Integrand size = 33, antiderivative size = 152 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^{3/2}} \, dx=\frac {2 (b d-a e)^5}{e^6 \sqrt {d+e x}}+\frac {10 b (b d-a e)^4 \sqrt {d+e x}}{e^6}-\frac {20 b^2 (b d-a e)^3 (d+e x)^{3/2}}{3 e^6}+\frac {4 b^3 (b d-a e)^2 (d+e x)^{5/2}}{e^6}-\frac {10 b^4 (b d-a e) (d+e x)^{7/2}}{7 e^6}+\frac {2 b^5 (d+e x)^{9/2}}{9 e^6} \] Output:
2*(-a*e+b*d)^5/e^6/(e*x+d)^(1/2)+10*b*(-a*e+b*d)^4*(e*x+d)^(1/2)/e^6-20/3* b^2*(-a*e+b*d)^3*(e*x+d)^(3/2)/e^6+4*b^3*(-a*e+b*d)^2*(e*x+d)^(5/2)/e^6-10 /7*b^4*(-a*e+b*d)*(e*x+d)^(7/2)/e^6+2/9*b^5*(e*x+d)^(9/2)/e^6
Time = 0.19 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.41 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^{3/2}} \, dx=\frac {2 \left (-63 a^5 e^5+315 a^4 b e^4 (2 d+e x)+210 a^3 b^2 e^3 \left (-8 d^2-4 d e x+e^2 x^2\right )+126 a^2 b^3 e^2 \left (16 d^3+8 d^2 e x-2 d e^2 x^2+e^3 x^3\right )+9 a b^4 e \left (-128 d^4-64 d^3 e x+16 d^2 e^2 x^2-8 d e^3 x^3+5 e^4 x^4\right )+b^5 \left (256 d^5+128 d^4 e x-32 d^3 e^2 x^2+16 d^2 e^3 x^3-10 d e^4 x^4+7 e^5 x^5\right )\right )}{63 e^6 \sqrt {d+e x}} \] Input:
Integrate[((a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^2)/(d + e*x)^(3/2),x]
Output:
(2*(-63*a^5*e^5 + 315*a^4*b*e^4*(2*d + e*x) + 210*a^3*b^2*e^3*(-8*d^2 - 4* d*e*x + e^2*x^2) + 126*a^2*b^3*e^2*(16*d^3 + 8*d^2*e*x - 2*d*e^2*x^2 + e^3 *x^3) + 9*a*b^4*e*(-128*d^4 - 64*d^3*e*x + 16*d^2*e^2*x^2 - 8*d*e^3*x^3 + 5*e^4*x^4) + b^5*(256*d^5 + 128*d^4*e*x - 32*d^3*e^2*x^2 + 16*d^2*e^3*x^3 - 10*d*e^4*x^4 + 7*e^5*x^5)))/(63*e^6*Sqrt[d + e*x])
Time = 0.50 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.121, Rules used = {1184, 27, 53, 2009}
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.
| \(\displaystyle \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^{3/2}} \, dx\) |
| \(\Big \downarrow \) 1184 |
| \(\displaystyle \frac {\int \frac {b^4 (a+b x)^5}{(d+e x)^{3/2}}dx}{b^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int \frac {(a+b x)^5}{(d+e x)^{3/2}}dx\) |
| \(\Big \downarrow \) 53 |
| \(\displaystyle \int \left (-\frac {5 b^4 (d+e x)^{5/2} (b d-a e)}{e^5}+\frac {10 b^3 (d+e x)^{3/2} (b d-a e)^2}{e^5}-\frac {10 b^2 \sqrt {d+e x} (b d-a e)^3}{e^5}+\frac {5 b (b d-a e)^4}{e^5 \sqrt {d+e x}}+\frac {(a e-b d)^5}{e^5 (d+e x)^{3/2}}+\frac {b^5 (d+e x)^{7/2}}{e^5}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {10 b^4 (d+e x)^{7/2} (b d-a e)}{7 e^6}+\frac {4 b^3 (d+e x)^{5/2} (b d-a e)^2}{e^6}-\frac {20 b^2 (d+e x)^{3/2} (b d-a e)^3}{3 e^6}+\frac {10 b \sqrt {d+e x} (b d-a e)^4}{e^6}+\frac {2 (b d-a e)^5}{e^6 \sqrt {d+e x}}+\frac {2 b^5 (d+e x)^{9/2}}{9 e^6}\) |
Input:
Int[((a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^2)/(d + e*x)^(3/2),x]
Output:
(2*(b*d - a*e)^5)/(e^6*Sqrt[d + e*x]) + (10*b*(b*d - a*e)^4*Sqrt[d + e*x]) /e^6 - (20*b^2*(b*d - a*e)^3*(d + e*x)^(3/2))/(3*e^6) + (4*b^3*(b*d - a*e) ^2*(d + e*x)^(5/2))/e^6 - (10*b^4*(b*d - a*e)*(d + e*x)^(7/2))/(7*e^6) + ( 2*b^5*(d + e*x)^(9/2))/(9*e^6)
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] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0] && LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_) + (b_.)*(x_ ) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[1/c^p Int[(d + e*x)^m*(f + g*x )^n*(b/2 + c*x)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n}, x] && E qQ[b^2 - 4*a*c, 0] && IntegerQ[p]
Time = 1.25 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.36
| method | result | size |
| pseudoelliptic | \(\frac {\frac {2 \left (7 e^{5} x^{5}-10 d \,e^{4} x^{4}+16 d^{2} e^{3} x^{3}-32 d^{3} e^{2} x^{2}+128 d^{4} e x +256 d^{5}\right ) b^{5}}{63}-\frac {256 e \left (-\frac {5}{128} e^{4} x^{4}+\frac {1}{16} d \,e^{3} x^{3}-\frac {1}{8} d^{2} e^{2} x^{2}+\frac {1}{2} d^{3} e x +d^{4}\right ) a \,b^{4}}{7}+64 \left (\frac {1}{16} e^{3} x^{3}-\frac {1}{8} d \,e^{2} x^{2}+\frac {1}{2} d^{2} e x +d^{3}\right ) e^{2} a^{2} b^{3}-\frac {160 e^{3} \left (-\frac {1}{8} e^{2} x^{2}+\frac {1}{2} d e x +d^{2}\right ) a^{3} b^{2}}{3}+20 e^{4} \left (\frac {e x}{2}+d \right ) a^{4} b -2 e^{5} a^{5}}{\sqrt {e x +d}\, e^{6}}\) | \(206\) |
| risch | \(\frac {2 b \left (7 b^{4} x^{4} e^{4}+45 x^{3} a \,b^{3} e^{4}-17 x^{3} b^{4} d \,e^{3}+126 x^{2} a^{2} b^{2} e^{4}-117 x^{2} a \,b^{3} d \,e^{3}+33 x^{2} b^{4} d^{2} e^{2}+210 x \,a^{3} b \,e^{4}-378 x \,a^{2} b^{2} d \,e^{3}+261 x a \,b^{3} d^{2} e^{2}-65 x \,b^{4} d^{3} e +315 a^{4} e^{4}-1050 a^{3} b d \,e^{3}+1386 a^{2} b^{2} d^{2} e^{2}-837 a \,b^{3} d^{3} e +193 b^{4} d^{4}\right ) \sqrt {e x +d}}{63 e^{6}}-\frac {2 \left (e^{5} a^{5}-5 a^{4} b d \,e^{4}+10 a^{3} b^{2} d^{2} e^{3}-10 a^{2} b^{3} d^{3} e^{2}+5 a \,b^{4} d^{4} e -b^{5} d^{5}\right )}{e^{6} \sqrt {e x +d}}\) | \(264\) |
| gosper | \(-\frac {2 \left (-7 x^{5} e^{5} b^{5}-45 x^{4} a \,b^{4} e^{5}+10 x^{4} b^{5} d \,e^{4}-126 x^{3} a^{2} b^{3} e^{5}+72 x^{3} a \,b^{4} d \,e^{4}-16 x^{3} b^{5} d^{2} e^{3}-210 x^{2} a^{3} b^{2} e^{5}+252 x^{2} a^{2} b^{3} d \,e^{4}-144 x^{2} a \,b^{4} d^{2} e^{3}+32 x^{2} b^{5} d^{3} e^{2}-315 a^{4} b \,e^{5} x +840 a^{3} b^{2} d \,e^{4} x -1008 x \,a^{2} b^{3} d^{2} e^{3}+576 x a \,b^{4} d^{3} e^{2}-128 b^{5} d^{4} e x +63 e^{5} a^{5}-630 a^{4} b d \,e^{4}+1680 a^{3} b^{2} d^{2} e^{3}-2016 a^{2} b^{3} d^{3} e^{2}+1152 a \,b^{4} d^{4} e -256 b^{5} d^{5}\right )}{63 \sqrt {e x +d}\, e^{6}}\) | \(273\) |
| trager | \(-\frac {2 \left (-7 x^{5} e^{5} b^{5}-45 x^{4} a \,b^{4} e^{5}+10 x^{4} b^{5} d \,e^{4}-126 x^{3} a^{2} b^{3} e^{5}+72 x^{3} a \,b^{4} d \,e^{4}-16 x^{3} b^{5} d^{2} e^{3}-210 x^{2} a^{3} b^{2} e^{5}+252 x^{2} a^{2} b^{3} d \,e^{4}-144 x^{2} a \,b^{4} d^{2} e^{3}+32 x^{2} b^{5} d^{3} e^{2}-315 a^{4} b \,e^{5} x +840 a^{3} b^{2} d \,e^{4} x -1008 x \,a^{2} b^{3} d^{2} e^{3}+576 x a \,b^{4} d^{3} e^{2}-128 b^{5} d^{4} e x +63 e^{5} a^{5}-630 a^{4} b d \,e^{4}+1680 a^{3} b^{2} d^{2} e^{3}-2016 a^{2} b^{3} d^{3} e^{2}+1152 a \,b^{4} d^{4} e -256 b^{5} d^{5}\right )}{63 \sqrt {e x +d}\, e^{6}}\) | \(273\) |
| orering | \(-\frac {2 \left (-7 x^{5} e^{5} b^{5}-45 x^{4} a \,b^{4} e^{5}+10 x^{4} b^{5} d \,e^{4}-126 x^{3} a^{2} b^{3} e^{5}+72 x^{3} a \,b^{4} d \,e^{4}-16 x^{3} b^{5} d^{2} e^{3}-210 x^{2} a^{3} b^{2} e^{5}+252 x^{2} a^{2} b^{3} d \,e^{4}-144 x^{2} a \,b^{4} d^{2} e^{3}+32 x^{2} b^{5} d^{3} e^{2}-315 a^{4} b \,e^{5} x +840 a^{3} b^{2} d \,e^{4} x -1008 x \,a^{2} b^{3} d^{2} e^{3}+576 x a \,b^{4} d^{3} e^{2}-128 b^{5} d^{4} e x +63 e^{5} a^{5}-630 a^{4} b d \,e^{4}+1680 a^{3} b^{2} d^{2} e^{3}-2016 a^{2} b^{3} d^{3} e^{2}+1152 a \,b^{4} d^{4} e -256 b^{5} d^{5}\right ) \left (b^{2} x^{2}+2 a b x +a^{2}\right )^{2}}{63 e^{6} \left (b x +a \right )^{4} \sqrt {e x +d}}\) | \(298\) |
| derivativedivides | \(\frac {\frac {2 b^{5} \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {10 a \,b^{4} e \left (e x +d \right )^{\frac {7}{2}}}{7}-\frac {10 b^{5} d \left (e x +d \right )^{\frac {7}{2}}}{7}+4 a^{2} b^{3} e^{2} \left (e x +d \right )^{\frac {5}{2}}-8 a \,b^{4} d e \left (e x +d \right )^{\frac {5}{2}}+4 b^{5} d^{2} \left (e x +d \right )^{\frac {5}{2}}+\frac {20 a^{3} b^{2} e^{3} \left (e x +d \right )^{\frac {3}{2}}}{3}-20 a^{2} b^{3} d \,e^{2} \left (e x +d \right )^{\frac {3}{2}}+20 a \,b^{4} d^{2} e \left (e x +d \right )^{\frac {3}{2}}-\frac {20 b^{5} d^{3} \left (e x +d \right )^{\frac {3}{2}}}{3}+10 a^{4} b \,e^{4} \sqrt {e x +d}-40 a^{3} b^{2} d \,e^{3} \sqrt {e x +d}+60 a^{2} b^{3} d^{2} e^{2} \sqrt {e x +d}-40 a \,b^{4} d^{3} e \sqrt {e x +d}+10 b^{5} d^{4} \sqrt {e x +d}-\frac {2 \left (e^{5} a^{5}-5 a^{4} b d \,e^{4}+10 a^{3} b^{2} d^{2} e^{3}-10 a^{2} b^{3} d^{3} e^{2}+5 a \,b^{4} d^{4} e -b^{5} d^{5}\right )}{\sqrt {e x +d}}}{e^{6}}\) | \(324\) |
| default | \(\frac {\frac {2 b^{5} \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {10 a \,b^{4} e \left (e x +d \right )^{\frac {7}{2}}}{7}-\frac {10 b^{5} d \left (e x +d \right )^{\frac {7}{2}}}{7}+4 a^{2} b^{3} e^{2} \left (e x +d \right )^{\frac {5}{2}}-8 a \,b^{4} d e \left (e x +d \right )^{\frac {5}{2}}+4 b^{5} d^{2} \left (e x +d \right )^{\frac {5}{2}}+\frac {20 a^{3} b^{2} e^{3} \left (e x +d \right )^{\frac {3}{2}}}{3}-20 a^{2} b^{3} d \,e^{2} \left (e x +d \right )^{\frac {3}{2}}+20 a \,b^{4} d^{2} e \left (e x +d \right )^{\frac {3}{2}}-\frac {20 b^{5} d^{3} \left (e x +d \right )^{\frac {3}{2}}}{3}+10 a^{4} b \,e^{4} \sqrt {e x +d}-40 a^{3} b^{2} d \,e^{3} \sqrt {e x +d}+60 a^{2} b^{3} d^{2} e^{2} \sqrt {e x +d}-40 a \,b^{4} d^{3} e \sqrt {e x +d}+10 b^{5} d^{4} \sqrt {e x +d}-\frac {2 \left (e^{5} a^{5}-5 a^{4} b d \,e^{4}+10 a^{3} b^{2} d^{2} e^{3}-10 a^{2} b^{3} d^{3} e^{2}+5 a \,b^{4} d^{4} e -b^{5} d^{5}\right )}{\sqrt {e x +d}}}{e^{6}}\) | \(324\) |
Input:
int((b*x+a)*(b^2*x^2+2*a*b*x+a^2)^2/(e*x+d)^(3/2),x,method=_RETURNVERBOSE)
Output:
2/63*((7*e^5*x^5-10*d*e^4*x^4+16*d^2*e^3*x^3-32*d^3*e^2*x^2+128*d^4*e*x+25 6*d^5)*b^5-1152*e*(-5/128*e^4*x^4+1/16*d*e^3*x^3-1/8*d^2*e^2*x^2+1/2*d^3*e *x+d^4)*a*b^4+2016*(1/16*e^3*x^3-1/8*d*e^2*x^2+1/2*d^2*e*x+d^3)*e^2*a^2*b^ 3-1680*e^3*(-1/8*e^2*x^2+1/2*d*e*x+d^2)*a^3*b^2+630*e^4*(1/2*e*x+d)*a^4*b- 63*e^5*a^5)/(e*x+d)^(1/2)/e^6
Leaf count of result is larger than twice the leaf count of optimal. 271 vs. \(2 (134) = 268\).
Time = 0.08 (sec) , antiderivative size = 271, normalized size of antiderivative = 1.78 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^{3/2}} \, dx=\frac {2 \, {\left (7 \, b^{5} e^{5} x^{5} + 256 \, b^{5} d^{5} - 1152 \, a b^{4} d^{4} e + 2016 \, a^{2} b^{3} d^{3} e^{2} - 1680 \, a^{3} b^{2} d^{2} e^{3} + 630 \, a^{4} b d e^{4} - 63 \, a^{5} e^{5} - 5 \, {\left (2 \, b^{5} d e^{4} - 9 \, a b^{4} e^{5}\right )} x^{4} + 2 \, {\left (8 \, b^{5} d^{2} e^{3} - 36 \, a b^{4} d e^{4} + 63 \, a^{2} b^{3} e^{5}\right )} x^{3} - 2 \, {\left (16 \, b^{5} d^{3} e^{2} - 72 \, a b^{4} d^{2} e^{3} + 126 \, a^{2} b^{3} d e^{4} - 105 \, a^{3} b^{2} e^{5}\right )} x^{2} + {\left (128 \, b^{5} d^{4} e - 576 \, a b^{4} d^{3} e^{2} + 1008 \, a^{2} b^{3} d^{2} e^{3} - 840 \, a^{3} b^{2} d e^{4} + 315 \, a^{4} b e^{5}\right )} x\right )} \sqrt {e x + d}}{63 \, {\left (e^{7} x + d e^{6}\right )}} \] Input:
integrate((b*x+a)*(b^2*x^2+2*a*b*x+a^2)^2/(e*x+d)^(3/2),x, algorithm="fric as")
Output:
2/63*(7*b^5*e^5*x^5 + 256*b^5*d^5 - 1152*a*b^4*d^4*e + 2016*a^2*b^3*d^3*e^ 2 - 1680*a^3*b^2*d^2*e^3 + 630*a^4*b*d*e^4 - 63*a^5*e^5 - 5*(2*b^5*d*e^4 - 9*a*b^4*e^5)*x^4 + 2*(8*b^5*d^2*e^3 - 36*a*b^4*d*e^4 + 63*a^2*b^3*e^5)*x^ 3 - 2*(16*b^5*d^3*e^2 - 72*a*b^4*d^2*e^3 + 126*a^2*b^3*d*e^4 - 105*a^3*b^2 *e^5)*x^2 + (128*b^5*d^4*e - 576*a*b^4*d^3*e^2 + 1008*a^2*b^3*d^2*e^3 - 84 0*a^3*b^2*d*e^4 + 315*a^4*b*e^5)*x)*sqrt(e*x + d)/(e^7*x + d*e^6)
Leaf count of result is larger than twice the leaf count of optimal. 284 vs. \(2 (141) = 282\).
Time = 9.18 (sec) , antiderivative size = 284, normalized size of antiderivative = 1.87 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^{3/2}} \, dx=\begin {cases} \frac {2 \left (\frac {b^{5} \left (d + e x\right )^{\frac {9}{2}}}{9 e^{5}} + \frac {\left (d + e x\right )^{\frac {7}{2}} \cdot \left (5 a b^{4} e - 5 b^{5} d\right )}{7 e^{5}} + \frac {\left (d + e x\right )^{\frac {5}{2}} \cdot \left (10 a^{2} b^{3} e^{2} - 20 a b^{4} d e + 10 b^{5} d^{2}\right )}{5 e^{5}} + \frac {\left (d + e x\right )^{\frac {3}{2}} \cdot \left (10 a^{3} b^{2} e^{3} - 30 a^{2} b^{3} d e^{2} + 30 a b^{4} d^{2} e - 10 b^{5} d^{3}\right )}{3 e^{5}} + \frac {\sqrt {d + e x} \left (5 a^{4} b e^{4} - 20 a^{3} b^{2} d e^{3} + 30 a^{2} b^{3} d^{2} e^{2} - 20 a b^{4} d^{3} e + 5 b^{5} d^{4}\right )}{e^{5}} - \frac {\left (a e - b d\right )^{5}}{e^{5} \sqrt {d + e x}}\right )}{e} & \text {for}\: e \neq 0 \\\frac {\begin {cases} a^{5} x & \text {for}\: b = 0 \\\frac {\left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{3}}{6 b} & \text {otherwise} \end {cases}}{d^{\frac {3}{2}}} & \text {otherwise} \end {cases} \] Input:
integrate((b*x+a)*(b**2*x**2+2*a*b*x+a**2)**2/(e*x+d)**(3/2),x)
Output:
Piecewise((2*(b**5*(d + e*x)**(9/2)/(9*e**5) + (d + e*x)**(7/2)*(5*a*b**4* e - 5*b**5*d)/(7*e**5) + (d + e*x)**(5/2)*(10*a**2*b**3*e**2 - 20*a*b**4*d *e + 10*b**5*d**2)/(5*e**5) + (d + e*x)**(3/2)*(10*a**3*b**2*e**3 - 30*a** 2*b**3*d*e**2 + 30*a*b**4*d**2*e - 10*b**5*d**3)/(3*e**5) + sqrt(d + e*x)* (5*a**4*b*e**4 - 20*a**3*b**2*d*e**3 + 30*a**2*b**3*d**2*e**2 - 20*a*b**4* d**3*e + 5*b**5*d**4)/e**5 - (a*e - b*d)**5/(e**5*sqrt(d + e*x)))/e, Ne(e, 0)), (Piecewise((a**5*x, Eq(b, 0)), ((a**2 + 2*a*b*x + b**2*x**2)**3/(6*b ), True))/d**(3/2), True))
Time = 0.05 (sec) , antiderivative size = 267, normalized size of antiderivative = 1.76 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^{3/2}} \, dx=\frac {2 \, {\left (\frac {7 \, {\left (e x + d\right )}^{\frac {9}{2}} b^{5} - 45 \, {\left (b^{5} d - a b^{4} e\right )} {\left (e x + d\right )}^{\frac {7}{2}} + 126 \, {\left (b^{5} d^{2} - 2 \, a b^{4} d e + a^{2} b^{3} e^{2}\right )} {\left (e x + d\right )}^{\frac {5}{2}} - 210 \, {\left (b^{5} d^{3} - 3 \, a b^{4} d^{2} e + 3 \, a^{2} b^{3} d e^{2} - a^{3} b^{2} e^{3}\right )} {\left (e x + d\right )}^{\frac {3}{2}} + 315 \, {\left (b^{5} d^{4} - 4 \, a b^{4} d^{3} e + 6 \, a^{2} b^{3} d^{2} e^{2} - 4 \, a^{3} b^{2} d e^{3} + a^{4} b e^{4}\right )} \sqrt {e x + d}}{e^{5}} + \frac {63 \, {\left (b^{5} d^{5} - 5 \, a b^{4} d^{4} e + 10 \, a^{2} b^{3} d^{3} e^{2} - 10 \, a^{3} b^{2} d^{2} e^{3} + 5 \, a^{4} b d e^{4} - a^{5} e^{5}\right )}}{\sqrt {e x + d} e^{5}}\right )}}{63 \, e} \] Input:
integrate((b*x+a)*(b^2*x^2+2*a*b*x+a^2)^2/(e*x+d)^(3/2),x, algorithm="maxi ma")
Output:
2/63*((7*(e*x + d)^(9/2)*b^5 - 45*(b^5*d - a*b^4*e)*(e*x + d)^(7/2) + 126* (b^5*d^2 - 2*a*b^4*d*e + a^2*b^3*e^2)*(e*x + d)^(5/2) - 210*(b^5*d^3 - 3*a *b^4*d^2*e + 3*a^2*b^3*d*e^2 - a^3*b^2*e^3)*(e*x + d)^(3/2) + 315*(b^5*d^4 - 4*a*b^4*d^3*e + 6*a^2*b^3*d^2*e^2 - 4*a^3*b^2*d*e^3 + a^4*b*e^4)*sqrt(e *x + d))/e^5 + 63*(b^5*d^5 - 5*a*b^4*d^4*e + 10*a^2*b^3*d^3*e^2 - 10*a^3*b ^2*d^2*e^3 + 5*a^4*b*d*e^4 - a^5*e^5)/(sqrt(e*x + d)*e^5))/e
Leaf count of result is larger than twice the leaf count of optimal. 350 vs. \(2 (134) = 268\).
Time = 0.18 (sec) , antiderivative size = 350, normalized size of antiderivative = 2.30 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^{3/2}} \, dx=\frac {2 \, {\left (b^{5} d^{5} - 5 \, a b^{4} d^{4} e + 10 \, a^{2} b^{3} d^{3} e^{2} - 10 \, a^{3} b^{2} d^{2} e^{3} + 5 \, a^{4} b d e^{4} - a^{5} e^{5}\right )}}{\sqrt {e x + d} e^{6}} + \frac {2 \, {\left (7 \, {\left (e x + d\right )}^{\frac {9}{2}} b^{5} e^{48} - 45 \, {\left (e x + d\right )}^{\frac {7}{2}} b^{5} d e^{48} + 126 \, {\left (e x + d\right )}^{\frac {5}{2}} b^{5} d^{2} e^{48} - 210 \, {\left (e x + d\right )}^{\frac {3}{2}} b^{5} d^{3} e^{48} + 315 \, \sqrt {e x + d} b^{5} d^{4} e^{48} + 45 \, {\left (e x + d\right )}^{\frac {7}{2}} a b^{4} e^{49} - 252 \, {\left (e x + d\right )}^{\frac {5}{2}} a b^{4} d e^{49} + 630 \, {\left (e x + d\right )}^{\frac {3}{2}} a b^{4} d^{2} e^{49} - 1260 \, \sqrt {e x + d} a b^{4} d^{3} e^{49} + 126 \, {\left (e x + d\right )}^{\frac {5}{2}} a^{2} b^{3} e^{50} - 630 \, {\left (e x + d\right )}^{\frac {3}{2}} a^{2} b^{3} d e^{50} + 1890 \, \sqrt {e x + d} a^{2} b^{3} d^{2} e^{50} + 210 \, {\left (e x + d\right )}^{\frac {3}{2}} a^{3} b^{2} e^{51} - 1260 \, \sqrt {e x + d} a^{3} b^{2} d e^{51} + 315 \, \sqrt {e x + d} a^{4} b e^{52}\right )}}{63 \, e^{54}} \] Input:
integrate((b*x+a)*(b^2*x^2+2*a*b*x+a^2)^2/(e*x+d)^(3/2),x, algorithm="giac ")
Output:
2*(b^5*d^5 - 5*a*b^4*d^4*e + 10*a^2*b^3*d^3*e^2 - 10*a^3*b^2*d^2*e^3 + 5*a ^4*b*d*e^4 - a^5*e^5)/(sqrt(e*x + d)*e^6) + 2/63*(7*(e*x + d)^(9/2)*b^5*e^ 48 - 45*(e*x + d)^(7/2)*b^5*d*e^48 + 126*(e*x + d)^(5/2)*b^5*d^2*e^48 - 21 0*(e*x + d)^(3/2)*b^5*d^3*e^48 + 315*sqrt(e*x + d)*b^5*d^4*e^48 + 45*(e*x + d)^(7/2)*a*b^4*e^49 - 252*(e*x + d)^(5/2)*a*b^4*d*e^49 + 630*(e*x + d)^( 3/2)*a*b^4*d^2*e^49 - 1260*sqrt(e*x + d)*a*b^4*d^3*e^49 + 126*(e*x + d)^(5 /2)*a^2*b^3*e^50 - 630*(e*x + d)^(3/2)*a^2*b^3*d*e^50 + 1890*sqrt(e*x + d) *a^2*b^3*d^2*e^50 + 210*(e*x + d)^(3/2)*a^3*b^2*e^51 - 1260*sqrt(e*x + d)* a^3*b^2*d*e^51 + 315*sqrt(e*x + d)*a^4*b*e^52)/e^54
Time = 11.01 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.26 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^{3/2}} \, dx=\frac {2\,b^5\,{\left (d+e\,x\right )}^{9/2}}{9\,e^6}-\frac {\left (10\,b^5\,d-10\,a\,b^4\,e\right )\,{\left (d+e\,x\right )}^{7/2}}{7\,e^6}-\frac {2\,a^5\,e^5-10\,a^4\,b\,d\,e^4+20\,a^3\,b^2\,d^2\,e^3-20\,a^2\,b^3\,d^3\,e^2+10\,a\,b^4\,d^4\,e-2\,b^5\,d^5}{e^6\,\sqrt {d+e\,x}}+\frac {20\,b^2\,{\left (a\,e-b\,d\right )}^3\,{\left (d+e\,x\right )}^{3/2}}{3\,e^6}+\frac {4\,b^3\,{\left (a\,e-b\,d\right )}^2\,{\left (d+e\,x\right )}^{5/2}}{e^6}+\frac {10\,b\,{\left (a\,e-b\,d\right )}^4\,\sqrt {d+e\,x}}{e^6} \] Input:
int(((a + b*x)*(a^2 + b^2*x^2 + 2*a*b*x)^2)/(d + e*x)^(3/2),x)
Output:
(2*b^5*(d + e*x)^(9/2))/(9*e^6) - ((10*b^5*d - 10*a*b^4*e)*(d + e*x)^(7/2) )/(7*e^6) - (2*a^5*e^5 - 2*b^5*d^5 - 20*a^2*b^3*d^3*e^2 + 20*a^3*b^2*d^2*e ^3 + 10*a*b^4*d^4*e - 10*a^4*b*d*e^4)/(e^6*(d + e*x)^(1/2)) + (20*b^2*(a*e - b*d)^3*(d + e*x)^(3/2))/(3*e^6) + (4*b^3*(a*e - b*d)^2*(d + e*x)^(5/2)) /e^6 + (10*b*(a*e - b*d)^4*(d + e*x)^(1/2))/e^6
Time = 0.27 (sec) , antiderivative size = 273, normalized size of antiderivative = 1.80 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^{3/2}} \, dx=\frac {\frac {2}{9} b^{5} e^{5} x^{5}+\frac {10}{7} a \,b^{4} e^{5} x^{4}-\frac {20}{63} b^{5} d \,e^{4} x^{4}+4 a^{2} b^{3} e^{5} x^{3}-\frac {16}{7} a \,b^{4} d \,e^{4} x^{3}+\frac {32}{63} b^{5} d^{2} e^{3} x^{3}+\frac {20}{3} a^{3} b^{2} e^{5} x^{2}-8 a^{2} b^{3} d \,e^{4} x^{2}+\frac {32}{7} a \,b^{4} d^{2} e^{3} x^{2}-\frac {64}{63} b^{5} d^{3} e^{2} x^{2}+10 a^{4} b \,e^{5} x -\frac {80}{3} a^{3} b^{2} d \,e^{4} x +32 a^{2} b^{3} d^{2} e^{3} x -\frac {128}{7} a \,b^{4} d^{3} e^{2} x +\frac {256}{63} b^{5} d^{4} e x -2 a^{5} e^{5}+20 a^{4} b d \,e^{4}-\frac {160}{3} a^{3} b^{2} d^{2} e^{3}+64 a^{2} b^{3} d^{3} e^{2}-\frac {256}{7} a \,b^{4} d^{4} e +\frac {512}{63} b^{5} d^{5}}{\sqrt {e x +d}\, e^{6}} \] Input:
int((b*x+a)*(b^2*x^2+2*a*b*x+a^2)^2/(e*x+d)^(3/2),x)
Output:
(2*( - 63*a**5*e**5 + 630*a**4*b*d*e**4 + 315*a**4*b*e**5*x - 1680*a**3*b* *2*d**2*e**3 - 840*a**3*b**2*d*e**4*x + 210*a**3*b**2*e**5*x**2 + 2016*a** 2*b**3*d**3*e**2 + 1008*a**2*b**3*d**2*e**3*x - 252*a**2*b**3*d*e**4*x**2 + 126*a**2*b**3*e**5*x**3 - 1152*a*b**4*d**4*e - 576*a*b**4*d**3*e**2*x + 144*a*b**4*d**2*e**3*x**2 - 72*a*b**4*d*e**4*x**3 + 45*a*b**4*e**5*x**4 + 256*b**5*d**5 + 128*b**5*d**4*e*x - 32*b**5*d**3*e**2*x**2 + 16*b**5*d**2* e**3*x**3 - 10*b**5*d*e**4*x**4 + 7*b**5*e**5*x**5))/(63*sqrt(d + e*x)*e** 6)