}+a}}d\sqrt {d+e x}}{3 e^4}-\frac {2 \sqrt {a+c x^2} \left (9 a B e^2+c e x (8 B d-5 A e)+4 c d (8 B d-5 A e)\right )}{3 e^2 \sqrt {d+e x}}\right )}{5 e^2}+\frac {2 \left (a+c x^2\right )^{3/2} (-5 A e+8 B d+3 B e x)}{15 e^2 (d+e x)^{3/2}}\) |
| \(\Big \downarrow \) 1511 |
| \(\displaystyle \frac {2 \left (\frac {4 c \left (\frac {\sqrt {a e^2+c d^2} \left (\frac {\sqrt {c} \left (5 A e \left (a e^2+4 c d^2\right )-B \left (17 a d e^2+32 c d^3\right )\right )}{\sqrt {a e^2+c d^2}}+9 a B e^2+4 c d (8 B d-5 A e)\right ) \int \frac {1}{\sqrt {\frac {c d^2}{e^2}-\frac {2 c (d+e x) d}{e^2}+\frac {c (d+e x)^2}{e^2}+a}}d\sqrt {d+e x}}{\sqrt {c}}-\frac {\sqrt {a e^2+c d^2} \left (9 a B e^2+4 c d (8 B d-5 A e)\right ) \int \frac {1-\frac {\sqrt {c} (d+e x)}{\sqrt {c d^2+a e^2}}}{\sqrt {\frac {c d^2}{e^2}-\frac {2 c (d+e x) d}{e^2}+\frac {c (d+e x)^2}{e^2}+a}}d\sqrt {d+e x}}{\sqrt {c}}\right )}{3 e^4}-\frac {2 \sqrt {a+c x^2} \left (9 a B e^2+c e x (8 B d-5 A e)+4 c d (8 B d-5 A e)\right )}{3 e^2 \sqrt {d+e x}}\right )}{5 e^2}+\frac {2 \left (a+c x^2\right )^{3/2} (-5 A e+8 B d+3 B e x)}{15 e^2 (d+e x)^{3/2}}\) |
| \(\Big \downarrow \) 1416 |
| \(\displaystyle \frac {2 \left (\frac {4 c \left (\frac {\left (a e^2+c d^2\right )^{3/4} \left (\frac {\sqrt {c} (d+e x)}{\sqrt {a e^2+c d^2}}+1\right ) \sqrt {\frac {a+\frac {c d^2}{e^2}-\frac {2 c d (d+e x)}{e^2}+\frac {c (d+e x)^2}{e^2}}{\left (a+\frac {c d^2}{e^2}\right ) \left (\frac {\sqrt {c} (d+e x)}{\sqrt {a e^2+c d^2}}+1\right )^2}} \left (\frac {\sqrt {c} \left (5 A e \left (a e^2+4 c d^2\right )-B \left (17 a d e^2+32 c d^3\right )\right )}{\sqrt {a e^2+c d^2}}+9 a B e^2+4 c d (8 B d-5 A e)\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt [4]{c d^2+a e^2}}\right ),\frac {1}{2} \left (\frac {\sqrt {c} d}{\sqrt {c d^2+a e^2}}+1\right )\right )}{2 c^{3/4} \sqrt {a+\frac {c d^2}{e^2}-\frac {2 c d (d+e x)}{e^2}+\frac {c (d+e x)^2}{e^2}}}-\frac {\sqrt {a e^2+c d^2} \left (9 a B e^2+4 c d (8 B d-5 A e)\right ) \int \frac {1-\frac {\sqrt {c} (d+e x)}{\sqrt {c d^2+a e^2}}}{\sqrt {\frac {c d^2}{e^2}-\frac {2 c (d+e x) d}{e^2}+\frac {c (d+e x)^2}{e^2}+a}}d\sqrt {d+e x}}{\sqrt {c}}\right )}{3 e^4}-\frac {2 \sqrt {a+c x^2} \left (9 a B e^2+c e x (8 B d-5 A e)+4 c d (8 B d-5 A e)\right )}{3 e^2 \sqrt {d+e x}}\right )}{5 e^2}+\frac {2 \left (a+c x^2\right )^{3/2} (-5 A e+8 B d+3 B e x)}{15 e^2 (d+e x)^{3/2}}\) |
| \(\Big \downarrow \) 1509 |
| \(\displaystyle \frac {2 \left (\frac {4 c \left (\frac {\left (a e^2+c d^2\right )^{3/4} \left (\frac {\sqrt {c} (d+e x)}{\sqrt {a e^2+c d^2}}+1\right ) \sqrt {\frac {a+\frac {c d^2}{e^2}-\frac {2 c d (d+e x)}{e^2}+\frac {c (d+e x)^2}{e^2}}{\left (a+\frac {c d^2}{e^2}\right ) \left (\frac {\sqrt {c} (d+e x)}{\sqrt {a e^2+c d^2}}+1\right )^2}} \left (\frac {\sqrt {c} \left (5 A e \left (a e^2+4 c d^2\right )-B \left (17 a d e^2+32 c d^3\right )\right )}{\sqrt {a e^2+c d^2}}+9 a B e^2+4 c d (8 B d-5 A e)\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt [4]{c d^2+a e^2}}\right ),\frac {1}{2} \left (\frac {\sqrt {c} d}{\sqrt {c d^2+a e^2}}+1\right )\right )}{2 c^{3/4} \sqrt {a+\frac {c d^2}{e^2}-\frac {2 c d (d+e x)}{e^2}+\frac {c (d+e x)^2}{e^2}}}-\frac {\sqrt {a e^2+c d^2} \left (9 a B e^2+4 c d (8 B d-5 A e)\right ) \left (\frac {\sqrt [4]{a e^2+c d^2} \left (\frac {\sqrt {c} (d+e x)}{\sqrt {a e^2+c d^2}}+1\right ) \sqrt {\frac {a+\frac {c d^2}{e^2}-\frac {2 c d (d+e x)}{e^2}+\frac {c (d+e x)^2}{e^2}}{\left (a+\frac {c d^2}{e^2}\right ) \left (\frac {\sqrt {c} (d+e x)}{\sqrt {a e^2+c d^2}}+1\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt [4]{c d^2+a e^2}}\right )|\frac {1}{2} \left (\frac {\sqrt {c} d}{\sqrt {c d^2+a e^2}}+1\right )\right )}{\sqrt [4]{c} \sqrt {a+\frac {c d^2}{e^2}-\frac {2 c d (d+e x)}{e^2}+\frac {c (d+e x)^2}{e^2}}}-\frac {\sqrt {d+e x} \sqrt {a+\frac {c d^2}{e^2}-\frac {2 c d (d+e x)}{e^2}+\frac {c (d+e x)^2}{e^2}}}{\left (a+\frac {c d^2}{e^2}\right ) \left (\frac {\sqrt {c} (d+e x)}{\sqrt {a e^2+c d^2}}+1\right )}\right )}{\sqrt {c}}\right )}{3 e^4}-\frac {2 \sqrt {a+c x^2} \left (9 a B e^2+c e x (8 B d-5 A e)+4 c d (8 B d-5 A e)\right )}{3 e^2 \sqrt {d+e x}}\right )}{5 e^2}+\frac {2 \left (a+c x^2\right )^{3/2} (-5 A e+8 B d+3 B e x)}{15 e^2 (d+e x)^{3/2}}\) |
(2*(8*B*d - 5*A*e + 3*B*e*x)*(a + c*x^2)^(3/2))/(15*e^2*(d + e*x)^(3/2)) + (2*((-2*(9*a*B*e^2 + 4*c*d*(8*B*d - 5*A*e) + c*e*(8*B*d - 5*A*e)*x)*Sqrt[ a + c*x^2])/(3*e^2*Sqrt[d + e*x]) + (4*c*(-((Sqrt[c*d^2 + a*e^2]*(9*a*B*e^ 2 + 4*c*d*(8*B*d - 5*A*e))*(-((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])))/Sqrt[c]) + (( c*d^2 + a*e^2)^(3/4)*(9*a*B*e^2 + 4*c*d*(8*B*d - 5*A*e) + (Sqrt[c]*(5*A*e* (4*c*d^2 + a*e^2) - B*(32*c*d^3 + 17*a*d*e^2)))/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^4)))/(5*e^2)
3.15.78.3.1 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)*(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. \(996\) vs. \(2(365)=730\).
Time = 4.99 (sec) , antiderivative size = 997, normalized size of antiderivative = 2.28
| method | result | size |
| elliptic | \(\frac {\sqrt {\left (e x +d \right ) \left (c \,x^{2}+a \right )}\, \left (-\frac {2 \left (A a \,e^{3}+A c \,d^{2} e -B a d \,e^{2}-B c \,d^{3}\right ) \sqrt {c e \,x^{3}+c d \,x^{2}+a e x +a d}}{3 e^{6} \left (x +\frac {d}{e}\right )^{2}}+\frac {2 \left (c e \,x^{2}+a e \right ) \left (8 A c d e -3 B a \,e^{2}-11 B c \,d^{2}\right )}{3 e^{5} \sqrt {\left (x +\frac {d}{e}\right ) \left (c e \,x^{2}+a e \right )}}+\frac {2 B c x \sqrt {c e \,x^{3}+c d \,x^{2}+a e x +a d}}{5 e^{3}}+\frac {2 \left (\frac {c^{2} \left (A e -2 B d \right )}{e^{3}}-\frac {4 B \,c^{2} d}{5 e^{3}}\right ) \sqrt {c e \,x^{3}+c d \,x^{2}+a e x +a d}}{3 c e}+\frac {2 \left (\frac {c \left (2 A a \,e^{3}+3 A c \,d^{2} e -4 B a d \,e^{2}-4 B c \,d^{3}\right )}{e^{5}}-\frac {\left (A a \,e^{3}+A c \,d^{2} e -B a d \,e^{2}-B c \,d^{3}\right ) c}{3 e^{5}}-\frac {\left (8 A c d e -3 B a \,e^{2}-11 B c \,d^{2}\right ) c d}{3 e^{5}}-\frac {2 B c a d}{5 e^{3}}-\frac {\left (\frac {c^{2} \left (A e -2 B d \right )}{e^{3}}-\frac {4 B \,c^{2} d}{5 e^{3}}\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 (-\frac {c \left (2 A c d e -2 B a \,e^{2}-3 B c \,d^{2}\right )}{e^{4}}-\frac {\left (8 A c d e -3 B a \,e^{2}-11 B c \,d^{2}\right ) c}{3 e^{4}}-\frac {3 B a c}{5 e^{2}}-\frac {2 \left (\frac {c^{2} \left (A e -2 B d \right )}{e^{3}}-\frac {4 B \,c^{2} d}{5 e^{3}}\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}}\) | \(997\) |
| risch | \(\text {Expression too large to display}\) | \(2664\) |
| default | \(\text {Expression too large to display}\) | \(3868\) |
((e*x+d)*(c*x^2+a))^(1/2)/(e*x+d)^(1/2)/(c*x^2+a)^(1/2)*(-2/3*(A*a*e^3+A*c *d^2*e-B*a*d*e^2-B*c*d^3)/e^6*(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)*(8*A*c*d*e-3*B*a*e^2-11*B*c*d^2)/e^5/((x+d/e)*(c*e*x^2+a *e))^(1/2)+2/5*B*c/e^3*x*(c*e*x^3+c*d*x^2+a*e*x+a*d)^(1/2)+2/3*(c^2/e^3*(A *e-2*B*d)-4/5*B*c^2/e^3*d)/c/e*(c*e*x^3+c*d*x^2+a*e*x+a*d)^(1/2)+2*(c*(2*A *a*e^3+3*A*c*d^2*e-4*B*a*d*e^2-4*B*c*d^3)/e^5-1/3*(A*a*e^3+A*c*d^2*e-B*a*d *e^2-B*c*d^3)*c/e^5-1/3*(8*A*c*d*e-3*B*a*e^2-11*B*c*d^2)*c/e^5*d-2/5*B*c/e ^3*a*d-1/3*(c^2/e^3*(A*e-2*B*d)-4/5*B*c^2/e^3*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*(-c/e^4*(2*A*c*d*e-2*B*a *e^2-3*B*c*d^2)-1/3*(8*A*c*d*e-3*B*a*e^2-11*B*c*d^2)*c/e^4-3/5*B*a*c/e^2-2 /3*(c^2/e^3*(A*e-2*B*d)-4/5*B*c^2/e^3*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 = 0.11 (sec) , antiderivative size = 510, normalized size of antiderivative = 1.17 \[ \int \frac {(A+B x) \left (a+c x^2\right )^{3/2}}{(d+e x)^{5/2}} \, dx=-\frac {2 \, {\left (4 \, {\left (32 \, B c d^{5} - 20 \, A c d^{4} e + 33 \, B a d^{3} e^{2} - 15 \, A a d^{2} e^{3} + {\left (32 \, B c d^{3} e^{2} - 20 \, A c d^{2} e^{3} + 33 \, B a d e^{4} - 15 \, A a e^{5}\right )} x^{2} + 2 \, {\left (32 \, B c d^{4} e - 20 \, A c d^{3} e^{2} + 33 \, B a d^{2} e^{3} - 15 \, A a d e^{4}\right )} x\right )} \sqrt {c e} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, \frac {3 \, e x + d}{3 \, e}\right ) + 12 \, {\left (32 \, B c d^{4} e - 20 \, A c d^{3} e^{2} + 9 \, B a d^{2} e^{3} + {\left (32 \, B c d^{2} e^{3} - 20 \, A c d e^{4} + 9 \, B a e^{5}\right )} x^{2} + 2 \, {\left (32 \, B c d^{3} e^{2} - 20 \, A c d^{2} e^{3} + 9 \, B a d e^{4}\right )} x\right )} \sqrt {c e} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, \frac {3 \, e x + d}{3 \, e}\right )\right ) - 3 \, {\left (3 \, B c e^{5} x^{3} - 64 \, B c d^{3} e^{2} + 40 \, A c d^{2} e^{3} - 10 \, B a d e^{4} - 5 \, A a e^{5} - {\left (8 \, B c d e^{4} - 5 \, A c e^{5}\right )} x^{2} - 5 \, {\left (16 \, B c d^{2} e^{3} - 10 \, A c d e^{4} + 3 \, B a e^{5}\right )} x\right )} \sqrt {c x^{2} + a} \sqrt {e x + d}\right )}}{45 \, {\left (e^{8} x^{2} + 2 \, d e^{7} x + d^{2} e^{6}\right )}} \]
-2/45*(4*(32*B*c*d^5 - 20*A*c*d^4*e + 33*B*a*d^3*e^2 - 15*A*a*d^2*e^3 + (3 2*B*c*d^3*e^2 - 20*A*c*d^2*e^3 + 33*B*a*d*e^4 - 15*A*a*e^5)*x^2 + 2*(32*B* c*d^4*e - 20*A*c*d^3*e^2 + 33*B*a*d^2*e^3 - 15*A*a*d*e^4)*x)*sqrt(c*e)*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) + 12*(32*B*c*d^4*e - 20*A*c*d^3*e^2 + 9*B*a*d^ 2*e^3 + (32*B*c*d^2*e^3 - 20*A*c*d*e^4 + 9*B*a*e^5)*x^2 + 2*(32*B*c*d^3*e^ 2 - 20*A*c*d^2*e^3 + 9*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*(3*B*c*e^5*x^3 - 64*B*c*d^3*e^2 + 40*A*c*d^2*e^3 - 10*B*a *d*e^4 - 5*A*a*e^5 - (8*B*c*d*e^4 - 5*A*c*e^5)*x^2 - 5*(16*B*c*d^2*e^3 - 1 0*A*c*d*e^4 + 3*B*a*e^5)*x)*sqrt(c*x^2 + a)*sqrt(e*x + d))/(e^8*x^2 + 2*d* e^7*x + d^2*e^6)
\[ \int \frac {(A+B x) \left (a+c x^2\right )^{3/2}}{(d+e x)^{5/2}} \, dx=\int \frac {\left (A + B x\right ) \left (a + c x^{2}\right )^{\frac {3}{2}}}{\left (d + e x\right )^{\frac {5}{2}}}\, dx \]
\[ \int \frac {(A+B x) \left (a+c x^2\right )^{3/2}}{(d+e x)^{5/2}} \, dx=\int { \frac {{\left (c x^{2} + a\right )}^{\frac {3}{2}} {\left (B x + A\right )}}{{\left (e x + d\right )}^{\frac {5}{2}}} \,d x } \]
\[ \int \frac {(A+B x) \left (a+c x^2\right )^{3/2}}{(d+e x)^{5/2}} \, dx=\int { \frac {{\left (c x^{2} + a\right )}^{\frac {3}{2}} {\left (B x + A\right )}}{{\left (e x + d\right )}^{\frac {5}{2}}} \,d x } \]
Timed out. \[ \int \frac {(A+B x) \left (a+c x^2\right )^{3/2}}{(d+e x)^{5/2}} \, dx=\int \frac {{\left (c\,x^2+a\right )}^{3/2}\,\left (A+B\,x\right )}{{\left (d+e\,x\right )}^{5/2}} \,d x \]