Integrand size = 26, antiderivative size = 498 \[ \int \frac {(A+B x) \left (a+c x^2\right )^{3/2}}{\sqrt {d+e x}} \, dx=-\frac {4 \sqrt {d+e x} \left (32 B c d^3-36 A c d^2 e+33 a B d e^2-45 a A e^3-3 e \left (8 B c d^2-9 A c d e+7 a B e^2\right ) x\right ) \sqrt {a+c x^2}}{315 e^4}-\frac {2 \sqrt {d+e x} (8 B d-9 A e-7 B e x) \left (a+c x^2\right )^{3/2}}{63 e^2}+\frac {8 \sqrt {-a} \left (36 A c d e \left (c d^2+2 a e^2\right )-B \left (32 c^2 d^4+57 a c d^2 e^2+21 a^2 e^4\right )\right ) \sqrt {d+e x} \sqrt {1+\frac {c x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{315 \sqrt {c} e^5 \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {a+c x^2}}+\frac {8 \sqrt {-a} \left (c d^2+a e^2\right ) \left (32 B c d^3-36 A c d^2 e+33 a B d e^2-45 a A e^3\right ) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {1+\frac {c x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right ),-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{315 \sqrt {c} e^5 \sqrt {d+e x} \sqrt {a+c x^2}} \]
-2/63*(-7*B*e*x-9*A*e+8*B*d)*(c*x^2+a)^(3/2)*(e*x+d)^(1/2)/e^2-4/315*(32*B *c*d^3-36*A*c*d^2*e+33*B*a*d*e^2-45*A*a*e^3-3*e*(-9*A*c*d*e+7*B*a*e^2+8*B* c*d^2)*x)*(e*x+d)^(1/2)*(c*x^2+a)^(1/2)/e^4+8/315*(36*A*c*d*e*(2*a*e^2+c*d ^2)-B*(21*a^2*e^4+57*a*c*d^2*e^2+32*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^5/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)+8/315*(a*e^2+c*d^2)*(-45*A*a*e^3- 36*A*c*d^2*e+33*B*a*d*e^2+32*B*c*d^3)*EllipticF(1/2*(1-x*c^(1/2)/(-a)^(1/2 ))^(1/2)*2^(1/2),(-2*a*e/(-a*e+d*(-a)^(1/2)*c^(1/2)))^(1/2))*(-a)^(1/2)*(1 +c*x^2/a)^(1/2)*((e*x+d)*c^(1/2)/(e*(-a)^(1/2)+d*c^(1/2)))^(1/2)/e^5/c^(1/ 2)/(e*x+d)^(1/2)/(c*x^2+a)^(1/2)
Result contains complex when optimal does not.
Time = 24.74 (sec) , antiderivative size = 730, normalized size of antiderivative = 1.47 \[ \int \frac {(A+B x) \left (a+c x^2\right )^{3/2}}{\sqrt {d+e x}} \, dx=\frac {\sqrt {d+e x} \left (\frac {2 \left (a+c x^2\right ) \left (135 a A e^3+a B e^2 (-106 d+77 e x)+9 A c e \left (8 d^2-6 d e x+5 e^2 x^2\right )+B c \left (-64 d^3+48 d^2 e x-40 d e^2 x^2+35 e^3 x^3\right )\right )}{e^4}-\frac {8 \left (e^2 \sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}} \left (36 A c d e \left (c d^2+2 a e^2\right )-B \left (32 c^2 d^4+57 a c d^2 e^2+21 a^2 e^4\right )\right ) \left (a+c x^2\right )+\sqrt {c} \left (-i \sqrt {c} d+\sqrt {a} e\right ) \left (36 A c d e \left (c d^2+2 a e^2\right )-B \left (32 c^2 d^4+57 a c d^2 e^2+21 a^2 e^4\right )\right ) \sqrt {\frac {e \left (\frac {i \sqrt {a}}{\sqrt {c}}+x\right )}{d+e x}} \sqrt {-\frac {\frac {i \sqrt {a} e}{\sqrt {c}}-e x}{d+e x}} (d+e x)^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right )|\frac {\sqrt {c} d-i \sqrt {a} e}{\sqrt {c} d+i \sqrt {a} e}\right )-\sqrt {a} \sqrt {c} e \left (\sqrt {c} d+i \sqrt {a} e\right ) \left (9 A \sqrt {c} e \left (4 c d^2-3 i \sqrt {a} \sqrt {c} d e+5 a e^2\right )+B \left (-32 c^{3/2} d^3+24 i \sqrt {a} c d^2 e-33 a \sqrt {c} d e^2+21 i a^{3/2} e^3\right )\right ) \sqrt {\frac {e \left (\frac {i \sqrt {a}}{\sqrt {c}}+x\right )}{d+e x}} \sqrt {-\frac {\frac {i \sqrt {a} e}{\sqrt {c}}-e x}{d+e x}} (d+e x)^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right ),\frac {\sqrt {c} d-i \sqrt {a} e}{\sqrt {c} d+i \sqrt {a} e}\right )\right )}{c e^6 \sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}} (d+e x)}\right )}{315 \sqrt {a+c x^2}} \]
(Sqrt[d + e*x]*((2*(a + c*x^2)*(135*a*A*e^3 + a*B*e^2*(-106*d + 77*e*x) + 9*A*c*e*(8*d^2 - 6*d*e*x + 5*e^2*x^2) + B*c*(-64*d^3 + 48*d^2*e*x - 40*d*e ^2*x^2 + 35*e^3*x^3)))/e^4 - (8*(e^2*Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]*(36* A*c*d*e*(c*d^2 + 2*a*e^2) - B*(32*c^2*d^4 + 57*a*c*d^2*e^2 + 21*a^2*e^4))* (a + c*x^2) + Sqrt[c]*((-I)*Sqrt[c]*d + Sqrt[a]*e)*(36*A*c*d*e*(c*d^2 + 2* a*e^2) - B*(32*c^2*d^4 + 57*a*c*d^2*e^2 + 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)*EllipticE[I*ArcSinh[Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]/Sqr t[d + e*x]], (Sqrt[c]*d - I*Sqrt[a]*e)/(Sqrt[c]*d + I*Sqrt[a]*e)] - Sqrt[a ]*Sqrt[c]*e*(Sqrt[c]*d + I*Sqrt[a]*e)*(9*A*Sqrt[c]*e*(4*c*d^2 - (3*I)*Sqrt [a]*Sqrt[c]*d*e + 5*a*e^2) + B*(-32*c^(3/2)*d^3 + (24*I)*Sqrt[a]*c*d^2*e - 33*a*Sqrt[c]*d*e^2 + (21*I)*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)*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^6*Sqrt[-d - ( I*Sqrt[a]*e)/Sqrt[c]]*(d + e*x))))/(315*Sqrt[a + c*x^2])
Time = 1.27 (sec) , antiderivative size = 872, normalized size of antiderivative = 1.75, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.346, Rules used = {682, 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.
| \(\displaystyle \int \frac {\left (a+c x^2\right )^{3/2} (A+B x)}{\sqrt {d+e x}} \, dx\) |
| \(\Big \downarrow \) 682 |
| \(\displaystyle \frac {4 \int -\frac {c \left (a e (B d-9 A e)-\left (8 B c d^2-9 A c e d+7 a B e^2\right ) x\right ) \sqrt {c x^2+a}}{2 \sqrt {d+e x}}dx}{21 c e^2}-\frac {2 \left (a+c x^2\right )^{3/2} \sqrt {d+e x} (-9 A e+8 B d-7 B e x)}{63 e^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {2 \int \frac {\left (a e (B d-9 A e)-\left (8 B c d^2-9 A c e d+7 a B e^2\right ) x\right ) \sqrt {c x^2+a}}{\sqrt {d+e x}}dx}{21 e^2}-\frac {2 \left (a+c x^2\right )^{3/2} \sqrt {d+e x} (-9 A e+8 B d-7 B e x)}{63 e^2}\) |
| \(\Big \downarrow \) 682 |
| \(\displaystyle -\frac {2 \left (\frac {4 \int \frac {c \left (a e \left (8 B c d^3-9 A c e d^2+12 a B e^2 d-45 a A e^3\right )+\left (36 A c d e \left (c d^2+2 a e^2\right )-B \left (32 c^2 d^4+57 a c e^2 d^2+21 a^2 e^4\right )\right ) x\right )}{2 \sqrt {d+e x} \sqrt {c x^2+a}}dx}{15 c e^2}+\frac {2 \sqrt {a+c x^2} \sqrt {d+e x} \left (-3 e x \left (7 a B e^2-9 A c d e+8 B c d^2\right )-45 a A e^3+33 a B d e^2-36 A c d^2 e+32 B c d^3\right )}{15 e^2}\right )}{21 e^2}-\frac {2 \left (a+c x^2\right )^{3/2} \sqrt {d+e x} (-9 A e+8 B d-7 B e x)}{63 e^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {2 \left (\frac {2 \int \frac {a e \left (8 B c d^3-9 A c e d^2+12 a B e^2 d-45 a A e^3\right )+\left (36 A c d e \left (c d^2+2 a e^2\right )-B \left (32 c^2 d^4+57 a c e^2 d^2+21 a^2 e^4\right )\right ) x}{\sqrt {d+e x} \sqrt {c x^2+a}}dx}{15 e^2}+\frac {2 \sqrt {a+c x^2} \sqrt {d+e x} \left (-3 e x \left (7 a B e^2-9 A c d e+8 B c d^2\right )-45 a A e^3+33 a B d e^2-36 A c d^2 e+32 B c d^3\right )}{15 e^2}\right )}{21 e^2}-\frac {2 \left (a+c x^2\right )^{3/2} \sqrt {d+e x} (-9 A e+8 B d-7 B e x)}{63 e^2}\) |
| \(\Big \downarrow \) 599 |
| \(\displaystyle -\frac {2 \left (\frac {2 \sqrt {a+c x^2} \sqrt {d+e x} \left (-3 e x \left (7 a B e^2-9 A c d e+8 B c d^2\right )-45 a A e^3+33 a B d e^2-36 A c d^2 e+32 B c d^3\right )}{15 e^2}-\frac {4 \int -\frac {\left (c d^2+a e^2\right ) \left (32 B c d^3-36 A c e d^2+33 a B e^2 d-45 a A e^3\right )+\left (36 A c d e \left (c d^2+2 a e^2\right )-B \left (32 c^2 d^4+57 a c e^2 d^2+21 a^2 e^4\right )\right ) (d+e x)}{\sqrt {\frac {c d^2}{e^2}-\frac {2 c (d+e x) d}{e^2}+\frac {c (d+e x)^2}{e^2}+a}}d\sqrt {d+e x}}{15 e^4}\right )}{21 e^2}-\frac {2 \left (a+c x^2\right )^{3/2} \sqrt {d+e x} (-9 A e+8 B d-7 B e x)}{63 e^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {2 \left (\frac {4 \int \frac {\left (c d^2+a e^2\right ) \left (32 B c d^3-36 A c e d^2+33 a B e^2 d-45 a A e^3\right )+\left (36 A c d e \left (c d^2+2 a e^2\right )-B \left (32 c^2 d^4+57 a c e^2 d^2+21 a^2 e^4\right )\right ) (d+e x)}{\sqrt {\frac {c d^2}{e^2}-\frac {2 c (d+e x) d}{e^2}+\frac {c (d+e x)^2}{e^2}+a}}d\sqrt {d+e x}}{15 e^4}+\frac {2 \sqrt {a+c x^2} \sqrt {d+e x} \left (-3 e x \left (7 a B e^2-9 A c d e+8 B c d^2\right )-45 a A e^3+33 a B d e^2-36 A c d^2 e+32 B c d^3\right )}{15 e^2}\right )}{21 e^2}-\frac {2 \left (a+c x^2\right )^{3/2} \sqrt {d+e x} (-9 A e+8 B d-7 B e x)}{63 e^2}\) |
| \(\Big \downarrow \) 1511 |
| \(\displaystyle -\frac {2 \left (\frac {2 \sqrt {a+c x^2} \sqrt {d+e x} \left (-3 e x \left (7 a B e^2-9 A c d e+8 B c d^2\right )-45 a A e^3+33 a B d e^2-36 A c d^2 e+32 B c d^3\right )}{15 e^2}-\frac {4 \left (\frac {\sqrt {a e^2+c d^2} \left (36 A c d e \left (2 a e^2+c d^2\right )-B \left (21 a^2 e^4+57 a c d^2 e^2+32 c^2 d^4\right )\right ) \int \frac {1-\frac {\sqrt {c} (d+e x)}{\sqrt {c d^2+a e^2}}}{\sqrt {\frac {c d^2}{e^2}-\frac {2 c (d+e x) d}{e^2}+\frac {c (d+e x)^2}{e^2}+a}}d\sqrt {d+e x}}{\sqrt {c}}-\frac {\sqrt {a e^2+c d^2} \left (-B \left (21 a^2 e^4+57 a c d^2 e^2+32 c^2 d^4\right )-\sqrt {c} \sqrt {a e^2+c d^2} \left (9 A e \left (5 a e^2+4 c d^2\right )-B \left (33 a d e^2+32 c d^3\right )\right )+36 A c d e \left (2 a e^2+c d^2\right )\right ) \int \frac {1}{\sqrt {\frac {c d^2}{e^2}-\frac {2 c (d+e x) d}{e^2}+\frac {c (d+e x)^2}{e^2}+a}}d\sqrt {d+e x}}{\sqrt {c}}\right )}{15 e^4}\right )}{21 e^2}-\frac {2 \left (a+c x^2\right )^{3/2} \sqrt {d+e x} (-9 A e+8 B d-7 B e x)}{63 e^2}\) |
| \(\Big \downarrow \) 1416 |
| \(\displaystyle -\frac {2 \left (\frac {2 \sqrt {a+c x^2} \sqrt {d+e x} \left (-3 e x \left (7 a B e^2-9 A c d e+8 B c d^2\right )-45 a A e^3+33 a B d e^2-36 A c d^2 e+32 B c d^3\right )}{15 e^2}-\frac {4 \left (\frac {\sqrt {a e^2+c d^2} \left (36 A c d e \left (2 a e^2+c d^2\right )-B \left (21 a^2 e^4+57 a c d^2 e^2+32 c^2 d^4\right )\right ) \int \frac {1-\frac {\sqrt {c} (d+e x)}{\sqrt {c d^2+a e^2}}}{\sqrt {\frac {c d^2}{e^2}-\frac {2 c (d+e x) d}{e^2}+\frac {c (d+e x)^2}{e^2}+a}}d\sqrt {d+e x}}{\sqrt {c}}-\frac {\left (a e^2+c d^2\right )^{3/4} \left (\frac {\sqrt {c} (d+e x)}{\sqrt {a e^2+c d^2}}+1\right ) \sqrt {\frac {a+\frac {c d^2}{e^2}-\frac {2 c d (d+e x)}{e^2}+\frac {c (d+e x)^2}{e^2}}{\left (a+\frac {c d^2}{e^2}\right ) \left (\frac {\sqrt {c} (d+e x)}{\sqrt {a e^2+c d^2}}+1\right )^2}} \left (-B \left (21 a^2 e^4+57 a c d^2 e^2+32 c^2 d^4\right )-\sqrt {c} \sqrt {a e^2+c d^2} \left (9 A e \left (5 a e^2+4 c d^2\right )-B \left (33 a d e^2+32 c d^3\right )\right )+36 A c d e \left (2 a e^2+c d^2\right )\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt [4]{c d^2+a e^2}}\right ),\frac {1}{2} \left (\frac {\sqrt {c} d}{\sqrt {c d^2+a e^2}}+1\right )\right )}{2 c^{3/4} \sqrt {a+\frac {c d^2}{e^2}-\frac {2 c d (d+e x)}{e^2}+\frac {c (d+e x)^2}{e^2}}}\right )}{15 e^4}\right )}{21 e^2}-\frac {2 \left (a+c x^2\right )^{3/2} \sqrt {d+e x} (-9 A e+8 B d-7 B e x)}{63 e^2}\) |
| \(\Big \downarrow \) 1509 |
| \(\displaystyle -\frac {2 \sqrt {d+e x} (8 B d-9 A e-7 B e x) \left (c x^2+a\right )^{3/2}}{63 e^2}-\frac {2 \left (\frac {2 \sqrt {d+e x} \left (32 B c d^3-36 A c e d^2+33 a B e^2 d-45 a A e^3-3 e \left (8 B c d^2-9 A c e d+7 a B e^2\right ) x\right ) \sqrt {c x^2+a}}{15 e^2}-\frac {4 \left (\frac {\sqrt {c d^2+a e^2} \left (36 A c d e \left (c d^2+2 a e^2\right )-B \left (32 c^2 d^4+57 a c e^2 d^2+21 a^2 e^4\right )\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}}-\frac {\left (c d^2+a e^2\right )^{3/4} \left (36 A c d e \left (c d^2+2 a e^2\right )-B \left (32 c^2 d^4+57 a c e^2 d^2+21 a^2 e^4\right )-\sqrt {c} \sqrt {c d^2+a e^2} \left (9 A e \left (4 c d^2+5 a e^2\right )-B \left (32 c d^3+33 a e^2 d\right )\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 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 )}{15 e^4}\right )}{21 e^2}\) |
(-2*Sqrt[d + e*x]*(8*B*d - 9*A*e - 7*B*e*x)*(a + c*x^2)^(3/2))/(63*e^2) - (2*((2*Sqrt[d + e*x]*(32*B*c*d^3 - 36*A*c*d^2*e + 33*a*B*d*e^2 - 45*a*A*e^ 3 - 3*e*(8*B*c*d^2 - 9*A*c*d*e + 7*a*B*e^2)*x)*Sqrt[a + c*x^2])/(15*e^2) - (4*((Sqrt[c*d^2 + a*e^2]*(36*A*c*d*e*(c*d^2 + 2*a*e^2) - B*(32*c^2*d^4 + 57*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 + (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))/Sq rt[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)*(36*A*c*d*e*(c*d^2 + 2*a*e^2) - B*(32*c^2*d^4 + 57* a*c*d^2*e^2 + 21*a^2*e^4) - Sqrt[c]*Sqrt[c*d^2 + a*e^2]*(9*A*e*(4*c*d^2 + 5*a*e^2) - B*(32*c*d^3 + 33*a*d*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)))/(21*e^2)
3.15.76.3.1 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)*(c*e*f*(m + 2*p + 2) - g*c*d*(2*p + 1) + g*c*e*(m + 2*p + 1)*x)*((a + c*x^2)^p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2))), x] + Simp[2*(p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2))) Int[(d + e*x) ^m*(a + c*x^2)^(p - 1)*Simp[f*a*c*e^2*(m + 2*p + 2) + a*c*d*e*g*m - (c^2*f* d*e*(m + 2*p + 2) - g*(c^2*d^2*(2*p + 1) + a*c*e^2*(m + 2*p + 1)))*x, x], x ], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && GtQ[p, 0] && (IntegerQ[p] || ! RationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])) && !ILtQ[m + 2*p, 0] && (Intege rQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
Int[1/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c /a, 4]}, Simp[(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2*x^2)^2)]/ (2*q*Sqrt[a + b*x^2 + c*x^4]))*EllipticF[2*ArcTan[q*x], 1/2 - b*(q^2/(4*c)) ], x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbo l] :> With[{q = Rt[c/a, 4]}, Simp[(-d)*x*(Sqrt[a + b*x^2 + c*x^4]/(a*(1 + q ^2*x^2))), x] + Simp[d*(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2* x^2)^2)]/(q*Sqrt[a + b*x^2 + c*x^4]))*EllipticE[2*ArcTan[q*x], 1/2 - b*(q^2 /(4*c))], x] /; EqQ[e + d*q^2, 0]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbo l] :> With[{q = Rt[c/a, 2]}, Simp[(e + d*q)/q Int[1/Sqrt[a + b*x^2 + c*x^ 4], x], x] - Simp[e/q Int[(1 - q*x^2)/Sqrt[a + b*x^2 + c*x^4], x], x] /; NeQ[e + d*q, 0]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && Pos Q[c/a]
Leaf count of result is larger than twice the leaf count of optimal. \(1058\) vs. \(2(426)=852\).
Time = 3.79 (sec) , antiderivative size = 1059, normalized size of antiderivative = 2.13
| method | result | size |
| elliptic | \(\text {Expression too large to display}\) | \(1059\) |
| risch | \(\text {Expression too large to display}\) | \(1378\) |
| default | \(\text {Expression too large to display}\) | \(3113\) |
((e*x+d)*(c*x^2+a))^(1/2)/(e*x+d)^(1/2)/(c*x^2+a)^(1/2)*(2/9*B/e*c*x^3*(c* e*x^3+c*d*x^2+a*e*x+a*d)^(1/2)+2/7*(A*c^2-8/9*c^2*d/e*B)/c/e*x^2*(c*e*x^3+ c*d*x^2+a*e*x+a*d)^(1/2)+2/5*(11/9*B*a*c-6/7*d/e*(A*c^2-8/9*c^2*d/e*B))/c/ e*x*(c*e*x^3+c*d*x^2+a*e*x+a*d)^(1/2)+2/3*(2*A*a*c-4/5*d/e*(11/9*B*a*c-6/7 *d/e*(A*c^2-8/9*c^2*d/e*B))-5/7*a/c*(A*c^2-8/9*c^2*d/e*B)-2/3*a*c*d/e*B)/c /e*(c*e*x^3+c*d*x^2+a*e*x+a*d)^(1/2)+2*(A*a^2-2/5*a/c*d/e*(11/9*B*a*c-6/7* d/e*(A*c^2-8/9*c^2*d/e*B))-1/3*a/c*(2*A*a*c-4/5*d/e*(11/9*B*a*c-6/7*d/e*(A *c^2-8/9*c^2*d/e*B))-5/7*a/c*(A*c^2-8/9*c^2*d/e*B)-2/3*a*c*d/e*B))*(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*(B*a^2-4/7* a/c*d/e*(A*c^2-8/9*c^2*d/e*B)-3/5*a/c*(11/9*B*a*c-6/7*d/e*(A*c^2-8/9*c^2*d /e*B))-2/3*d/e*(2*A*a*c-4/5*d/e*(11/9*B*a*c-6/7*d/e*(A*c^2-8/9*c^2*d/e*B)) -5/7*a/c*(A*c^2-8/9*c^2*d/e*B)-2/3*a*c*d/e*B))*(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)...
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.12 (sec) , antiderivative size = 414, normalized size of antiderivative = 0.83 \[ \int \frac {(A+B x) \left (a+c x^2\right )^{3/2}}{\sqrt {d+e x}} \, dx=-\frac {2 \, {\left (4 \, {\left (32 \, B c^{2} d^{5} - 36 \, A c^{2} d^{4} e + 81 \, B a c d^{3} e^{2} - 99 \, A a c d^{2} e^{3} + 57 \, B a^{2} d e^{4} - 135 \, A a^{2} e^{5}\right )} \sqrt {c e} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, \frac {3 \, e x + d}{3 \, e}\right ) + 12 \, {\left (32 \, B c^{2} d^{4} e - 36 \, A c^{2} d^{3} e^{2} + 57 \, B a c d^{2} e^{3} - 72 \, A a c d e^{4} + 21 \, B a^{2} e^{5}\right )} \sqrt {c e} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, \frac {3 \, e x + d}{3 \, e}\right )\right ) - 3 \, {\left (35 \, B c^{2} e^{5} x^{3} - 64 \, B c^{2} d^{3} e^{2} + 72 \, A c^{2} d^{2} e^{3} - 106 \, B a c d e^{4} + 135 \, A a c e^{5} - 5 \, {\left (8 \, B c^{2} d e^{4} - 9 \, A c^{2} e^{5}\right )} x^{2} + {\left (48 \, B c^{2} d^{2} e^{3} - 54 \, A c^{2} d e^{4} + 77 \, B a c e^{5}\right )} x\right )} \sqrt {c x^{2} + a} \sqrt {e x + d}\right )}}{945 \, c e^{6}} \]
-2/945*(4*(32*B*c^2*d^5 - 36*A*c^2*d^4*e + 81*B*a*c*d^3*e^2 - 99*A*a*c*d^2 *e^3 + 57*B*a^2*d*e^4 - 135*A*a^2*e^5)*sqrt(c*e)*weierstrassPInverse(4/3*( c*d^2 - 3*a*e^2)/(c*e^2), -8/27*(c*d^3 + 9*a*d*e^2)/(c*e^3), 1/3*(3*e*x + d)/e) + 12*(32*B*c^2*d^4*e - 36*A*c^2*d^3*e^2 + 57*B*a*c*d^2*e^3 - 72*A*a* c*d*e^4 + 21*B*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*B*c^2*e^5*x^3 - 64*B*c^2*d^3*e^2 + 72*A*c^2*d^2*e^3 - 106*B*a*c*d* e^4 + 135*A*a*c*e^5 - 5*(8*B*c^2*d*e^4 - 9*A*c^2*e^5)*x^2 + (48*B*c^2*d^2* e^3 - 54*A*c^2*d*e^4 + 77*B*a*c*e^5)*x)*sqrt(c*x^2 + a)*sqrt(e*x + d))/(c* e^6)
\[ \int \frac {(A+B x) \left (a+c x^2\right )^{3/2}}{\sqrt {d+e x}} \, dx=\int \frac {\left (A + B x\right ) \left (a + c x^{2}\right )^{\frac {3}{2}}}{\sqrt {d + e x}}\, dx \]
\[ \int \frac {(A+B x) \left (a+c x^2\right )^{3/2}}{\sqrt {d+e x}} \, dx=\int { \frac {{\left (c x^{2} + a\right )}^{\frac {3}{2}} {\left (B x + A\right )}}{\sqrt {e x + d}} \,d x } \]
\[ \int \frac {(A+B x) \left (a+c x^2\right )^{3/2}}{\sqrt {d+e x}} \, dx=\int { \frac {{\left (c x^{2} + a\right )}^{\frac {3}{2}} {\left (B x + A\right )}}{\sqrt {e x + d}} \,d x } \]
Timed out. \[ \int \frac {(A+B x) \left (a+c x^2\right )^{3/2}}{\sqrt {d+e x}} \, dx=\int \frac {{\left (c\,x^2+a\right )}^{3/2}\,\left (A+B\,x\right )}{\sqrt {d+e\,x}} \,d x \]