Integrand size = 21, antiderivative size = 447 \[ \int \frac {1}{(d+e x)^{7/2} \sqrt {a+c x^2}} \, dx=-\frac {2 e \sqrt {a+c x^2}}{5 \left (c d^2+a e^2\right ) (d+e x)^{5/2}}-\frac {16 c d e \sqrt {a+c x^2}}{15 \left (c d^2+a e^2\right )^2 (d+e x)^{3/2}}-\frac {2 c e \left (23 c d^2-9 a e^2\right ) \sqrt {a+c x^2}}{15 \left (c d^2+a e^2\right )^3 \sqrt {d+e x}}-\frac {2 \sqrt {-a} c^{3/2} \left (23 c d^2-9 a e^2\right ) \sqrt {d+e x} \sqrt {1+\frac {c x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{15 \left (c d^2+a e^2\right )^3 \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {a+c x^2}}+\frac {16 \sqrt {-a} c^{3/2} d \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {1+\frac {c x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right ),-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{15 \left (c d^2+a e^2\right )^2 \sqrt {d+e x} \sqrt {a+c x^2}} \]
-2/5*e*(c*x^2+a)^(1/2)/(a*e^2+c*d^2)/(e*x+d)^(5/2)-16/15*c*d*e*(c*x^2+a)^( 1/2)/(a*e^2+c*d^2)^2/(e*x+d)^(3/2)-2/15*c*e*(-9*a*e^2+23*c*d^2)*(c*x^2+a)^ (1/2)/(a*e^2+c*d^2)^3/(e*x+d)^(1/2)-2/15*c^(3/2)*(-9*a*e^2+23*c*d^2)*Ellip ticE(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)/(a*e^2+c*d^2) ^3/(c*x^2+a)^(1/2)/((e*x+d)*c^(1/2)/(e*(-a)^(1/2)+d*c^(1/2)))^(1/2)+16/15* c^(3/2)*d*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)/(a*e^2+c*d^2)^2/(e*x+d)^(1/2)/(c*x^2+ a)^(1/2)
Result contains complex when optimal does not.
Time = 13.08 (sec) , antiderivative size = 599, normalized size of antiderivative = 1.34 \[ \int \frac {1}{(d+e x)^{7/2} \sqrt {a+c x^2}} \, dx=\frac {2 \left (-e^2 \left (a+c x^2\right ) \left (3 \left (c d^2+a e^2\right )^2+8 c d \left (c d^2+a e^2\right ) (d+e x)+c \left (23 c d^2-9 a e^2\right ) (d+e x)^2\right )+\frac {c (d+e x)^2 \left (e^2 \sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}} \left (23 c d^2-9 a e^2\right ) \left (a+c x^2\right )+\sqrt {c} \left (-23 i c^{3/2} d^3+23 \sqrt {a} c d^2 e+9 i a \sqrt {c} d e^2-9 a^{3/2} e^3\right ) \sqrt {\frac {e \left (\frac {i \sqrt {a}}{\sqrt {c}}+x\right )}{d+e x}} \sqrt {-\frac {\frac {i \sqrt {a} e}{\sqrt {c}}-e x}{d+e x}} (d+e x)^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right )|\frac {\sqrt {c} d-i \sqrt {a} e}{\sqrt {c} d+i \sqrt {a} e}\right )+\sqrt {c} \left (15 i c^{3/2} d^3-23 \sqrt {a} c d^2 e-17 i a \sqrt {c} d e^2+9 a^{3/2} e^3\right ) \sqrt {\frac {e \left (\frac {i \sqrt {a}}{\sqrt {c}}+x\right )}{d+e x}} \sqrt {-\frac {\frac {i \sqrt {a} e}{\sqrt {c}}-e x}{d+e x}} (d+e x)^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right ),\frac {\sqrt {c} d-i \sqrt {a} e}{\sqrt {c} d+i \sqrt {a} e}\right )\right )}{\sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}\right )}{15 e \left (c d^2+a e^2\right )^3 (d+e x)^{5/2} \sqrt {a+c x^2}} \]
(2*(-(e^2*(a + c*x^2)*(3*(c*d^2 + a*e^2)^2 + 8*c*d*(c*d^2 + a*e^2)*(d + e* x) + c*(23*c*d^2 - 9*a*e^2)*(d + e*x)^2)) + (c*(d + e*x)^2*(e^2*Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]*(23*c*d^2 - 9*a*e^2)*(a + c*x^2) + Sqrt[c]*((-23*I) *c^(3/2)*d^3 + 23*Sqrt[a]*c*d^2*e + (9*I)*a*Sqrt[c]*d*e^2 - 9*a^(3/2)*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*Sqr t[a]*e)/Sqrt[c]]/Sqrt[d + e*x]], (Sqrt[c]*d - I*Sqrt[a]*e)/(Sqrt[c]*d + I* Sqrt[a]*e)] + Sqrt[c]*((15*I)*c^(3/2)*d^3 - 23*Sqrt[a]*c*d^2*e - (17*I)*a* Sqrt[c]*d*e^2 + 9*a^(3/2)*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)*Ellipt icF[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[-d - (I*Sqrt[a]*e)/Sqrt[c ]]))/(15*e*(c*d^2 + a*e^2)^3*(d + e*x)^(5/2)*Sqrt[a + c*x^2])
Time = 0.91 (sec) , antiderivative size = 809, normalized size of antiderivative = 1.81, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.524, Rules used = {498, 27, 688, 27, 688, 27, 599, 27, 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.
| \(\displaystyle \int \frac {1}{\sqrt {a+c x^2} (d+e x)^{7/2}} \, dx\) |
| \(\Big \downarrow \) 498 |
| \(\displaystyle -\frac {2 c \int -\frac {5 d-3 e x}{2 (d+e x)^{5/2} \sqrt {c x^2+a}}dx}{5 \left (a e^2+c d^2\right )}-\frac {2 e \sqrt {a+c x^2}}{5 (d+e x)^{5/2} \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {c \int \frac {5 d-3 e x}{(d+e x)^{5/2} \sqrt {c x^2+a}}dx}{5 \left (a e^2+c d^2\right )}-\frac {2 e \sqrt {a+c x^2}}{5 (d+e x)^{5/2} \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 688 |
| \(\displaystyle \frac {c \left (-\frac {2 \int -\frac {3 \left (5 c d^2-3 a e^2\right )-8 c d e x}{2 (d+e x)^{3/2} \sqrt {c x^2+a}}dx}{3 \left (a e^2+c d^2\right )}-\frac {16 d e \sqrt {a+c x^2}}{3 (d+e x)^{3/2} \left (a e^2+c d^2\right )}\right )}{5 \left (a e^2+c d^2\right )}-\frac {2 e \sqrt {a+c x^2}}{5 (d+e x)^{5/2} \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {c \left (\frac {\int \frac {3 \left (5 c d^2-3 a e^2\right )-8 c d e x}{(d+e x)^{3/2} \sqrt {c x^2+a}}dx}{3 \left (a e^2+c d^2\right )}-\frac {16 d e \sqrt {a+c x^2}}{3 (d+e x)^{3/2} \left (a e^2+c d^2\right )}\right )}{5 \left (a e^2+c d^2\right )}-\frac {2 e \sqrt {a+c x^2}}{5 (d+e x)^{5/2} \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 688 |
| \(\displaystyle \frac {c \left (\frac {-\frac {2 \int -\frac {c \left (d \left (15 c d^2-17 a e^2\right )+e \left (23 c d^2-9 a e^2\right ) x\right )}{2 \sqrt {d+e x} \sqrt {c x^2+a}}dx}{a e^2+c d^2}-\frac {2 e \sqrt {a+c x^2} \left (23 c d^2-9 a e^2\right )}{\sqrt {d+e x} \left (a e^2+c d^2\right )}}{3 \left (a e^2+c d^2\right )}-\frac {16 d e \sqrt {a+c x^2}}{3 (d+e x)^{3/2} \left (a e^2+c d^2\right )}\right )}{5 \left (a e^2+c d^2\right )}-\frac {2 e \sqrt {a+c x^2}}{5 (d+e x)^{5/2} \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {c \left (\frac {\frac {c \int \frac {d \left (15 c d^2-17 a e^2\right )+e \left (23 c d^2-9 a e^2\right ) x}{\sqrt {d+e x} \sqrt {c x^2+a}}dx}{a e^2+c d^2}-\frac {2 e \sqrt {a+c x^2} \left (23 c d^2-9 a e^2\right )}{\sqrt {d+e x} \left (a e^2+c d^2\right )}}{3 \left (a e^2+c d^2\right )}-\frac {16 d e \sqrt {a+c x^2}}{3 (d+e x)^{3/2} \left (a e^2+c d^2\right )}\right )}{5 \left (a e^2+c d^2\right )}-\frac {2 e \sqrt {a+c x^2}}{5 (d+e x)^{5/2} \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 599 |
| \(\displaystyle \frac {c \left (\frac {-\frac {2 c \int \frac {e \left (8 d \left (c d^2+a e^2\right )-\left (23 c d^2-9 a e^2\right ) (d+e x)\right )}{\sqrt {\frac {c d^2}{e^2}-\frac {2 c (d+e x) d}{e^2}+\frac {c (d+e x)^2}{e^2}+a}}d\sqrt {d+e x}}{e^2 \left (a e^2+c d^2\right )}-\frac {2 e \sqrt {a+c x^2} \left (23 c d^2-9 a e^2\right )}{\sqrt {d+e x} \left (a e^2+c d^2\right )}}{3 \left (a e^2+c d^2\right )}-\frac {16 d e \sqrt {a+c x^2}}{3 (d+e x)^{3/2} \left (a e^2+c d^2\right )}\right )}{5 \left (a e^2+c d^2\right )}-\frac {2 e \sqrt {a+c x^2}}{5 (d+e x)^{5/2} \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {c \left (\frac {-\frac {2 c \int \frac {8 d \left (c d^2+a e^2\right )-\left (23 c d^2-9 a e^2\right ) (d+e x)}{\sqrt {\frac {c d^2}{e^2}-\frac {2 c (d+e x) d}{e^2}+\frac {c (d+e x)^2}{e^2}+a}}d\sqrt {d+e x}}{e \left (a e^2+c d^2\right )}-\frac {2 e \sqrt {a+c x^2} \left (23 c d^2-9 a e^2\right )}{\sqrt {d+e x} \left (a e^2+c d^2\right )}}{3 \left (a e^2+c d^2\right )}-\frac {16 d e \sqrt {a+c x^2}}{3 (d+e x)^{3/2} \left (a e^2+c d^2\right )}\right )}{5 \left (a e^2+c d^2\right )}-\frac {2 e \sqrt {a+c x^2}}{5 (d+e x)^{5/2} \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 1511 |
| \(\displaystyle \frac {c \left (\frac {-\frac {2 c \left (\frac {\left (23 c d^2-9 a e^2\right ) \sqrt {a e^2+c d^2} \int \frac {1-\frac {\sqrt {c} (d+e x)}{\sqrt {c d^2+a e^2}}}{\sqrt {\frac {c d^2}{e^2}-\frac {2 c (d+e x) d}{e^2}+\frac {c (d+e x)^2}{e^2}+a}}d\sqrt {d+e x}}{\sqrt {c}}-\frac {\sqrt {a e^2+c d^2} \left (-8 \sqrt {c} d \sqrt {a e^2+c d^2}-9 a e^2+23 c d^2\right ) \int \frac {1}{\sqrt {\frac {c d^2}{e^2}-\frac {2 c (d+e x) d}{e^2}+\frac {c (d+e x)^2}{e^2}+a}}d\sqrt {d+e x}}{\sqrt {c}}\right )}{e \left (a e^2+c d^2\right )}-\frac {2 e \sqrt {a+c x^2} \left (23 c d^2-9 a e^2\right )}{\sqrt {d+e x} \left (a e^2+c d^2\right )}}{3 \left (a e^2+c d^2\right )}-\frac {16 d e \sqrt {a+c x^2}}{3 (d+e x)^{3/2} \left (a e^2+c d^2\right )}\right )}{5 \left (a e^2+c d^2\right )}-\frac {2 e \sqrt {a+c x^2}}{5 (d+e x)^{5/2} \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 1416 |
| \(\displaystyle \frac {c \left (\frac {-\frac {2 c \left (\frac {\left (23 c d^2-9 a e^2\right ) \sqrt {a e^2+c d^2} \int \frac {1-\frac {\sqrt {c} (d+e x)}{\sqrt {c d^2+a e^2}}}{\sqrt {\frac {c d^2}{e^2}-\frac {2 c (d+e x) d}{e^2}+\frac {c (d+e x)^2}{e^2}+a}}d\sqrt {d+e x}}{\sqrt {c}}-\frac {\left (a e^2+c d^2\right )^{3/4} \left (-8 \sqrt {c} d \sqrt {a e^2+c d^2}-9 a e^2+23 c d^2\right ) \left (\frac {\sqrt {c} (d+e x)}{\sqrt {a e^2+c d^2}}+1\right ) \sqrt {\frac {a+\frac {c d^2}{e^2}-\frac {2 c d (d+e x)}{e^2}+\frac {c (d+e x)^2}{e^2}}{\left (a+\frac {c d^2}{e^2}\right ) \left (\frac {\sqrt {c} (d+e x)}{\sqrt {a e^2+c d^2}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt [4]{c d^2+a e^2}}\right ),\frac {1}{2} \left (\frac {\sqrt {c} d}{\sqrt {c d^2+a e^2}}+1\right )\right )}{2 c^{3/4} \sqrt {a+\frac {c d^2}{e^2}-\frac {2 c d (d+e x)}{e^2}+\frac {c (d+e x)^2}{e^2}}}\right )}{e \left (a e^2+c d^2\right )}-\frac {2 e \sqrt {a+c x^2} \left (23 c d^2-9 a e^2\right )}{\sqrt {d+e x} \left (a e^2+c d^2\right )}}{3 \left (a e^2+c d^2\right )}-\frac {16 d e \sqrt {a+c x^2}}{3 (d+e x)^{3/2} \left (a e^2+c d^2\right )}\right )}{5 \left (a e^2+c d^2\right )}-\frac {2 e \sqrt {a+c x^2}}{5 (d+e x)^{5/2} \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 1509 |
| \(\displaystyle \frac {c \left (\frac {-\frac {2 e \sqrt {c x^2+a} \left (23 c d^2-9 a e^2\right )}{\left (c d^2+a e^2\right ) \sqrt {d+e x}}-\frac {2 c \left (\frac {\left (23 c d^2-9 a e^2\right ) \sqrt {c d^2+a e^2} \left (\frac {\sqrt [4]{c d^2+a e^2} \left (\frac {\sqrt {c} (d+e x)}{\sqrt {c d^2+a e^2}}+1\right ) \sqrt {\frac {\frac {c d^2}{e^2}-\frac {2 c (d+e x) d}{e^2}+\frac {c (d+e x)^2}{e^2}+a}{\left (\frac {c d^2}{e^2}+a\right ) \left (\frac {\sqrt {c} (d+e x)}{\sqrt {c d^2+a e^2}}+1\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt [4]{c d^2+a e^2}}\right )|\frac {1}{2} \left (\frac {\sqrt {c} d}{\sqrt {c d^2+a e^2}}+1\right )\right )}{\sqrt [4]{c} \sqrt {\frac {c d^2}{e^2}-\frac {2 c (d+e x) d}{e^2}+\frac {c (d+e x)^2}{e^2}+a}}-\frac {\sqrt {d+e x} \sqrt {\frac {c d^2}{e^2}-\frac {2 c (d+e x) d}{e^2}+\frac {c (d+e x)^2}{e^2}+a}}{\left (\frac {c d^2}{e^2}+a\right ) \left (\frac {\sqrt {c} (d+e x)}{\sqrt {c d^2+a e^2}}+1\right )}\right )}{\sqrt {c}}-\frac {\left (c d^2+a e^2\right )^{3/4} \left (23 c d^2-8 \sqrt {c} \sqrt {c d^2+a e^2} d-9 a e^2\right ) \left (\frac {\sqrt {c} (d+e x)}{\sqrt {c d^2+a e^2}}+1\right ) \sqrt {\frac {\frac {c d^2}{e^2}-\frac {2 c (d+e x) d}{e^2}+\frac {c (d+e x)^2}{e^2}+a}{\left (\frac {c d^2}{e^2}+a\right ) \left (\frac {\sqrt {c} (d+e x)}{\sqrt {c d^2+a e^2}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt [4]{c d^2+a e^2}}\right ),\frac {1}{2} \left (\frac {\sqrt {c} d}{\sqrt {c d^2+a e^2}}+1\right )\right )}{2 c^{3/4} \sqrt {\frac {c d^2}{e^2}-\frac {2 c (d+e x) d}{e^2}+\frac {c (d+e x)^2}{e^2}+a}}\right )}{e \left (c d^2+a e^2\right )}}{3 \left (c d^2+a e^2\right )}-\frac {16 d e \sqrt {c x^2+a}}{3 \left (c d^2+a e^2\right ) (d+e x)^{3/2}}\right )}{5 \left (c d^2+a e^2\right )}-\frac {2 e \sqrt {c x^2+a}}{5 \left (c d^2+a e^2\right ) (d+e x)^{5/2}}\) |
(-2*e*Sqrt[a + c*x^2])/(5*(c*d^2 + a*e^2)*(d + e*x)^(5/2)) + (c*((-16*d*e* Sqrt[a + c*x^2])/(3*(c*d^2 + a*e^2)*(d + e*x)^(3/2)) + ((-2*e*(23*c*d^2 - 9*a*e^2)*Sqrt[a + c*x^2])/((c*d^2 + a*e^2)*Sqrt[d + e*x]) - (2*c*(((23*c*d ^2 - 9*a*e^2)*Sqrt[c*d^2 + a*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 + (Sq rt[c]*(d + e*x))/Sqrt[c*d^2 + a*e^2]))) + ((c*d^2 + a*e^2)^(1/4)*(1 + (Sqr t[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)*Sqr t[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)*(23*c*d^2 - 9*a*e^2 - 8*Sqrt[c]*d*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)*Sq rt[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])))/(e*(c*d^2 + a*e^2)))/(3*(c*d^2 + a*e^2))))/(5*(c*d^2 + a*e^2) )
3.7.84.3.1 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[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ d*(c + d*x)^(n + 1)*((a + b*x^2)^(p + 1)/((n + 1)*(b*c^2 + a*d^2))), x] + S imp[b/((n + 1)*(b*c^2 + a*d^2)) Int[(c + d*x)^(n + 1)*(a + b*x^2)^p*(c*(n + 1) - d*(n + 2*p + 3)*x), x], x] /; FreeQ[{a, b, c, d, n, p}, x] && NeQ[n , -1] && ((LtQ[n, -1] && IntQuadraticQ[a, 0, b, c, d, n, p, x]) || (SumSimp lerQ[n, 1] && IntegerQ[p]) || ILtQ[Simplify[n + 2*p + 3], 0])
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[(e*f - d*g)*(d + e*x)^(m + 1)*((a + c*x^2)^(p + 1)/( (m + 1)*(c*d^2 + a*e^2))), x] + Simp[1/((m + 1)*(c*d^2 + a*e^2)) Int[(d + e*x)^(m + 1)*(a + c*x^2)^p*Simp[(c*d*f + a*e*g)*(m + 1) - c*(e*f - d*g)*(m + 2*p + 3)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g, p}, x] && LtQ[m, -1] && (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. \(794\) vs. \(2(369)=738\).
Time = 3.59 (sec) , antiderivative size = 795, normalized size of antiderivative = 1.78
| method | result | size |
| elliptic | \(\frac {\sqrt {\left (e x +d \right ) \left (c \,x^{2}+a \right )}\, \left (-\frac {2 \sqrt {c e \,x^{3}+c d \,x^{2}+a e x +a d}}{5 e^{2} \left (e^{2} a +c \,d^{2}\right ) \left (x +\frac {d}{e}\right )^{3}}-\frac {16 c d \sqrt {c e \,x^{3}+c d \,x^{2}+a e x +a d}}{15 \left (e^{2} a +c \,d^{2}\right )^{2} e \left (x +\frac {d}{e}\right )^{2}}+\frac {2 \left (c e \,x^{2}+a e \right ) c \left (9 e^{2} a -23 c \,d^{2}\right )}{15 \left (e^{2} a +c \,d^{2}\right )^{3} \sqrt {\left (x +\frac {d}{e}\right ) \left (c e \,x^{2}+a e \right )}}+\frac {2 \left (-\frac {8 c^{2} d}{15 \left (e^{2} a +c \,d^{2}\right )^{2}}-\frac {c^{2} d \left (9 e^{2} a -23 c \,d^{2}\right )}{15 \left (e^{2} a +c \,d^{2}\right )^{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 c^{2} e \left (9 e^{2} a -23 c \,d^{2}\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 )}{15 \left (e^{2} a +c \,d^{2}\right )^{3} \sqrt {c e \,x^{3}+c d \,x^{2}+a e x +a d}}\right )}{\sqrt {e x +d}\, \sqrt {c \,x^{2}+a}}\) | \(795\) |
| default | \(\text {Expression too large to display}\) | \(3863\) |
((e*x+d)*(c*x^2+a))^(1/2)/(e*x+d)^(1/2)/(c*x^2+a)^(1/2)*(-2/5/e^2/(a*e^2+c *d^2)*(c*e*x^3+c*d*x^2+a*e*x+a*d)^(1/2)/(x+d/e)^3-16/15*c/(a*e^2+c*d^2)^2* d/e*(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)^3*c*(9*a*e^2-23*c*d^2)/((x+d/e)*(c*e*x^2+a*e))^(1/2)+2*(-8/15*c^2*d /(a*e^2+c*d^2)^2-1/15*c^2*d*(9*a*e^2-23*c*d^2)/(a*e^2+c*d^2)^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))-2/15*c^2*e*(9*a* e^2-23*c*d^2)/(a*e^2+c*d^2)^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)*(( -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 = 632, normalized size of antiderivative = 1.41 \[ \int \frac {1}{(d+e x)^{7/2} \sqrt {a+c x^2}} \, dx=\frac {2 \, {\left (2 \, {\left (11 \, c^{2} d^{6} - 21 \, a c d^{4} e^{2} + {\left (11 \, c^{2} d^{3} e^{3} - 21 \, a c d e^{5}\right )} x^{3} + 3 \, {\left (11 \, c^{2} d^{4} e^{2} - 21 \, a c d^{2} e^{4}\right )} x^{2} + 3 \, {\left (11 \, c^{2} d^{5} e - 21 \, a c d^{3} e^{3}\right )} x\right )} \sqrt {c e} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, \frac {3 \, e x + d}{3 \, e}\right ) - 3 \, {\left (23 \, c^{2} d^{5} e - 9 \, a c d^{3} e^{3} + {\left (23 \, c^{2} d^{2} e^{4} - 9 \, a c e^{6}\right )} x^{3} + 3 \, {\left (23 \, c^{2} d^{3} e^{3} - 9 \, a c d e^{5}\right )} x^{2} + 3 \, {\left (23 \, c^{2} d^{4} e^{2} - 9 \, a c d^{2} e^{4}\right )} x\right )} \sqrt {c e} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, \frac {3 \, e x + d}{3 \, e}\right )\right ) - 3 \, {\left (34 \, c^{2} d^{4} e^{2} + 5 \, a c d^{2} e^{4} + 3 \, a^{2} e^{6} + {\left (23 \, c^{2} d^{2} e^{4} - 9 \, a c e^{6}\right )} x^{2} + 2 \, {\left (27 \, c^{2} d^{3} e^{3} - 5 \, a c d e^{5}\right )} x\right )} \sqrt {c x^{2} + a} \sqrt {e x + d}\right )}}{45 \, {\left (c^{3} d^{9} e + 3 \, a c^{2} d^{7} e^{3} + 3 \, a^{2} c d^{5} e^{5} + a^{3} d^{3} e^{7} + {\left (c^{3} d^{6} e^{4} + 3 \, a c^{2} d^{4} e^{6} + 3 \, a^{2} c d^{2} e^{8} + a^{3} e^{10}\right )} x^{3} + 3 \, {\left (c^{3} d^{7} e^{3} + 3 \, a c^{2} d^{5} e^{5} + 3 \, a^{2} c d^{3} e^{7} + a^{3} d e^{9}\right )} x^{2} + 3 \, {\left (c^{3} d^{8} e^{2} + 3 \, a c^{2} d^{6} e^{4} + 3 \, a^{2} c d^{4} e^{6} + a^{3} d^{2} e^{8}\right )} x\right )}} \]
2/45*(2*(11*c^2*d^6 - 21*a*c*d^4*e^2 + (11*c^2*d^3*e^3 - 21*a*c*d*e^5)*x^3 + 3*(11*c^2*d^4*e^2 - 21*a*c*d^2*e^4)*x^2 + 3*(11*c^2*d^5*e - 21*a*c*d^3* e^3)*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) - 3*(23*c^2*d^5*e - 9*a*c *d^3*e^3 + (23*c^2*d^2*e^4 - 9*a*c*e^6)*x^3 + 3*(23*c^2*d^3*e^3 - 9*a*c*d* e^5)*x^2 + 3*(23*c^2*d^4*e^2 - 9*a*c*d^2*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), weierst rassPInverse(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*(34*c^2*d^4*e^2 + 5*a*c*d^2*e^4 + 3*a^2*e^6 + (23*c^2*d^2*e^4 - 9*a*c*e^6)*x^2 + 2*(27*c^2*d^3*e^3 - 5*a*c*d*e^5)*x)*sq rt(c*x^2 + a)*sqrt(e*x + d))/(c^3*d^9*e + 3*a*c^2*d^7*e^3 + 3*a^2*c*d^5*e^ 5 + a^3*d^3*e^7 + (c^3*d^6*e^4 + 3*a*c^2*d^4*e^6 + 3*a^2*c*d^2*e^8 + a^3*e ^10)*x^3 + 3*(c^3*d^7*e^3 + 3*a*c^2*d^5*e^5 + 3*a^2*c*d^3*e^7 + a^3*d*e^9) *x^2 + 3*(c^3*d^8*e^2 + 3*a*c^2*d^6*e^4 + 3*a^2*c*d^4*e^6 + a^3*d^2*e^8)*x )
\[ \int \frac {1}{(d+e x)^{7/2} \sqrt {a+c x^2}} \, dx=\int \frac {1}{\sqrt {a + c x^{2}} \left (d + e x\right )^{\frac {7}{2}}}\, dx \]
\[ \int \frac {1}{(d+e x)^{7/2} \sqrt {a+c x^2}} \, dx=\int { \frac {1}{\sqrt {c x^{2} + a} {\left (e x + d\right )}^{\frac {7}{2}}} \,d x } \]
\[ \int \frac {1}{(d+e x)^{7/2} \sqrt {a+c x^2}} \, dx=\int { \frac {1}{\sqrt {c x^{2} + a} {\left (e x + d\right )}^{\frac {7}{2}}} \,d x } \]
Timed out. \[ \int \frac {1}{(d+e x)^{7/2} \sqrt {a+c x^2}} \, dx=\int \frac {1}{\sqrt {c\,x^2+a}\,{\left (d+e\,x\right )}^{7/2}} \,d x \]