Integrand size = 21, antiderivative size = 410 \[ \int \frac {\left (a+c x^2\right )^{3/2}}{(d+e x)^{7/2}} \, dx=-\frac {4 c \left (2 d \left (2 c d^2+a e^2\right )+e \left (5 c d^2+3 a e^2\right ) x\right ) \sqrt {a+c x^2}}{5 e^3 \left (c d^2+a e^2\right ) (d+e x)^{3/2}}-\frac {2 \left (a+c x^2\right )^{3/2}}{5 e (d+e x)^{5/2}}-\frac {8 \sqrt {-a} c^{3/2} \left (4 c d^2+3 a e^2\right ) \sqrt {d+e x} \sqrt {1+\frac {c x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{5 e^4 \left (c d^2+a e^2\right ) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {a+c x^2}}+\frac {32 \sqrt {-a} c^{3/2} d \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {1+\frac {c x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right ),-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{5 e^4 \sqrt {d+e x} \sqrt {a+c x^2}} \]
-2/5*(c*x^2+a)^(3/2)/e/(e*x+d)^(5/2)-4/5*c*(2*d*(a*e^2+2*c*d^2)+e*(3*a*e^2 +5*c*d^2)*x)*(c*x^2+a)^(1/2)/e^3/(a*e^2+c*d^2)/(e*x+d)^(3/2)-8/5*c^(3/2)*( 3*a*e^2+4*c*d^2)*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/(a*e^2+c*d^2)/(c*x^2+a)^(1/2)/((e*x+d)*c^(1/2)/(e*(-a)^(1/2)+ d*c^(1/2)))^(1/2)+32/5*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)/e^4/(e*x+d)^(1 /2)/(c*x^2+a)^(1/2)
Result contains complex when optimal does not.
Time = 12.18 (sec) , antiderivative size = 585, normalized size of antiderivative = 1.43 \[ \int \frac {\left (a+c x^2\right )^{3/2}}{(d+e x)^{7/2}} \, dx=\frac {2 \left (-e^2 \left (a+c x^2\right ) \left (\left (c d^2+a e^2\right )^2-4 c d \left (c d^2+a e^2\right ) (d+e x)+c \left (11 c d^2+7 a e^2\right ) (d+e x)^2\right )+\frac {4 c (d+e x)^2 \left (e^2 \sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}} \left (4 c d^2+3 a e^2\right ) \left (a+c x^2\right )+\sqrt {c} \left (-4 i c^{3/2} d^3+4 \sqrt {a} c d^2 e-3 i a \sqrt {c} d e^2+3 a^{3/2} e^3\right ) \sqrt {\frac {e \left (\frac {i \sqrt {a}}{\sqrt {c}}+x\right )}{d+e x}} \sqrt {-\frac {\frac {i \sqrt {a} e}{\sqrt {c}}-e x}{d+e x}} (d+e x)^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right )|\frac {\sqrt {c} d-i \sqrt {a} e}{\sqrt {c} d+i \sqrt {a} e}\right )-\sqrt {a} \sqrt {c} e \left (4 c d^2+i \sqrt {a} \sqrt {c} d e+3 a e^2\right ) \sqrt {\frac {e \left (\frac {i \sqrt {a}}{\sqrt {c}}+x\right )}{d+e x}} \sqrt {-\frac {\frac {i \sqrt {a} e}{\sqrt {c}}-e x}{d+e x}} (d+e x)^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right ),\frac {\sqrt {c} d-i \sqrt {a} e}{\sqrt {c} d+i \sqrt {a} e}\right )\right )}{\sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}\right )}{5 e^5 \left (c d^2+a e^2\right ) (d+e x)^{5/2} \sqrt {a+c x^2}} \]
(2*(-(e^2*(a + c*x^2)*((c*d^2 + a*e^2)^2 - 4*c*d*(c*d^2 + a*e^2)*(d + e*x) + c*(11*c*d^2 + 7*a*e^2)*(d + e*x)^2)) + (4*c*(d + e*x)^2*(e^2*Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]*(4*c*d^2 + 3*a*e^2)*(a + c*x^2) + Sqrt[c]*((-4*I)*c ^(3/2)*d^3 + 4*Sqrt[a]*c*d^2*e - (3*I)*a*Sqrt[c]*d*e^2 + 3*a^(3/2)*e^3)*Sq rt[(e*((I*Sqrt[a])/Sqrt[c] + x))/(d + e*x)]*Sqrt[-(((I*Sqrt[a]*e)/Sqrt[c] - e*x)/(d + e*x))]*(d + e*x)^(3/2)*EllipticE[I*ArcSinh[Sqrt[-d - (I*Sqrt[a ]*e)/Sqrt[c]]/Sqrt[d + e*x]], (Sqrt[c]*d - I*Sqrt[a]*e)/(Sqrt[c]*d + I*Sqr t[a]*e)] - Sqrt[a]*Sqrt[c]*e*(4*c*d^2 + I*Sqrt[a]*Sqrt[c]*d*e + 3*a*e^2)*S qrt[(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*Sq rt[a]*e)]))/Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]))/(5*e^5*(c*d^2 + a*e^2)*(d + e*x)^(5/2)*Sqrt[a + c*x^2])
Time = 0.81 (sec) , antiderivative size = 757, normalized size of antiderivative = 1.85, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {492, 589, 25, 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.
| \(\displaystyle \int \frac {\left (a+c x^2\right )^{3/2}}{(d+e x)^{7/2}} \, dx\) |
| \(\Big \downarrow \) 492 |
| \(\displaystyle \frac {6 c \int \frac {x \sqrt {c x^2+a}}{(d+e x)^{5/2}}dx}{5 e}-\frac {2 \left (a+c x^2\right )^{3/2}}{5 e (d+e x)^{5/2}}\) |
| \(\Big \downarrow \) 589 |
| \(\displaystyle \frac {6 c \left (\frac {2 c \int -\frac {a d e-\left (4 c d^2+3 a e^2\right ) x}{\sqrt {d+e x} \sqrt {c x^2+a}}dx}{3 e^2 \left (a e^2+c d^2\right )}-\frac {2 \sqrt {a+c x^2} \left (e x \left (3 a e^2+5 c d^2\right )+2 d \left (a e^2+2 c d^2\right )\right )}{3 e^2 (d+e x)^{3/2} \left (a e^2+c d^2\right )}\right )}{5 e}-\frac {2 \left (a+c x^2\right )^{3/2}}{5 e (d+e x)^{5/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {6 c \left (-\frac {2 c \int \frac {a d e-\left (4 c d^2+3 a e^2\right ) x}{\sqrt {d+e x} \sqrt {c x^2+a}}dx}{3 e^2 \left (a e^2+c d^2\right )}-\frac {2 \sqrt {a+c x^2} \left (e x \left (3 a e^2+5 c d^2\right )+2 d \left (a e^2+2 c d^2\right )\right )}{3 e^2 (d+e x)^{3/2} \left (a e^2+c d^2\right )}\right )}{5 e}-\frac {2 \left (a+c x^2\right )^{3/2}}{5 e (d+e x)^{5/2}}\) |
| \(\Big \downarrow \) 599 |
| \(\displaystyle \frac {6 c \left (\frac {4 c \int -\frac {4 d \left (c d^2+a e^2\right )-\left (4 c d^2+3 a e^2\right ) (d+e x)}{\sqrt {\frac {c d^2}{e^2}-\frac {2 c (d+e x) d}{e^2}+\frac {c (d+e x)^2}{e^2}+a}}d\sqrt {d+e x}}{3 e^4 \left (a e^2+c d^2\right )}-\frac {2 \sqrt {a+c x^2} \left (e x \left (3 a e^2+5 c d^2\right )+2 d \left (a e^2+2 c d^2\right )\right )}{3 e^2 (d+e x)^{3/2} \left (a e^2+c d^2\right )}\right )}{5 e}-\frac {2 \left (a+c x^2\right )^{3/2}}{5 e (d+e x)^{5/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {6 c \left (-\frac {4 c \int \frac {4 d \left (c d^2+a e^2\right )-\left (4 c d^2+3 a e^2\right ) (d+e x)}{\sqrt {\frac {c d^2}{e^2}-\frac {2 c (d+e x) d}{e^2}+\frac {c (d+e x)^2}{e^2}+a}}d\sqrt {d+e x}}{3 e^4 \left (a e^2+c d^2\right )}-\frac {2 \sqrt {a+c x^2} \left (e x \left (3 a e^2+5 c d^2\right )+2 d \left (a e^2+2 c d^2\right )\right )}{3 e^2 (d+e x)^{3/2} \left (a e^2+c d^2\right )}\right )}{5 e}-\frac {2 \left (a+c x^2\right )^{3/2}}{5 e (d+e x)^{5/2}}\) |
| \(\Big \downarrow \) 1511 |
| \(\displaystyle \frac {6 c \left (\frac {4 c \left (\frac {\sqrt {a e^2+c d^2} \left (-4 \sqrt {c} d \sqrt {a e^2+c d^2}+3 a e^2+4 c d^2\right ) \int \frac {1}{\sqrt {\frac {c d^2}{e^2}-\frac {2 c (d+e x) d}{e^2}+\frac {c (d+e x)^2}{e^2}+a}}d\sqrt {d+e x}}{\sqrt {c}}-\frac {\sqrt {a e^2+c d^2} \left (3 a e^2+4 c d^2\right ) \int \frac {1-\frac {\sqrt {c} (d+e x)}{\sqrt {c d^2+a e^2}}}{\sqrt {\frac {c d^2}{e^2}-\frac {2 c (d+e x) d}{e^2}+\frac {c (d+e x)^2}{e^2}+a}}d\sqrt {d+e x}}{\sqrt {c}}\right )}{3 e^4 \left (a e^2+c d^2\right )}-\frac {2 \sqrt {a+c x^2} \left (e x \left (3 a e^2+5 c d^2\right )+2 d \left (a e^2+2 c d^2\right )\right )}{3 e^2 (d+e x)^{3/2} \left (a e^2+c d^2\right )}\right )}{5 e}-\frac {2 \left (a+c x^2\right )^{3/2}}{5 e (d+e x)^{5/2}}\) |
| \(\Big \downarrow \) 1416 |
| \(\displaystyle \frac {6 c \left (\frac {4 c \left (\frac {\left (a e^2+c d^2\right )^{3/4} \left (-4 \sqrt {c} d \sqrt {a e^2+c d^2}+3 a e^2+4 c d^2\right ) \left (\frac {\sqrt {c} (d+e x)}{\sqrt {a e^2+c d^2}}+1\right ) \sqrt {\frac {a+\frac {c d^2}{e^2}-\frac {2 c d (d+e x)}{e^2}+\frac {c (d+e x)^2}{e^2}}{\left (a+\frac {c d^2}{e^2}\right ) \left (\frac {\sqrt {c} (d+e x)}{\sqrt {a e^2+c d^2}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt [4]{c d^2+a e^2}}\right ),\frac {1}{2} \left (\frac {\sqrt {c} d}{\sqrt {c d^2+a e^2}}+1\right )\right )}{2 c^{3/4} \sqrt {a+\frac {c d^2}{e^2}-\frac {2 c d (d+e x)}{e^2}+\frac {c (d+e x)^2}{e^2}}}-\frac {\sqrt {a e^2+c d^2} \left (3 a e^2+4 c d^2\right ) \int \frac {1-\frac {\sqrt {c} (d+e x)}{\sqrt {c d^2+a e^2}}}{\sqrt {\frac {c d^2}{e^2}-\frac {2 c (d+e x) d}{e^2}+\frac {c (d+e x)^2}{e^2}+a}}d\sqrt {d+e x}}{\sqrt {c}}\right )}{3 e^4 \left (a e^2+c d^2\right )}-\frac {2 \sqrt {a+c x^2} \left (e x \left (3 a e^2+5 c d^2\right )+2 d \left (a e^2+2 c d^2\right )\right )}{3 e^2 (d+e x)^{3/2} \left (a e^2+c d^2\right )}\right )}{5 e}-\frac {2 \left (a+c x^2\right )^{3/2}}{5 e (d+e x)^{5/2}}\) |
| \(\Big \downarrow \) 1509 |
| \(\displaystyle \frac {6 c \left (\frac {4 c \left (\frac {\left (a e^2+c d^2\right )^{3/4} \left (-4 \sqrt {c} d \sqrt {a e^2+c d^2}+3 a e^2+4 c d^2\right ) \left (\frac {\sqrt {c} (d+e x)}{\sqrt {a e^2+c d^2}}+1\right ) \sqrt {\frac {a+\frac {c d^2}{e^2}-\frac {2 c d (d+e x)}{e^2}+\frac {c (d+e x)^2}{e^2}}{\left (a+\frac {c d^2}{e^2}\right ) \left (\frac {\sqrt {c} (d+e x)}{\sqrt {a e^2+c d^2}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt [4]{c d^2+a e^2}}\right ),\frac {1}{2} \left (\frac {\sqrt {c} d}{\sqrt {c d^2+a e^2}}+1\right )\right )}{2 c^{3/4} \sqrt {a+\frac {c d^2}{e^2}-\frac {2 c d (d+e x)}{e^2}+\frac {c (d+e x)^2}{e^2}}}-\frac {\sqrt {a e^2+c d^2} \left (3 a e^2+4 c d^2\right ) \left (\frac {\sqrt [4]{a e^2+c d^2} \left (\frac {\sqrt {c} (d+e x)}{\sqrt {a e^2+c d^2}}+1\right ) \sqrt {\frac {a+\frac {c d^2}{e^2}-\frac {2 c d (d+e x)}{e^2}+\frac {c (d+e x)^2}{e^2}}{\left (a+\frac {c d^2}{e^2}\right ) \left (\frac {\sqrt {c} (d+e x)}{\sqrt {a e^2+c d^2}}+1\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt [4]{c d^2+a e^2}}\right )|\frac {1}{2} \left (\frac {\sqrt {c} d}{\sqrt {c d^2+a e^2}}+1\right )\right )}{\sqrt [4]{c} \sqrt {a+\frac {c d^2}{e^2}-\frac {2 c d (d+e x)}{e^2}+\frac {c (d+e x)^2}{e^2}}}-\frac {\sqrt {d+e x} \sqrt {a+\frac {c d^2}{e^2}-\frac {2 c d (d+e x)}{e^2}+\frac {c (d+e x)^2}{e^2}}}{\left (a+\frac {c d^2}{e^2}\right ) \left (\frac {\sqrt {c} (d+e x)}{\sqrt {a e^2+c d^2}}+1\right )}\right )}{\sqrt {c}}\right )}{3 e^4 \left (a e^2+c d^2\right )}-\frac {2 \sqrt {a+c x^2} \left (e x \left (3 a e^2+5 c d^2\right )+2 d \left (a e^2+2 c d^2\right )\right )}{3 e^2 (d+e x)^{3/2} \left (a e^2+c d^2\right )}\right )}{5 e}-\frac {2 \left (a+c x^2\right )^{3/2}}{5 e (d+e x)^{5/2}}\) |
(-2*(a + c*x^2)^(3/2))/(5*e*(d + e*x)^(5/2)) + (6*c*((-2*(2*d*(2*c*d^2 + a *e^2) + e*(5*c*d^2 + 3*a*e^2)*x)*Sqrt[a + c*x^2])/(3*e^2*(c*d^2 + a*e^2)*( d + e*x)^(3/2)) + (4*c*(-((Sqrt[c*d^2 + a*e^2]*(4*c*d^2 + 3*a*e^2)*(-((Sqr t[d + e*x]*Sqrt[a + (c*d^2)/e^2 - (2*c*d*(d + e*x))/e^2 + (c*(d + e*x)^2)/ e^2])/((a + (c*d^2)/e^2)*(1 + (Sqrt[c]*(d + e*x))/Sqrt[c*d^2 + a*e^2]))) + ((c*d^2 + a*e^2)^(1/4)*(1 + (Sqrt[c]*(d + e*x))/Sqrt[c*d^2 + a*e^2])*Sqrt [(a + (c*d^2)/e^2 - (2*c*d*(d + e*x))/e^2 + (c*(d + e*x)^2)/e^2)/((a + (c* d^2)/e^2)*(1 + (Sqrt[c]*(d + e*x))/Sqrt[c*d^2 + a*e^2])^2)]*EllipticE[2*Ar cTan[(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*d^2 + 3* a*e^2 - 4*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)*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])))/(3*e^4*(c*d^2 + a*e^2))))/(5 *e)
3.7.68.3.1 Defintions of rubi rules used
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 + 1))), x] - Simp[2*b*(p/(d*(n + 1)) ) Int[x*(c + d*x)^(n + 1)*(a + b*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d, n}, x] && GtQ[p, 0] && (IntegerQ[p] || LtQ[n, -1]) && NeQ[n, -1] && !IL tQ[n + 2*p + 1, 0] && IntQuadraticQ[a, 0, b, c, d, n, p, x]
Int[(x_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-(c + d*x)^(n + 1))*(a + b*x^2)^p*((c*(a*d^2 + b*c^2*(2*p + 1)) - d*( a*d^2*(n + 1) + b*c^2*(n - 2*p + 1))*x)/(d^2*(n + 1)*(n + 2)*(b*c^2 + a*d^2 ))), x] + Simp[b*(p/(d^2*(n + 1)*(n + 2)*(b*c^2 + a*d^2))) Int[(c + d*x)^ (n + 2)*(a + b*x^2)^(p - 1)*Simp[2*a*c*d*(n + 2) - (2*a*d^2*(n + 1) - 2*b*c ^2*(2*p + 1))*x, x], x], x] /; FreeQ[{a, b, c, d}, x] && GtQ[p, 0] && LtQ[n , -2] && LtQ[n + 2*p, 0] && !ILtQ[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[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. \(787\) vs. \(2(338)=676\).
Time = 2.64 (sec) , antiderivative size = 788, normalized size of antiderivative = 1.92
| method | result | size |
| elliptic | \(\frac {\sqrt {\left (e x +d \right ) \left (c \,x^{2}+a \right )}\, \left (-\frac {2 \left (e^{2} a +c \,d^{2}\right ) \sqrt {c e \,x^{3}+c d \,x^{2}+a e x +a d}}{5 e^{6} \left (x +\frac {d}{e}\right )^{3}}+\frac {8 c d \sqrt {c e \,x^{3}+c d \,x^{2}+a e x +a d}}{5 e^{5} \left (x +\frac {d}{e}\right )^{2}}-\frac {2 \left (c e \,x^{2}+a e \right ) c \left (7 e^{2} a +11 c \,d^{2}\right )}{5 e^{4} \left (e^{2} a +c \,d^{2}\right ) \sqrt {\left (x +\frac {d}{e}\right ) \left (c e \,x^{2}+a e \right )}}+\frac {2 \left (-\frac {11 c^{2} d}{5 e^{4}}+\frac {c^{2} d \left (7 e^{2} a +11 c \,d^{2}\right )}{5 e^{4} \left (e^{2} a +c \,d^{2}\right )}\right ) \left (\frac {d}{e}-\frac {\sqrt {-a c}}{c}\right ) \sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x -\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x +\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}+\frac {\sqrt {-a c}}{c}}}\, F\left (\sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}, \sqrt {\frac {-\frac {d}{e}+\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\right )}{\sqrt {c e \,x^{3}+c d \,x^{2}+a e x +a d}}+\frac {2 \left (\frac {c^{2}}{e^{3}}+\frac {c^{2} \left (7 e^{2} a +11 c \,d^{2}\right )}{5 e^{3} \left (e^{2} a +c \,d^{2}\right )}\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}}\) | \(788\) |
| default | \(\text {Expression too large to display}\) | \(3410\) |
((e*x+d)*(c*x^2+a))^(1/2)/(e*x+d)^(1/2)/(c*x^2+a)^(1/2)*(-2/5*(a*e^2+c*d^2 )/e^6*(c*e*x^3+c*d*x^2+a*e*x+a*d)^(1/2)/(x+d/e)^3+8/5*c/e^5*d*(c*e*x^3+c*d *x^2+a*e*x+a*d)^(1/2)/(x+d/e)^2-2/5*(c*e*x^2+a*e)/e^4/(a*e^2+c*d^2)*c*(7*a *e^2+11*c*d^2)/((x+d/e)*(c*e*x^2+a*e))^(1/2)+2*(-11/5*c^2*d/e^4+1/5*c^2*d/ e^4*(7*a*e^2+11*c*d^2)/(a*e^2+c*d^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*(c^2/e^3+1/5*c^2/e^3*(7*a*e^2+11*c*d^2)/ (a*e^2+c*d^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 = 0.11 (sec) , antiderivative size = 521, normalized size of antiderivative = 1.27 \[ \int \frac {\left (a+c x^2\right )^{3/2}}{(d+e x)^{7/2}} \, dx=-\frac {2 \, {\left (8 \, {\left (2 \, c^{2} d^{6} + 3 \, a c d^{4} e^{2} + {\left (2 \, c^{2} d^{3} e^{3} + 3 \, a c d e^{5}\right )} x^{3} + 3 \, {\left (2 \, c^{2} d^{4} e^{2} + 3 \, a c d^{2} e^{4}\right )} x^{2} + 3 \, {\left (2 \, c^{2} d^{5} e + 3 \, a c d^{3} e^{3}\right )} x\right )} \sqrt {c e} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, \frac {3 \, e x + d}{3 \, e}\right ) + 12 \, {\left (4 \, c^{2} d^{5} e + 3 \, a c d^{3} e^{3} + {\left (4 \, c^{2} d^{2} e^{4} + 3 \, a c e^{6}\right )} x^{3} + 3 \, {\left (4 \, c^{2} d^{3} e^{3} + 3 \, a c d e^{5}\right )} x^{2} + 3 \, {\left (4 \, c^{2} d^{4} e^{2} + 3 \, a c d^{2} e^{4}\right )} x\right )} \sqrt {c e} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, \frac {3 \, e x + d}{3 \, e}\right )\right ) + 3 \, {\left (8 \, c^{2} d^{4} e^{2} + 5 \, a c d^{2} e^{4} + a^{2} e^{6} + {\left (11 \, c^{2} d^{2} e^{4} + 7 \, a c e^{6}\right )} x^{2} + 2 \, {\left (9 \, c^{2} d^{3} e^{3} + 5 \, a c d e^{5}\right )} x\right )} \sqrt {c x^{2} + a} \sqrt {e x + d}\right )}}{15 \, {\left (c d^{5} e^{5} + a d^{3} e^{7} + {\left (c d^{2} e^{8} + a e^{10}\right )} x^{3} + 3 \, {\left (c d^{3} e^{7} + a d e^{9}\right )} x^{2} + 3 \, {\left (c d^{4} e^{6} + a d^{2} e^{8}\right )} x\right )}} \]
-2/15*(8*(2*c^2*d^6 + 3*a*c*d^4*e^2 + (2*c^2*d^3*e^3 + 3*a*c*d*e^5)*x^3 + 3*(2*c^2*d^4*e^2 + 3*a*c*d^2*e^4)*x^2 + 3*(2*c^2*d^5*e + 3*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) + 12*(4*c^2*d^5*e + 3*a*c*d^3*e^ 3 + (4*c^2*d^2*e^4 + 3*a*c*e^6)*x^3 + 3*(4*c^2*d^3*e^3 + 3*a*c*d*e^5)*x^2 + 3*(4*c^2*d^4*e^2 + 3*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), weierstrassPInver se(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*(8*c^2*d^4*e^2 + 5*a*c*d^2*e^4 + a^2*e^6 + (11*c^2*d^2* e^4 + 7*a*c*e^6)*x^2 + 2*(9*c^2*d^3*e^3 + 5*a*c*d*e^5)*x)*sqrt(c*x^2 + a)* sqrt(e*x + d))/(c*d^5*e^5 + a*d^3*e^7 + (c*d^2*e^8 + a*e^10)*x^3 + 3*(c*d^ 3*e^7 + a*d*e^9)*x^2 + 3*(c*d^4*e^6 + a*d^2*e^8)*x)
\[ \int \frac {\left (a+c x^2\right )^{3/2}}{(d+e x)^{7/2}} \, dx=\int \frac {\left (a + c x^{2}\right )^{\frac {3}{2}}}{\left (d + e x\right )^{\frac {7}{2}}}\, dx \]
\[ \int \frac {\left (a+c x^2\right )^{3/2}}{(d+e x)^{7/2}} \, dx=\int { \frac {{\left (c x^{2} + a\right )}^{\frac {3}{2}}}{{\left (e x + d\right )}^{\frac {7}{2}}} \,d x } \]
\[ \int \frac {\left (a+c x^2\right )^{3/2}}{(d+e x)^{7/2}} \, dx=\int { \frac {{\left (c x^{2} + a\right )}^{\frac {3}{2}}}{{\left (e x + d\right )}^{\frac {7}{2}}} \,d x } \]
Timed out. \[ \int \frac {\left (a+c x^2\right )^{3/2}}{(d+e x)^{7/2}} \, dx=\int \frac {{\left (c\,x^2+a\right )}^{3/2}}{{\left (d+e\,x\right )}^{7/2}} \,d x \]