Integrand size = 30, antiderivative size = 400 \[ \int \frac {(d+e x)^{13/2}}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\frac {3003 e^4 (b d-a e)^2 (a+b x) \sqrt {d+e x}}{64 b^7 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {1001 e^4 (b d-a e) (a+b x) (d+e x)^{3/2}}{64 b^6 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {3003 e^4 (a+b x) (d+e x)^{5/2}}{320 b^5 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {429 e^3 (d+e x)^{7/2}}{64 b^4 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {143 e^2 (d+e x)^{9/2}}{96 b^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {13 e (d+e x)^{11/2}}{24 b^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(d+e x)^{13/2}}{4 b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {3003 e^4 (b d-a e)^{5/2} (a+b x) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{64 b^{15/2} \sqrt {a^2+2 a b x+b^2 x^2}} \]
1001/64*e^4*(-a*e+b*d)*(b*x+a)*(e*x+d)^(3/2)/b^6/((b*x+a)^2)^(1/2)+3003/32 0*e^4*(b*x+a)*(e*x+d)^(5/2)/b^5/((b*x+a)^2)^(1/2)-429/64*e^3*(e*x+d)^(7/2) /b^4/((b*x+a)^2)^(1/2)-143/96*e^2*(e*x+d)^(9/2)/b^3/(b*x+a)/((b*x+a)^2)^(1 /2)-13/24*e*(e*x+d)^(11/2)/b^2/(b*x+a)^2/((b*x+a)^2)^(1/2)-1/4*(e*x+d)^(13 /2)/b/(b*x+a)^3/((b*x+a)^2)^(1/2)-3003/64*e^4*(-a*e+b*d)^(5/2)*(b*x+a)*arc tanh(b^(1/2)*(e*x+d)^(1/2)/(-a*e+b*d)^(1/2))/b^(15/2)/((b*x+a)^2)^(1/2)+30 03/64*e^4*(-a*e+b*d)^2*(b*x+a)*(e*x+d)^(1/2)/b^7/((b*x+a)^2)^(1/2)
Time = 1.41 (sec) , antiderivative size = 374, normalized size of antiderivative = 0.94 \[ \int \frac {(d+e x)^{13/2}}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\frac {e^4 (a+b x)^5 \left (-\frac {\sqrt {b} \sqrt {d+e x} \left (-45045 a^6 e^6+15015 a^5 b e^5 (7 d-11 e x)-3003 a^4 b^2 e^4 \left (23 d^2-129 d e x+73 e^2 x^2\right )+429 a^3 b^3 e^3 \left (15 d^3-599 d^2 e x+1207 d e^2 x^2-279 e^3 x^3\right )+143 a^2 b^4 e^2 \left (10 d^4+175 d^3 e x-2433 d^2 e^2 x^2+1999 d e^3 x^3-128 e^4 x^4\right )+13 a b^5 e \left (40 d^5+420 d^4 e x+2765 d^3 e^2 x^2-15077 d^2 e^3 x^3+3456 d e^4 x^4+128 e^5 x^5\right )+b^6 \left (240 d^6+1960 d^5 e x+7630 d^4 e^2 x^2+22155 d^3 e^3 x^3-32384 d^2 e^4 x^4-3968 d e^5 x^5-384 e^6 x^6\right )\right )}{e^4 (a+b x)^4}-45045 (-b d+a e)^{5/2} \arctan \left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {-b d+a e}}\right )\right )}{960 b^{15/2} \left ((a+b x)^2\right )^{5/2}} \]
(e^4*(a + b*x)^5*(-((Sqrt[b]*Sqrt[d + e*x]*(-45045*a^6*e^6 + 15015*a^5*b*e ^5*(7*d - 11*e*x) - 3003*a^4*b^2*e^4*(23*d^2 - 129*d*e*x + 73*e^2*x^2) + 4 29*a^3*b^3*e^3*(15*d^3 - 599*d^2*e*x + 1207*d*e^2*x^2 - 279*e^3*x^3) + 143 *a^2*b^4*e^2*(10*d^4 + 175*d^3*e*x - 2433*d^2*e^2*x^2 + 1999*d*e^3*x^3 - 1 28*e^4*x^4) + 13*a*b^5*e*(40*d^5 + 420*d^4*e*x + 2765*d^3*e^2*x^2 - 15077* d^2*e^3*x^3 + 3456*d*e^4*x^4 + 128*e^5*x^5) + b^6*(240*d^6 + 1960*d^5*e*x + 7630*d^4*e^2*x^2 + 22155*d^3*e^3*x^3 - 32384*d^2*e^4*x^4 - 3968*d*e^5*x^ 5 - 384*e^6*x^6)))/(e^4*(a + b*x)^4)) - 45045*(-(b*d) + a*e)^(5/2)*ArcTan[ (Sqrt[b]*Sqrt[d + e*x])/Sqrt[-(b*d) + a*e]]))/(960*b^(15/2)*((a + b*x)^2)^ (5/2))
Time = 0.35 (sec) , antiderivative size = 272, normalized size of antiderivative = 0.68, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.367, Rules used = {1102, 27, 51, 51, 51, 51, 60, 60, 60, 73, 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.
| \(\displaystyle \int \frac {(d+e x)^{13/2}}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 1102 |
| \(\displaystyle \frac {b^5 (a+b x) \int \frac {(d+e x)^{13/2}}{b^5 (a+b x)^5}dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {(a+b x) \int \frac {(d+e x)^{13/2}}{(a+b x)^5}dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
| \(\Big \downarrow \) 51 |
| \(\displaystyle \frac {(a+b x) \left (\frac {13 e \int \frac {(d+e x)^{11/2}}{(a+b x)^4}dx}{8 b}-\frac {(d+e x)^{13/2}}{4 b (a+b x)^4}\right )}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
| \(\Big \downarrow \) 51 |
| \(\displaystyle \frac {(a+b x) \left (\frac {13 e \left (\frac {11 e \int \frac {(d+e x)^{9/2}}{(a+b x)^3}dx}{6 b}-\frac {(d+e x)^{11/2}}{3 b (a+b x)^3}\right )}{8 b}-\frac {(d+e x)^{13/2}}{4 b (a+b x)^4}\right )}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
| \(\Big \downarrow \) 51 |
| \(\displaystyle \frac {(a+b x) \left (\frac {13 e \left (\frac {11 e \left (\frac {9 e \int \frac {(d+e x)^{7/2}}{(a+b x)^2}dx}{4 b}-\frac {(d+e x)^{9/2}}{2 b (a+b x)^2}\right )}{6 b}-\frac {(d+e x)^{11/2}}{3 b (a+b x)^3}\right )}{8 b}-\frac {(d+e x)^{13/2}}{4 b (a+b x)^4}\right )}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
| \(\Big \downarrow \) 51 |
| \(\displaystyle \frac {(a+b x) \left (\frac {13 e \left (\frac {11 e \left (\frac {9 e \left (\frac {7 e \int \frac {(d+e x)^{5/2}}{a+b x}dx}{2 b}-\frac {(d+e x)^{7/2}}{b (a+b x)}\right )}{4 b}-\frac {(d+e x)^{9/2}}{2 b (a+b x)^2}\right )}{6 b}-\frac {(d+e x)^{11/2}}{3 b (a+b x)^3}\right )}{8 b}-\frac {(d+e x)^{13/2}}{4 b (a+b x)^4}\right )}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {(a+b x) \left (\frac {13 e \left (\frac {11 e \left (\frac {9 e \left (\frac {7 e \left (\frac {(b d-a e) \int \frac {(d+e x)^{3/2}}{a+b x}dx}{b}+\frac {2 (d+e x)^{5/2}}{5 b}\right )}{2 b}-\frac {(d+e x)^{7/2}}{b (a+b x)}\right )}{4 b}-\frac {(d+e x)^{9/2}}{2 b (a+b x)^2}\right )}{6 b}-\frac {(d+e x)^{11/2}}{3 b (a+b x)^3}\right )}{8 b}-\frac {(d+e x)^{13/2}}{4 b (a+b x)^4}\right )}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {(a+b x) \left (\frac {13 e \left (\frac {11 e \left (\frac {9 e \left (\frac {7 e \left (\frac {(b d-a e) \left (\frac {(b d-a e) \int \frac {\sqrt {d+e x}}{a+b x}dx}{b}+\frac {2 (d+e x)^{3/2}}{3 b}\right )}{b}+\frac {2 (d+e x)^{5/2}}{5 b}\right )}{2 b}-\frac {(d+e x)^{7/2}}{b (a+b x)}\right )}{4 b}-\frac {(d+e x)^{9/2}}{2 b (a+b x)^2}\right )}{6 b}-\frac {(d+e x)^{11/2}}{3 b (a+b x)^3}\right )}{8 b}-\frac {(d+e x)^{13/2}}{4 b (a+b x)^4}\right )}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {(a+b x) \left (\frac {13 e \left (\frac {11 e \left (\frac {9 e \left (\frac {7 e \left (\frac {(b d-a e) \left (\frac {(b d-a e) \left (\frac {(b d-a e) \int \frac {1}{(a+b x) \sqrt {d+e x}}dx}{b}+\frac {2 \sqrt {d+e x}}{b}\right )}{b}+\frac {2 (d+e x)^{3/2}}{3 b}\right )}{b}+\frac {2 (d+e x)^{5/2}}{5 b}\right )}{2 b}-\frac {(d+e x)^{7/2}}{b (a+b x)}\right )}{4 b}-\frac {(d+e x)^{9/2}}{2 b (a+b x)^2}\right )}{6 b}-\frac {(d+e x)^{11/2}}{3 b (a+b x)^3}\right )}{8 b}-\frac {(d+e x)^{13/2}}{4 b (a+b x)^4}\right )}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {(a+b x) \left (\frac {13 e \left (\frac {11 e \left (\frac {9 e \left (\frac {7 e \left (\frac {(b d-a e) \left (\frac {(b d-a e) \left (\frac {2 (b d-a e) \int \frac {1}{a+\frac {b (d+e x)}{e}-\frac {b d}{e}}d\sqrt {d+e x}}{b e}+\frac {2 \sqrt {d+e x}}{b}\right )}{b}+\frac {2 (d+e x)^{3/2}}{3 b}\right )}{b}+\frac {2 (d+e x)^{5/2}}{5 b}\right )}{2 b}-\frac {(d+e x)^{7/2}}{b (a+b x)}\right )}{4 b}-\frac {(d+e x)^{9/2}}{2 b (a+b x)^2}\right )}{6 b}-\frac {(d+e x)^{11/2}}{3 b (a+b x)^3}\right )}{8 b}-\frac {(d+e x)^{13/2}}{4 b (a+b x)^4}\right )}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {(a+b x) \left (\frac {13 e \left (\frac {11 e \left (\frac {9 e \left (\frac {7 e \left (\frac {(b d-a e) \left (\frac {(b d-a e) \left (\frac {2 \sqrt {d+e x}}{b}-\frac {2 \sqrt {b d-a e} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{b^{3/2}}\right )}{b}+\frac {2 (d+e x)^{3/2}}{3 b}\right )}{b}+\frac {2 (d+e x)^{5/2}}{5 b}\right )}{2 b}-\frac {(d+e x)^{7/2}}{b (a+b x)}\right )}{4 b}-\frac {(d+e x)^{9/2}}{2 b (a+b x)^2}\right )}{6 b}-\frac {(d+e x)^{11/2}}{3 b (a+b x)^3}\right )}{8 b}-\frac {(d+e x)^{13/2}}{4 b (a+b x)^4}\right )}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
((a + b*x)*(-1/4*(d + e*x)^(13/2)/(b*(a + b*x)^4) + (13*e*(-1/3*(d + e*x)^ (11/2)/(b*(a + b*x)^3) + (11*e*(-1/2*(d + e*x)^(9/2)/(b*(a + b*x)^2) + (9* e*(-((d + e*x)^(7/2)/(b*(a + b*x))) + (7*e*((2*(d + e*x)^(5/2))/(5*b) + (( b*d - a*e)*((2*(d + e*x)^(3/2))/(3*b) + ((b*d - a*e)*((2*Sqrt[d + e*x])/b - (2*Sqrt[b*d - a*e]*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/b^( 3/2)))/b))/b))/(2*b)))/(4*b)))/(6*b)))/(8*b)))/Sqrt[a^2 + 2*a*b*x + b^2*x^ 2]
3.18.20.3.1 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_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + 1))), x] - Simp[d*(n/(b*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d, n}, x ] && ILtQ[m, -1] && FractionQ[n] && GtQ[n, 0]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( b*(m + n + 1))) Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] && !(IGtQ[m, 0] && ( !Integer Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) && !ILtQ[m + n + 2, 0] && IntLinear Q[a, b, c, d, m, n, x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, 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[((d_.) + (e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> Simp[(a + b*x + c*x^2)^FracPart[p]/(c^IntPart[p]*(b/2 + c*x)^(2*F racPart[p])) Int[(d + e*x)^m*(b/2 + c*x)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && EqQ[b^2 - 4*a*c, 0]
Time = 2.32 (sec) , antiderivative size = 311, normalized size of antiderivative = 0.78
| method | result | size |
| risch | \(\frac {2 e^{4} \left (3 x^{2} b^{2} e^{2}-25 x a b \,e^{2}+31 b^{2} d e x +225 a^{2} e^{2}-475 a b d e +253 b^{2} d^{2}\right ) \sqrt {e x +d}\, \sqrt {\left (b x +a \right )^{2}}}{15 b^{7} \left (b x +a \right )}-\frac {\left (2 a^{3} e^{3}-6 a^{2} b d \,e^{2}+6 a \,b^{2} d^{2} e -2 b^{3} d^{3}\right ) e^{4} \left (\frac {-\frac {1477 \left (e x +d \right )^{\frac {7}{2}} b^{3}}{128}-\frac {11767 \left (a e -b d \right ) b^{2} \left (e x +d \right )^{\frac {5}{2}}}{384}+\left (-\frac {10633}{384} e^{2} a^{2} b +\frac {10633}{192} a d e \,b^{2}-\frac {10633}{384} b^{3} d^{2}\right ) \left (e x +d \right )^{\frac {3}{2}}+\left (-\frac {1083}{128} a^{3} e^{3}+\frac {3249}{128} a^{2} b d \,e^{2}-\frac {3249}{128} a \,b^{2} d^{2} e +\frac {1083}{128} b^{3} d^{3}\right ) \sqrt {e x +d}}{\left (b \left (e x +d \right )+a e -b d \right )^{4}}+\frac {3003 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{128 \sqrt {\left (a e -b d \right ) b}}\right ) \sqrt {\left (b x +a \right )^{2}}}{b^{7} \left (b x +a \right )}\) | \(311\) |
| default | \(\text {Expression too large to display}\) | \(2192\) |
2/15*e^4*(3*b^2*e^2*x^2-25*a*b*e^2*x+31*b^2*d*e*x+225*a^2*e^2-475*a*b*d*e+ 253*b^2*d^2)*(e*x+d)^(1/2)/b^7*((b*x+a)^2)^(1/2)/(b*x+a)-1/b^7*(2*a^3*e^3- 6*a^2*b*d*e^2+6*a*b^2*d^2*e-2*b^3*d^3)*e^4*((-1477/128*(e*x+d)^(7/2)*b^3-1 1767/384*(a*e-b*d)*b^2*(e*x+d)^(5/2)+(-10633/384*e^2*a^2*b+10633/192*a*d*e *b^2-10633/384*b^3*d^2)*(e*x+d)^(3/2)+(-1083/128*a^3*e^3+3249/128*a^2*b*d* e^2-3249/128*a*b^2*d^2*e+1083/128*b^3*d^3)*(e*x+d)^(1/2))/(b*(e*x+d)+a*e-b *d)^4+3003/128/((a*e-b*d)*b)^(1/2)*arctan(b*(e*x+d)^(1/2)/((a*e-b*d)*b)^(1 /2)))*((b*x+a)^2)^(1/2)/(b*x+a)
Leaf count of result is larger than twice the leaf count of optimal. 638 vs. \(2 (272) = 544\).
Time = 0.57 (sec) , antiderivative size = 1286, normalized size of antiderivative = 3.22 \[ \int \frac {(d+e x)^{13/2}}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\text {Too large to display} \]
[1/1920*(45045*(a^4*b^2*d^2*e^4 - 2*a^5*b*d*e^5 + a^6*e^6 + (b^6*d^2*e^4 - 2*a*b^5*d*e^5 + a^2*b^4*e^6)*x^4 + 4*(a*b^5*d^2*e^4 - 2*a^2*b^4*d*e^5 + a ^3*b^3*e^6)*x^3 + 6*(a^2*b^4*d^2*e^4 - 2*a^3*b^3*d*e^5 + a^4*b^2*e^6)*x^2 + 4*(a^3*b^3*d^2*e^4 - 2*a^4*b^2*d*e^5 + a^5*b*e^6)*x)*sqrt((b*d - a*e)/b) *log((b*e*x + 2*b*d - a*e - 2*sqrt(e*x + d)*b*sqrt((b*d - a*e)/b))/(b*x + a)) + 2*(384*b^6*e^6*x^6 - 240*b^6*d^6 - 520*a*b^5*d^5*e - 1430*a^2*b^4*d^ 4*e^2 - 6435*a^3*b^3*d^3*e^3 + 69069*a^4*b^2*d^2*e^4 - 105105*a^5*b*d*e^5 + 45045*a^6*e^6 + 128*(31*b^6*d*e^5 - 13*a*b^5*e^6)*x^5 + 128*(253*b^6*d^2 *e^4 - 351*a*b^5*d*e^5 + 143*a^2*b^4*e^6)*x^4 - (22155*b^6*d^3*e^3 - 19600 1*a*b^5*d^2*e^4 + 285857*a^2*b^4*d*e^5 - 119691*a^3*b^3*e^6)*x^3 - (7630*b ^6*d^4*e^2 + 35945*a*b^5*d^3*e^3 - 347919*a^2*b^4*d^2*e^4 + 517803*a^3*b^3 *d*e^5 - 219219*a^4*b^2*e^6)*x^2 - (1960*b^6*d^5*e + 5460*a*b^5*d^4*e^2 + 25025*a^2*b^4*d^3*e^3 - 256971*a^3*b^3*d^2*e^4 + 387387*a^4*b^2*d*e^5 - 16 5165*a^5*b*e^6)*x)*sqrt(e*x + d))/(b^11*x^4 + 4*a*b^10*x^3 + 6*a^2*b^9*x^2 + 4*a^3*b^8*x + a^4*b^7), -1/960*(45045*(a^4*b^2*d^2*e^4 - 2*a^5*b*d*e^5 + a^6*e^6 + (b^6*d^2*e^4 - 2*a*b^5*d*e^5 + a^2*b^4*e^6)*x^4 + 4*(a*b^5*d^2 *e^4 - 2*a^2*b^4*d*e^5 + a^3*b^3*e^6)*x^3 + 6*(a^2*b^4*d^2*e^4 - 2*a^3*b^3 *d*e^5 + a^4*b^2*e^6)*x^2 + 4*(a^3*b^3*d^2*e^4 - 2*a^4*b^2*d*e^5 + a^5*b*e ^6)*x)*sqrt(-(b*d - a*e)/b)*arctan(-sqrt(e*x + d)*b*sqrt(-(b*d - a*e)/b)/( b*d - a*e)) - (384*b^6*e^6*x^6 - 240*b^6*d^6 - 520*a*b^5*d^5*e - 1430*a...
Timed out. \[ \int \frac {(d+e x)^{13/2}}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\text {Timed out} \]
\[ \int \frac {(d+e x)^{13/2}}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\int { \frac {{\left (e x + d\right )}^{\frac {13}{2}}}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}}} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 651 vs. \(2 (272) = 544\).
Time = 0.31 (sec) , antiderivative size = 651, normalized size of antiderivative = 1.63 \[ \int \frac {(d+e x)^{13/2}}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\frac {3003 \, {\left (b^{3} d^{3} e^{4} - 3 \, a b^{2} d^{2} e^{5} + 3 \, a^{2} b d e^{6} - a^{3} e^{7}\right )} \arctan \left (\frac {\sqrt {e x + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{64 \, \sqrt {-b^{2} d + a b e} b^{7} \mathrm {sgn}\left (b x + a\right )} - \frac {4431 \, {\left (e x + d\right )}^{\frac {7}{2}} b^{6} d^{3} e^{4} - 11767 \, {\left (e x + d\right )}^{\frac {5}{2}} b^{6} d^{4} e^{4} + 10633 \, {\left (e x + d\right )}^{\frac {3}{2}} b^{6} d^{5} e^{4} - 3249 \, \sqrt {e x + d} b^{6} d^{6} e^{4} - 13293 \, {\left (e x + d\right )}^{\frac {7}{2}} a b^{5} d^{2} e^{5} + 47068 \, {\left (e x + d\right )}^{\frac {5}{2}} a b^{5} d^{3} e^{5} - 53165 \, {\left (e x + d\right )}^{\frac {3}{2}} a b^{5} d^{4} e^{5} + 19494 \, \sqrt {e x + d} a b^{5} d^{5} e^{5} + 13293 \, {\left (e x + d\right )}^{\frac {7}{2}} a^{2} b^{4} d e^{6} - 70602 \, {\left (e x + d\right )}^{\frac {5}{2}} a^{2} b^{4} d^{2} e^{6} + 106330 \, {\left (e x + d\right )}^{\frac {3}{2}} a^{2} b^{4} d^{3} e^{6} - 48735 \, \sqrt {e x + d} a^{2} b^{4} d^{4} e^{6} - 4431 \, {\left (e x + d\right )}^{\frac {7}{2}} a^{3} b^{3} e^{7} + 47068 \, {\left (e x + d\right )}^{\frac {5}{2}} a^{3} b^{3} d e^{7} - 106330 \, {\left (e x + d\right )}^{\frac {3}{2}} a^{3} b^{3} d^{2} e^{7} + 64980 \, \sqrt {e x + d} a^{3} b^{3} d^{3} e^{7} - 11767 \, {\left (e x + d\right )}^{\frac {5}{2}} a^{4} b^{2} e^{8} + 53165 \, {\left (e x + d\right )}^{\frac {3}{2}} a^{4} b^{2} d e^{8} - 48735 \, \sqrt {e x + d} a^{4} b^{2} d^{2} e^{8} - 10633 \, {\left (e x + d\right )}^{\frac {3}{2}} a^{5} b e^{9} + 19494 \, \sqrt {e x + d} a^{5} b d e^{9} - 3249 \, \sqrt {e x + d} a^{6} e^{10}}{192 \, {\left ({\left (e x + d\right )} b - b d + a e\right )}^{4} b^{7} \mathrm {sgn}\left (b x + a\right )} + \frac {2 \, {\left (3 \, {\left (e x + d\right )}^{\frac {5}{2}} b^{20} e^{4} + 25 \, {\left (e x + d\right )}^{\frac {3}{2}} b^{20} d e^{4} + 225 \, \sqrt {e x + d} b^{20} d^{2} e^{4} - 25 \, {\left (e x + d\right )}^{\frac {3}{2}} a b^{19} e^{5} - 450 \, \sqrt {e x + d} a b^{19} d e^{5} + 225 \, \sqrt {e x + d} a^{2} b^{18} e^{6}\right )}}{15 \, b^{25} \mathrm {sgn}\left (b x + a\right )} \]
3003/64*(b^3*d^3*e^4 - 3*a*b^2*d^2*e^5 + 3*a^2*b*d*e^6 - a^3*e^7)*arctan(s qrt(e*x + d)*b/sqrt(-b^2*d + a*b*e))/(sqrt(-b^2*d + a*b*e)*b^7*sgn(b*x + a )) - 1/192*(4431*(e*x + d)^(7/2)*b^6*d^3*e^4 - 11767*(e*x + d)^(5/2)*b^6*d ^4*e^4 + 10633*(e*x + d)^(3/2)*b^6*d^5*e^4 - 3249*sqrt(e*x + d)*b^6*d^6*e^ 4 - 13293*(e*x + d)^(7/2)*a*b^5*d^2*e^5 + 47068*(e*x + d)^(5/2)*a*b^5*d^3* e^5 - 53165*(e*x + d)^(3/2)*a*b^5*d^4*e^5 + 19494*sqrt(e*x + d)*a*b^5*d^5* e^5 + 13293*(e*x + d)^(7/2)*a^2*b^4*d*e^6 - 70602*(e*x + d)^(5/2)*a^2*b^4* d^2*e^6 + 106330*(e*x + d)^(3/2)*a^2*b^4*d^3*e^6 - 48735*sqrt(e*x + d)*a^2 *b^4*d^4*e^6 - 4431*(e*x + d)^(7/2)*a^3*b^3*e^7 + 47068*(e*x + d)^(5/2)*a^ 3*b^3*d*e^7 - 106330*(e*x + d)^(3/2)*a^3*b^3*d^2*e^7 + 64980*sqrt(e*x + d) *a^3*b^3*d^3*e^7 - 11767*(e*x + d)^(5/2)*a^4*b^2*e^8 + 53165*(e*x + d)^(3/ 2)*a^4*b^2*d*e^8 - 48735*sqrt(e*x + d)*a^4*b^2*d^2*e^8 - 10633*(e*x + d)^( 3/2)*a^5*b*e^9 + 19494*sqrt(e*x + d)*a^5*b*d*e^9 - 3249*sqrt(e*x + d)*a^6* e^10)/(((e*x + d)*b - b*d + a*e)^4*b^7*sgn(b*x + a)) + 2/15*(3*(e*x + d)^( 5/2)*b^20*e^4 + 25*(e*x + d)^(3/2)*b^20*d*e^4 + 225*sqrt(e*x + d)*b^20*d^2 *e^4 - 25*(e*x + d)^(3/2)*a*b^19*e^5 - 450*sqrt(e*x + d)*a*b^19*d*e^5 + 22 5*sqrt(e*x + d)*a^2*b^18*e^6)/(b^25*sgn(b*x + a))
Timed out. \[ \int \frac {(d+e x)^{13/2}}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\int \frac {{\left (d+e\,x\right )}^{13/2}}{{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2}} \,d x \]