Integrand size = 21, antiderivative size = 448 \[ \int \sqrt {d+e x} \left (a+c x^2\right )^{3/2} \, dx=\frac {4 \sqrt {d+e x} \left (4 d \left (c d^2+3 a e^2\right )-3 e \left (c d^2-7 a e^2\right ) x\right ) \sqrt {a+c x^2}}{315 e^3}-\frac {4 d \sqrt {d+e x} \left (a+c x^2\right )^{3/2}}{21 e}+\frac {2 (d+e x)^{3/2} \left (a+c x^2\right )^{3/2}}{9 e}+\frac {8 \sqrt {-a} \left (4 c^2 d^4+15 a c d^2 e^2-21 a^2 e^4\right ) \sqrt {d+e x} \sqrt {1+\frac {c x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{315 \sqrt {c} e^4 \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {a+c x^2}}-\frac {32 \sqrt {-a} d \left (c d^2+a e^2\right ) \left (c d^2+3 a e^2\right ) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {1+\frac {c x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right ),-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{315 \sqrt {c} e^4 \sqrt {d+e x} \sqrt {a+c x^2}} \]
2/9*(e*x+d)^(3/2)*(c*x^2+a)^(3/2)/e-4/21*d*(c*x^2+a)^(3/2)*(e*x+d)^(1/2)/e +4/315*(4*d*(3*a*e^2+c*d^2)-3*e*(-7*a*e^2+c*d^2)*x)*(e*x+d)^(1/2)*(c*x^2+a )^(1/2)/e^3+8/315*(-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))*(-a)^(1/2)*(e*x+d)^(1/2)*(1+c*x^2/a)^(1/2)/e^4/c^(1/2)/(c*x^2+a)^(1/ 2)/((e*x+d)*c^(1/2)/(e*(-a)^(1/2)+d*c^(1/2)))^(1/2)-32/315*d*(a*e^2+c*d^2) *(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))*(-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^4/c^(1/2)/(e*x+d)^(1/2)/(c*x ^2+a)^(1/2)
Result contains complex when optimal does not.
Time = 25.54 (sec) , antiderivative size = 646, normalized size of antiderivative = 1.44 \[ \int \sqrt {d+e x} \left (a+c x^2\right )^{3/2} \, dx=\frac {\sqrt {d+e x} \left (\frac {2 \left (a+c x^2\right ) \left (a e^2 (29 d+77 e x)+c \left (8 d^3-6 d^2 e x+5 d e^2 x^2+35 e^3 x^3\right )\right )}{e^3}+\frac {8 \left (-e^2 \sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}} \left (4 c^2 d^4+15 a c d^2 e^2-21 a^2 e^4\right ) \left (a+c x^2\right )+\sqrt {c} \left (4 i c^{5/2} d^5-4 \sqrt {a} c^2 d^4 e+15 i a c^{3/2} d^3 e^2-15 a^{3/2} c d^2 e^3-21 i a^2 \sqrt {c} d e^4+21 a^{5/2} e^5\right ) \sqrt {\frac {e \left (\frac {i \sqrt {a}}{\sqrt {c}}+x\right )}{d+e x}} \sqrt {-\frac {\frac {i \sqrt {a} e}{\sqrt {c}}-e x}{d+e x}} (d+e x)^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right )|\frac {\sqrt {c} d-i \sqrt {a} e}{\sqrt {c} d+i \sqrt {a} e}\right )+\sqrt {a} \sqrt {c} e \left (4 c^2 d^4+i \sqrt {a} c^{3/2} d^3 e+15 a c d^2 e^2+33 i a^{3/2} \sqrt {c} d e^3-21 a^2 e^4\right ) \sqrt {\frac {e \left (\frac {i \sqrt {a}}{\sqrt {c}}+x\right )}{d+e x}} \sqrt {-\frac {\frac {i \sqrt {a} e}{\sqrt {c}}-e x}{d+e x}} (d+e x)^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right ),\frac {\sqrt {c} d-i \sqrt {a} e}{\sqrt {c} d+i \sqrt {a} e}\right )\right )}{c e^5 \sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}} (d+e x)}\right )}{315 \sqrt {a+c x^2}} \]
(Sqrt[d + e*x]*((2*(a + c*x^2)*(a*e^2*(29*d + 77*e*x) + c*(8*d^3 - 6*d^2*e *x + 5*d*e^2*x^2 + 35*e^3*x^3)))/e^3 + (8*(-(e^2*Sqrt[-d - (I*Sqrt[a]*e)/S qrt[c]]*(4*c^2*d^4 + 15*a*c*d^2*e^2 - 21*a^2*e^4)*(a + c*x^2)) + Sqrt[c]*( (4*I)*c^(5/2)*d^5 - 4*Sqrt[a]*c^2*d^4*e + (15*I)*a*c^(3/2)*d^3*e^2 - 15*a^ (3/2)*c*d^2*e^3 - (21*I)*a^2*Sqrt[c]*d*e^4 + 21*a^(5/2)*e^5)*Sqrt[(e*((I*S qrt[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]*Sqrt[c]*e*(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)*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)]))/(c*e^5*Sqrt[-d - ( I*Sqrt[a]*e)/Sqrt[c]]*(d + e*x))))/(315*Sqrt[a + c*x^2])
Time = 1.04 (sec) , antiderivative size = 799, normalized size of antiderivative = 1.78, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.476, Rules used = {493, 687, 27, 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.
| \(\displaystyle \int \left (a+c x^2\right )^{3/2} \sqrt {d+e x} \, dx\) |
| \(\Big \downarrow \) 493 |
| \(\displaystyle \frac {2 \int (a e-c d x) \sqrt {d+e x} \sqrt {c x^2+a}dx}{3 e}+\frac {2 \left (a+c x^2\right )^{3/2} (d+e x)^{3/2}}{9 e}\) |
| \(\Big \downarrow \) 687 |
| \(\displaystyle \frac {2 \left (\frac {2 \int \frac {c \left (8 a d e-\left (c d^2-7 a e^2\right ) x\right ) \sqrt {c x^2+a}}{2 \sqrt {d+e x}}dx}{7 c}-\frac {2}{7} d \left (a+c x^2\right )^{3/2} \sqrt {d+e x}\right )}{3 e}+\frac {2 \left (a+c x^2\right )^{3/2} (d+e x)^{3/2}}{9 e}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 \left (\frac {1}{7} \int \frac {\left (8 a d e-\left (c d^2-7 a e^2\right ) x\right ) \sqrt {c x^2+a}}{\sqrt {d+e x}}dx-\frac {2}{7} d \left (a+c x^2\right )^{3/2} \sqrt {d+e x}\right )}{3 e}+\frac {2 \left (a+c x^2\right )^{3/2} (d+e x)^{3/2}}{9 e}\) |
| \(\Big \downarrow \) 682 |
| \(\displaystyle \frac {2 \left (\frac {1}{7} \left (\frac {4 \int \frac {c \left (a d e \left (c d^2+33 a e^2\right )-\left (4 c^2 d^4+15 a c e^2 d^2-21 a^2 e^4\right ) x\right )}{2 \sqrt {d+e x} \sqrt {c x^2+a}}dx}{15 c e^2}+\frac {2 \sqrt {a+c x^2} \sqrt {d+e x} \left (4 d \left (3 a e^2+c d^2\right )-3 e x \left (c d^2-7 a e^2\right )\right )}{15 e^2}\right )-\frac {2}{7} d \left (a+c x^2\right )^{3/2} \sqrt {d+e x}\right )}{3 e}+\frac {2 \left (a+c x^2\right )^{3/2} (d+e x)^{3/2}}{9 e}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 \left (\frac {1}{7} \left (\frac {2 \int \frac {a d e \left (c d^2+33 a e^2\right )-\left (4 c^2 d^4+15 a c e^2 d^2-21 a^2 e^4\right ) x}{\sqrt {d+e x} \sqrt {c x^2+a}}dx}{15 e^2}+\frac {2 \sqrt {a+c x^2} \sqrt {d+e x} \left (4 d \left (3 a e^2+c d^2\right )-3 e x \left (c d^2-7 a e^2\right )\right )}{15 e^2}\right )-\frac {2}{7} d \left (a+c x^2\right )^{3/2} \sqrt {d+e x}\right )}{3 e}+\frac {2 \left (a+c x^2\right )^{3/2} (d+e x)^{3/2}}{9 e}\) |
| \(\Big \downarrow \) 599 |
| \(\displaystyle \frac {2 \left (\frac {1}{7} \left (\frac {2 \sqrt {a+c x^2} \sqrt {d+e x} \left (4 d \left (3 a e^2+c d^2\right )-3 e x \left (c d^2-7 a e^2\right )\right )}{15 e^2}-\frac {4 \int -\frac {4 d \left (c d^2+a e^2\right ) \left (c d^2+3 a e^2\right )-\left (4 c^2 d^4+15 a c e^2 d^2-21 a^2 e^4\right ) (d+e x)}{\sqrt {\frac {c d^2}{e^2}-\frac {2 c (d+e x) d}{e^2}+\frac {c (d+e x)^2}{e^2}+a}}d\sqrt {d+e x}}{15 e^4}\right )-\frac {2}{7} d \left (a+c x^2\right )^{3/2} \sqrt {d+e x}\right )}{3 e}+\frac {2 \left (a+c x^2\right )^{3/2} (d+e x)^{3/2}}{9 e}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 \left (\frac {1}{7} \left (\frac {4 \int \frac {4 d \left (c d^2+a e^2\right ) \left (c d^2+3 a e^2\right )-\left (4 c^2 d^4+15 a c e^2 d^2-21 a^2 e^4\right ) (d+e x)}{\sqrt {\frac {c d^2}{e^2}-\frac {2 c (d+e x) d}{e^2}+\frac {c (d+e x)^2}{e^2}+a}}d\sqrt {d+e x}}{15 e^4}+\frac {2 \sqrt {a+c x^2} \sqrt {d+e x} \left (4 d \left (3 a e^2+c d^2\right )-3 e x \left (c d^2-7 a e^2\right )\right )}{15 e^2}\right )-\frac {2}{7} d \left (a+c x^2\right )^{3/2} \sqrt {d+e x}\right )}{3 e}+\frac {2 \left (a+c x^2\right )^{3/2} (d+e x)^{3/2}}{9 e}\) |
| \(\Big \downarrow \) 1511 |
| \(\displaystyle \frac {2 \left (\frac {1}{7} \left (\frac {2 \sqrt {a+c x^2} \sqrt {d+e x} \left (4 d \left (3 a e^2+c d^2\right )-3 e x \left (c d^2-7 a e^2\right )\right )}{15 e^2}-\frac {4 \left (\frac {\sqrt {a e^2+c d^2} \left (-21 a^2 e^4+15 a c d^2 e^2-4 \sqrt {c} d \sqrt {a e^2+c d^2} \left (3 a e^2+c d^2\right )+4 c^2 d^4\right ) \int \frac {1}{\sqrt {\frac {c d^2}{e^2}-\frac {2 c (d+e x) d}{e^2}+\frac {c (d+e x)^2}{e^2}+a}}d\sqrt {d+e x}}{\sqrt {c}}-\frac {\sqrt {a e^2+c d^2} \left (-21 a^2 e^4+15 a c d^2 e^2+4 c^2 d^4\right ) \int \frac {1-\frac {\sqrt {c} (d+e x)}{\sqrt {c d^2+a e^2}}}{\sqrt {\frac {c d^2}{e^2}-\frac {2 c (d+e x) d}{e^2}+\frac {c (d+e x)^2}{e^2}+a}}d\sqrt {d+e x}}{\sqrt {c}}\right )}{15 e^4}\right )-\frac {2}{7} d \left (a+c x^2\right )^{3/2} \sqrt {d+e x}\right )}{3 e}+\frac {2 \left (a+c x^2\right )^{3/2} (d+e x)^{3/2}}{9 e}\) |
| \(\Big \downarrow \) 1416 |
| \(\displaystyle \frac {2 \left (\frac {1}{7} \left (\frac {2 \sqrt {a+c x^2} \sqrt {d+e x} \left (4 d \left (3 a e^2+c d^2\right )-3 e x \left (c d^2-7 a e^2\right )\right )}{15 e^2}-\frac {4 \left (\frac {\left (a e^2+c d^2\right )^{3/4} \left (-21 a^2 e^4+15 a c d^2 e^2-4 \sqrt {c} d \sqrt {a e^2+c d^2} \left (3 a e^2+c d^2\right )+4 c^2 d^4\right ) \left (\frac {\sqrt {c} (d+e x)}{\sqrt {a e^2+c d^2}}+1\right ) \sqrt {\frac {a+\frac {c d^2}{e^2}-\frac {2 c d (d+e x)}{e^2}+\frac {c (d+e x)^2}{e^2}}{\left (a+\frac {c d^2}{e^2}\right ) \left (\frac {\sqrt {c} (d+e x)}{\sqrt {a e^2+c d^2}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt [4]{c d^2+a e^2}}\right ),\frac {1}{2} \left (\frac {\sqrt {c} d}{\sqrt {c d^2+a e^2}}+1\right )\right )}{2 c^{3/4} \sqrt {a+\frac {c d^2}{e^2}-\frac {2 c d (d+e x)}{e^2}+\frac {c (d+e x)^2}{e^2}}}-\frac {\sqrt {a e^2+c d^2} \left (-21 a^2 e^4+15 a c d^2 e^2+4 c^2 d^4\right ) \int \frac {1-\frac {\sqrt {c} (d+e x)}{\sqrt {c d^2+a e^2}}}{\sqrt {\frac {c d^2}{e^2}-\frac {2 c (d+e x) d}{e^2}+\frac {c (d+e x)^2}{e^2}+a}}d\sqrt {d+e x}}{\sqrt {c}}\right )}{15 e^4}\right )-\frac {2}{7} d \left (a+c x^2\right )^{3/2} \sqrt {d+e x}\right )}{3 e}+\frac {2 \left (a+c x^2\right )^{3/2} (d+e x)^{3/2}}{9 e}\) |
| \(\Big \downarrow \) 1509 |
| \(\displaystyle \frac {2 (d+e x)^{3/2} \left (c x^2+a\right )^{3/2}}{9 e}+\frac {2 \left (\frac {1}{7} \left (\frac {2 \sqrt {d+e x} \left (4 d \left (c d^2+3 a e^2\right )-3 e \left (c d^2-7 a e^2\right ) x\right ) \sqrt {c x^2+a}}{15 e^2}-\frac {4 \left (\frac {\left (c d^2+a e^2\right )^{3/4} \left (4 c^2 d^4+15 a c e^2 d^2-4 \sqrt {c} \sqrt {c d^2+a e^2} \left (c d^2+3 a e^2\right ) d-21 a^2 e^4\right ) \left (\frac {\sqrt {c} (d+e x)}{\sqrt {c d^2+a e^2}}+1\right ) \sqrt {\frac {\frac {c d^2}{e^2}-\frac {2 c (d+e x) d}{e^2}+\frac {c (d+e x)^2}{e^2}+a}{\left (\frac {c d^2}{e^2}+a\right ) \left (\frac {\sqrt {c} (d+e x)}{\sqrt {c d^2+a e^2}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt [4]{c d^2+a e^2}}\right ),\frac {1}{2} \left (\frac {\sqrt {c} d}{\sqrt {c d^2+a e^2}}+1\right )\right )}{2 c^{3/4} \sqrt {\frac {c d^2}{e^2}-\frac {2 c (d+e x) d}{e^2}+\frac {c (d+e x)^2}{e^2}+a}}-\frac {\sqrt {c d^2+a e^2} \left (4 c^2 d^4+15 a c e^2 d^2-21 a^2 e^4\right ) \left (\frac {\sqrt [4]{c d^2+a e^2} \left (\frac {\sqrt {c} (d+e x)}{\sqrt {c d^2+a e^2}}+1\right ) \sqrt {\frac {\frac {c d^2}{e^2}-\frac {2 c (d+e x) d}{e^2}+\frac {c (d+e x)^2}{e^2}+a}{\left (\frac {c d^2}{e^2}+a\right ) \left (\frac {\sqrt {c} (d+e x)}{\sqrt {c d^2+a e^2}}+1\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt [4]{c d^2+a e^2}}\right )|\frac {1}{2} \left (\frac {\sqrt {c} d}{\sqrt {c d^2+a e^2}}+1\right )\right )}{\sqrt [4]{c} \sqrt {\frac {c d^2}{e^2}-\frac {2 c (d+e x) d}{e^2}+\frac {c (d+e x)^2}{e^2}+a}}-\frac {\sqrt {d+e x} \sqrt {\frac {c d^2}{e^2}-\frac {2 c (d+e x) d}{e^2}+\frac {c (d+e x)^2}{e^2}+a}}{\left (\frac {c d^2}{e^2}+a\right ) \left (\frac {\sqrt {c} (d+e x)}{\sqrt {c d^2+a e^2}}+1\right )}\right )}{\sqrt {c}}\right )}{15 e^4}\right )-\frac {2}{7} d \sqrt {d+e x} \left (c x^2+a\right )^{3/2}\right )}{3 e}\) |
(2*(d + e*x)^(3/2)*(a + c*x^2)^(3/2))/(9*e) + (2*((-2*d*Sqrt[d + e*x]*(a + c*x^2)^(3/2))/7 + ((2*Sqrt[d + e*x]*(4*d*(c*d^2 + 3*a*e^2) - 3*e*(c*d^2 - 7*a*e^2)*x)*Sqrt[a + c*x^2])/(15*e^2) - (4*(-((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]*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 + (S qrt[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)*S qrt[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])))/(15*e^4))/7))/(3*e)
3.7.64.3.1 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[ (c + d*x)^(n + 1)*((a + b*x^2)^p/(d*(n + 2*p + 1))), x] + Simp[2*(p/(d*(n + 2*p + 1))) Int[(c + d*x)^n*(a + b*x^2)^(p - 1)*(a*d - b*c*x), x], x] /; FreeQ[{a, b, c, d, n}, x] && GtQ[p, 0] && NeQ[n + 2*p + 1, 0] && ( !Rationa lQ[n] || LtQ[n, 1]) && !ILtQ[n + 2*p, 0] && IntQuadraticQ[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)*(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[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p _.), x_Symbol] :> Simp[g*(d + e*x)^m*((a + c*x^2)^(p + 1)/(c*(m + 2*p + 2)) ), x] + Simp[1/(c*(m + 2*p + 2)) Int[(d + e*x)^(m - 1)*(a + c*x^2)^p*Simp [c*d*f*(m + 2*p + 2) - a*e*g*m + c*(e*f*(m + 2*p + 2) + d*g*m)*x, x], x], x ] /; FreeQ[{a, c, d, e, f, g, p}, x] && GtQ[m, 0] && NeQ[m + 2*p + 2, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p]) && !(IGtQ[m, 0] && Eq Q[f, 0])
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. \(862\) vs. \(2(370)=740\).
Time = 4.50 (sec) , antiderivative size = 863, normalized size of antiderivative = 1.93
| method | result | size |
| elliptic | \(\frac {\sqrt {\left (e x +d \right ) \left (c \,x^{2}+a \right )}\, \left (\frac {2 c \,x^{3} \sqrt {c e \,x^{3}+c d \,x^{2}+a e x +a d}}{9}+\frac {2 c d \,x^{2} \sqrt {c e \,x^{3}+c d \,x^{2}+a e x +a d}}{63 e}+\frac {2 \left (\frac {11 a c e}{9}-\frac {2 d^{2} c^{2}}{21 e}\right ) x \sqrt {c e \,x^{3}+c d \,x^{2}+a e x +a d}}{5 c e}+\frac {2 \left (\frac {79 a c d}{63}-\frac {4 d \left (\frac {11 a c e}{9}-\frac {2 d^{2} c^{2}}{21 e}\right )}{5 e}\right ) \sqrt {c e \,x^{3}+c d \,x^{2}+a e x +a d}}{3 c e}+\frac {2 \left (a^{2} d -\frac {2 a d \left (\frac {11 a c e}{9}-\frac {2 d^{2} c^{2}}{21 e}\right )}{5 c e}-\frac {a \left (\frac {79 a c d}{63}-\frac {4 d \left (\frac {11 a c e}{9}-\frac {2 d^{2} c^{2}}{21 e}\right )}{5 e}\right )}{3 c}\right ) \left (\frac {d}{e}-\frac {\sqrt {-a c}}{c}\right ) \sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x -\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x +\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}+\frac {\sqrt {-a c}}{c}}}\, F\left (\sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}, \sqrt {\frac {-\frac {d}{e}+\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\right )}{\sqrt {c e \,x^{3}+c d \,x^{2}+a e x +a d}}+\frac {2 \left (a^{2} e -\frac {4 a c \,d^{2}}{63 e}-\frac {3 a \left (\frac {11 a c e}{9}-\frac {2 d^{2} c^{2}}{21 e}\right )}{5 c}-\frac {2 d \left (\frac {79 a c d}{63}-\frac {4 d \left (\frac {11 a c e}{9}-\frac {2 d^{2} c^{2}}{21 e}\right )}{5 e}\right )}{3 e}\right ) \left (\frac {d}{e}-\frac {\sqrt {-a c}}{c}\right ) \sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x -\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x +\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}+\frac {\sqrt {-a c}}{c}}}\, \left (\left (-\frac {d}{e}-\frac {\sqrt {-a c}}{c}\right ) E\left (\sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}, \sqrt {\frac {-\frac {d}{e}+\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\right )+\frac {\sqrt {-a c}\, F\left (\sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}, \sqrt {\frac {-\frac {d}{e}+\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\right )}{c}\right )}{\sqrt {c e \,x^{3}+c d \,x^{2}+a e x +a d}}\right )}{\sqrt {e x +d}\, \sqrt {c \,x^{2}+a}}\) | \(863\) |
| risch | \(\frac {2 \left (35 c \,x^{3} e^{3}+5 c d \,x^{2} e^{2}+77 a \,e^{3} x -6 c \,d^{2} e x +29 a d \,e^{2}+8 d^{3} c \right ) \sqrt {e x +d}\, \sqrt {c \,x^{2}+a}}{315 e^{3}}+\frac {4 \left (\frac {2 a c \,d^{3} e \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 {66 a^{2} d \,e^{3} \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 (21 a^{2} e^{4}-15 a c \,d^{2} e^{2}-4 c^{2} d^{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 {\left (e x +d \right ) \left (c \,x^{2}+a \right )}}{315 e^{3} \sqrt {e x +d}\, \sqrt {c \,x^{2}+a}}\) | \(878\) |
| default | \(\text {Expression too large to display}\) | \(1731\) |
((e*x+d)*(c*x^2+a))^(1/2)/(e*x+d)^(1/2)/(c*x^2+a)^(1/2)*(2/9*c*x^3*(c*e*x^ 3+c*d*x^2+a*e*x+a*d)^(1/2)+2/63*c*d/e*x^2*(c*e*x^3+c*d*x^2+a*e*x+a*d)^(1/2 )+2/5*(11/9*a*c*e-2/21*d^2/e*c^2)/c/e*x*(c*e*x^3+c*d*x^2+a*e*x+a*d)^(1/2)+ 2/3*(79/63*a*c*d-4/5*d/e*(11/9*a*c*e-2/21*d^2/e*c^2))/c/e*(c*e*x^3+c*d*x^2 +a*e*x+a*d)^(1/2)+2*(a^2*d-2/5*a/c*d/e*(11/9*a*c*e-2/21*d^2/e*c^2)-1/3*a/c *(79/63*a*c*d-4/5*d/e*(11/9*a*c*e-2/21*d^2/e*c^2)))*(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*(a^2*e-4/63*a*c*d^2/e-3/5* a/c*(11/9*a*c*e-2/21*d^2/e*c^2)-2/3*d/e*(79/63*a*c*d-4/5*d/e*(11/9*a*c*e-2 /21*d^2/e*c^2)))*(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 = 0.19 (sec) , antiderivative size = 310, normalized size of antiderivative = 0.69 \[ \int \sqrt {d+e x} \left (a+c x^2\right )^{3/2} \, dx=\frac {2 \, {\left (8 \, {\left (2 \, c^{2} d^{5} + 9 \, a c d^{3} e^{2} + 39 \, a^{2} d e^{4}\right )} \sqrt {c e} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, \frac {3 \, e x + d}{3 \, e}\right ) + 12 \, {\left (4 \, c^{2} d^{4} e + 15 \, a c d^{2} e^{3} - 21 \, a^{2} e^{5}\right )} \sqrt {c e} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, \frac {3 \, e x + d}{3 \, e}\right )\right ) + 3 \, {\left (35 \, c^{2} e^{5} x^{3} + 5 \, c^{2} d e^{4} x^{2} + 8 \, c^{2} d^{3} e^{2} + 29 \, a c d e^{4} - {\left (6 \, c^{2} d^{2} e^{3} - 77 \, a c e^{5}\right )} x\right )} \sqrt {c x^{2} + a} \sqrt {e x + d}\right )}}{945 \, c e^{5}} \]
2/945*(8*(2*c^2*d^5 + 9*a*c*d^3*e^2 + 39*a^2*d*e^4)*sqrt(c*e)*weierstrassP Inverse(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*(4*c^2*d^4*e + 15*a*c*d^2*e^3 - 21*a^2*e^5)*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*(35*c^2*e^5*x^3 + 5*c^2*d* e^4*x^2 + 8*c^2*d^3*e^2 + 29*a*c*d*e^4 - (6*c^2*d^2*e^3 - 77*a*c*e^5)*x)*s qrt(c*x^2 + a)*sqrt(e*x + d))/(c*e^5)
\[ \int \sqrt {d+e x} \left (a+c x^2\right )^{3/2} \, dx=\int \left (a + c x^{2}\right )^{\frac {3}{2}} \sqrt {d + e x}\, dx \]
\[ \int \sqrt {d+e x} \left (a+c x^2\right )^{3/2} \, dx=\int { {\left (c x^{2} + a\right )}^{\frac {3}{2}} \sqrt {e x + d} \,d x } \]
\[ \int \sqrt {d+e x} \left (a+c x^2\right )^{3/2} \, dx=\int { {\left (c x^{2} + a\right )}^{\frac {3}{2}} \sqrt {e x + d} \,d x } \]
Timed out. \[ \int \sqrt {d+e x} \left (a+c x^2\right )^{3/2} \, dx=\int {\left (c\,x^2+a\right )}^{3/2}\,\sqrt {d+e\,x} \,d x \]