Integrand size = 26, antiderivative size = 448 \[ \int \frac {(A+B x) \left (a+c x^2\right )^{3/2}}{(d+e x)^{3/2}} \, dx=\frac {4 \sqrt {d+e x} \left (5 a B e^2+4 c d (8 B d-7 A e)-3 c e (8 B d-7 A e) x\right ) \sqrt {a+c x^2}}{35 e^4}+\frac {2 (8 B d-7 A e+B e x) \left (a+c x^2\right )^{3/2}}{7 e^2 \sqrt {d+e x}}+\frac {8 \sqrt {-a} \sqrt {c} \left (32 B c d^3-28 A c d^2 e+29 a B d e^2-21 a A e^3\right ) \sqrt {d+e x} \sqrt {1+\frac {c x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{35 e^5 \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {a+c x^2}}-\frac {8 \sqrt {-a} \left (c d^2+a e^2\right ) \left (32 B c d^2-28 A c d e+5 a B e^2\right ) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {1+\frac {c x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right ),-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{35 \sqrt {c} e^5 \sqrt {d+e x} \sqrt {a+c x^2}} \]
2/7*(B*e*x-7*A*e+8*B*d)*(c*x^2+a)^(3/2)/e^2/(e*x+d)^(1/2)+4/35*(5*B*a*e^2+ 4*c*d*(-7*A*e+8*B*d)-3*c*e*(-7*A*e+8*B*d)*x)*(e*x+d)^(1/2)*(c*x^2+a)^(1/2) /e^4+8/35*(-21*A*a*e^3-28*A*c*d^2*e+29*B*a*d*e^2+32*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)*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)-8/35*(a*e^2+c*d^2) *(-28*A*c*d*e+5*B*a*e^2+32*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^5/c^(1/2) /(e*x+d)^(1/2)/(c*x^2+a)^(1/2)
Result contains complex when optimal does not.
Time = 24.36 (sec) , antiderivative size = 661, normalized size of antiderivative = 1.48 \[ \int \frac {(A+B x) \left (a+c x^2\right )^{3/2}}{(d+e x)^{3/2}} \, dx=\frac {\sqrt {d+e x} \left (\frac {2 \left (a+c x^2\right ) \left (-7 A e \left (5 a e^2+c \left (8 d^2+2 d e x-e^2 x^2\right )\right )+B \left (5 a e^2 (10 d+3 e x)+c \left (64 d^3+16 d^2 e x-8 d e^2 x^2+5 e^3 x^3\right )\right )\right )}{e^4 (d+e x)}+\frac {8 \left (e^2 \sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}} \left (-32 B c d^3+28 A c d^2 e-29 a B d e^2+21 a A e^3\right ) \left (a+c x^2\right )+\sqrt {c} \left (-i \sqrt {c} d+\sqrt {a} e\right ) \left (-32 B c d^3+28 A c d^2 e-29 a B d e^2+21 a A e^3\right ) \sqrt {\frac {e \left (\frac {i \sqrt {a}}{\sqrt {c}}+x\right )}{d+e x}} \sqrt {-\frac {\frac {i \sqrt {a} e}{\sqrt {c}}-e x}{d+e x}} (d+e x)^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right )|\frac {\sqrt {c} d-i \sqrt {a} e}{\sqrt {c} d+i \sqrt {a} e}\right )+\sqrt {a} e \left (\sqrt {c} d+i \sqrt {a} e\right ) \left (32 B c d^2-24 i \sqrt {a} B \sqrt {c} d e-28 A c d e+5 a B e^2+21 i \sqrt {a} A \sqrt {c} e^2\right ) \sqrt {\frac {e \left (\frac {i \sqrt {a}}{\sqrt {c}}+x\right )}{d+e x}} \sqrt {-\frac {\frac {i \sqrt {a} e}{\sqrt {c}}-e x}{d+e x}} (d+e x)^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right ),\frac {\sqrt {c} d-i \sqrt {a} e}{\sqrt {c} d+i \sqrt {a} e}\right )\right )}{e^6 \sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}} (d+e x)}\right )}{35 \sqrt {a+c x^2}} \]
(Sqrt[d + e*x]*((2*(a + c*x^2)*(-7*A*e*(5*a*e^2 + c*(8*d^2 + 2*d*e*x - e^2 *x^2)) + B*(5*a*e^2*(10*d + 3*e*x) + c*(64*d^3 + 16*d^2*e*x - 8*d*e^2*x^2 + 5*e^3*x^3))))/(e^4*(d + e*x)) + (8*(e^2*Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]] *(-32*B*c*d^3 + 28*A*c*d^2*e - 29*a*B*d*e^2 + 21*a*A*e^3)*(a + c*x^2) + Sq rt[c]*((-I)*Sqrt[c]*d + Sqrt[a]*e)*(-32*B*c*d^3 + 28*A*c*d^2*e - 29*a*B*d* e^2 + 21*a*A*e^3)*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]*e*(Sqrt[c]*d + I*Sqrt[a]*e)*(32*B*c *d^2 - (24*I)*Sqrt[a]*B*Sqrt[c]*d*e - 28*A*c*d*e + 5*a*B*e^2 + (21*I)*Sqrt [a]*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*ArcSin h[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))))/(35*Sqrt[a + c*x^2])
Time = 1.08 (sec) , antiderivative size = 803, normalized size of antiderivative = 1.79, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.346, Rules used = {681, 25, 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.
| \(\displaystyle \int \frac {\left (a+c x^2\right )^{3/2} (A+B x)}{(d+e x)^{3/2}} \, dx\) |
| \(\Big \downarrow \) 681 |
| \(\displaystyle \frac {2 \left (a+c x^2\right )^{3/2} (-7 A e+8 B d+B e x)}{7 e^2 \sqrt {d+e x}}-\frac {6 \int -\frac {(a B e-c (8 B d-7 A e) x) \sqrt {c x^2+a}}{\sqrt {d+e x}}dx}{7 e^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {6 \int \frac {(a B e-c (8 B d-7 A e) x) \sqrt {c x^2+a}}{\sqrt {d+e x}}dx}{7 e^2}+\frac {2 \left (a+c x^2\right )^{3/2} (-7 A e+8 B d+B e x)}{7 e^2 \sqrt {d+e x}}\) |
| \(\Big \downarrow \) 682 |
| \(\displaystyle \frac {6 \left (\frac {4 \int \frac {c \left (a e \left (5 a B e^2+c d (8 B d-7 A e)\right )-c \left (32 B c d^3-28 A c e d^2+29 a B e^2 d-21 a A e^3\right ) x\right )}{2 \sqrt {d+e x} \sqrt {c x^2+a}}dx}{15 c e^2}+\frac {2 \sqrt {a+c x^2} \sqrt {d+e x} \left (5 a B e^2-3 c e x (8 B d-7 A e)+4 c d (8 B d-7 A e)\right )}{15 e^2}\right )}{7 e^2}+\frac {2 \left (a+c x^2\right )^{3/2} (-7 A e+8 B d+B e x)}{7 e^2 \sqrt {d+e x}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {6 \left (\frac {2 \int \frac {a e \left (5 a B e^2+c d (8 B d-7 A e)\right )-c \left (32 B c d^3-28 A c e d^2+29 a B e^2 d-21 a A e^3\right ) x}{\sqrt {d+e x} \sqrt {c x^2+a}}dx}{15 e^2}+\frac {2 \sqrt {a+c x^2} \sqrt {d+e x} \left (5 a B e^2-3 c e x (8 B d-7 A e)+4 c d (8 B d-7 A e)\right )}{15 e^2}\right )}{7 e^2}+\frac {2 \left (a+c x^2\right )^{3/2} (-7 A e+8 B d+B e x)}{7 e^2 \sqrt {d+e x}}\) |
| \(\Big \downarrow \) 599 |
| \(\displaystyle \frac {6 \left (\frac {2 \sqrt {a+c x^2} \sqrt {d+e x} \left (5 a B e^2-3 c e x (8 B d-7 A e)+4 c d (8 B d-7 A e)\right )}{15 e^2}-\frac {4 \int -\frac {\left (c d^2+a e^2\right ) \left (32 B c d^2-28 A c e d+5 a B e^2\right )-c \left (32 B c d^3-28 A c e d^2+29 a B e^2 d-21 a A e^3\right ) (d+e x)}{\sqrt {\frac {c d^2}{e^2}-\frac {2 c (d+e x) d}{e^2}+\frac {c (d+e x)^2}{e^2}+a}}d\sqrt {d+e x}}{15 e^4}\right )}{7 e^2}+\frac {2 \left (a+c x^2\right )^{3/2} (-7 A e+8 B d+B e x)}{7 e^2 \sqrt {d+e x}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {6 \left (\frac {4 \int \frac {\left (c d^2+a e^2\right ) \left (32 B c d^2-28 A c e d+5 a B e^2\right )-c \left (32 B c d^3-28 A c e d^2+29 a B e^2 d-21 a A e^3\right ) (d+e x)}{\sqrt {\frac {c d^2}{e^2}-\frac {2 c (d+e x) d}{e^2}+\frac {c (d+e x)^2}{e^2}+a}}d\sqrt {d+e x}}{15 e^4}+\frac {2 \sqrt {a+c x^2} \sqrt {d+e x} \left (5 a B e^2-3 c e x (8 B d-7 A e)+4 c d (8 B d-7 A e)\right )}{15 e^2}\right )}{7 e^2}+\frac {2 \left (a+c x^2\right )^{3/2} (-7 A e+8 B d+B e x)}{7 e^2 \sqrt {d+e x}}\) |
| \(\Big \downarrow \) 1511 |
| \(\displaystyle \frac {6 \left (\frac {2 \sqrt {a+c x^2} \sqrt {d+e x} \left (5 a B e^2-3 c e x (8 B d-7 A e)+4 c d (8 B d-7 A e)\right )}{15 e^2}-\frac {4 \left (-\sqrt {a e^2+c d^2} \left (\sqrt {a e^2+c d^2} \left (5 a B e^2-28 A c d e+32 B c d^2\right )-\sqrt {c} \left (-21 a A e^3+29 a B d e^2-28 A c d^2 e+32 B c d^3\right )\right ) \int \frac {1}{\sqrt {\frac {c d^2}{e^2}-\frac {2 c (d+e x) d}{e^2}+\frac {c (d+e x)^2}{e^2}+a}}d\sqrt {d+e x}-\sqrt {c} \sqrt {a e^2+c d^2} \left (-21 a A e^3+29 a B d e^2-28 A c d^2 e+32 B c d^3\right ) \int \frac {1-\frac {\sqrt {c} (d+e x)}{\sqrt {c d^2+a e^2}}}{\sqrt {\frac {c d^2}{e^2}-\frac {2 c (d+e x) d}{e^2}+\frac {c (d+e x)^2}{e^2}+a}}d\sqrt {d+e x}\right )}{15 e^4}\right )}{7 e^2}+\frac {2 \left (a+c x^2\right )^{3/2} (-7 A e+8 B d+B e x)}{7 e^2 \sqrt {d+e x}}\) |
| \(\Big \downarrow \) 1416 |
| \(\displaystyle \frac {6 \left (\frac {2 \sqrt {a+c x^2} \sqrt {d+e x} \left (5 a B e^2-3 c e x (8 B d-7 A e)+4 c d (8 B d-7 A e)\right )}{15 e^2}-\frac {4 \left (-\sqrt {c} \sqrt {a e^2+c d^2} \left (-21 a A e^3+29 a B d e^2-28 A c d^2 e+32 B c d^3\right ) \int \frac {1-\frac {\sqrt {c} (d+e x)}{\sqrt {c d^2+a e^2}}}{\sqrt {\frac {c d^2}{e^2}-\frac {2 c (d+e x) d}{e^2}+\frac {c (d+e x)^2}{e^2}+a}}d\sqrt {d+e x}-\frac {\left (a e^2+c d^2\right )^{3/4} \left (\frac {\sqrt {c} (d+e x)}{\sqrt {a e^2+c d^2}}+1\right ) \sqrt {\frac {a+\frac {c d^2}{e^2}-\frac {2 c d (d+e x)}{e^2}+\frac {c (d+e x)^2}{e^2}}{\left (a+\frac {c d^2}{e^2}\right ) \left (\frac {\sqrt {c} (d+e x)}{\sqrt {a e^2+c d^2}}+1\right )^2}} \left (\sqrt {a e^2+c d^2} \left (5 a B e^2-28 A c d e+32 B c d^2\right )-\sqrt {c} \left (-21 a A e^3+29 a B d e^2-28 A c d^2 e+32 B c d^3\right )\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt [4]{c d^2+a e^2}}\right ),\frac {1}{2} \left (\frac {\sqrt {c} d}{\sqrt {c d^2+a e^2}}+1\right )\right )}{2 \sqrt [4]{c} \sqrt {a+\frac {c d^2}{e^2}-\frac {2 c d (d+e x)}{e^2}+\frac {c (d+e x)^2}{e^2}}}\right )}{15 e^4}\right )}{7 e^2}+\frac {2 \left (a+c x^2\right )^{3/2} (-7 A e+8 B d+B e x)}{7 e^2 \sqrt {d+e x}}\) |
| \(\Big \downarrow \) 1509 |
| \(\displaystyle \frac {2 (8 B d-7 A e+B e x) \left (c x^2+a\right )^{3/2}}{7 e^2 \sqrt {d+e x}}+\frac {6 \left (\frac {2 \sqrt {d+e x} \left (5 a B e^2-3 c (8 B d-7 A e) x e+4 c d (8 B d-7 A e)\right ) \sqrt {c x^2+a}}{15 e^2}-\frac {4 \left (-\sqrt {c} \sqrt {c d^2+a e^2} \left (32 B c d^3-28 A c e d^2+29 a B e^2 d-21 a A e^3\right ) \left (\frac {\sqrt [4]{c d^2+a e^2} \left (\frac {\sqrt {c} (d+e x)}{\sqrt {c d^2+a e^2}}+1\right ) \sqrt {\frac {\frac {c d^2}{e^2}-\frac {2 c (d+e x) d}{e^2}+\frac {c (d+e x)^2}{e^2}+a}{\left (\frac {c d^2}{e^2}+a\right ) \left (\frac {\sqrt {c} (d+e x)}{\sqrt {c d^2+a e^2}}+1\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt [4]{c d^2+a e^2}}\right )|\frac {1}{2} \left (\frac {\sqrt {c} d}{\sqrt {c d^2+a e^2}}+1\right )\right )}{\sqrt [4]{c} \sqrt {\frac {c d^2}{e^2}-\frac {2 c (d+e x) d}{e^2}+\frac {c (d+e x)^2}{e^2}+a}}-\frac {\sqrt {d+e x} \sqrt {\frac {c d^2}{e^2}-\frac {2 c (d+e x) d}{e^2}+\frac {c (d+e x)^2}{e^2}+a}}{\left (\frac {c d^2}{e^2}+a\right ) \left (\frac {\sqrt {c} (d+e x)}{\sqrt {c d^2+a e^2}}+1\right )}\right )-\frac {\left (c d^2+a e^2\right )^{3/4} \left (\sqrt {c d^2+a e^2} \left (32 B c d^2-28 A c e d+5 a B e^2\right )-\sqrt {c} \left (32 B c d^3-28 A c e d^2+29 a B e^2 d-21 a A e^3\right )\right ) \left (\frac {\sqrt {c} (d+e x)}{\sqrt {c d^2+a e^2}}+1\right ) \sqrt {\frac {\frac {c d^2}{e^2}-\frac {2 c (d+e x) d}{e^2}+\frac {c (d+e x)^2}{e^2}+a}{\left (\frac {c d^2}{e^2}+a\right ) \left (\frac {\sqrt {c} (d+e x)}{\sqrt {c d^2+a e^2}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt [4]{c d^2+a e^2}}\right ),\frac {1}{2} \left (\frac {\sqrt {c} d}{\sqrt {c d^2+a e^2}}+1\right )\right )}{2 \sqrt [4]{c} \sqrt {\frac {c d^2}{e^2}-\frac {2 c (d+e x) d}{e^2}+\frac {c (d+e x)^2}{e^2}+a}}\right )}{15 e^4}\right )}{7 e^2}\) |
(2*(8*B*d - 7*A*e + B*e*x)*(a + c*x^2)^(3/2))/(7*e^2*Sqrt[d + e*x]) + (6*( (2*Sqrt[d + e*x]*(5*a*B*e^2 + 4*c*d*(8*B*d - 7*A*e) - 3*c*e*(8*B*d - 7*A*e )*x)*Sqrt[a + c*x^2])/(15*e^2) - (4*(-(Sqrt[c]*Sqrt[c*d^2 + a*e^2]*(32*B*c *d^3 - 28*A*c*d^2*e + 29*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 + (Sq rt[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]*(32*B*c*d^2 - 28*A* c*d*e + 5*a*B*e^2) - Sqrt[c]*(32*B*c*d^3 - 28*A*c*d^2*e + 29*a*B*d*e^2 - 2 1*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*e^2)
3.15.77.3.1 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[((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[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. \(1185\) vs. \(2(376)=752\).
Time = 3.74 (sec) , antiderivative size = 1186, normalized size of antiderivative = 2.65
| method | result | size |
| elliptic | \(\text {Expression too large to display}\) | \(1186\) |
| risch | \(\text {Expression too large to display}\) | \(1371\) |
| default | \(\text {Expression too large to display}\) | \(2561\) |
((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*a*e^3+A*c*d^2*e-B*a*d*e^2-B*c*d^3)/e^5/((x+d/e)*(c*e*x^2+a*e))^(1/2)+2/ 7*B*c/e^2*x^2*(c*e*x^3+c*d*x^2+a*e*x+a*d)^(1/2)+2/5*(c^2/e^2*(A*e-B*d)-6/7 *B*c^2/e^2*d)/c/e*x*(c*e*x^3+c*d*x^2+a*e*x+a*d)^(1/2)+2/3*(-c/e^3*(A*c*d*e -2*B*a*e^2-B*c*d^2)-5/7*B*c/e*a-4/5*(c^2/e^2*(A*e-B*d)-6/7*B*c^2/e^2*d)/e* d)/c/e*(c*e*x^3+c*d*x^2+a*e*x+a*d)^(1/2)+2*(-(2*A*a*c*d*e^3+A*c^2*d^3*e-B* a^2*e^4-2*B*a*c*d^2*e^2-B*c^2*d^4)/e^5+(A*a*e^3+A*c*d^2*e-B*a*d*e^2-B*c*d^ 3)*c/e^5*d-2/5*(c^2/e^2*(A*e-B*d)-6/7*B*c^2/e^2*d)/c/e*a*d-1/3*(-c/e^3*(A* c*d*e-2*B*a*e^2-B*c*d^2)-5/7*B*c/e*a-4/5*(c^2/e^2*(A*e-B*d)-6/7*B*c^2/e^2* 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*(c/e^4*(2*A*a*e^3+A*c*d^2*e-2*B*a*d*e^2-B*c*d^3)+(A*a*e^3+A*c*d^2 *e-B*a*d*e^2-B*c*d^3)*c/e^4-4/7*B*c/e^2*a*d-3/5*(c^2/e^2*(A*e-B*d)-6/7*B*c ^2/e^2*d)/c*a-2/3*(-c/e^3*(A*c*d*e-2*B*a*e^2-B*c*d^2)-5/7*B*c/e*a-4/5*(c^2 /e^2*(A*e-B*d)-6/7*B*c^2/e^2*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...
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.11 (sec) , antiderivative size = 503, normalized size of antiderivative = 1.12 \[ \int \frac {(A+B x) \left (a+c x^2\right )^{3/2}}{(d+e x)^{3/2}} \, dx=\frac {2 \, {\left (4 \, {\left (32 \, B c^{2} d^{5} - 28 \, A c^{2} d^{4} e + 53 \, B a c d^{3} e^{2} - 42 \, A a c d^{2} e^{3} + 15 \, B a^{2} d e^{4} + {\left (32 \, B c^{2} d^{4} e - 28 \, A c^{2} d^{3} e^{2} + 53 \, B a c d^{2} e^{3} - 42 \, A a c d e^{4} + 15 \, B a^{2} e^{5}\right )} x\right )} \sqrt {c e} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, \frac {3 \, e x + d}{3 \, e}\right ) + 12 \, {\left (32 \, B c^{2} d^{4} e - 28 \, A c^{2} d^{3} e^{2} + 29 \, B a c d^{2} e^{3} - 21 \, A a c d e^{4} + {\left (32 \, B c^{2} d^{3} e^{2} - 28 \, A c^{2} d^{2} e^{3} + 29 \, B a c d e^{4} - 21 \, A a c e^{5}\right )} x\right )} \sqrt {c e} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, \frac {3 \, e x + d}{3 \, e}\right )\right ) + 3 \, {\left (5 \, B c^{2} e^{5} x^{3} + 64 \, B c^{2} d^{3} e^{2} - 56 \, A c^{2} d^{2} e^{3} + 50 \, B a c d e^{4} - 35 \, A a c e^{5} - {\left (8 \, B c^{2} d e^{4} - 7 \, A c^{2} e^{5}\right )} x^{2} + {\left (16 \, B c^{2} d^{2} e^{3} - 14 \, A c^{2} d e^{4} + 15 \, B a c e^{5}\right )} x\right )} \sqrt {c x^{2} + a} \sqrt {e x + d}\right )}}{105 \, {\left (c e^{7} x + c d e^{6}\right )}} \]
2/105*(4*(32*B*c^2*d^5 - 28*A*c^2*d^4*e + 53*B*a*c*d^3*e^2 - 42*A*a*c*d^2* e^3 + 15*B*a^2*d*e^4 + (32*B*c^2*d^4*e - 28*A*c^2*d^3*e^2 + 53*B*a*c*d^2*e ^3 - 42*A*a*c*d*e^4 + 15*B*a^2*e^5)*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) + 12*(32*B*c^2*d^4*e - 28*A*c^2*d^3*e^2 + 29*B*a*c*d^2*e^3 - 21*A*a* c*d*e^4 + (32*B*c^2*d^3*e^2 - 28*A*c^2*d^2*e^3 + 29*B*a*c*d*e^4 - 21*A*a*c *e^5)*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*(5*B*c^2*e^ 5*x^3 + 64*B*c^2*d^3*e^2 - 56*A*c^2*d^2*e^3 + 50*B*a*c*d*e^4 - 35*A*a*c*e^ 5 - (8*B*c^2*d*e^4 - 7*A*c^2*e^5)*x^2 + (16*B*c^2*d^2*e^3 - 14*A*c^2*d*e^4 + 15*B*a*c*e^5)*x)*sqrt(c*x^2 + a)*sqrt(e*x + d))/(c*e^7*x + c*d*e^6)
\[ \int \frac {(A+B x) \left (a+c x^2\right )^{3/2}}{(d+e x)^{3/2}} \, dx=\int \frac {\left (A + B x\right ) \left (a + c x^{2}\right )^{\frac {3}{2}}}{\left (d + e x\right )^{\frac {3}{2}}}\, dx \]
\[ \int \frac {(A+B x) \left (a+c x^2\right )^{3/2}}{(d+e x)^{3/2}} \, dx=\int { \frac {{\left (c x^{2} + a\right )}^{\frac {3}{2}} {\left (B x + A\right )}}{{\left (e x + d\right )}^{\frac {3}{2}}} \,d x } \]
\[ \int \frac {(A+B x) \left (a+c x^2\right )^{3/2}}{(d+e x)^{3/2}} \, dx=\int { \frac {{\left (c x^{2} + a\right )}^{\frac {3}{2}} {\left (B x + A\right )}}{{\left (e x + d\right )}^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {(A+B x) \left (a+c x^2\right )^{3/2}}{(d+e x)^{3/2}} \, dx=\int \frac {{\left (c\,x^2+a\right )}^{3/2}\,\left (A+B\,x\right )}{{\left (d+e\,x\right )}^{3/2}} \,d x \]