Integrand size = 20, antiderivative size = 167 \[ \int \frac {(d+e x)^{5/2}}{a-c x^2} \, dx=-\frac {4 d e \sqrt {d+e x}}{c}-\frac {2 e (d+e x)^{3/2}}{3 c}-\frac {\left (\sqrt {c} d-\sqrt {a} e\right )^{5/2} \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{\sqrt {a} c^{7/4}}+\frac {\left (\sqrt {c} d+\sqrt {a} e\right )^{5/2} \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d+\sqrt {a} e}}\right )}{\sqrt {a} c^{7/4}} \]
-2/3*e*(e*x+d)^(3/2)/c-arctanh(c^(1/4)*(e*x+d)^(1/2)/(-e*a^(1/2)+d*c^(1/2) )^(1/2))*(-e*a^(1/2)+d*c^(1/2))^(5/2)/c^(7/4)/a^(1/2)+arctanh(c^(1/4)*(e*x +d)^(1/2)/(e*a^(1/2)+d*c^(1/2))^(1/2))*(e*a^(1/2)+d*c^(1/2))^(5/2)/c^(7/4) /a^(1/2)-4*d*e*(e*x+d)^(1/2)/c
Time = 0.68 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.35 \[ \int \frac {(d+e x)^{5/2}}{a-c x^2} \, dx=-\frac {2 c e \sqrt {d+e x} (7 d+e x)+\frac {3 \left (\sqrt {c} d+\sqrt {a} e\right )^2 \sqrt {-c d-\sqrt {a} \sqrt {c} e} \arctan \left (\frac {\sqrt {-c d-\sqrt {a} \sqrt {c} e} \sqrt {d+e x}}{\sqrt {c} d+\sqrt {a} e}\right )}{\sqrt {a}}+\frac {3 \sqrt {c} \left (\sqrt {c} d-\sqrt {a} e\right )^3 \arctan \left (\frac {\sqrt {-c d+\sqrt {a} \sqrt {c} e} \sqrt {d+e x}}{\sqrt {c} d-\sqrt {a} e}\right )}{\sqrt {a} \sqrt {-c d+\sqrt {a} \sqrt {c} e}}}{3 c^2} \]
-1/3*(2*c*e*Sqrt[d + e*x]*(7*d + e*x) + (3*(Sqrt[c]*d + Sqrt[a]*e)^2*Sqrt[ -(c*d) - Sqrt[a]*Sqrt[c]*e]*ArcTan[(Sqrt[-(c*d) - Sqrt[a]*Sqrt[c]*e]*Sqrt[ d + e*x])/(Sqrt[c]*d + Sqrt[a]*e)])/Sqrt[a] + (3*Sqrt[c]*(Sqrt[c]*d - Sqrt [a]*e)^3*ArcTan[(Sqrt[-(c*d) + Sqrt[a]*Sqrt[c]*e]*Sqrt[d + e*x])/(Sqrt[c]* d - Sqrt[a]*e)])/(Sqrt[a]*Sqrt[-(c*d) + Sqrt[a]*Sqrt[c]*e]))/c^2
Time = 0.45 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.10, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.450, Rules used = {481, 25, 653, 25, 27, 654, 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.
| \(\displaystyle \int \frac {(d+e x)^{5/2}}{a-c x^2} \, dx\) |
| \(\Big \downarrow \) 481 |
| \(\displaystyle -\frac {\int -\frac {\sqrt {d+e x} \left (c d^2+2 c e x d+a e^2\right )}{a-c x^2}dx}{c}-\frac {2 e (d+e x)^{3/2}}{3 c}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {\sqrt {d+e x} \left (c d^2+2 c e x d+a e^2\right )}{a-c x^2}dx}{c}-\frac {2 e (d+e x)^{3/2}}{3 c}\) |
| \(\Big \downarrow \) 653 |
| \(\displaystyle \frac {-\frac {\int -\frac {c \left (d \left (c d^2+3 a e^2\right )+e \left (3 c d^2+a e^2\right ) x\right )}{\sqrt {d+e x} \left (a-c x^2\right )}dx}{c}-4 d e \sqrt {d+e x}}{c}-\frac {2 e (d+e x)^{3/2}}{3 c}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int \frac {c \left (d \left (c d^2+3 a e^2\right )+e \left (3 c d^2+a e^2\right ) x\right )}{\sqrt {d+e x} \left (a-c x^2\right )}dx}{c}-4 d e \sqrt {d+e x}}{c}-\frac {2 e (d+e x)^{3/2}}{3 c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {d \left (c d^2+3 a e^2\right )+e \left (3 c d^2+a e^2\right ) x}{\sqrt {d+e x} \left (a-c x^2\right )}dx-4 d e \sqrt {d+e x}}{c}-\frac {2 e (d+e x)^{3/2}}{3 c}\) |
| \(\Big \downarrow \) 654 |
| \(\displaystyle \frac {2 \int \frac {e \left (2 d \left (c d^2-a e^2\right )-\left (3 c d^2+a e^2\right ) (d+e x)\right )}{c d^2-2 c (d+e x) d-a e^2+c (d+e x)^2}d\sqrt {d+e x}-4 d e \sqrt {d+e x}}{c}-\frac {2 e (d+e x)^{3/2}}{3 c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 e \int \frac {2 d \left (c d^2-a e^2\right )-\left (3 c d^2+a e^2\right ) (d+e x)}{c d^2-2 c (d+e x) d-a e^2+c (d+e x)^2}d\sqrt {d+e x}-4 d e \sqrt {d+e x}}{c}-\frac {2 e (d+e x)^{3/2}}{3 c}\) |
| \(\Big \downarrow \) 1480 |
| \(\displaystyle \frac {2 e \left (\frac {\left (\sqrt {c} d-\sqrt {a} e\right )^3 \int \frac {1}{c (d+e x)-\sqrt {c} \left (\sqrt {c} d-\sqrt {a} e\right )}d\sqrt {d+e x}}{2 \sqrt {a} e}-\frac {\left (\sqrt {a} e+\sqrt {c} d\right )^3 \int \frac {1}{c (d+e x)-\sqrt {c} \left (\sqrt {c} d+\sqrt {a} e\right )}d\sqrt {d+e x}}{2 \sqrt {a} e}\right )-4 d e \sqrt {d+e x}}{c}-\frac {2 e (d+e x)^{3/2}}{3 c}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {2 e \left (\frac {\left (\sqrt {a} e+\sqrt {c} d\right )^{5/2} \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {a} e+\sqrt {c} d}}\right )}{2 \sqrt {a} c^{3/4} e}-\frac {\left (\sqrt {c} d-\sqrt {a} e\right )^{5/2} \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{2 \sqrt {a} c^{3/4} e}\right )-4 d e \sqrt {d+e x}}{c}-\frac {2 e (d+e x)^{3/2}}{3 c}\) |
(-2*e*(d + e*x)^(3/2))/(3*c) + (-4*d*e*Sqrt[d + e*x] + 2*e*(-1/2*((Sqrt[c] *d - Sqrt[a]*e)^(5/2)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d - Sqr t[a]*e]])/(Sqrt[a]*c^(3/4)*e) + ((Sqrt[c]*d + Sqrt[a]*e)^(5/2)*ArcTanh[(c^ (1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d + Sqrt[a]*e]])/(2*Sqrt[a]*c^(3/4)*e))) /c
3.7.12.3.1 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), x_Symbol] :> Simp[d*((c + d*x)^(n - 1)/(b*(n - 1))), x] + Simp[1/b Int[(c + d*x)^(n - 2)*(Simp[b *c^2 - a*d^2 + 2*b*c*d*x, x]/(a + b*x^2)), x], x] /; FreeQ[{a, b, c, d}, x] && GtQ[n, 1]
Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Simp[g*((d + e*x)^m/(c*m)), x] + Simp[1/c Int[(d + e*x)^(m - 1)*(Simp[c*d*f - a*e*g + (g*c*d + c*e*f)*x, x]/(a + c*x^2)), x], x] /; Fr eeQ[{a, c, d, e, f, g}, x] && FractionQ[m] && GtQ[m, 0]
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_)^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 = 2.84 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.29
| method | result | size |
| pseudoelliptic | \(-\frac {e \left (\frac {2 \sqrt {e x +d}\, \left (e x +7 d \right )}{3}+\frac {\left (-3 d \,e^{2} a c -c^{2} d^{3}+\sqrt {a c \,e^{2}}\, a \,e^{2}+3 \sqrt {a c \,e^{2}}\, c \,d^{2}\right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{\sqrt {a c \,e^{2}}\, \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}-\frac {\left (3 d \,e^{2} a c +c^{2} d^{3}+\sqrt {a c \,e^{2}}\, a \,e^{2}+3 \sqrt {a c \,e^{2}}\, c \,d^{2}\right ) \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{\sqrt {a c \,e^{2}}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{c}\) | \(216\) |
| risch | \(-\frac {2 \left (e x +7 d \right ) \sqrt {e x +d}\, e}{3 c}-2 e \left (\frac {\left (-3 d \,e^{2} a c -c^{2} d^{3}+\sqrt {a c \,e^{2}}\, a \,e^{2}+3 \sqrt {a c \,e^{2}}\, c \,d^{2}\right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 c \sqrt {a c \,e^{2}}\, \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}-\frac {\left (3 d \,e^{2} a c +c^{2} d^{3}+\sqrt {a c \,e^{2}}\, a \,e^{2}+3 \sqrt {a c \,e^{2}}\, c \,d^{2}\right ) \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 c \sqrt {a c \,e^{2}}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )\) | \(225\) |
| derivativedivides | \(-2 e \left (\frac {\frac {\left (e x +d \right )^{\frac {3}{2}}}{3}+2 d \sqrt {e x +d}}{c}-\frac {\left (3 d \,e^{2} a c +c^{2} d^{3}+\sqrt {a c \,e^{2}}\, a \,e^{2}+3 \sqrt {a c \,e^{2}}\, c \,d^{2}\right ) \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 c \sqrt {a c \,e^{2}}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}+\frac {\left (-3 d \,e^{2} a c -c^{2} d^{3}+\sqrt {a c \,e^{2}}\, a \,e^{2}+3 \sqrt {a c \,e^{2}}\, c \,d^{2}\right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 c \sqrt {a c \,e^{2}}\, \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )\) | \(228\) |
| default | \(2 e \left (-\frac {\frac {\left (e x +d \right )^{\frac {3}{2}}}{3}+2 d \sqrt {e x +d}}{c}-\frac {\left (-3 d \,e^{2} a c -c^{2} d^{3}-\sqrt {a c \,e^{2}}\, a \,e^{2}-3 \sqrt {a c \,e^{2}}\, c \,d^{2}\right ) \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 c \sqrt {a c \,e^{2}}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}+\frac {\left (3 d \,e^{2} a c +c^{2} d^{3}-\sqrt {a c \,e^{2}}\, a \,e^{2}-3 \sqrt {a c \,e^{2}}\, c \,d^{2}\right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 c \sqrt {a c \,e^{2}}\, \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )\) | \(231\) |
-e/c*(2/3*(e*x+d)^(1/2)*(e*x+7*d)+(-3*d*e^2*a*c-c^2*d^3+(a*c*e^2)^(1/2)*a* e^2+3*(a*c*e^2)^(1/2)*c*d^2)/(a*c*e^2)^(1/2)/((-c*d+(a*c*e^2)^(1/2))*c)^(1 /2)*arctan(c*(e*x+d)^(1/2)/((-c*d+(a*c*e^2)^(1/2))*c)^(1/2))-(3*d*e^2*a*c+ c^2*d^3+(a*c*e^2)^(1/2)*a*e^2+3*(a*c*e^2)^(1/2)*c*d^2)/(a*c*e^2)^(1/2)/((c *d+(a*c*e^2)^(1/2))*c)^(1/2)*arctanh(c*(e*x+d)^(1/2)/((c*d+(a*c*e^2)^(1/2) )*c)^(1/2)))
Leaf count of result is larger than twice the leaf count of optimal. 1617 vs. \(2 (121) = 242\).
Time = 0.33 (sec) , antiderivative size = 1617, normalized size of antiderivative = 9.68 \[ \int \frac {(d+e x)^{5/2}}{a-c x^2} \, dx=\text {Too large to display} \]
-1/6*(3*c*sqrt((c^2*d^5 + 10*a*c*d^3*e^2 + 5*a^2*d*e^4 + a*c^3*sqrt((25*c^ 4*d^8*e^2 + 100*a*c^3*d^6*e^4 + 110*a^2*c^2*d^4*e^6 + 20*a^3*c*d^2*e^8 + a ^4*e^10)/(a*c^7)))/(a*c^3))*log((5*c^4*d^8*e - 14*a^2*c^2*d^4*e^5 + 8*a^3* c*d^2*e^7 + a^4*e^9)*sqrt(e*x + d) + (10*a*c^4*d^5*e^2 + 20*a^2*c^3*d^3*e^ 4 + 2*a^3*c^2*d*e^6 - (a*c^6*d^2 + a^2*c^5*e^2)*sqrt((25*c^4*d^8*e^2 + 100 *a*c^3*d^6*e^4 + 110*a^2*c^2*d^4*e^6 + 20*a^3*c*d^2*e^8 + a^4*e^10)/(a*c^7 )))*sqrt((c^2*d^5 + 10*a*c*d^3*e^2 + 5*a^2*d*e^4 + a*c^3*sqrt((25*c^4*d^8* e^2 + 100*a*c^3*d^6*e^4 + 110*a^2*c^2*d^4*e^6 + 20*a^3*c*d^2*e^8 + a^4*e^1 0)/(a*c^7)))/(a*c^3))) - 3*c*sqrt((c^2*d^5 + 10*a*c*d^3*e^2 + 5*a^2*d*e^4 + a*c^3*sqrt((25*c^4*d^8*e^2 + 100*a*c^3*d^6*e^4 + 110*a^2*c^2*d^4*e^6 + 2 0*a^3*c*d^2*e^8 + a^4*e^10)/(a*c^7)))/(a*c^3))*log((5*c^4*d^8*e - 14*a^2*c ^2*d^4*e^5 + 8*a^3*c*d^2*e^7 + a^4*e^9)*sqrt(e*x + d) - (10*a*c^4*d^5*e^2 + 20*a^2*c^3*d^3*e^4 + 2*a^3*c^2*d*e^6 - (a*c^6*d^2 + a^2*c^5*e^2)*sqrt((2 5*c^4*d^8*e^2 + 100*a*c^3*d^6*e^4 + 110*a^2*c^2*d^4*e^6 + 20*a^3*c*d^2*e^8 + a^4*e^10)/(a*c^7)))*sqrt((c^2*d^5 + 10*a*c*d^3*e^2 + 5*a^2*d*e^4 + a*c^ 3*sqrt((25*c^4*d^8*e^2 + 100*a*c^3*d^6*e^4 + 110*a^2*c^2*d^4*e^6 + 20*a^3* c*d^2*e^8 + a^4*e^10)/(a*c^7)))/(a*c^3))) + 3*c*sqrt((c^2*d^5 + 10*a*c*d^3 *e^2 + 5*a^2*d*e^4 - a*c^3*sqrt((25*c^4*d^8*e^2 + 100*a*c^3*d^6*e^4 + 110* a^2*c^2*d^4*e^6 + 20*a^3*c*d^2*e^8 + a^4*e^10)/(a*c^7)))/(a*c^3))*log((5*c ^4*d^8*e - 14*a^2*c^2*d^4*e^5 + 8*a^3*c*d^2*e^7 + a^4*e^9)*sqrt(e*x + d...
\[ \int \frac {(d+e x)^{5/2}}{a-c x^2} \, dx=- \int \frac {d^{2} \sqrt {d + e x}}{- a + c x^{2}}\, dx - \int \frac {e^{2} x^{2} \sqrt {d + e x}}{- a + c x^{2}}\, dx - \int \frac {2 d e x \sqrt {d + e x}}{- a + c x^{2}}\, dx \]
-Integral(d**2*sqrt(d + e*x)/(-a + c*x**2), x) - Integral(e**2*x**2*sqrt(d + e*x)/(-a + c*x**2), x) - Integral(2*d*e*x*sqrt(d + e*x)/(-a + c*x**2), x)
\[ \int \frac {(d+e x)^{5/2}}{a-c x^2} \, dx=\int { -\frac {{\left (e x + d\right )}^{\frac {5}{2}}}{c x^{2} - a} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 416 vs. \(2 (121) = 242\).
Time = 0.33 (sec) , antiderivative size = 416, normalized size of antiderivative = 2.49 \[ \int \frac {(d+e x)^{5/2}}{a-c x^2} \, dx=-\frac {{\left (\sqrt {a c} c^{4} d^{4} e + 3 \, \sqrt {a c} a c^{3} d^{2} e^{3} - {\left (3 \, \sqrt {a c} a c d^{2} e + \sqrt {a c} a^{2} e^{3}\right )} c^{2} e^{2} + 2 \, {\left (a c^{3} d^{3} e - a^{2} c^{2} d e^{3}\right )} {\left | c \right |} {\left | e \right |}\right )} \arctan \left (\frac {\sqrt {e x + d}}{\sqrt {-\frac {c^{4} d + \sqrt {c^{8} d^{2} - {\left (c^{4} d^{2} - a c^{3} e^{2}\right )} c^{4}}}{c^{4}}}}\right )}{{\left (a c^{4} d - \sqrt {a c} a c^{3} e\right )} \sqrt {-c^{2} d - \sqrt {a c} c e} {\left | e \right |}} + \frac {{\left (\sqrt {a c} c^{4} d^{4} e + 3 \, \sqrt {a c} a c^{3} d^{2} e^{3} - {\left (3 \, \sqrt {a c} a c d^{2} e + \sqrt {a c} a^{2} e^{3}\right )} c^{2} e^{2} - 2 \, {\left (a c^{3} d^{3} e - a^{2} c^{2} d e^{3}\right )} {\left | c \right |} {\left | e \right |}\right )} \arctan \left (\frac {\sqrt {e x + d}}{\sqrt {-\frac {c^{4} d - \sqrt {c^{8} d^{2} - {\left (c^{4} d^{2} - a c^{3} e^{2}\right )} c^{4}}}{c^{4}}}}\right )}{{\left (a c^{4} d + \sqrt {a c} a c^{3} e\right )} \sqrt {-c^{2} d + \sqrt {a c} c e} {\left | e \right |}} - \frac {2 \, {\left ({\left (e x + d\right )}^{\frac {3}{2}} c^{2} e + 6 \, \sqrt {e x + d} c^{2} d e\right )}}{3 \, c^{3}} \]
-(sqrt(a*c)*c^4*d^4*e + 3*sqrt(a*c)*a*c^3*d^2*e^3 - (3*sqrt(a*c)*a*c*d^2*e + sqrt(a*c)*a^2*e^3)*c^2*e^2 + 2*(a*c^3*d^3*e - a^2*c^2*d*e^3)*abs(c)*abs (e))*arctan(sqrt(e*x + d)/sqrt(-(c^4*d + sqrt(c^8*d^2 - (c^4*d^2 - a*c^3*e ^2)*c^4))/c^4))/((a*c^4*d - sqrt(a*c)*a*c^3*e)*sqrt(-c^2*d - sqrt(a*c)*c*e )*abs(e)) + (sqrt(a*c)*c^4*d^4*e + 3*sqrt(a*c)*a*c^3*d^2*e^3 - (3*sqrt(a*c )*a*c*d^2*e + sqrt(a*c)*a^2*e^3)*c^2*e^2 - 2*(a*c^3*d^3*e - a^2*c^2*d*e^3) *abs(c)*abs(e))*arctan(sqrt(e*x + d)/sqrt(-(c^4*d - sqrt(c^8*d^2 - (c^4*d^ 2 - a*c^3*e^2)*c^4))/c^4))/((a*c^4*d + sqrt(a*c)*a*c^3*e)*sqrt(-c^2*d + sq rt(a*c)*c*e)*abs(e)) - 2/3*((e*x + d)^(3/2)*c^2*e + 6*sqrt(e*x + d)*c^2*d* e)/c^3
Time = 9.80 (sec) , antiderivative size = 3385, normalized size of antiderivative = 20.27 \[ \int \frac {(d+e x)^{5/2}}{a-c x^2} \, dx=\text {Too large to display} \]
- atan((a^3*e^8*(d + e*x)^(1/2)*((e^5*(a^3*c^7)^(1/2))/(4*c^7) + d^5/(4*a* c) + (5*d^3*e^2)/(2*c^2) + (5*a*d*e^4)/(4*c^3) + (5*d^4*e*(a^3*c^7)^(1/2)) /(4*a^2*c^5) + (5*d^2*e^3*(a^3*c^7)^(1/2))/(2*a*c^6))^(1/2)*32i)/((16*a^4* e^11)/c^2 + 64*a^2*d^4*e^7 - 80*c^2*d^8*e^3 + (160*a^3*d^2*e^9)/c - 160*a* c*d^6*e^5 - (160*d^5*e^6*(a^3*c^7)^(1/2))/c^3 + (288*a*d^3*e^8*(a^3*c^7)^( 1/2))/c^4 + (32*a^2*d*e^10*(a^3*c^7)^(1/2))/c^5 - (160*d^7*e^4*(a^3*c^7)^( 1/2))/(a*c^2)) - (d^5*e^3*(a^3*c^7)^(1/2)*(d + e*x)^(1/2)*((e^5*(a^3*c^7)^ (1/2))/(4*c^7) + d^5/(4*a*c) + (5*d^3*e^2)/(2*c^2) + (5*a*d*e^4)/(4*c^3) + (5*d^4*e*(a^3*c^7)^(1/2))/(4*a^2*c^5) + (5*d^2*e^3*(a^3*c^7)^(1/2))/(2*a* c^6))^(1/2)*160i)/((16*a^5*e^11)/c + 160*a^4*d^2*e^9 - 80*a*c^3*d^8*e^3 + 64*a^3*c*d^4*e^7 - 160*a^2*c^2*d^6*e^5 - (160*d^7*e^4*(a^3*c^7)^(1/2))/c - (160*a*d^5*e^6*(a^3*c^7)^(1/2))/c^2 + (32*a^3*d*e^10*(a^3*c^7)^(1/2))/c^4 + (288*a^2*d^3*e^8*(a^3*c^7)^(1/2))/c^3) - (d^3*e^5*(a^3*c^7)^(1/2)*(d + e*x)^(1/2)*((e^5*(a^3*c^7)^(1/2))/(4*c^7) + d^5/(4*a*c) + (5*d^3*e^2)/(2*c ^2) + (5*a*d*e^4)/(4*c^3) + (5*d^4*e*(a^3*c^7)^(1/2))/(4*a^2*c^5) + (5*d^2 *e^3*(a^3*c^7)^(1/2))/(2*a*c^6))^(1/2)*320i)/(16*a^4*e^11 - 80*c^4*d^8*e^3 - 160*a*c^3*d^6*e^5 + 160*a^3*c*d^2*e^9 + 64*a^2*c^2*d^4*e^7 - (160*d^7*e ^4*(a^3*c^7)^(1/2))/a - (160*d^5*e^6*(a^3*c^7)^(1/2))/c + (288*a*d^3*e^8*( a^3*c^7)^(1/2))/c^2 + (32*a^2*d*e^10*(a^3*c^7)^(1/2))/c^3) - (a*d*e^7*(a^3 *c^7)^(1/2)*(d + e*x)^(1/2)*((e^5*(a^3*c^7)^(1/2))/(4*c^7) + d^5/(4*a*c...