Integrand size = 33, antiderivative size = 138 \[ \int \frac {(a+b x) (d+e x)^{7/2}}{a^2+2 a b x+b^2 x^2} \, dx=\frac {2 (b d-a e)^3 \sqrt {d+e x}}{b^4}+\frac {2 (b d-a e)^2 (d+e x)^{3/2}}{3 b^3}+\frac {2 (b d-a e) (d+e x)^{5/2}}{5 b^2}+\frac {2 (d+e x)^{7/2}}{7 b}-\frac {2 (b d-a e)^{7/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{b^{9/2}} \]
2/3*(-a*e+b*d)^2*(e*x+d)^(3/2)/b^3+2/5*(-a*e+b*d)*(e*x+d)^(5/2)/b^2+2/7*(e *x+d)^(7/2)/b-2*(-a*e+b*d)^(7/2)*arctanh(b^(1/2)*(e*x+d)^(1/2)/(-a*e+b*d)^ (1/2))/b^(9/2)+2*(-a*e+b*d)^3*(e*x+d)^(1/2)/b^4
Time = 0.06 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.08 \[ \int \frac {(a+b x) (d+e x)^{7/2}}{a^2+2 a b x+b^2 x^2} \, dx=\frac {2 \sqrt {d+e x} \left (-105 a^3 e^3+35 a^2 b e^2 (10 d+e x)-7 a b^2 e \left (58 d^2+16 d e x+3 e^2 x^2\right )+b^3 \left (176 d^3+122 d^2 e x+66 d e^2 x^2+15 e^3 x^3\right )\right )}{105 b^4}+\frac {2 (-b d+a e)^{7/2} \arctan \left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {-b d+a e}}\right )}{b^{9/2}} \]
(2*Sqrt[d + e*x]*(-105*a^3*e^3 + 35*a^2*b*e^2*(10*d + e*x) - 7*a*b^2*e*(58 *d^2 + 16*d*e*x + 3*e^2*x^2) + b^3*(176*d^3 + 122*d^2*e*x + 66*d*e^2*x^2 + 15*e^3*x^3)))/(105*b^4) + (2*(-(b*d) + a*e)^(7/2)*ArcTan[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[-(b*d) + a*e]])/b^(9/2)
Time = 0.29 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.08, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.242, Rules used = {1184, 27, 60, 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.
| \(\displaystyle \int \frac {(a+b x) (d+e x)^{7/2}}{a^2+2 a b x+b^2 x^2} \, dx\) |
| \(\Big \downarrow \) 1184 |
| \(\displaystyle b^2 \int \frac {(d+e x)^{7/2}}{b^2 (a+b x)}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int \frac {(d+e x)^{7/2}}{a+b x}dx\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {(b d-a e) \int \frac {(d+e x)^{5/2}}{a+b x}dx}{b}+\frac {2 (d+e x)^{7/2}}{7 b}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {(b d-a e) \left (\frac {(b d-a e) \int \frac {(d+e x)^{3/2}}{a+b x}dx}{b}+\frac {2 (d+e x)^{5/2}}{5 b}\right )}{b}+\frac {2 (d+e x)^{7/2}}{7 b}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {(b d-a e) \left (\frac {(b d-a e) \left (\frac {(b d-a e) \int \frac {\sqrt {d+e x}}{a+b x}dx}{b}+\frac {2 (d+e x)^{3/2}}{3 b}\right )}{b}+\frac {2 (d+e x)^{5/2}}{5 b}\right )}{b}+\frac {2 (d+e x)^{7/2}}{7 b}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {(b d-a e) \left (\frac {(b d-a e) \left (\frac {(b d-a e) \left (\frac {(b d-a e) \int \frac {1}{(a+b x) \sqrt {d+e x}}dx}{b}+\frac {2 \sqrt {d+e x}}{b}\right )}{b}+\frac {2 (d+e x)^{3/2}}{3 b}\right )}{b}+\frac {2 (d+e x)^{5/2}}{5 b}\right )}{b}+\frac {2 (d+e x)^{7/2}}{7 b}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {(b d-a e) \left (\frac {(b d-a e) \left (\frac {(b d-a e) \left (\frac {2 (b d-a e) \int \frac {1}{a+\frac {b (d+e x)}{e}-\frac {b d}{e}}d\sqrt {d+e x}}{b e}+\frac {2 \sqrt {d+e x}}{b}\right )}{b}+\frac {2 (d+e x)^{3/2}}{3 b}\right )}{b}+\frac {2 (d+e x)^{5/2}}{5 b}\right )}{b}+\frac {2 (d+e x)^{7/2}}{7 b}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {(b d-a e) \left (\frac {(b d-a e) \left (\frac {(b d-a e) \left (\frac {2 \sqrt {d+e x}}{b}-\frac {2 \sqrt {b d-a e} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{b^{3/2}}\right )}{b}+\frac {2 (d+e x)^{3/2}}{3 b}\right )}{b}+\frac {2 (d+e x)^{5/2}}{5 b}\right )}{b}+\frac {2 (d+e x)^{7/2}}{7 b}\) |
(2*(d + e*x)^(7/2))/(7*b) + ((b*d - a*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))/b
3.21.66.3.1 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 + 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_.)*((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 = 0.37 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.11
| method | result | size |
| pseudoelliptic | \(\frac {-2 \sqrt {\left (a e -b d \right ) b}\, \left (\left (-\frac {1}{7} x^{3} b^{3}+\frac {1}{5} a \,b^{2} x^{2}-\frac {1}{3} b \,a^{2} x +a^{3}\right ) e^{3}-\frac {10 b \left (\frac {33}{175} b^{2} x^{2}-\frac {8}{25} a b x +a^{2}\right ) d \,e^{2}}{3}+\frac {58 b^{2} \left (-\frac {61 b x}{203}+a \right ) d^{2} e}{15}-\frac {176 b^{3} d^{3}}{105}\right ) \sqrt {e x +d}+2 \left (a e -b d \right )^{4} \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{b^{4} \sqrt {\left (a e -b d \right ) b}}\) | \(153\) |
| derivativedivides | \(-\frac {2 \left (-\frac {\left (e x +d \right )^{\frac {7}{2}} b^{3}}{7}+\frac {\left (a e -b d \right ) \left (e x +d \right )^{\frac {5}{2}} b^{2}}{5}-\frac {\left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) \left (e x +d \right )^{\frac {3}{2}} b}{3}+\left (a e -b d \right ) \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) \sqrt {e x +d}\right )}{b^{4}}+\frac {2 \left (e^{4} a^{4}-4 b d \,e^{3} a^{3}+6 b^{2} d^{2} e^{2} a^{2}-4 b^{3} d^{3} e a +b^{4} d^{4}\right ) \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{b^{4} \sqrt {\left (a e -b d \right ) b}}\) | \(196\) |
| default | \(-\frac {2 \left (-\frac {\left (e x +d \right )^{\frac {7}{2}} b^{3}}{7}+\frac {\left (a e -b d \right ) \left (e x +d \right )^{\frac {5}{2}} b^{2}}{5}-\frac {\left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) \left (e x +d \right )^{\frac {3}{2}} b}{3}+\left (a e -b d \right ) \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) \sqrt {e x +d}\right )}{b^{4}}+\frac {2 \left (e^{4} a^{4}-4 b d \,e^{3} a^{3}+6 b^{2} d^{2} e^{2} a^{2}-4 b^{3} d^{3} e a +b^{4} d^{4}\right ) \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{b^{4} \sqrt {\left (a e -b d \right ) b}}\) | \(196\) |
| risch | \(-\frac {2 \left (-15 b^{3} x^{3} e^{3}+21 x^{2} a \,b^{2} e^{3}-66 x^{2} b^{3} d \,e^{2}-35 x \,a^{2} b \,e^{3}+112 x a \,b^{2} d \,e^{2}-122 x \,b^{3} d^{2} e +105 a^{3} e^{3}-350 a^{2} b d \,e^{2}+406 a \,b^{2} d^{2} e -176 b^{3} d^{3}\right ) \sqrt {e x +d}}{105 b^{4}}+\frac {2 \left (e^{4} a^{4}-4 b d \,e^{3} a^{3}+6 b^{2} d^{2} e^{2} a^{2}-4 b^{3} d^{3} e a +b^{4} d^{4}\right ) \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{b^{4} \sqrt {\left (a e -b d \right ) b}}\) | \(205\) |
2/((a*e-b*d)*b)^(1/2)*(-((a*e-b*d)*b)^(1/2)*((-1/7*x^3*b^3+1/5*a*b^2*x^2-1 /3*b*a^2*x+a^3)*e^3-10/3*b*(33/175*b^2*x^2-8/25*a*b*x+a^2)*d*e^2+58/15*b^2 *(-61/203*b*x+a)*d^2*e-176/105*b^3*d^3)*(e*x+d)^(1/2)+(a*e-b*d)^4*arctan(b *(e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)))/b^4
Time = 0.33 (sec) , antiderivative size = 424, normalized size of antiderivative = 3.07 \[ \int \frac {(a+b x) (d+e x)^{7/2}}{a^2+2 a b x+b^2 x^2} \, dx=\left [-\frac {105 \, {\left (b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - a^{3} e^{3}\right )} \sqrt {\frac {b d - a e}{b}} \log \left (\frac {b e x + 2 \, b d - a e + 2 \, \sqrt {e x + d} b \sqrt {\frac {b d - a e}{b}}}{b x + a}\right ) - 2 \, {\left (15 \, b^{3} e^{3} x^{3} + 176 \, b^{3} d^{3} - 406 \, a b^{2} d^{2} e + 350 \, a^{2} b d e^{2} - 105 \, a^{3} e^{3} + 3 \, {\left (22 \, b^{3} d e^{2} - 7 \, a b^{2} e^{3}\right )} x^{2} + {\left (122 \, b^{3} d^{2} e - 112 \, a b^{2} d e^{2} + 35 \, a^{2} b e^{3}\right )} x\right )} \sqrt {e x + d}}{105 \, b^{4}}, -\frac {2 \, {\left (105 \, {\left (b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - a^{3} e^{3}\right )} \sqrt {-\frac {b d - a e}{b}} \arctan \left (-\frac {\sqrt {e x + d} b \sqrt {-\frac {b d - a e}{b}}}{b d - a e}\right ) - {\left (15 \, b^{3} e^{3} x^{3} + 176 \, b^{3} d^{3} - 406 \, a b^{2} d^{2} e + 350 \, a^{2} b d e^{2} - 105 \, a^{3} e^{3} + 3 \, {\left (22 \, b^{3} d e^{2} - 7 \, a b^{2} e^{3}\right )} x^{2} + {\left (122 \, b^{3} d^{2} e - 112 \, a b^{2} d e^{2} + 35 \, a^{2} b e^{3}\right )} x\right )} \sqrt {e x + d}\right )}}{105 \, b^{4}}\right ] \]
[-1/105*(105*(b^3*d^3 - 3*a*b^2*d^2*e + 3*a^2*b*d*e^2 - a^3*e^3)*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*(15*b^3*e^3*x^3 + 176*b^3*d^3 - 406*a*b^2*d^2*e + 350*a^ 2*b*d*e^2 - 105*a^3*e^3 + 3*(22*b^3*d*e^2 - 7*a*b^2*e^3)*x^2 + (122*b^3*d^ 2*e - 112*a*b^2*d*e^2 + 35*a^2*b*e^3)*x)*sqrt(e*x + d))/b^4, -2/105*(105*( b^3*d^3 - 3*a*b^2*d^2*e + 3*a^2*b*d*e^2 - a^3*e^3)*sqrt(-(b*d - a*e)/b)*ar ctan(-sqrt(e*x + d)*b*sqrt(-(b*d - a*e)/b)/(b*d - a*e)) - (15*b^3*e^3*x^3 + 176*b^3*d^3 - 406*a*b^2*d^2*e + 350*a^2*b*d*e^2 - 105*a^3*e^3 + 3*(22*b^ 3*d*e^2 - 7*a*b^2*e^3)*x^2 + (122*b^3*d^2*e - 112*a*b^2*d*e^2 + 35*a^2*b*e ^3)*x)*sqrt(e*x + d))/b^4]
Time = 18.72 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.54 \[ \int \frac {(a+b x) (d+e x)^{7/2}}{a^2+2 a b x+b^2 x^2} \, dx=\begin {cases} \frac {2 \left (\frac {e \left (d + e x\right )^{\frac {7}{2}}}{7 b} + \frac {\left (d + e x\right )^{\frac {5}{2}} \left (- a e^{2} + b d e\right )}{5 b^{2}} + \frac {\left (d + e x\right )^{\frac {3}{2}} \left (a^{2} e^{3} - 2 a b d e^{2} + b^{2} d^{2} e\right )}{3 b^{3}} + \frac {\sqrt {d + e x} \left (- a^{3} e^{4} + 3 a^{2} b d e^{3} - 3 a b^{2} d^{2} e^{2} + b^{3} d^{3} e\right )}{b^{4}} + \frac {e \left (a e - b d\right )^{4} \operatorname {atan}{\left (\frac {\sqrt {d + e x}}{\sqrt {\frac {a e - b d}{b}}} \right )}}{b^{5} \sqrt {\frac {a e - b d}{b}}}\right )}{e} & \text {for}\: e \neq 0 \\\frac {d^{\frac {7}{2}} \log {\left (a^{2} + 2 a b x + b^{2} x^{2} \right )}}{2 b} & \text {otherwise} \end {cases} \]
Piecewise((2*(e*(d + e*x)**(7/2)/(7*b) + (d + e*x)**(5/2)*(-a*e**2 + b*d*e )/(5*b**2) + (d + e*x)**(3/2)*(a**2*e**3 - 2*a*b*d*e**2 + b**2*d**2*e)/(3* b**3) + sqrt(d + e*x)*(-a**3*e**4 + 3*a**2*b*d*e**3 - 3*a*b**2*d**2*e**2 + b**3*d**3*e)/b**4 + e*(a*e - b*d)**4*atan(sqrt(d + e*x)/sqrt((a*e - b*d)/ b))/(b**5*sqrt((a*e - b*d)/b)))/e, Ne(e, 0)), (d**(7/2)*log(a**2 + 2*a*b*x + b**2*x**2)/(2*b), True))
Exception generated. \[ \int \frac {(a+b x) (d+e x)^{7/2}}{a^2+2 a b x+b^2 x^2} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(a*e-b*d>0)', see `assume?` for m ore detail
Leaf count of result is larger than twice the leaf count of optimal. 253 vs. \(2 (114) = 228\).
Time = 0.28 (sec) , antiderivative size = 253, normalized size of antiderivative = 1.83 \[ \int \frac {(a+b x) (d+e x)^{7/2}}{a^2+2 a b x+b^2 x^2} \, dx=\frac {2 \, {\left (b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4}\right )} \arctan \left (\frac {\sqrt {e x + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{\sqrt {-b^{2} d + a b e} b^{4}} + \frac {2 \, {\left (15 \, {\left (e x + d\right )}^{\frac {7}{2}} b^{6} + 21 \, {\left (e x + d\right )}^{\frac {5}{2}} b^{6} d + 35 \, {\left (e x + d\right )}^{\frac {3}{2}} b^{6} d^{2} + 105 \, \sqrt {e x + d} b^{6} d^{3} - 21 \, {\left (e x + d\right )}^{\frac {5}{2}} a b^{5} e - 70 \, {\left (e x + d\right )}^{\frac {3}{2}} a b^{5} d e - 315 \, \sqrt {e x + d} a b^{5} d^{2} e + 35 \, {\left (e x + d\right )}^{\frac {3}{2}} a^{2} b^{4} e^{2} + 315 \, \sqrt {e x + d} a^{2} b^{4} d e^{2} - 105 \, \sqrt {e x + d} a^{3} b^{3} e^{3}\right )}}{105 \, b^{7}} \]
2*(b^4*d^4 - 4*a*b^3*d^3*e + 6*a^2*b^2*d^2*e^2 - 4*a^3*b*d*e^3 + a^4*e^4)* arctan(sqrt(e*x + d)*b/sqrt(-b^2*d + a*b*e))/(sqrt(-b^2*d + a*b*e)*b^4) + 2/105*(15*(e*x + d)^(7/2)*b^6 + 21*(e*x + d)^(5/2)*b^6*d + 35*(e*x + d)^(3 /2)*b^6*d^2 + 105*sqrt(e*x + d)*b^6*d^3 - 21*(e*x + d)^(5/2)*a*b^5*e - 70* (e*x + d)^(3/2)*a*b^5*d*e - 315*sqrt(e*x + d)*a*b^5*d^2*e + 35*(e*x + d)^( 3/2)*a^2*b^4*e^2 + 315*sqrt(e*x + d)*a^2*b^4*d*e^2 - 105*sqrt(e*x + d)*a^3 *b^3*e^3)/b^7
Time = 10.94 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.20 \[ \int \frac {(a+b x) (d+e x)^{7/2}}{a^2+2 a b x+b^2 x^2} \, dx=\frac {2\,{\left (d+e\,x\right )}^{7/2}}{7\,b}-\frac {2\,\left (a\,e-b\,d\right )\,{\left (d+e\,x\right )}^{5/2}}{5\,b^2}+\frac {2\,\mathrm {atan}\left (\frac {\sqrt {b}\,{\left (a\,e-b\,d\right )}^{7/2}\,\sqrt {d+e\,x}}{a^4\,e^4-4\,a^3\,b\,d\,e^3+6\,a^2\,b^2\,d^2\,e^2-4\,a\,b^3\,d^3\,e+b^4\,d^4}\right )\,{\left (a\,e-b\,d\right )}^{7/2}}{b^{9/2}}+\frac {2\,{\left (a\,e-b\,d\right )}^2\,{\left (d+e\,x\right )}^{3/2}}{3\,b^3}-\frac {2\,{\left (a\,e-b\,d\right )}^3\,\sqrt {d+e\,x}}{b^4} \]
(2*(d + e*x)^(7/2))/(7*b) - (2*(a*e - b*d)*(d + e*x)^(5/2))/(5*b^2) + (2*a tan((b^(1/2)*(a*e - b*d)^(7/2)*(d + e*x)^(1/2))/(a^4*e^4 + b^4*d^4 + 6*a^2 *b^2*d^2*e^2 - 4*a*b^3*d^3*e - 4*a^3*b*d*e^3))*(a*e - b*d)^(7/2))/b^(9/2) + (2*(a*e - b*d)^2*(d + e*x)^(3/2))/(3*b^3) - (2*(a*e - b*d)^3*(d + e*x)^( 1/2))/b^4