Integrand size = 20, antiderivative size = 268 \[ \int \frac {(d+e x)^{3/2}}{\left (a-c x^2\right )^3} \, dx=\frac {(a e+c d x) \sqrt {d+e x}}{4 a c \left (a-c x^2\right )^2}-\frac {(a e-6 c d x) \sqrt {d+e x}}{16 a^2 c \left (a-c x^2\right )}-\frac {3 \left (4 c d^2-2 \sqrt {a} \sqrt {c} d e-a e^2\right ) \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{32 a^{5/2} c^{5/4} \sqrt {\sqrt {c} d-\sqrt {a} e}}+\frac {3 \left (4 c d^2+2 \sqrt {a} \sqrt {c} d e-a e^2\right ) \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d+\sqrt {a} e}}\right )}{32 a^{5/2} c^{5/4} \sqrt {\sqrt {c} d+\sqrt {a} e}} \]
1/4*(c*d*x+a*e)*(e*x+d)^(1/2)/a/c/(-c*x^2+a)^2-1/16*(-6*c*d*x+a*e)*(e*x+d) ^(1/2)/a^2/c/(-c*x^2+a)-3/32*arctanh(c^(1/4)*(e*x+d)^(1/2)/(-e*a^(1/2)+d*c ^(1/2))^(1/2))*(4*c*d^2-a*e^2-2*d*e*a^(1/2)*c^(1/2))/a^(5/2)/c^(5/4)/(-e*a ^(1/2)+d*c^(1/2))^(1/2)+3/32*arctanh(c^(1/4)*(e*x+d)^(1/2)/(e*a^(1/2)+d*c^ (1/2))^(1/2))*(4*c*d^2-a*e^2+2*d*e*a^(1/2)*c^(1/2))/a^(5/2)/c^(5/4)/(e*a^( 1/2)+d*c^(1/2))^(1/2)
Time = 1.99 (sec) , antiderivative size = 268, normalized size of antiderivative = 1.00 \[ \int \frac {(d+e x)^{3/2}}{\left (a-c x^2\right )^3} \, dx=\frac {-\frac {2 \sqrt {a} \sqrt {d+e x} \left (-3 a^2 e+6 c^2 d x^3-a c x (10 d+e x)\right )}{\left (a-c x^2\right )^2}+\frac {3 \left (4 c d^2+2 \sqrt {a} \sqrt {c} d e-a e^2\right ) \arctan \left (\frac {\sqrt {-c d-\sqrt {a} \sqrt {c} e} \sqrt {d+e x}}{\sqrt {c} d+\sqrt {a} e}\right )}{\sqrt {-c d-\sqrt {a} \sqrt {c} e}}-\frac {3 \left (4 c d^2-2 \sqrt {a} \sqrt {c} d e-a e^2\right ) \arctan \left (\frac {\sqrt {-c d+\sqrt {a} \sqrt {c} e} \sqrt {d+e x}}{\sqrt {c} d-\sqrt {a} e}\right )}{\sqrt {-c d+\sqrt {a} \sqrt {c} e}}}{32 a^{5/2} c} \]
((-2*Sqrt[a]*Sqrt[d + e*x]*(-3*a^2*e + 6*c^2*d*x^3 - a*c*x*(10*d + e*x)))/ (a - c*x^2)^2 + (3*(4*c*d^2 + 2*Sqrt[a]*Sqrt[c]*d*e - a*e^2)*ArcTan[(Sqrt[ -(c*d) - Sqrt[a]*Sqrt[c]*e]*Sqrt[d + e*x])/(Sqrt[c]*d + Sqrt[a]*e)])/Sqrt[ -(c*d) - Sqrt[a]*Sqrt[c]*e] - (3*(4*c*d^2 - 2*Sqrt[a]*Sqrt[c]*d*e - a*e^2) *ArcTan[(Sqrt[-(c*d) + Sqrt[a]*Sqrt[c]*e]*Sqrt[d + e*x])/(Sqrt[c]*d - Sqrt [a]*e)])/Sqrt[-(c*d) + Sqrt[a]*Sqrt[c]*e])/(32*a^(5/2)*c)
Time = 0.51 (sec) , antiderivative size = 284, normalized size of antiderivative = 1.06, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.450, Rules used = {495, 27, 686, 27, 654, 25, 27, 1480, 221}
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.
| \(\displaystyle \int \frac {(d+e x)^{3/2}}{\left (a-c x^2\right )^3} \, dx\) |
| \(\Big \downarrow \) 495 |
| \(\displaystyle \frac {\sqrt {d+e x} (a e+c d x)}{4 a c \left (a-c x^2\right )^2}-\frac {\int -\frac {6 c d^2+5 c e x d-a e^2}{2 \sqrt {d+e x} \left (a-c x^2\right )^2}dx}{4 a c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {6 c d^2+5 c e x d-a e^2}{\sqrt {d+e x} \left (a-c x^2\right )^2}dx}{8 a c}+\frac {\sqrt {d+e x} (a e+c d x)}{4 a c \left (a-c x^2\right )^2}\) |
| \(\Big \downarrow \) 686 |
| \(\displaystyle \frac {-\frac {\int -\frac {3 c \left (c d^2-a e^2\right ) \left (4 c d^2+2 c e x d-a e^2\right )}{2 \sqrt {d+e x} \left (a-c x^2\right )}dx}{2 a c \left (c d^2-a e^2\right )}-\frac {\sqrt {d+e x} (a e-6 c d x)}{2 a \left (a-c x^2\right )}}{8 a c}+\frac {\sqrt {d+e x} (a e+c d x)}{4 a c \left (a-c x^2\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {3 \int \frac {4 c d^2+2 c e x d-a e^2}{\sqrt {d+e x} \left (a-c x^2\right )}dx}{4 a}-\frac {\sqrt {d+e x} (a e-6 c d x)}{2 a \left (a-c x^2\right )}}{8 a c}+\frac {\sqrt {d+e x} (a e+c d x)}{4 a c \left (a-c x^2\right )^2}\) |
| \(\Big \downarrow \) 654 |
| \(\displaystyle \frac {\frac {3 \int -\frac {e \left (2 c d^2+2 c (d+e x) d-a e^2\right )}{c d^2-2 c (d+e x) d-a e^2+c (d+e x)^2}d\sqrt {d+e x}}{2 a}-\frac {\sqrt {d+e x} (a e-6 c d x)}{2 a \left (a-c x^2\right )}}{8 a c}+\frac {\sqrt {d+e x} (a e+c d x)}{4 a c \left (a-c x^2\right )^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-\frac {3 \int \frac {e \left (2 c d^2+2 c (d+e x) d-a e^2\right )}{c d^2-2 c (d+e x) d-a e^2+c (d+e x)^2}d\sqrt {d+e x}}{2 a}-\frac {\sqrt {d+e x} (a e-6 c d x)}{2 a \left (a-c x^2\right )}}{8 a c}+\frac {\sqrt {d+e x} (a e+c d x)}{4 a c \left (a-c x^2\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {3 e \int \frac {2 c d^2+2 c (d+e x) d-a e^2}{c d^2-2 c (d+e x) d-a e^2+c (d+e x)^2}d\sqrt {d+e x}}{2 a}-\frac {\sqrt {d+e x} (a e-6 c d x)}{2 a \left (a-c x^2\right )}}{8 a c}+\frac {\sqrt {d+e x} (a e+c d x)}{4 a c \left (a-c x^2\right )^2}\) |
| \(\Big \downarrow \) 1480 |
| \(\displaystyle \frac {-\frac {3 e \left (\frac {1}{2} \sqrt {c} \left (2 \sqrt {c} d-\frac {4 c d^2-a e^2}{\sqrt {a} e}\right ) \int \frac {1}{c (d+e x)-\sqrt {c} \left (\sqrt {c} d-\sqrt {a} e\right )}d\sqrt {d+e x}+\frac {1}{2} \sqrt {c} \left (\frac {4 c d^2-a e^2}{\sqrt {a} e}+2 \sqrt {c} d\right ) \int \frac {1}{c (d+e x)-\sqrt {c} \left (\sqrt {c} d+\sqrt {a} e\right )}d\sqrt {d+e x}\right )}{2 a}-\frac {\sqrt {d+e x} (a e-6 c d x)}{2 a \left (a-c x^2\right )}}{8 a c}+\frac {\sqrt {d+e x} (a e+c d x)}{4 a c \left (a-c x^2\right )^2}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {-\frac {3 e \left (-\frac {\left (2 \sqrt {c} d-\frac {4 c d^2-a e^2}{\sqrt {a} e}\right ) \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{2 \sqrt [4]{c} \sqrt {\sqrt {c} d-\sqrt {a} e}}-\frac {\left (\frac {4 c d^2-a e^2}{\sqrt {a} e}+2 \sqrt {c} d\right ) \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {a} e+\sqrt {c} d}}\right )}{2 \sqrt [4]{c} \sqrt {\sqrt {a} e+\sqrt {c} d}}\right )}{2 a}-\frac {\sqrt {d+e x} (a e-6 c d x)}{2 a \left (a-c x^2\right )}}{8 a c}+\frac {\sqrt {d+e x} (a e+c d x)}{4 a c \left (a-c x^2\right )^2}\) |
((a*e + c*d*x)*Sqrt[d + e*x])/(4*a*c*(a - c*x^2)^2) + (-1/2*((a*e - 6*c*d* x)*Sqrt[d + e*x])/(a*(a - c*x^2)) - (3*e*(-1/2*((2*Sqrt[c]*d - (4*c*d^2 - a*e^2)/(Sqrt[a]*e))*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[ a]*e]])/(c^(1/4)*Sqrt[Sqrt[c]*d - Sqrt[a]*e]) - ((2*Sqrt[c]*d + (4*c*d^2 - a*e^2)/(Sqrt[a]*e))*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d + Sqrt [a]*e]])/(2*c^(1/4)*Sqrt[Sqrt[c]*d + Sqrt[a]*e])))/(2*a))/(8*a*c)
3.7.40.3.1 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_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
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*b*(p + 1))), x] - Simp[1/(2*a*b*(p + 1)) Int[(c + d*x)^(n - 2)*(a + b*x^2)^(p + 1)*Simp[a* d^2*(n - 1) - b*c^2*(2*p + 3) - b*c*d*(n + 2*p + 2)*x, x], x], x] /; FreeQ[ {a, b, c, d}, x] && LtQ[p, -1] && GtQ[n, 1] && IntQuadraticQ[a, 0, b, c, d, n, p, x]
Int[((f_.) + (g_.)*(x_))/(Sqrt[(d_.) + (e_.)*(x_)]*((a_) + (c_.)*(x_)^2)), x_Symbol] :> Simp[2 Subst[Int[(e*f - d*g + g*x^2)/(c*d^2 + a*e^2 - 2*c*d* x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /; FreeQ[{a, c, d, e, f, g}, x]
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_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] : > With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(e/2 + (2*c*d - b*e)/(2*q)) Int[1/( b/2 - q/2 + c*x^2), x], x] + Simp[(e/2 - (2*c*d - b*e)/(2*q)) Int[1/(b/2 + q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^2 - 4*a*c]
Time = 2.44 (sec) , antiderivative size = 281, normalized size of antiderivative = 1.05
| method | result | size |
| pseudoelliptic | \(\frac {-\frac {3 \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}\, c e \left (-c \,x^{2}+a \right )^{2} \left (e^{2} a -4 c \,d^{2}+2 \sqrt {a c \,e^{2}}\, d \right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{32}+\frac {3 \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}\, \left (-\frac {c e \left (-c \,x^{2}+a \right )^{2} \left (e^{2} a -4 c \,d^{2}-2 \sqrt {a c \,e^{2}}\, d \right ) \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2}+\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}\, \left (-2 c^{2} d \,x^{3}+\frac {10 x \left (\frac {e x}{10}+d \right ) a c}{3}+a^{2} e \right ) \sqrt {a c \,e^{2}}\, \sqrt {e x +d}\right )}{16}}{a^{2} c \left (-c \,x^{2}+a \right )^{2} \sqrt {a c \,e^{2}}\, \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\) | \(281\) |
| default | \(2 e^{5} \left (\frac {-\frac {3 c d \left (e x +d \right )^{\frac {7}{2}}}{16 a^{2} e^{4}}+\frac {\left (e^{2} a +18 c \,d^{2}\right ) \left (e x +d \right )^{\frac {5}{2}}}{32 a^{2} e^{4}}+\frac {d \left (4 e^{2} a -9 c \,d^{2}\right ) \left (e x +d \right )^{\frac {3}{2}}}{16 a^{2} e^{4}}+\frac {3 \left (e^{2} a -c \,d^{2}\right ) \left (e^{2} a -2 c \,d^{2}\right ) \sqrt {e x +d}}{32 a^{2} e^{4} c}}{\left (-c \left (e x +d \right )^{2}+2 c d \left (e x +d \right )+e^{2} a -c \,d^{2}\right )^{2}}+\frac {\frac {3 \left (-e^{2} a +4 c \,d^{2}-2 \sqrt {a c \,e^{2}}\, d \right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{64 \sqrt {a c \,e^{2}}\, \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}-\frac {3 \left (e^{2} a -4 c \,d^{2}-2 \sqrt {a c \,e^{2}}\, d \right ) \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{64 \sqrt {a c \,e^{2}}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}}{a^{2} e^{4}}\right )\) | \(319\) |
| derivativedivides | \(-2 e^{5} \left (-\frac {-\frac {3 c d \left (e x +d \right )^{\frac {7}{2}}}{16 a^{2} e^{4}}+\frac {\left (e^{2} a +18 c \,d^{2}\right ) \left (e x +d \right )^{\frac {5}{2}}}{32 a^{2} e^{4}}+\frac {d \left (4 e^{2} a -9 c \,d^{2}\right ) \left (e x +d \right )^{\frac {3}{2}}}{16 a^{2} e^{4}}+\frac {3 \left (e^{2} a -c \,d^{2}\right ) \left (e^{2} a -2 c \,d^{2}\right ) \sqrt {e x +d}}{32 a^{2} e^{4} c}}{\left (-c \left (e x +d \right )^{2}+2 c d \left (e x +d \right )+e^{2} a -c \,d^{2}\right )^{2}}-\frac {3 \left (\frac {\left (-e^{2} a +4 c \,d^{2}-2 \sqrt {a c \,e^{2}}\, d \right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 \sqrt {a c \,e^{2}}\, \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}-\frac {\left (e^{2} a -4 c \,d^{2}-2 \sqrt {a c \,e^{2}}\, d \right ) \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 \sqrt {a c \,e^{2}}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{32 a^{2} e^{4}}\right )\) | \(320\) |
3/16*(-1/2*((c*d+(a*c*e^2)^(1/2))*c)^(1/2)*c*e*(-c*x^2+a)^2*(e^2*a-4*c*d^2 +2*(a*c*e^2)^(1/2)*d)*arctan(c*(e*x+d)^(1/2)/((-c*d+(a*c*e^2)^(1/2))*c)^(1 /2))+((-c*d+(a*c*e^2)^(1/2))*c)^(1/2)*(-1/2*c*e*(-c*x^2+a)^2*(e^2*a-4*c*d^ 2-2*(a*c*e^2)^(1/2)*d)*arctanh(c*(e*x+d)^(1/2)/((c*d+(a*c*e^2)^(1/2))*c)^( 1/2))+((c*d+(a*c*e^2)^(1/2))*c)^(1/2)*(-2*c^2*d*x^3+10/3*x*(1/10*e*x+d)*a* c+a^2*e)*(a*c*e^2)^(1/2)*(e*x+d)^(1/2)))/((-c*d+(a*c*e^2)^(1/2))*c)^(1/2)/ ((c*d+(a*c*e^2)^(1/2))*c)^(1/2)/(a*c*e^2)^(1/2)/a^2/c/(-c*x^2+a)^2
Leaf count of result is larger than twice the leaf count of optimal. 1753 vs. \(2 (211) = 422\).
Time = 0.58 (sec) , antiderivative size = 1753, normalized size of antiderivative = 6.54 \[ \int \frac {(d+e x)^{3/2}}{\left (a-c x^2\right )^3} \, dx=\text {Too large to display} \]
1/64*(3*(a^2*c^3*x^4 - 2*a^3*c^2*x^2 + a^4*c)*sqrt((16*c^2*d^5 - 20*a*c*d^ 3*e^2 + 5*a^2*d*e^4 + (a^5*c^3*d^2 - a^6*c^2*e^2)*sqrt(e^10/(a^5*c^7*d^4 - 2*a^6*c^6*d^2*e^2 + a^7*c^5*e^4)))/(a^5*c^3*d^2 - a^6*c^2*e^2))*log(27*(1 6*c^2*d^4*e^5 - 12*a*c*d^2*e^7 + a^2*e^9)*sqrt(e*x + d) + 27*(2*a^3*c^2*d^ 2*e^6 - a^4*c*e^8 - (4*a^5*c^6*d^5 - 7*a^6*c^5*d^3*e^2 + 3*a^7*c^4*d*e^4)* sqrt(e^10/(a^5*c^7*d^4 - 2*a^6*c^6*d^2*e^2 + a^7*c^5*e^4)))*sqrt((16*c^2*d ^5 - 20*a*c*d^3*e^2 + 5*a^2*d*e^4 + (a^5*c^3*d^2 - a^6*c^2*e^2)*sqrt(e^10/ (a^5*c^7*d^4 - 2*a^6*c^6*d^2*e^2 + a^7*c^5*e^4)))/(a^5*c^3*d^2 - a^6*c^2*e ^2))) - 3*(a^2*c^3*x^4 - 2*a^3*c^2*x^2 + a^4*c)*sqrt((16*c^2*d^5 - 20*a*c* d^3*e^2 + 5*a^2*d*e^4 + (a^5*c^3*d^2 - a^6*c^2*e^2)*sqrt(e^10/(a^5*c^7*d^4 - 2*a^6*c^6*d^2*e^2 + a^7*c^5*e^4)))/(a^5*c^3*d^2 - a^6*c^2*e^2))*log(27* (16*c^2*d^4*e^5 - 12*a*c*d^2*e^7 + a^2*e^9)*sqrt(e*x + d) - 27*(2*a^3*c^2* d^2*e^6 - a^4*c*e^8 - (4*a^5*c^6*d^5 - 7*a^6*c^5*d^3*e^2 + 3*a^7*c^4*d*e^4 )*sqrt(e^10/(a^5*c^7*d^4 - 2*a^6*c^6*d^2*e^2 + a^7*c^5*e^4)))*sqrt((16*c^2 *d^5 - 20*a*c*d^3*e^2 + 5*a^2*d*e^4 + (a^5*c^3*d^2 - a^6*c^2*e^2)*sqrt(e^1 0/(a^5*c^7*d^4 - 2*a^6*c^6*d^2*e^2 + a^7*c^5*e^4)))/(a^5*c^3*d^2 - a^6*c^2 *e^2))) + 3*(a^2*c^3*x^4 - 2*a^3*c^2*x^2 + a^4*c)*sqrt((16*c^2*d^5 - 20*a* c*d^3*e^2 + 5*a^2*d*e^4 - (a^5*c^3*d^2 - a^6*c^2*e^2)*sqrt(e^10/(a^5*c^7*d ^4 - 2*a^6*c^6*d^2*e^2 + a^7*c^5*e^4)))/(a^5*c^3*d^2 - a^6*c^2*e^2))*log(2 7*(16*c^2*d^4*e^5 - 12*a*c*d^2*e^7 + a^2*e^9)*sqrt(e*x + d) + 27*(2*a^3...
Timed out. \[ \int \frac {(d+e x)^{3/2}}{\left (a-c x^2\right )^3} \, dx=\text {Timed out} \]
\[ \int \frac {(d+e x)^{3/2}}{\left (a-c x^2\right )^3} \, dx=\int { -\frac {{\left (e x + d\right )}^{\frac {3}{2}}}{{\left (c x^{2} - a\right )}^{3}} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 500 vs. \(2 (211) = 422\).
Time = 0.43 (sec) , antiderivative size = 500, normalized size of antiderivative = 1.87 \[ \int \frac {(d+e x)^{3/2}}{\left (a-c x^2\right )^3} \, dx=\frac {3 \, {\left (4 \, c^{3} d^{3} e - 3 \, a c^{2} d e^{3} - {\left (2 \, \sqrt {a c} c d^{2} e - \sqrt {a c} a e^{3}\right )} {\left | c \right |} {\left | e \right |}\right )} \arctan \left (\frac {\sqrt {e x + d}}{\sqrt {-\frac {a^{2} c^{2} d + \sqrt {a^{4} c^{4} d^{2} - {\left (a^{2} c^{2} d^{2} - a^{3} c e^{2}\right )} a^{2} c^{2}}}{a^{2} c^{2}}}}\right )}{32 \, {\left (a^{3} c^{2} e - \sqrt {a c} a^{2} c^{2} d\right )} \sqrt {-c^{2} d - \sqrt {a c} c e} {\left | e \right |}} + \frac {3 \, {\left (4 \, c^{3} d^{3} e - 3 \, a c^{2} d e^{3} + {\left (2 \, \sqrt {a c} c d^{2} e - \sqrt {a c} a e^{3}\right )} {\left | c \right |} {\left | e \right |}\right )} \arctan \left (\frac {\sqrt {e x + d}}{\sqrt {-\frac {a^{2} c^{2} d - \sqrt {a^{4} c^{4} d^{2} - {\left (a^{2} c^{2} d^{2} - a^{3} c e^{2}\right )} a^{2} c^{2}}}{a^{2} c^{2}}}}\right )}{32 \, {\left (a^{3} c^{2} e + \sqrt {a c} a^{2} c^{2} d\right )} \sqrt {-c^{2} d + \sqrt {a c} c e} {\left | e \right |}} - \frac {6 \, {\left (e x + d\right )}^{\frac {7}{2}} c^{2} d e - 18 \, {\left (e x + d\right )}^{\frac {5}{2}} c^{2} d^{2} e + 18 \, {\left (e x + d\right )}^{\frac {3}{2}} c^{2} d^{3} e - 6 \, \sqrt {e x + d} c^{2} d^{4} e - {\left (e x + d\right )}^{\frac {5}{2}} a c e^{3} - 8 \, {\left (e x + d\right )}^{\frac {3}{2}} a c d e^{3} + 9 \, \sqrt {e x + d} a c d^{2} e^{3} - 3 \, \sqrt {e x + d} a^{2} e^{5}}{16 \, {\left ({\left (e x + d\right )}^{2} c - 2 \, {\left (e x + d\right )} c d + c d^{2} - a e^{2}\right )}^{2} a^{2} c} \]
3/32*(4*c^3*d^3*e - 3*a*c^2*d*e^3 - (2*sqrt(a*c)*c*d^2*e - sqrt(a*c)*a*e^3 )*abs(c)*abs(e))*arctan(sqrt(e*x + d)/sqrt(-(a^2*c^2*d + sqrt(a^4*c^4*d^2 - (a^2*c^2*d^2 - a^3*c*e^2)*a^2*c^2))/(a^2*c^2)))/((a^3*c^2*e - sqrt(a*c)* a^2*c^2*d)*sqrt(-c^2*d - sqrt(a*c)*c*e)*abs(e)) + 3/32*(4*c^3*d^3*e - 3*a* c^2*d*e^3 + (2*sqrt(a*c)*c*d^2*e - sqrt(a*c)*a*e^3)*abs(c)*abs(e))*arctan( sqrt(e*x + d)/sqrt(-(a^2*c^2*d - sqrt(a^4*c^4*d^2 - (a^2*c^2*d^2 - a^3*c*e ^2)*a^2*c^2))/(a^2*c^2)))/((a^3*c^2*e + sqrt(a*c)*a^2*c^2*d)*sqrt(-c^2*d + sqrt(a*c)*c*e)*abs(e)) - 1/16*(6*(e*x + d)^(7/2)*c^2*d*e - 18*(e*x + d)^( 5/2)*c^2*d^2*e + 18*(e*x + d)^(3/2)*c^2*d^3*e - 6*sqrt(e*x + d)*c^2*d^4*e - (e*x + d)^(5/2)*a*c*e^3 - 8*(e*x + d)^(3/2)*a*c*d*e^3 + 9*sqrt(e*x + d)* a*c*d^2*e^3 - 3*sqrt(e*x + d)*a^2*e^5)/(((e*x + d)^2*c - 2*(e*x + d)*c*d + c*d^2 - a*e^2)^2*a^2*c)
Time = 11.96 (sec) , antiderivative size = 3191, normalized size of antiderivative = 11.91 \[ \int \frac {(d+e x)^{3/2}}{\left (a-c x^2\right )^3} \, dx=\text {Too large to display} \]
(((4*a*d*e^3 - 9*c*d^3*e)*(d + e*x)^(3/2))/(8*a^2) + (e*(a*e^2 + 18*c*d^2) *(d + e*x)^(5/2))/(16*a^2) + (3*(d + e*x)^(1/2)*(a^2*e^5 + 2*c^2*d^4*e - 3 *a*c*d^2*e^3))/(16*a^2*c) - (3*c*d*e*(d + e*x)^(7/2))/(8*a^2))/(c^2*(d + e *x)^4 + a^2*e^4 + c^2*d^4 + (6*c^2*d^2 - 2*a*c*e^2)*(d + e*x)^2 - (4*c^2*d ^3 - 4*a*c*d*e^2)*(d + e*x) - 4*c^2*d*(d + e*x)^3 - 2*a*c*d^2*e^2) + atan( ((((3*(2048*a^6*c^2*e^5 - 4096*a^5*c^3*d^2*e^3))/(2048*a^6) - 64*a*c^4*d*e ^2*(d + e*x)^(1/2)*(-(9*(e^5*(a^15*c^5)^(1/2) - 16*a^5*c^5*d^5 - 5*a^7*c^3 *d*e^4 + 20*a^6*c^4*d^3*e^2))/(4096*(a^10*c^6*d^2 - a^11*c^5*e^2)))^(1/2)) *(-(9*(e^5*(a^15*c^5)^(1/2) - 16*a^5*c^5*d^5 - 5*a^7*c^3*d*e^4 + 20*a^6*c^ 4*d^3*e^2))/(4096*(a^10*c^6*d^2 - a^11*c^5*e^2)))^(1/2) + ((d + e*x)^(1/2) *(9*a^2*c*e^6 + 144*c^3*d^4*e^2 - 36*a*c^2*d^2*e^4))/(64*a^4))*(-(9*(e^5*( a^15*c^5)^(1/2) - 16*a^5*c^5*d^5 - 5*a^7*c^3*d*e^4 + 20*a^6*c^4*d^3*e^2))/ (4096*(a^10*c^6*d^2 - a^11*c^5*e^2)))^(1/2)*1i - (((3*(2048*a^6*c^2*e^5 - 4096*a^5*c^3*d^2*e^3))/(2048*a^6) + 64*a*c^4*d*e^2*(d + e*x)^(1/2)*(-(9*(e ^5*(a^15*c^5)^(1/2) - 16*a^5*c^5*d^5 - 5*a^7*c^3*d*e^4 + 20*a^6*c^4*d^3*e^ 2))/(4096*(a^10*c^6*d^2 - a^11*c^5*e^2)))^(1/2))*(-(9*(e^5*(a^15*c^5)^(1/2 ) - 16*a^5*c^5*d^5 - 5*a^7*c^3*d*e^4 + 20*a^6*c^4*d^3*e^2))/(4096*(a^10*c^ 6*d^2 - a^11*c^5*e^2)))^(1/2) - ((d + e*x)^(1/2)*(9*a^2*c*e^6 + 144*c^3*d^ 4*e^2 - 36*a*c^2*d^2*e^4))/(64*a^4))*(-(9*(e^5*(a^15*c^5)^(1/2) - 16*a^5*c ^5*d^5 - 5*a^7*c^3*d*e^4 + 20*a^6*c^4*d^3*e^2))/(4096*(a^10*c^6*d^2 - a...