Integrand size = 21, antiderivative size = 532 \[ \int \frac {1}{(d+e x)^{3/2} \left (a+c x^2\right )^{5/2}} \, dx=\frac {a e+c d x}{3 a \left (c d^2+a e^2\right ) \sqrt {d+e x} \left (a+c x^2\right )^{3/2}}-\frac {a e \left (c d^2-7 a e^2\right )-4 c d \left (c d^2+3 a e^2\right ) x}{6 a^2 \left (c d^2+a e^2\right )^2 \sqrt {d+e x} \sqrt {a+c x^2}}+\frac {e \left (4 c^2 d^4+15 a c d^2 e^2-21 a^2 e^4\right ) \sqrt {a+c x^2}}{6 a^2 \left (c d^2+a e^2\right )^3 \sqrt {d+e x}}+\frac {\sqrt {c} \left (4 c^2 d^4+15 a c d^2 e^2-21 a^2 e^4\right ) \sqrt {d+e x} \sqrt {1+\frac {c x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{6 (-a)^{3/2} \left (c d^2+a e^2\right )^3 \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {a+c x^2}}-\frac {2 \sqrt {c} d \left (c d^2+3 a e^2\right ) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {1+\frac {c x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right ),-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{3 (-a)^{3/2} \left (c d^2+a e^2\right )^2 \sqrt {d+e x} \sqrt {a+c x^2}} \]
1/3*(c*d*x+a*e)/a/(a*e^2+c*d^2)/(c*x^2+a)^(3/2)/(e*x+d)^(1/2)+1/6*(-a*e*(- 7*a*e^2+c*d^2)+4*c*d*(3*a*e^2+c*d^2)*x)/a^2/(a*e^2+c*d^2)^2/(e*x+d)^(1/2)/ (c*x^2+a)^(1/2)+1/6*e*(-21*a^2*e^4+15*a*c*d^2*e^2+4*c^2*d^4)*(c*x^2+a)^(1/ 2)/a^2/(a*e^2+c*d^2)^3/(e*x+d)^(1/2)+1/6*(-21*a^2*e^4+15*a*c*d^2*e^2+4*c^2 *d^4)*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))*c^(1/2)*(e*x+d)^(1/2)*(1+c*x^2/a)^(1/2)/(-a)^ (3/2)/(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)-2/3*d*(3*a*e^2+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))*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)/(-a)^(3/2)/(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 = 25.16 (sec) , antiderivative size = 638, normalized size of antiderivative = 1.20 \[ \int \frac {1}{(d+e x)^{3/2} \left (a+c x^2\right )^{5/2}} \, dx=\frac {-12 a^2 e^5 \left (a+c x^2\right )+e \left (-4 c^2 d^4-15 a c d^2 e^2+21 a^2 e^4\right ) \left (a+c x^2\right )+\frac {2 a c \left (c d^2+a e^2\right ) (d+e x) \left (c d^2 x+a e (2 d-e x)\right )}{a+c x^2}+c (d+e x) \left (4 c^2 d^4 x+3 a^2 e^3 (7 d-3 e x)+a c d^2 e (d+15 e x)\right )-\frac {i c \sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}} \left (4 c^2 d^4+15 a c d^2 e^2-21 a^2 e^4\right ) \sqrt {\frac {e \left (\frac {i \sqrt {a}}{\sqrt {c}}+x\right )}{d+e x}} \sqrt {-\frac {\frac {i \sqrt {a} e}{\sqrt {c}}-e x}{d+e x}} (d+e x)^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right )|\frac {\sqrt {c} d-i \sqrt {a} e}{\sqrt {c} d+i \sqrt {a} e}\right )}{e}+\frac {\sqrt {a} \sqrt {c} \left (4 c^2 d^4+i \sqrt {a} c^{3/2} d^3 e+15 a c d^2 e^2+33 i a^{3/2} \sqrt {c} d e^3-21 a^2 e^4\right ) \sqrt {\frac {e \left (\frac {i \sqrt {a}}{\sqrt {c}}+x\right )}{d+e x}} \sqrt {-\frac {\frac {i \sqrt {a} e}{\sqrt {c}}-e x}{d+e x}} (d+e x)^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right ),\frac {\sqrt {c} d-i \sqrt {a} e}{\sqrt {c} d+i \sqrt {a} e}\right )}{\sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}}{6 a^2 \left (c d^2+a e^2\right )^3 \sqrt {d+e x} \sqrt {a+c x^2}} \]
(-12*a^2*e^5*(a + c*x^2) + e*(-4*c^2*d^4 - 15*a*c*d^2*e^2 + 21*a^2*e^4)*(a + c*x^2) + (2*a*c*(c*d^2 + a*e^2)*(d + e*x)*(c*d^2*x + a*e*(2*d - e*x)))/ (a + c*x^2) + c*(d + e*x)*(4*c^2*d^4*x + 3*a^2*e^3*(7*d - 3*e*x) + a*c*d^2 *e*(d + 15*e*x)) - (I*c*Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]*(4*c^2*d^4 + 15*a *c*d^2*e^2 - 21*a^2*e^4)*Sqrt[(e*((I*Sqrt[a])/Sqrt[c] + x))/(d + e*x)]*Sqr t[-(((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*Sq rt[a]*e)/(Sqrt[c]*d + I*Sqrt[a]*e)])/e + (Sqrt[a]*Sqrt[c]*(4*c^2*d^4 + I*S qrt[a]*c^(3/2)*d^3*e + 15*a*c*d^2*e^2 + (33*I)*a^(3/2)*Sqrt[c]*d*e^3 - 21* a^2*e^4)*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)])/Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]])/(6*a^2*(c*d^2 + a*e ^2)^3*Sqrt[d + e*x]*Sqrt[a + c*x^2])
Time = 1.08 (sec) , antiderivative size = 911, normalized size of antiderivative = 1.71, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.524, Rules used = {496, 27, 686, 27, 688, 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.
| \(\displaystyle \int \frac {1}{\left (a+c x^2\right )^{5/2} (d+e x)^{3/2}} \, dx\) |
| \(\Big \downarrow \) 496 |
| \(\displaystyle \frac {a e+c d x}{3 a \left (a+c x^2\right )^{3/2} \sqrt {d+e x} \left (a e^2+c d^2\right )}-\frac {\int -\frac {4 c d^2+5 c e x d+7 a e^2}{2 (d+e x)^{3/2} \left (c x^2+a\right )^{3/2}}dx}{3 a \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {4 c d^2+5 c e x d+7 a e^2}{(d+e x)^{3/2} \left (c x^2+a\right )^{3/2}}dx}{6 a \left (a e^2+c d^2\right )}+\frac {a e+c d x}{3 a \left (a+c x^2\right )^{3/2} \sqrt {d+e x} \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 686 |
| \(\displaystyle \frac {-\frac {\int \frac {c e \left (3 a e \left (c d^2-7 a e^2\right )-4 c d \left (c d^2+3 a e^2\right ) x\right )}{2 (d+e x)^{3/2} \sqrt {c x^2+a}}dx}{a c \left (a e^2+c d^2\right )}-\frac {a e \left (c d^2-7 a e^2\right )-4 c d x \left (3 a e^2+c d^2\right )}{a \sqrt {a+c x^2} \sqrt {d+e x} \left (a e^2+c d^2\right )}}{6 a \left (a e^2+c d^2\right )}+\frac {a e+c d x}{3 a \left (a+c x^2\right )^{3/2} \sqrt {d+e x} \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {e \int \frac {3 a e \left (c d^2-7 a e^2\right )-4 c d \left (c d^2+3 a e^2\right ) x}{(d+e x)^{3/2} \sqrt {c x^2+a}}dx}{2 a \left (a e^2+c d^2\right )}-\frac {a e \left (c d^2-7 a e^2\right )-4 c d x \left (3 a e^2+c d^2\right )}{a \sqrt {a+c x^2} \sqrt {d+e x} \left (a e^2+c d^2\right )}}{6 a \left (a e^2+c d^2\right )}+\frac {a e+c d x}{3 a \left (a+c x^2\right )^{3/2} \sqrt {d+e x} \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 688 |
| \(\displaystyle \frac {-\frac {e \left (-\frac {2 \int \frac {c \left (a d e \left (c d^2+33 a e^2\right )-\left (4 c^2 d^4+15 a c e^2 d^2-21 a^2 e^4\right ) x\right )}{2 \sqrt {d+e x} \sqrt {c x^2+a}}dx}{a e^2+c d^2}-\frac {2 \sqrt {a+c x^2} \left (-21 a^2 e^4+15 a c d^2 e^2+4 c^2 d^4\right )}{\sqrt {d+e x} \left (a e^2+c d^2\right )}\right )}{2 a \left (a e^2+c d^2\right )}-\frac {a e \left (c d^2-7 a e^2\right )-4 c d x \left (3 a e^2+c d^2\right )}{a \sqrt {a+c x^2} \sqrt {d+e x} \left (a e^2+c d^2\right )}}{6 a \left (a e^2+c d^2\right )}+\frac {a e+c d x}{3 a \left (a+c x^2\right )^{3/2} \sqrt {d+e x} \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {e \left (-\frac {c \int \frac {a d e \left (c d^2+33 a e^2\right )-\left (4 c^2 d^4+15 a c e^2 d^2-21 a^2 e^4\right ) x}{\sqrt {d+e x} \sqrt {c x^2+a}}dx}{a e^2+c d^2}-\frac {2 \sqrt {a+c x^2} \left (-21 a^2 e^4+15 a c d^2 e^2+4 c^2 d^4\right )}{\sqrt {d+e x} \left (a e^2+c d^2\right )}\right )}{2 a \left (a e^2+c d^2\right )}-\frac {a e \left (c d^2-7 a e^2\right )-4 c d x \left (3 a e^2+c d^2\right )}{a \sqrt {a+c x^2} \sqrt {d+e x} \left (a e^2+c d^2\right )}}{6 a \left (a e^2+c d^2\right )}+\frac {a e+c d x}{3 a \left (a+c x^2\right )^{3/2} \sqrt {d+e x} \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 599 |
| \(\displaystyle \frac {-\frac {e \left (\frac {2 c \int -\frac {4 d \left (c d^2+a e^2\right ) \left (c d^2+3 a e^2\right )-\left (4 c^2 d^4+15 a c e^2 d^2-21 a^2 e^4\right ) (d+e x)}{\sqrt {\frac {c d^2}{e^2}-\frac {2 c (d+e x) d}{e^2}+\frac {c (d+e x)^2}{e^2}+a}}d\sqrt {d+e x}}{e^2 \left (a e^2+c d^2\right )}-\frac {2 \sqrt {a+c x^2} \left (-21 a^2 e^4+15 a c d^2 e^2+4 c^2 d^4\right )}{\sqrt {d+e x} \left (a e^2+c d^2\right )}\right )}{2 a \left (a e^2+c d^2\right )}-\frac {a e \left (c d^2-7 a e^2\right )-4 c d x \left (3 a e^2+c d^2\right )}{a \sqrt {a+c x^2} \sqrt {d+e x} \left (a e^2+c d^2\right )}}{6 a \left (a e^2+c d^2\right )}+\frac {a e+c d x}{3 a \left (a+c x^2\right )^{3/2} \sqrt {d+e x} \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-\frac {e \left (-\frac {2 c \int \frac {4 d \left (c d^2+a e^2\right ) \left (c d^2+3 a e^2\right )-\left (4 c^2 d^4+15 a c e^2 d^2-21 a^2 e^4\right ) (d+e x)}{\sqrt {\frac {c d^2}{e^2}-\frac {2 c (d+e x) d}{e^2}+\frac {c (d+e x)^2}{e^2}+a}}d\sqrt {d+e x}}{e^2 \left (a e^2+c d^2\right )}-\frac {2 \sqrt {a+c x^2} \left (-21 a^2 e^4+15 a c d^2 e^2+4 c^2 d^4\right )}{\sqrt {d+e x} \left (a e^2+c d^2\right )}\right )}{2 a \left (a e^2+c d^2\right )}-\frac {a e \left (c d^2-7 a e^2\right )-4 c d x \left (3 a e^2+c d^2\right )}{a \sqrt {a+c x^2} \sqrt {d+e x} \left (a e^2+c d^2\right )}}{6 a \left (a e^2+c d^2\right )}+\frac {a e+c d x}{3 a \left (a+c x^2\right )^{3/2} \sqrt {d+e x} \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 1511 |
| \(\displaystyle \frac {-\frac {e \left (\frac {2 c \left (\frac {\sqrt {a e^2+c d^2} \left (-21 a^2 e^4+15 a c d^2 e^2-4 \sqrt {c} d \sqrt {a e^2+c d^2} \left (3 a e^2+c d^2\right )+4 c^2 d^4\right ) \int \frac {1}{\sqrt {\frac {c d^2}{e^2}-\frac {2 c (d+e x) d}{e^2}+\frac {c (d+e x)^2}{e^2}+a}}d\sqrt {d+e x}}{\sqrt {c}}-\frac {\sqrt {a e^2+c d^2} \left (-21 a^2 e^4+15 a c d^2 e^2+4 c^2 d^4\right ) \int \frac {1-\frac {\sqrt {c} (d+e x)}{\sqrt {c d^2+a e^2}}}{\sqrt {\frac {c d^2}{e^2}-\frac {2 c (d+e x) d}{e^2}+\frac {c (d+e x)^2}{e^2}+a}}d\sqrt {d+e x}}{\sqrt {c}}\right )}{e^2 \left (a e^2+c d^2\right )}-\frac {2 \sqrt {a+c x^2} \left (-21 a^2 e^4+15 a c d^2 e^2+4 c^2 d^4\right )}{\sqrt {d+e x} \left (a e^2+c d^2\right )}\right )}{2 a \left (a e^2+c d^2\right )}-\frac {a e \left (c d^2-7 a e^2\right )-4 c d x \left (3 a e^2+c d^2\right )}{a \sqrt {a+c x^2} \sqrt {d+e x} \left (a e^2+c d^2\right )}}{6 a \left (a e^2+c d^2\right )}+\frac {a e+c d x}{3 a \left (a+c x^2\right )^{3/2} \sqrt {d+e x} \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 1416 |
| \(\displaystyle \frac {-\frac {e \left (\frac {2 c \left (\frac {\left (a e^2+c d^2\right )^{3/4} \left (-21 a^2 e^4+15 a c d^2 e^2-4 \sqrt {c} d \sqrt {a e^2+c d^2} \left (3 a e^2+c d^2\right )+4 c^2 d^4\right ) \left (\frac {\sqrt {c} (d+e x)}{\sqrt {a e^2+c d^2}}+1\right ) \sqrt {\frac {a+\frac {c d^2}{e^2}-\frac {2 c d (d+e x)}{e^2}+\frac {c (d+e x)^2}{e^2}}{\left (a+\frac {c d^2}{e^2}\right ) \left (\frac {\sqrt {c} (d+e x)}{\sqrt {a e^2+c d^2}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt [4]{c d^2+a e^2}}\right ),\frac {1}{2} \left (\frac {\sqrt {c} d}{\sqrt {c d^2+a e^2}}+1\right )\right )}{2 c^{3/4} \sqrt {a+\frac {c d^2}{e^2}-\frac {2 c d (d+e x)}{e^2}+\frac {c (d+e x)^2}{e^2}}}-\frac {\sqrt {a e^2+c d^2} \left (-21 a^2 e^4+15 a c d^2 e^2+4 c^2 d^4\right ) \int \frac {1-\frac {\sqrt {c} (d+e x)}{\sqrt {c d^2+a e^2}}}{\sqrt {\frac {c d^2}{e^2}-\frac {2 c (d+e x) d}{e^2}+\frac {c (d+e x)^2}{e^2}+a}}d\sqrt {d+e x}}{\sqrt {c}}\right )}{e^2 \left (a e^2+c d^2\right )}-\frac {2 \sqrt {a+c x^2} \left (-21 a^2 e^4+15 a c d^2 e^2+4 c^2 d^4\right )}{\sqrt {d+e x} \left (a e^2+c d^2\right )}\right )}{2 a \left (a e^2+c d^2\right )}-\frac {a e \left (c d^2-7 a e^2\right )-4 c d x \left (3 a e^2+c d^2\right )}{a \sqrt {a+c x^2} \sqrt {d+e x} \left (a e^2+c d^2\right )}}{6 a \left (a e^2+c d^2\right )}+\frac {a e+c d x}{3 a \left (a+c x^2\right )^{3/2} \sqrt {d+e x} \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 1509 |
| \(\displaystyle \frac {a e+c d x}{3 a \left (c d^2+a e^2\right ) \sqrt {d+e x} \left (c x^2+a\right )^{3/2}}+\frac {-\frac {a e \left (c d^2-7 a e^2\right )-4 c d \left (c d^2+3 a e^2\right ) x}{a \left (c d^2+a e^2\right ) \sqrt {d+e x} \sqrt {c x^2+a}}-\frac {e \left (\frac {2 c \left (\frac {\left (c d^2+a e^2\right )^{3/4} \left (4 c^2 d^4+15 a c e^2 d^2-4 \sqrt {c} \sqrt {c d^2+a e^2} \left (c d^2+3 a e^2\right ) d-21 a^2 e^4\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}}-\frac {\sqrt {c d^2+a e^2} \left (4 c^2 d^4+15 a c e^2 d^2-21 a^2 e^4\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 )}{\sqrt {c}}\right )}{e^2 \left (c d^2+a e^2\right )}-\frac {2 \left (4 c^2 d^4+15 a c e^2 d^2-21 a^2 e^4\right ) \sqrt {c x^2+a}}{\left (c d^2+a e^2\right ) \sqrt {d+e x}}\right )}{2 a \left (c d^2+a e^2\right )}}{6 a \left (c d^2+a e^2\right )}\) |
(a*e + c*d*x)/(3*a*(c*d^2 + a*e^2)*Sqrt[d + e*x]*(a + c*x^2)^(3/2)) + (-(( a*e*(c*d^2 - 7*a*e^2) - 4*c*d*(c*d^2 + 3*a*e^2)*x)/(a*(c*d^2 + a*e^2)*Sqrt [d + e*x]*Sqrt[a + c*x^2])) - (e*((-2*(4*c^2*d^4 + 15*a*c*d^2*e^2 - 21*a^2 *e^4)*Sqrt[a + c*x^2])/((c*d^2 + a*e^2)*Sqrt[d + e*x]) + (2*c*(-((Sqrt[c*d ^2 + a*e^2]*(4*c^2*d^4 + 15*a*c*d^2*e^2 - 21*a^2*e^4)*(-((Sqrt[d + e*x]*Sq rt[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)*(4*c^2*d^4 + 15*a*c*d^2*e^ 2 - 21*a^2*e^4 - 4*Sqrt[c]*d*Sqrt[c*d^2 + a*e^2]*(c*d^2 + 3*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])))/( e^2*(c*d^2 + a*e^2))))/(2*a*(c*d^2 + a*e^2)))/(6*a*(c*d^2 + a*e^2))
3.7.98.3.1 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[ (-(a*d + b*c*x))*(c + d*x)^(n + 1)*((a + b*x^2)^(p + 1)/(2*a*(p + 1)*(b*c^2 + a*d^2))), x] + Simp[1/(2*a*(p + 1)*(b*c^2 + a*d^2)) Int[(c + d*x)^n*(a + b*x^2)^(p + 1)*Simp[b*c^2*(2*p + 3) + a*d^2*(n + 2*p + 3) + b*c*d*(n + 2 *p + 4)*x, x], x], x] /; FreeQ[{a, b, c, d, n}, x] && LtQ[p, -1] && IntQuad raticQ[a, 0, b, c, d, n, p, 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))*(f*a*c*e - a*g*c*d + c*(c*d*f + a*e*g)*x)*((a + c*x^2)^(p + 1)/(2*a*c*(p + 1)*(c*d^2 + a*e^2))), x] + Simp[ 1/(2*a*c*(p + 1)*(c*d^2 + a*e^2)) Int[(d + e*x)^m*(a + c*x^2)^(p + 1)*Sim p[f*(c^2*d^2*(2*p + 3) + a*c*e^2*(m + 2*p + 3)) - a*c*d*e*g*m + c*e*(c*d*f + a*e*g)*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g}, x] && LtQ [p, -1] && (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[(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. \(999\) vs. \(2(455)=910\).
Time = 4.07 (sec) , antiderivative size = 1000, normalized size of antiderivative = 1.88
| method | result | size |
| elliptic | \(\frac {\sqrt {\left (e x +d \right ) \left (c \,x^{2}+a \right )}\, \left (\frac {\left (-\frac {\left (e^{2} a -c \,d^{2}\right ) x}{3 a \left (e^{2} a +c \,d^{2}\right )^{2} c}+\frac {2 d e}{3 \left (e^{2} a +c \,d^{2}\right )^{2} c}\right ) \sqrt {c e \,x^{3}+c d \,x^{2}+a e x +a d}}{\left (x^{2}+\frac {a}{c}\right )^{2}}-\frac {2 \left (c e x +c d \right ) \left (\frac {\left (9 a^{2} e^{4}-15 a c \,d^{2} e^{2}-4 c^{2} d^{4}\right ) x}{12 a^{2} \left (e^{2} a +c \,d^{2}\right )^{3}}-\frac {d e \left (21 e^{2} a +c \,d^{2}\right )}{12 a \left (e^{2} a +c \,d^{2}\right )^{3}}\right )}{\sqrt {\left (x^{2}+\frac {a}{c}\right ) \left (c e x +c d \right )}}-\frac {2 \left (c e \,x^{2}+a e \right ) e^{4}}{\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 {2 c d \left (3 e^{2} a +c \,d^{2}\right )}{3 \left (e^{2} a +c \,d^{2}\right )^{2} a^{2}}-\frac {c \,e^{2} d \left (21 e^{2} a +c \,d^{2}\right )}{12 a \left (e^{2} a +c \,d^{2}\right )^{3}}+\frac {c d \left (9 a^{2} e^{4}-15 a c \,d^{2} e^{2}-4 c^{2} d^{4}\right )}{6 a^{2} \left (e^{2} a +c \,d^{2}\right )^{3}}+\frac {c d \,e^{4}}{\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 \left (\frac {c e \left (9 a^{2} e^{4}-15 a c \,d^{2} e^{2}-4 c^{2} d^{4}\right )}{12 a^{2} \left (e^{2} a +c \,d^{2}\right )^{3}}+\frac {c \,e^{5}}{\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}}}\, \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}}\) | \(1000\) |
| default | \(\text {Expression too large to display}\) | \(3322\) |
((e*x+d)*(c*x^2+a))^(1/2)/(e*x+d)^(1/2)/(c*x^2+a)^(1/2)*((-1/3*(a*e^2-c*d^ 2)/a/(a*e^2+c*d^2)^2/c*x+2/3*d*e/(a*e^2+c*d^2)^2/c)*(c*e*x^3+c*d*x^2+a*e*x +a*d)^(1/2)/(x^2+1/c*a)^2-2*(c*e*x+c*d)*(1/12*(9*a^2*e^4-15*a*c*d^2*e^2-4* c^2*d^4)/a^2/(a*e^2+c*d^2)^3*x-1/12*d*e*(21*a*e^2+c*d^2)/a/(a*e^2+c*d^2)^3 )/((x^2+1/c*a)*(c*e*x+c*d))^(1/2)-2*(c*e*x^2+a*e)*e^4/(a*e^2+c*d^2)^3/((x+ d/e)*(c*e*x^2+a*e))^(1/2)+2*(2/3*c*d*(3*a*e^2+c*d^2)/(a*e^2+c*d^2)^2/a^2-1 /12*c*e^2*d*(21*a*e^2+c*d^2)/a/(a*e^2+c*d^2)^3+1/6*c*d*(9*a^2*e^4-15*a*c*d ^2*e^2-4*c^2*d^4)/a^2/(a*e^2+c*d^2)^3+c*d*e^4/(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*(1/12*c*e*(9*a^ 2*e^4-15*a*c*d^2*e^2-4*c^2*d^4)/a^2/(a*e^2+c*d^2)^3+c*e^5/(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)*Elliptic E(((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 = 0.16 (sec) , antiderivative size = 1148, normalized size of antiderivative = 2.16 \[ \int \frac {1}{(d+e x)^{3/2} \left (a+c x^2\right )^{5/2}} \, dx=\text {Too large to display} \]
1/18*(2*(2*a^2*c^2*d^6 + 9*a^3*c*d^4*e^2 + 39*a^4*d^2*e^4 + (2*c^4*d^5*e + 9*a*c^3*d^3*e^3 + 39*a^2*c^2*d*e^5)*x^5 + (2*c^4*d^6 + 9*a*c^3*d^4*e^2 + 39*a^2*c^2*d^2*e^4)*x^4 + 2*(2*a*c^3*d^5*e + 9*a^2*c^2*d^3*e^3 + 39*a^3*c* d*e^5)*x^3 + 2*(2*a*c^3*d^6 + 9*a^2*c^2*d^4*e^2 + 39*a^3*c*d^2*e^4)*x^2 + (2*a^2*c^2*d^5*e + 9*a^3*c*d^3*e^3 + 39*a^4*d*e^5)*x)*sqrt(c*e)*weierstras sPInverse(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*(4*a^2*c^2*d^5*e + 15*a^3*c*d^3*e^3 - 21*a^4*d*e^ 5 + (4*c^4*d^4*e^2 + 15*a*c^3*d^2*e^4 - 21*a^2*c^2*e^6)*x^5 + (4*c^4*d^5*e + 15*a*c^3*d^3*e^3 - 21*a^2*c^2*d*e^5)*x^4 + 2*(4*a*c^3*d^4*e^2 + 15*a^2* c^2*d^2*e^4 - 21*a^3*c*e^6)*x^3 + 2*(4*a*c^3*d^5*e + 15*a^2*c^2*d^3*e^3 - 21*a^3*c*d*e^5)*x^2 + (4*a^2*c^2*d^4*e^2 + 15*a^3*c*d^2*e^4 - 21*a^4*e^6)* 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*a^2*c^2*d^4*e^ 2 + 25*a^3*c*d^2*e^4 - 12*a^4*e^6 + (4*c^4*d^4*e^2 + 15*a*c^3*d^2*e^4 - 21 *a^2*c^2*e^6)*x^4 + 4*(c^4*d^5*e + 4*a*c^3*d^3*e^3 + 3*a^2*c^2*d*e^5)*x^3 + (7*a*c^3*d^4*e^2 + 36*a^2*c^2*d^2*e^4 - 35*a^3*c*e^6)*x^2 + 2*(3*a*c^3*d ^5*e + 10*a^2*c^2*d^3*e^3 + 7*a^3*c*d*e^5)*x)*sqrt(c*x^2 + a)*sqrt(e*x + d ))/(a^4*c^3*d^7*e + 3*a^5*c^2*d^5*e^3 + 3*a^6*c*d^3*e^5 + a^7*d*e^7 + (a^2 *c^5*d^6*e^2 + 3*a^3*c^4*d^4*e^4 + 3*a^4*c^3*d^2*e^6 + a^5*c^2*e^8)*x^5...
\[ \int \frac {1}{(d+e x)^{3/2} \left (a+c x^2\right )^{5/2}} \, dx=\int \frac {1}{\left (a + c x^{2}\right )^{\frac {5}{2}} \left (d + e x\right )^{\frac {3}{2}}}\, dx \]
\[ \int \frac {1}{(d+e x)^{3/2} \left (a+c x^2\right )^{5/2}} \, dx=\int { \frac {1}{{\left (c x^{2} + a\right )}^{\frac {5}{2}} {\left (e x + d\right )}^{\frac {3}{2}}} \,d x } \]
\[ \int \frac {1}{(d+e x)^{3/2} \left (a+c x^2\right )^{5/2}} \, dx=\int { \frac {1}{{\left (c x^{2} + a\right )}^{\frac {5}{2}} {\left (e x + d\right )}^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {1}{(d+e x)^{3/2} \left (a+c x^2\right )^{5/2}} \, dx=\int \frac {1}{{\left (c\,x^2+a\right )}^{5/2}\,{\left (d+e\,x\right )}^{3/2}} \,d x \]