Integrand size = 26, antiderivative size = 541 \[ \int \frac {(A+B x) \left (a+c x^2\right )^{3/2}}{(d+e x)^{7/2}} \, dx=\frac {4 c \left (32 B c d^3-12 A c d^2 e+29 a B d e^2-9 a A e^3+e \left (8 B c d^2-3 A c d e+5 a B e^2\right ) x\right ) \sqrt {a+c x^2}}{15 e^4 \left (c d^2+a e^2\right ) \sqrt {d+e x}}-\frac {2 \left (2 B \left (4 c d^3+a d e^2\right )-3 A \left (c d^2 e-a e^3\right )+e \left (11 B c d^2-6 A c d e+5 a B e^2\right ) x\right ) \left (a+c x^2\right )^{3/2}}{15 e^2 \left (c d^2+a e^2\right ) (d+e x)^{5/2}}+\frac {8 \sqrt {-a} c^{3/2} \left (32 B c d^3-12 A c d^2 e+29 a B d e^2-9 a A e^3\right ) \sqrt {d+e x} \sqrt {1+\frac {c x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{15 e^5 \left (c d^2+a e^2\right ) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {a+c x^2}}-\frac {8 \sqrt {-a} \sqrt {c} \left (32 B c d^2-12 A c d e+5 a B e^2\right ) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {1+\frac {c x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right ),-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{15 e^5 \sqrt {d+e x} \sqrt {a+c x^2}} \]
-2/15*(2*B*(a*d*e^2+4*c*d^3)-3*A*(-a*e^3+c*d^2*e)+e*(-6*A*c*d*e+5*B*a*e^2+ 11*B*c*d^2)*x)*(c*x^2+a)^(3/2)/e^2/(a*e^2+c*d^2)/(e*x+d)^(5/2)+4/15*c*(32* B*c*d^3-12*A*c*d^2*e+29*B*a*d*e^2-9*A*a*e^3+e*(-3*A*c*d*e+5*B*a*e^2+8*B*c* d^2)*x)*(c*x^2+a)^(1/2)/e^4/(a*e^2+c*d^2)/(e*x+d)^(1/2)+8/15*c^(3/2)*(-9*A *a*e^3-12*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)*(e*x+d)^(1/2)*(1+c*x^2/a)^(1/2)/e^5/(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)-8/15*(-12*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)*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 = 24.99 (sec) , antiderivative size = 705, normalized size of antiderivative = 1.30 \[ \int \frac {(A+B x) \left (a+c x^2\right )^{3/2}}{(d+e x)^{7/2}} \, dx=\frac {\sqrt {d+e x} \left (\frac {2 \left (a+c x^2\right ) \left (5 B c+\frac {3 (B d-A e) \left (c d^2+a e^2\right )}{(d+e x)^3}+\frac {-17 B c d^2+12 A c d e-5 a B e^2}{(d+e x)^2}+\frac {c \left (73 B c d^3-33 A c d^2 e+61 a B d e^2-21 a A e^3\right )}{\left (c d^2+a e^2\right ) (d+e x)}\right )}{e^4}-\frac {8 c \left (e^2 \sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}} \left (32 B c d^3-12 A c d^2 e+29 a B d e^2-9 a A e^3\right ) \left (a+c x^2\right )+i \sqrt {c} \left (\sqrt {c} d+i \sqrt {a} e\right ) \left (-32 B c d^3+12 A c d^2 e-29 a B d e^2+9 a A e^3\right ) \sqrt {\frac {e \left (\frac {i \sqrt {a}}{\sqrt {c}}+x\right )}{d+e x}} \sqrt {-\frac {\frac {i \sqrt {a} e}{\sqrt {c}}-e x}{d+e x}} (d+e x)^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right )|\frac {\sqrt {c} d-i \sqrt {a} e}{\sqrt {c} d+i \sqrt {a} e}\right )-\sqrt {a} e \left (\sqrt {c} d+i \sqrt {a} e\right ) \left (32 B c d^2-24 i \sqrt {a} B \sqrt {c} d e-12 A c d e+5 a B e^2+9 i \sqrt {a} A \sqrt {c} e^2\right ) \sqrt {\frac {e \left (\frac {i \sqrt {a}}{\sqrt {c}}+x\right )}{d+e x}} \sqrt {-\frac {\frac {i \sqrt {a} e}{\sqrt {c}}-e x}{d+e x}} (d+e x)^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right ),\frac {\sqrt {c} d-i \sqrt {a} e}{\sqrt {c} d+i \sqrt {a} e}\right )\right )}{e^6 \sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}} \left (c d^2+a e^2\right ) (d+e x)}\right )}{15 \sqrt {a+c x^2}} \]
(Sqrt[d + e*x]*((2*(a + c*x^2)*(5*B*c + (3*(B*d - A*e)*(c*d^2 + a*e^2))/(d + e*x)^3 + (-17*B*c*d^2 + 12*A*c*d*e - 5*a*B*e^2)/(d + e*x)^2 + (c*(73*B* c*d^3 - 33*A*c*d^2*e + 61*a*B*d*e^2 - 21*a*A*e^3))/((c*d^2 + a*e^2)*(d + e *x))))/e^4 - (8*c*(e^2*Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]*(32*B*c*d^3 - 12*A *c*d^2*e + 29*a*B*d*e^2 - 9*a*A*e^3)*(a + c*x^2) + I*Sqrt[c]*(Sqrt[c]*d + I*Sqrt[a]*e)*(-32*B*c*d^3 + 12*A*c*d^2*e - 29*a*B*d*e^2 + 9*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 - 12*A*c*d*e + 5*a*B*e^2 + (9*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*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]]*(c*d^2 + a*e^2)*(d + e*x)))) /(15*Sqrt[a + c*x^2])
Time = 1.17 (sec) , antiderivative size = 894, normalized size of antiderivative = 1.65, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {680, 27, 681, 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.
| \(\displaystyle \int \frac {\left (a+c x^2\right )^{3/2} (A+B x)}{(d+e x)^{7/2}} \, dx\) |
| \(\Big \downarrow \) 680 |
| \(\displaystyle -\frac {2 \int \frac {c \left (3 a e (B d-A e)-\left (8 B c d^2-3 A c e d+5 a B e^2\right ) x\right ) \sqrt {c x^2+a}}{(d+e x)^{3/2}}dx}{5 e^2 \left (a e^2+c d^2\right )}-\frac {2 \left (a+c x^2\right )^{3/2} \left (e x \left (5 a B e^2-6 A c d e+11 B c d^2\right )-3 A \left (c d^2 e-a e^3\right )+2 B \left (a d e^2+4 c d^3\right )\right )}{15 e^2 (d+e x)^{5/2} \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {2 c \int \frac {\left (3 a e (B d-A e)-\left (8 B c d^2-3 A c e d+5 a B e^2\right ) x\right ) \sqrt {c x^2+a}}{(d+e x)^{3/2}}dx}{5 e^2 \left (a e^2+c d^2\right )}-\frac {2 \left (a+c x^2\right )^{3/2} \left (e x \left (5 a B e^2-6 A c d e+11 B c d^2\right )-3 A \left (c d^2 e-a e^3\right )+2 B \left (a d e^2+4 c d^3\right )\right )}{15 e^2 (d+e x)^{5/2} \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 681 |
| \(\displaystyle -\frac {2 c \left (-\frac {2 \int \frac {a e \left (8 B c d^2-3 A c e d+5 a B e^2\right )-c \left (32 B c d^3-12 A c e d^2+29 a B e^2 d-9 a A e^3\right ) x}{\sqrt {d+e x} \sqrt {c x^2+a}}dx}{3 e^2}-\frac {2 \sqrt {a+c x^2} \left (e x \left (5 a B e^2-3 A c d e+8 B c d^2\right )-9 a A e^3+29 a B d e^2-12 A c d^2 e+32 B c d^3\right )}{3 e^2 \sqrt {d+e x}}\right )}{5 e^2 \left (a e^2+c d^2\right )}-\frac {2 \left (a+c x^2\right )^{3/2} \left (e x \left (5 a B e^2-6 A c d e+11 B c d^2\right )-3 A \left (c d^2 e-a e^3\right )+2 B \left (a d e^2+4 c d^3\right )\right )}{15 e^2 (d+e x)^{5/2} \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 599 |
| \(\displaystyle -\frac {2 c \left (\frac {4 \int -\frac {\left (c d^2+a e^2\right ) \left (32 B c d^2-12 A c e d+5 a B e^2\right )-c \left (32 B c d^3-12 A c e d^2+29 a B e^2 d-9 a A e^3\right ) (d+e x)}{\sqrt {\frac {c d^2}{e^2}-\frac {2 c (d+e x) d}{e^2}+\frac {c (d+e x)^2}{e^2}+a}}d\sqrt {d+e x}}{3 e^4}-\frac {2 \sqrt {a+c x^2} \left (e x \left (5 a B e^2-3 A c d e+8 B c d^2\right )-9 a A e^3+29 a B d e^2-12 A c d^2 e+32 B c d^3\right )}{3 e^2 \sqrt {d+e x}}\right )}{5 e^2 \left (a e^2+c d^2\right )}-\frac {2 \left (a+c x^2\right )^{3/2} \left (e x \left (5 a B e^2-6 A c d e+11 B c d^2\right )-3 A \left (c d^2 e-a e^3\right )+2 B \left (a d e^2+4 c d^3\right )\right )}{15 e^2 (d+e x)^{5/2} \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {2 c \left (-\frac {4 \int \frac {\left (c d^2+a e^2\right ) \left (32 B c d^2-12 A c e d+5 a B e^2\right )-c \left (32 B c d^3-12 A c e d^2+29 a B e^2 d-9 a A e^3\right ) (d+e x)}{\sqrt {\frac {c d^2}{e^2}-\frac {2 c (d+e x) d}{e^2}+\frac {c (d+e x)^2}{e^2}+a}}d\sqrt {d+e x}}{3 e^4}-\frac {2 \sqrt {a+c x^2} \left (e x \left (5 a B e^2-3 A c d e+8 B c d^2\right )-9 a A e^3+29 a B d e^2-12 A c d^2 e+32 B c d^3\right )}{3 e^2 \sqrt {d+e x}}\right )}{5 e^2 \left (a e^2+c d^2\right )}-\frac {2 \left (a+c x^2\right )^{3/2} \left (e x \left (5 a B e^2-6 A c d e+11 B c d^2\right )-3 A \left (c d^2 e-a e^3\right )+2 B \left (a d e^2+4 c d^3\right )\right )}{15 e^2 (d+e x)^{5/2} \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 1511 |
| \(\displaystyle -\frac {2 c \left (\frac {4 \left (-\sqrt {a e^2+c d^2} \left (\sqrt {a e^2+c d^2} \left (5 a B e^2-12 A c d e+32 B c d^2\right )-\sqrt {c} \left (-9 a A e^3+29 a B d e^2-12 A c d^2 e+32 B c d^3\right )\right ) \int \frac {1}{\sqrt {\frac {c d^2}{e^2}-\frac {2 c (d+e x) d}{e^2}+\frac {c (d+e x)^2}{e^2}+a}}d\sqrt {d+e x}-\sqrt {c} \sqrt {a e^2+c d^2} \left (-9 a A e^3+29 a B d e^2-12 A c d^2 e+32 B c d^3\right ) \int \frac {1-\frac {\sqrt {c} (d+e x)}{\sqrt {c d^2+a e^2}}}{\sqrt {\frac {c d^2}{e^2}-\frac {2 c (d+e x) d}{e^2}+\frac {c (d+e x)^2}{e^2}+a}}d\sqrt {d+e x}\right )}{3 e^4}-\frac {2 \sqrt {a+c x^2} \left (e x \left (5 a B e^2-3 A c d e+8 B c d^2\right )-9 a A e^3+29 a B d e^2-12 A c d^2 e+32 B c d^3\right )}{3 e^2 \sqrt {d+e x}}\right )}{5 e^2 \left (a e^2+c d^2\right )}-\frac {2 \left (a+c x^2\right )^{3/2} \left (e x \left (5 a B e^2-6 A c d e+11 B c d^2\right )-3 A \left (c d^2 e-a e^3\right )+2 B \left (a d e^2+4 c d^3\right )\right )}{15 e^2 (d+e x)^{5/2} \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 1416 |
| \(\displaystyle -\frac {2 c \left (\frac {4 \left (-\sqrt {c} \sqrt {a e^2+c d^2} \left (-9 a A e^3+29 a B d e^2-12 A c d^2 e+32 B c d^3\right ) \int \frac {1-\frac {\sqrt {c} (d+e x)}{\sqrt {c d^2+a e^2}}}{\sqrt {\frac {c d^2}{e^2}-\frac {2 c (d+e x) d}{e^2}+\frac {c (d+e x)^2}{e^2}+a}}d\sqrt {d+e x}-\frac {\left (a e^2+c d^2\right )^{3/4} \left (\frac {\sqrt {c} (d+e x)}{\sqrt {a e^2+c d^2}}+1\right ) \sqrt {\frac {a+\frac {c d^2}{e^2}-\frac {2 c d (d+e x)}{e^2}+\frac {c (d+e x)^2}{e^2}}{\left (a+\frac {c d^2}{e^2}\right ) \left (\frac {\sqrt {c} (d+e x)}{\sqrt {a e^2+c d^2}}+1\right )^2}} \left (\sqrt {a e^2+c d^2} \left (5 a B e^2-12 A c d e+32 B c d^2\right )-\sqrt {c} \left (-9 a A e^3+29 a B d e^2-12 A c d^2 e+32 B c d^3\right )\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt [4]{c d^2+a e^2}}\right ),\frac {1}{2} \left (\frac {\sqrt {c} d}{\sqrt {c d^2+a e^2}}+1\right )\right )}{2 \sqrt [4]{c} \sqrt {a+\frac {c d^2}{e^2}-\frac {2 c d (d+e x)}{e^2}+\frac {c (d+e x)^2}{e^2}}}\right )}{3 e^4}-\frac {2 \sqrt {a+c x^2} \left (e x \left (5 a B e^2-3 A c d e+8 B c d^2\right )-9 a A e^3+29 a B d e^2-12 A c d^2 e+32 B c d^3\right )}{3 e^2 \sqrt {d+e x}}\right )}{5 e^2 \left (a e^2+c d^2\right )}-\frac {2 \left (a+c x^2\right )^{3/2} \left (e x \left (5 a B e^2-6 A c d e+11 B c d^2\right )-3 A \left (c d^2 e-a e^3\right )+2 B \left (a d e^2+4 c d^3\right )\right )}{15 e^2 (d+e x)^{5/2} \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 1509 |
| \(\displaystyle -\frac {2 \left (2 B \left (4 c d^3+a e^2 d\right )-3 A \left (c d^2 e-a e^3\right )+e \left (11 B c d^2-6 A c e d+5 a B e^2\right ) x\right ) \left (c x^2+a\right )^{3/2}}{15 e^2 \left (c d^2+a e^2\right ) (d+e x)^{5/2}}-\frac {2 c \left (\frac {4 \left (-\sqrt {c} \sqrt {c d^2+a e^2} \left (32 B c d^3-12 A c e d^2+29 a B e^2 d-9 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^2-12 A c e d+5 a B e^2\right )-\sqrt {c} \left (32 B c d^3-12 A c e d^2+29 a B e^2 d-9 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 )}{3 e^4}-\frac {2 \left (32 B c d^3-12 A c e d^2+29 a B e^2 d-9 a A e^3+e \left (8 B c d^2-3 A c e d+5 a B e^2\right ) x\right ) \sqrt {c x^2+a}}{3 e^2 \sqrt {d+e x}}\right )}{5 e^2 \left (c d^2+a e^2\right )}\) |
(-2*(2*B*(4*c*d^3 + a*d*e^2) - 3*A*(c*d^2*e - a*e^3) + e*(11*B*c*d^2 - 6*A *c*d*e + 5*a*B*e^2)*x)*(a + c*x^2)^(3/2))/(15*e^2*(c*d^2 + a*e^2)*(d + e*x )^(5/2)) - (2*c*((-2*(32*B*c*d^3 - 12*A*c*d^2*e + 29*a*B*d*e^2 - 9*a*A*e^3 + e*(8*B*c*d^2 - 3*A*c*d*e + 5*a*B*e^2)*x)*Sqrt[a + c*x^2])/(3*e^2*Sqrt[d + e*x]) + (4*(-(Sqrt[c]*Sqrt[c*d^2 + a*e^2]*(32*B*c*d^3 - 12*A*c*d^2*e + 29*a*B*d*e^2 - 9*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]*(32*B*c*d^2 - 12*A*c*d*e + 5*a*B*e^2) - Sq rt[c]*(32*B*c*d^3 - 12*A*c*d^2*e + 29*a*B*d*e^2 - 9*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))/S qrt[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])))/(3*e...
3.15.79.3.1 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[((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. \(978\) vs. \(2(469)=938\).
Time = 5.78 (sec) , antiderivative size = 979, 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 \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}}{5 e^{7} \left (x +\frac {d}{e}\right )^{3}}+\frac {2 \left (12 A c d e -5 B a \,e^{2}-17 B c \,d^{2}\right ) \sqrt {c e \,x^{3}+c d \,x^{2}+a e x +a d}}{15 e^{6} \left (x +\frac {d}{e}\right )^{2}}-\frac {2 \left (c e \,x^{2}+a e \right ) c \left (21 A a \,e^{3}+33 A c \,d^{2} e -61 B a d \,e^{2}-73 B c \,d^{3}\right )}{15 \left (e^{2} a +c \,d^{2}\right ) e^{5} \sqrt {\left (x +\frac {d}{e}\right ) \left (c e \,x^{2}+a e \right )}}+\frac {2 B c \sqrt {c e \,x^{3}+c d \,x^{2}+a e x +a d}}{3 e^{4}}+\frac {2 \left (-\frac {c \left (3 A c d e -2 B a \,e^{2}-6 B c \,d^{2}\right )}{e^{5}}+\frac {c \left (12 A c d e -5 B a \,e^{2}-17 B c \,d^{2}\right )}{15 e^{5}}+\frac {c^{2} d \left (21 A a \,e^{3}+33 A c \,d^{2} e -61 B a d \,e^{2}-73 B c \,d^{3}\right )}{15 e^{5} \left (e^{2} a +c \,d^{2}\right )}-\frac {B a c}{3 e^{3}}\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^{2} \left (A e -3 B d \right )}{e^{4}}+\frac {c^{2} \left (21 A a \,e^{3}+33 A c \,d^{2} e -61 B a d \,e^{2}-73 B c \,d^{3}\right )}{15 e^{4} \left (e^{2} a +c \,d^{2}\right )}-\frac {2 B \,c^{2} d}{3 e^{4}}\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}}\) | \(979\) |
| risch | \(\text {Expression too large to display}\) | \(2804\) |
| default | \(\text {Expression too large to display}\) | \(7383\) |
((e*x+d)*(c*x^2+a))^(1/2)/(e*x+d)^(1/2)/(c*x^2+a)^(1/2)*(-2/5*(A*a*e^3+A*c *d^2*e-B*a*d*e^2-B*c*d^3)/e^7*(c*e*x^3+c*d*x^2+a*e*x+a*d)^(1/2)/(x+d/e)^3+ 2/15*(12*A*c*d*e-5*B*a*e^2-17*B*c*d^2)/e^6*(c*e*x^3+c*d*x^2+a*e*x+a*d)^(1/ 2)/(x+d/e)^2-2/15*(c*e*x^2+a*e)/(a*e^2+c*d^2)/e^5*c*(21*A*a*e^3+33*A*c*d^2 *e-61*B*a*d*e^2-73*B*c*d^3)/((x+d/e)*(c*e*x^2+a*e))^(1/2)+2/3*B*c/e^4*(c*e *x^3+c*d*x^2+a*e*x+a*d)^(1/2)+2*(-c*(3*A*c*d*e-2*B*a*e^2-6*B*c*d^2)/e^5+1/ 15*c*(12*A*c*d*e-5*B*a*e^2-17*B*c*d^2)/e^5+1/15*c^2/e^5*d*(21*A*a*e^3+33*A *c*d^2*e-61*B*a*d*e^2-73*B*c*d^3)/(a*e^2+c*d^2)-1/3*B*a*c/e^3)*(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^2/e^4*(A*e-3 *B*d)+1/15*c^2/e^4*(21*A*a*e^3+33*A*c*d^2*e-61*B*a*d*e^2-73*B*c*d^3)/(a*e^ 2+c*d^2)-2/3*B*c^2/e^4*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 = 877, normalized size of antiderivative = 1.62 \[ \int \frac {(A+B x) \left (a+c x^2\right )^{3/2}}{(d+e x)^{7/2}} \, dx=\frac {2 \, {\left (4 \, {\left (32 \, B c^{2} d^{7} - 12 \, A c^{2} d^{6} e + 53 \, B a c d^{5} e^{2} - 18 \, A a c d^{4} e^{3} + 15 \, B a^{2} d^{3} e^{4} + {\left (32 \, B c^{2} d^{4} e^{3} - 12 \, A c^{2} d^{3} e^{4} + 53 \, B a c d^{2} e^{5} - 18 \, A a c d e^{6} + 15 \, B a^{2} e^{7}\right )} x^{3} + 3 \, {\left (32 \, B c^{2} d^{5} e^{2} - 12 \, A c^{2} d^{4} e^{3} + 53 \, B a c d^{3} e^{4} - 18 \, A a c d^{2} e^{5} + 15 \, B a^{2} d e^{6}\right )} x^{2} + 3 \, {\left (32 \, B c^{2} d^{6} e - 12 \, A c^{2} d^{5} e^{2} + 53 \, B a c d^{4} e^{3} - 18 \, A a c d^{3} e^{4} + 15 \, B a^{2} d^{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^{6} e - 12 \, A c^{2} d^{5} e^{2} + 29 \, B a c d^{4} e^{3} - 9 \, A a c d^{3} e^{4} + {\left (32 \, B c^{2} d^{3} e^{4} - 12 \, A c^{2} d^{2} e^{5} + 29 \, B a c d e^{6} - 9 \, A a c e^{7}\right )} x^{3} + 3 \, {\left (32 \, B c^{2} d^{4} e^{3} - 12 \, A c^{2} d^{3} e^{4} + 29 \, B a c d^{2} e^{5} - 9 \, A a c d e^{6}\right )} x^{2} + 3 \, {\left (32 \, B c^{2} d^{5} e^{2} - 12 \, A c^{2} d^{4} e^{3} + 29 \, B a c d^{3} e^{4} - 9 \, A a c d^{2} 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 (64 \, B c^{2} d^{5} e^{2} - 24 \, A c^{2} d^{4} e^{3} + 50 \, B a c d^{3} e^{4} - 15 \, A a c d^{2} e^{5} - 2 \, B a^{2} d e^{6} - 3 \, A a^{2} e^{7} + 5 \, {\left (B c^{2} d^{2} e^{5} + B a c e^{7}\right )} x^{3} + {\left (88 \, B c^{2} d^{3} e^{4} - 33 \, A c^{2} d^{2} e^{5} + 76 \, B a c d e^{6} - 21 \, A a c e^{7}\right )} x^{2} + {\left (144 \, B c^{2} d^{4} e^{3} - 54 \, A c^{2} d^{3} e^{4} + 115 \, B a c d^{2} e^{5} - 30 \, A a c d e^{6} - 5 \, B a^{2} e^{7}\right )} x\right )} \sqrt {c x^{2} + a} \sqrt {e x + d}\right )}}{45 \, {\left (c d^{5} e^{6} + a d^{3} e^{8} + {\left (c d^{2} e^{9} + a e^{11}\right )} x^{3} + 3 \, {\left (c d^{3} e^{8} + a d e^{10}\right )} x^{2} + 3 \, {\left (c d^{4} e^{7} + a d^{2} e^{9}\right )} x\right )}} \]
2/45*(4*(32*B*c^2*d^7 - 12*A*c^2*d^6*e + 53*B*a*c*d^5*e^2 - 18*A*a*c*d^4*e ^3 + 15*B*a^2*d^3*e^4 + (32*B*c^2*d^4*e^3 - 12*A*c^2*d^3*e^4 + 53*B*a*c*d^ 2*e^5 - 18*A*a*c*d*e^6 + 15*B*a^2*e^7)*x^3 + 3*(32*B*c^2*d^5*e^2 - 12*A*c^ 2*d^4*e^3 + 53*B*a*c*d^3*e^4 - 18*A*a*c*d^2*e^5 + 15*B*a^2*d*e^6)*x^2 + 3* (32*B*c^2*d^6*e - 12*A*c^2*d^5*e^2 + 53*B*a*c*d^4*e^3 - 18*A*a*c*d^3*e^4 + 15*B*a^2*d^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^6*e - 12*A*c^2*d^5*e^2 + 29*B*a*c*d^4*e^3 - 9*A*a*c*d^3*e^4 + (32*B* c^2*d^3*e^4 - 12*A*c^2*d^2*e^5 + 29*B*a*c*d*e^6 - 9*A*a*c*e^7)*x^3 + 3*(32 *B*c^2*d^4*e^3 - 12*A*c^2*d^3*e^4 + 29*B*a*c*d^2*e^5 - 9*A*a*c*d*e^6)*x^2 + 3*(32*B*c^2*d^5*e^2 - 12*A*c^2*d^4*e^3 + 29*B*a*c*d^3*e^4 - 9*A*a*c*d^2* 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*(64*B*c^2*d^ 5*e^2 - 24*A*c^2*d^4*e^3 + 50*B*a*c*d^3*e^4 - 15*A*a*c*d^2*e^5 - 2*B*a^2*d *e^6 - 3*A*a^2*e^7 + 5*(B*c^2*d^2*e^5 + B*a*c*e^7)*x^3 + (88*B*c^2*d^3*e^4 - 33*A*c^2*d^2*e^5 + 76*B*a*c*d*e^6 - 21*A*a*c*e^7)*x^2 + (144*B*c^2*d^4* e^3 - 54*A*c^2*d^3*e^4 + 115*B*a*c*d^2*e^5 - 30*A*a*c*d*e^6 - 5*B*a^2*e^7) *x)*sqrt(c*x^2 + a)*sqrt(e*x + d))/(c*d^5*e^6 + a*d^3*e^8 + (c*d^2*e^9 + a *e^11)*x^3 + 3*(c*d^3*e^8 + a*d*e^10)*x^2 + 3*(c*d^4*e^7 + a*d^2*e^9)*x...
\[ \int \frac {(A+B x) \left (a+c x^2\right )^{3/2}}{(d+e x)^{7/2}} \, dx=\int \frac {\left (A + B x\right ) \left (a + c x^{2}\right )^{\frac {3}{2}}}{\left (d + e x\right )^{\frac {7}{2}}}\, dx \]
\[ \int \frac {(A+B x) \left (a+c x^2\right )^{3/2}}{(d+e x)^{7/2}} \, dx=\int { \frac {{\left (c x^{2} + a\right )}^{\frac {3}{2}} {\left (B x + A\right )}}{{\left (e x + d\right )}^{\frac {7}{2}}} \,d x } \]
\[ \int \frac {(A+B x) \left (a+c x^2\right )^{3/2}}{(d+e x)^{7/2}} \, dx=\int { \frac {{\left (c x^{2} + a\right )}^{\frac {3}{2}} {\left (B x + A\right )}}{{\left (e x + d\right )}^{\frac {7}{2}}} \,d x } \]
Timed out. \[ \int \frac {(A+B x) \left (a+c x^2\right )^{3/2}}{(d+e x)^{7/2}} \, dx=\int \frac {{\left (c\,x^2+a\right )}^{3/2}\,\left (A+B\,x\right )}{{\left (d+e\,x\right )}^{7/2}} \,d x \]