\(\int \frac {(A+B x) (d+e x)^{7/2}}{a+b x} \, dx\) [1742]

   Optimal result
   Rubi [A] (verified)
   Mathematica [A] (verified)
   Maple [A] (verified)
   Fricas [B] (verification not implemented)
   Sympy [A] (verification not implemented)
   Maxima [F(-2)]
   Giac [B] (verification not implemented)
   Mupad [B] (verification not implemented)

Optimal result

Integrand size = 22, antiderivative size = 198 \[ \int \frac {(A+B x) (d+e x)^{7/2}}{a+b x} \, dx=\frac {2 (A b-a B) (b d-a e)^3 \sqrt {d+e x}}{b^5}+\frac {2 (A b-a B) (b d-a e)^2 (d+e x)^{3/2}}{3 b^4}+\frac {2 (A b-a B) (b d-a e) (d+e x)^{5/2}}{5 b^3}+\frac {2 (A b-a B) (d+e x)^{7/2}}{7 b^2}+\frac {2 B (d+e x)^{9/2}}{9 b e}-\frac {2 (A b-a B) (b d-a e)^{7/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{b^{11/2}} \]

[Out]

2/3*(A*b-B*a)*(-a*e+b*d)^2*(e*x+d)^(3/2)/b^4+2/5*(A*b-B*a)*(-a*e+b*d)*(e*x+d)^(5/2)/b^3+2/7*(A*b-B*a)*(e*x+d)^
(7/2)/b^2+2/9*B*(e*x+d)^(9/2)/b/e-2*(A*b-B*a)*(-a*e+b*d)^(7/2)*arctanh(b^(1/2)*(e*x+d)^(1/2)/(-a*e+b*d)^(1/2))
/b^(11/2)+2*(A*b-B*a)*(-a*e+b*d)^3*(e*x+d)^(1/2)/b^5

Rubi [A] (verified)

Time = 0.17 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {81, 52, 65, 214} \[ \int \frac {(A+B x) (d+e x)^{7/2}}{a+b x} \, dx=-\frac {2 (A b-a B) (b d-a e)^{7/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{b^{11/2}}+\frac {2 \sqrt {d+e x} (A b-a B) (b d-a e)^3}{b^5}+\frac {2 (d+e x)^{3/2} (A b-a B) (b d-a e)^2}{3 b^4}+\frac {2 (d+e x)^{5/2} (A b-a B) (b d-a e)}{5 b^3}+\frac {2 (d+e x)^{7/2} (A b-a B)}{7 b^2}+\frac {2 B (d+e x)^{9/2}}{9 b e} \]

[In]

Int[((A + B*x)*(d + e*x)^(7/2))/(a + b*x),x]

[Out]

(2*(A*b - a*B)*(b*d - a*e)^3*Sqrt[d + e*x])/b^5 + (2*(A*b - a*B)*(b*d - a*e)^2*(d + e*x)^(3/2))/(3*b^4) + (2*(
A*b - a*B)*(b*d - a*e)*(d + e*x)^(5/2))/(5*b^3) + (2*(A*b - a*B)*(d + e*x)^(7/2))/(7*b^2) + (2*B*(d + e*x)^(9/
2))/(9*b*e) - (2*(A*b - a*B)*(b*d - a*e)^(7/2)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/b^(11/2)

Rule 52

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] + Dist[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] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[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] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 81

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[b*(c + d*x)^
(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), x] + Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(
d*f*(n + p + 2)), Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2,
0]

Rule 214

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]

Rubi steps \begin{align*} \text {integral}& = \frac {2 B (d+e x)^{9/2}}{9 b e}+\frac {\left (2 \left (\frac {9 A b e}{2}-\frac {9 a B e}{2}\right )\right ) \int \frac {(d+e x)^{7/2}}{a+b x} \, dx}{9 b e} \\ & = \frac {2 (A b-a B) (d+e x)^{7/2}}{7 b^2}+\frac {2 B (d+e x)^{9/2}}{9 b e}+\frac {((A b-a B) (b d-a e)) \int \frac {(d+e x)^{5/2}}{a+b x} \, dx}{b^2} \\ & = \frac {2 (A b-a B) (b d-a e) (d+e x)^{5/2}}{5 b^3}+\frac {2 (A b-a B) (d+e x)^{7/2}}{7 b^2}+\frac {2 B (d+e x)^{9/2}}{9 b e}+\frac {\left ((A b-a B) (b d-a e)^2\right ) \int \frac {(d+e x)^{3/2}}{a+b x} \, dx}{b^3} \\ & = \frac {2 (A b-a B) (b d-a e)^2 (d+e x)^{3/2}}{3 b^4}+\frac {2 (A b-a B) (b d-a e) (d+e x)^{5/2}}{5 b^3}+\frac {2 (A b-a B) (d+e x)^{7/2}}{7 b^2}+\frac {2 B (d+e x)^{9/2}}{9 b e}+\frac {\left ((A b-a B) (b d-a e)^3\right ) \int \frac {\sqrt {d+e x}}{a+b x} \, dx}{b^4} \\ & = \frac {2 (A b-a B) (b d-a e)^3 \sqrt {d+e x}}{b^5}+\frac {2 (A b-a B) (b d-a e)^2 (d+e x)^{3/2}}{3 b^4}+\frac {2 (A b-a B) (b d-a e) (d+e x)^{5/2}}{5 b^3}+\frac {2 (A b-a B) (d+e x)^{7/2}}{7 b^2}+\frac {2 B (d+e x)^{9/2}}{9 b e}+\frac {\left ((A b-a B) (b d-a e)^4\right ) \int \frac {1}{(a+b x) \sqrt {d+e x}} \, dx}{b^5} \\ & = \frac {2 (A b-a B) (b d-a e)^3 \sqrt {d+e x}}{b^5}+\frac {2 (A b-a B) (b d-a e)^2 (d+e x)^{3/2}}{3 b^4}+\frac {2 (A b-a B) (b d-a e) (d+e x)^{5/2}}{5 b^3}+\frac {2 (A b-a B) (d+e x)^{7/2}}{7 b^2}+\frac {2 B (d+e x)^{9/2}}{9 b e}+\frac {\left (2 (A b-a B) (b d-a e)^4\right ) \text {Subst}\left (\int \frac {1}{a-\frac {b d}{e}+\frac {b x^2}{e}} \, dx,x,\sqrt {d+e x}\right )}{b^5 e} \\ & = \frac {2 (A b-a B) (b d-a e)^3 \sqrt {d+e x}}{b^5}+\frac {2 (A b-a B) (b d-a e)^2 (d+e x)^{3/2}}{3 b^4}+\frac {2 (A b-a B) (b d-a e) (d+e x)^{5/2}}{5 b^3}+\frac {2 (A b-a B) (d+e x)^{7/2}}{7 b^2}+\frac {2 B (d+e x)^{9/2}}{9 b e}-\frac {2 (A b-a B) (b d-a e)^{7/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{b^{11/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.38 (sec) , antiderivative size = 263, normalized size of antiderivative = 1.33 \[ \int \frac {(A+B x) (d+e x)^{7/2}}{a+b x} \, dx=\frac {2 \sqrt {d+e x} \left (315 a^4 B e^4-105 a^3 b e^3 (10 B d+3 A e+B e x)+21 a^2 b^2 e^2 \left (5 A e (10 d+e x)+B \left (58 d^2+16 d e x+3 e^2 x^2\right )\right )-3 a b^3 e \left (7 A e \left (58 d^2+16 d e x+3 e^2 x^2\right )+B \left (176 d^3+122 d^2 e x+66 d e^2 x^2+15 e^3 x^3\right )\right )+b^4 \left (35 B (d+e x)^4+3 A e \left (176 d^3+122 d^2 e x+66 d e^2 x^2+15 e^3 x^3\right )\right )\right )}{315 b^5 e}+\frac {2 (A b-a B) (-b d+a e)^{7/2} \arctan \left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {-b d+a e}}\right )}{b^{11/2}} \]

[In]

Integrate[((A + B*x)*(d + e*x)^(7/2))/(a + b*x),x]

[Out]

(2*Sqrt[d + e*x]*(315*a^4*B*e^4 - 105*a^3*b*e^3*(10*B*d + 3*A*e + B*e*x) + 21*a^2*b^2*e^2*(5*A*e*(10*d + e*x)
+ B*(58*d^2 + 16*d*e*x + 3*e^2*x^2)) - 3*a*b^3*e*(7*A*e*(58*d^2 + 16*d*e*x + 3*e^2*x^2) + B*(176*d^3 + 122*d^2
*e*x + 66*d*e^2*x^2 + 15*e^3*x^3)) + b^4*(35*B*(d + e*x)^4 + 3*A*e*(176*d^3 + 122*d^2*e*x + 66*d*e^2*x^2 + 15*
e^3*x^3))))/(315*b^5*e) + (2*(A*b - a*B)*(-(b*d) + a*e)^(7/2)*ArcTan[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[-(b*d) + a*e
]])/b^(11/2)

Maple [A] (verified)

Time = 1.26 (sec) , antiderivative size = 269, normalized size of antiderivative = 1.36

method result size
pseudoelliptic \(\frac {-2 \left (\frac {\left (-\frac {\left (e x +d \right )^{4} B}{3}-\frac {176 \left (\frac {15}{176} e^{3} x^{3}+\frac {3}{8} d \,e^{2} x^{2}+\frac {61}{88} d^{2} e x +d^{3}\right ) e A}{35}\right ) b^{4}}{3}+\frac {58 e a \left (\frac {\left (\frac {15}{2} e^{3} x^{3}+33 d \,e^{2} x^{2}+61 d^{2} e x +88 d^{3}\right ) B}{203}+A e \left (\frac {3}{58} x^{2} e^{2}+\frac {8}{29} d e x +d^{2}\right )\right ) b^{3}}{15}-\frac {10 \left (\frac {\left (\frac {3}{2} x^{2} e^{2}+8 d e x +29 d^{2}\right ) B}{25}+A e \left (\frac {e x}{10}+d \right )\right ) e^{2} a^{2} b^{2}}{3}+e^{3} a^{3} \left (\frac {\left (e x +10 d \right ) B}{3}+A e \right ) b -B \,a^{4} e^{4}\right ) \sqrt {\left (a e -b d \right ) b}\, \sqrt {e x +d}+2 e \left (a e -b d \right )^{4} \left (A b -B a \right ) \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{e \,b^{5} \sqrt {\left (a e -b d \right ) b}}\) \(269\)
risch \(-\frac {2 \left (-35 B \,e^{4} b^{4} x^{4}-45 A \,b^{4} e^{4} x^{3}+45 B a \,b^{3} e^{4} x^{3}-140 B \,b^{4} d \,e^{3} x^{3}+63 A a \,b^{3} e^{4} x^{2}-198 A \,b^{4} d \,e^{3} x^{2}-63 B \,a^{2} b^{2} e^{4} x^{2}+198 B a \,b^{3} d \,e^{3} x^{2}-210 B \,b^{4} d^{2} e^{2} x^{2}-105 A \,a^{2} b^{2} e^{4} x +336 A a \,b^{3} d \,e^{3} x -366 A \,b^{4} d^{2} e^{2} x +105 B \,a^{3} b \,e^{4} x -336 B \,a^{2} b^{2} d \,e^{3} x +366 B a \,b^{3} d^{2} e^{2} x -140 B \,b^{4} d^{3} e x +315 A \,a^{3} b \,e^{4}-1050 A \,a^{2} b^{2} d \,e^{3}+1218 A a \,b^{3} d^{2} e^{2}-528 A \,b^{4} d^{3} e -315 B \,a^{4} e^{4}+1050 B \,a^{3} b d \,e^{3}-1218 B \,a^{2} b^{2} d^{2} e^{2}+528 B a \,b^{3} d^{3} e -35 B \,b^{4} d^{4}\right ) \sqrt {e x +d}}{315 e \,b^{5}}+\frac {2 \left (A \,a^{4} b \,e^{4}-4 A \,a^{3} b^{2} d \,e^{3}+6 A \,a^{2} b^{3} d^{2} e^{2}-4 A a \,b^{4} d^{3} e +A \,d^{4} b^{5}-a^{5} B \,e^{4}+4 B \,a^{4} b d \,e^{3}-6 B \,a^{3} b^{2} d^{2} e^{2}+4 B \,a^{2} b^{3} d^{3} e -B a \,b^{4} d^{4}\right ) \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{b^{5} \sqrt {\left (a e -b d \right ) b}}\) \(481\)
derivativedivides \(\frac {-\frac {2 \left (-\frac {B \left (e x +d \right )^{\frac {9}{2}} b^{4}}{9}-\frac {A \,b^{4} e \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {B a \,b^{3} e \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {A a \,b^{3} e^{2} \left (e x +d \right )^{\frac {5}{2}}}{5}-\frac {A \,b^{4} d e \left (e x +d \right )^{\frac {5}{2}}}{5}-\frac {B \,a^{2} b^{2} e^{2} \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {B a \,b^{3} d e \left (e x +d \right )^{\frac {5}{2}}}{5}-\frac {A \,a^{2} b^{2} e^{3} \left (e x +d \right )^{\frac {3}{2}}}{3}+\frac {2 A a \,b^{3} d \,e^{2} \left (e x +d \right )^{\frac {3}{2}}}{3}-\frac {A \,b^{4} d^{2} e \left (e x +d \right )^{\frac {3}{2}}}{3}+\frac {B \,a^{3} b \,e^{3} \left (e x +d \right )^{\frac {3}{2}}}{3}-\frac {2 B \,a^{2} b^{2} d \,e^{2} \left (e x +d \right )^{\frac {3}{2}}}{3}+\frac {B a \,b^{3} d^{2} e \left (e x +d \right )^{\frac {3}{2}}}{3}+A \,a^{3} b \,e^{4} \sqrt {e x +d}-3 A \,a^{2} b^{2} d \,e^{3} \sqrt {e x +d}+3 A a \,b^{3} d^{2} e^{2} \sqrt {e x +d}-A \,b^{4} d^{3} e \sqrt {e x +d}-B \,a^{4} e^{4} \sqrt {e x +d}+3 B \,a^{3} b d \,e^{3} \sqrt {e x +d}-3 B \,a^{2} b^{2} d^{2} e^{2} \sqrt {e x +d}+B a \,b^{3} d^{3} e \sqrt {e x +d}\right )}{b^{5}}+\frac {2 e \left (A \,a^{4} b \,e^{4}-4 A \,a^{3} b^{2} d \,e^{3}+6 A \,a^{2} b^{3} d^{2} e^{2}-4 A a \,b^{4} d^{3} e +A \,d^{4} b^{5}-a^{5} B \,e^{4}+4 B \,a^{4} b d \,e^{3}-6 B \,a^{3} b^{2} d^{2} e^{2}+4 B \,a^{2} b^{3} d^{3} e -B a \,b^{4} d^{4}\right ) \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{b^{5} \sqrt {\left (a e -b d \right ) b}}}{e}\) \(531\)
default \(\frac {-\frac {2 \left (-\frac {B \left (e x +d \right )^{\frac {9}{2}} b^{4}}{9}-\frac {A \,b^{4} e \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {B a \,b^{3} e \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {A a \,b^{3} e^{2} \left (e x +d \right )^{\frac {5}{2}}}{5}-\frac {A \,b^{4} d e \left (e x +d \right )^{\frac {5}{2}}}{5}-\frac {B \,a^{2} b^{2} e^{2} \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {B a \,b^{3} d e \left (e x +d \right )^{\frac {5}{2}}}{5}-\frac {A \,a^{2} b^{2} e^{3} \left (e x +d \right )^{\frac {3}{2}}}{3}+\frac {2 A a \,b^{3} d \,e^{2} \left (e x +d \right )^{\frac {3}{2}}}{3}-\frac {A \,b^{4} d^{2} e \left (e x +d \right )^{\frac {3}{2}}}{3}+\frac {B \,a^{3} b \,e^{3} \left (e x +d \right )^{\frac {3}{2}}}{3}-\frac {2 B \,a^{2} b^{2} d \,e^{2} \left (e x +d \right )^{\frac {3}{2}}}{3}+\frac {B a \,b^{3} d^{2} e \left (e x +d \right )^{\frac {3}{2}}}{3}+A \,a^{3} b \,e^{4} \sqrt {e x +d}-3 A \,a^{2} b^{2} d \,e^{3} \sqrt {e x +d}+3 A a \,b^{3} d^{2} e^{2} \sqrt {e x +d}-A \,b^{4} d^{3} e \sqrt {e x +d}-B \,a^{4} e^{4} \sqrt {e x +d}+3 B \,a^{3} b d \,e^{3} \sqrt {e x +d}-3 B \,a^{2} b^{2} d^{2} e^{2} \sqrt {e x +d}+B a \,b^{3} d^{3} e \sqrt {e x +d}\right )}{b^{5}}+\frac {2 e \left (A \,a^{4} b \,e^{4}-4 A \,a^{3} b^{2} d \,e^{3}+6 A \,a^{2} b^{3} d^{2} e^{2}-4 A a \,b^{4} d^{3} e +A \,d^{4} b^{5}-a^{5} B \,e^{4}+4 B \,a^{4} b d \,e^{3}-6 B \,a^{3} b^{2} d^{2} e^{2}+4 B \,a^{2} b^{3} d^{3} e -B a \,b^{4} d^{4}\right ) \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{b^{5} \sqrt {\left (a e -b d \right ) b}}}{e}\) \(531\)

[In]

int((B*x+A)*(e*x+d)^(7/2)/(b*x+a),x,method=_RETURNVERBOSE)

[Out]

2/((a*e-b*d)*b)^(1/2)*(-(1/3*(-1/3*(e*x+d)^4*B-176/35*(15/176*e^3*x^3+3/8*d*e^2*x^2+61/88*d^2*e*x+d^3)*e*A)*b^
4+58/15*e*a*(1/203*(15/2*e^3*x^3+33*d*e^2*x^2+61*d^2*e*x+88*d^3)*B+A*e*(3/58*x^2*e^2+8/29*d*e*x+d^2))*b^3-10/3
*(1/25*(3/2*x^2*e^2+8*d*e*x+29*d^2)*B+A*e*(1/10*e*x+d))*e^2*a^2*b^2+e^3*a^3*(1/3*(e*x+10*d)*B+A*e)*b-B*a^4*e^4
)*((a*e-b*d)*b)^(1/2)*(e*x+d)^(1/2)+e*(a*e-b*d)^4*(A*b-B*a)*arctan(b*(e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)))/e/b^5

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 427 vs. \(2 (170) = 340\).

Time = 0.24 (sec) , antiderivative size = 865, normalized size of antiderivative = 4.37 \[ \int \frac {(A+B x) (d+e x)^{7/2}}{a+b x} \, dx=\left [-\frac {315 \, {\left ({\left (B a b^{3} - A b^{4}\right )} d^{3} e - 3 \, {\left (B a^{2} b^{2} - A a b^{3}\right )} d^{2} e^{2} + 3 \, {\left (B a^{3} b - A a^{2} b^{2}\right )} d e^{3} - {\left (B a^{4} - A a^{3} b\right )} e^{4}\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 (35 \, B b^{4} e^{4} x^{4} + 35 \, B b^{4} d^{4} - 528 \, {\left (B a b^{3} - A b^{4}\right )} d^{3} e + 1218 \, {\left (B a^{2} b^{2} - A a b^{3}\right )} d^{2} e^{2} - 1050 \, {\left (B a^{3} b - A a^{2} b^{2}\right )} d e^{3} + 315 \, {\left (B a^{4} - A a^{3} b\right )} e^{4} + 5 \, {\left (28 \, B b^{4} d e^{3} - 9 \, {\left (B a b^{3} - A b^{4}\right )} e^{4}\right )} x^{3} + 3 \, {\left (70 \, B b^{4} d^{2} e^{2} - 66 \, {\left (B a b^{3} - A b^{4}\right )} d e^{3} + 21 \, {\left (B a^{2} b^{2} - A a b^{3}\right )} e^{4}\right )} x^{2} + {\left (140 \, B b^{4} d^{3} e - 366 \, {\left (B a b^{3} - A b^{4}\right )} d^{2} e^{2} + 336 \, {\left (B a^{2} b^{2} - A a b^{3}\right )} d e^{3} - 105 \, {\left (B a^{3} b - A a^{2} b^{2}\right )} e^{4}\right )} x\right )} \sqrt {e x + d}}{315 \, b^{5} e}, \frac {2 \, {\left (315 \, {\left ({\left (B a b^{3} - A b^{4}\right )} d^{3} e - 3 \, {\left (B a^{2} b^{2} - A a b^{3}\right )} d^{2} e^{2} + 3 \, {\left (B a^{3} b - A a^{2} b^{2}\right )} d e^{3} - {\left (B a^{4} - A a^{3} b\right )} e^{4}\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 (35 \, B b^{4} e^{4} x^{4} + 35 \, B b^{4} d^{4} - 528 \, {\left (B a b^{3} - A b^{4}\right )} d^{3} e + 1218 \, {\left (B a^{2} b^{2} - A a b^{3}\right )} d^{2} e^{2} - 1050 \, {\left (B a^{3} b - A a^{2} b^{2}\right )} d e^{3} + 315 \, {\left (B a^{4} - A a^{3} b\right )} e^{4} + 5 \, {\left (28 \, B b^{4} d e^{3} - 9 \, {\left (B a b^{3} - A b^{4}\right )} e^{4}\right )} x^{3} + 3 \, {\left (70 \, B b^{4} d^{2} e^{2} - 66 \, {\left (B a b^{3} - A b^{4}\right )} d e^{3} + 21 \, {\left (B a^{2} b^{2} - A a b^{3}\right )} e^{4}\right )} x^{2} + {\left (140 \, B b^{4} d^{3} e - 366 \, {\left (B a b^{3} - A b^{4}\right )} d^{2} e^{2} + 336 \, {\left (B a^{2} b^{2} - A a b^{3}\right )} d e^{3} - 105 \, {\left (B a^{3} b - A a^{2} b^{2}\right )} e^{4}\right )} x\right )} \sqrt {e x + d}\right )}}{315 \, b^{5} e}\right ] \]

[In]

integrate((B*x+A)*(e*x+d)^(7/2)/(b*x+a),x, algorithm="fricas")

[Out]

[-1/315*(315*((B*a*b^3 - A*b^4)*d^3*e - 3*(B*a^2*b^2 - A*a*b^3)*d^2*e^2 + 3*(B*a^3*b - A*a^2*b^2)*d*e^3 - (B*a
^4 - A*a^3*b)*e^4)*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*(35*B*b^4*e^4*x^4 + 35*B*b^4*d^4 - 528*(B*a*b^3 - A*b^4)*d^3*e + 1218*(B*a^2*b^2 - A*a*b^3)*d^2*e^2
- 1050*(B*a^3*b - A*a^2*b^2)*d*e^3 + 315*(B*a^4 - A*a^3*b)*e^4 + 5*(28*B*b^4*d*e^3 - 9*(B*a*b^3 - A*b^4)*e^4)*
x^3 + 3*(70*B*b^4*d^2*e^2 - 66*(B*a*b^3 - A*b^4)*d*e^3 + 21*(B*a^2*b^2 - A*a*b^3)*e^4)*x^2 + (140*B*b^4*d^3*e
- 366*(B*a*b^3 - A*b^4)*d^2*e^2 + 336*(B*a^2*b^2 - A*a*b^3)*d*e^3 - 105*(B*a^3*b - A*a^2*b^2)*e^4)*x)*sqrt(e*x
 + d))/(b^5*e), 2/315*(315*((B*a*b^3 - A*b^4)*d^3*e - 3*(B*a^2*b^2 - A*a*b^3)*d^2*e^2 + 3*(B*a^3*b - A*a^2*b^2
)*d*e^3 - (B*a^4 - A*a^3*b)*e^4)*sqrt(-(b*d - a*e)/b)*arctan(-sqrt(e*x + d)*b*sqrt(-(b*d - a*e)/b)/(b*d - a*e)
) + (35*B*b^4*e^4*x^4 + 35*B*b^4*d^4 - 528*(B*a*b^3 - A*b^4)*d^3*e + 1218*(B*a^2*b^2 - A*a*b^3)*d^2*e^2 - 1050
*(B*a^3*b - A*a^2*b^2)*d*e^3 + 315*(B*a^4 - A*a^3*b)*e^4 + 5*(28*B*b^4*d*e^3 - 9*(B*a*b^3 - A*b^4)*e^4)*x^3 +
3*(70*B*b^4*d^2*e^2 - 66*(B*a*b^3 - A*b^4)*d*e^3 + 21*(B*a^2*b^2 - A*a*b^3)*e^4)*x^2 + (140*B*b^4*d^3*e - 366*
(B*a*b^3 - A*b^4)*d^2*e^2 + 336*(B*a^2*b^2 - A*a*b^3)*d*e^3 - 105*(B*a^3*b - A*a^2*b^2)*e^4)*x)*sqrt(e*x + d))
/(b^5*e)]

Sympy [A] (verification not implemented)

Time = 4.15 (sec) , antiderivative size = 376, normalized size of antiderivative = 1.90 \[ \int \frac {(A+B x) (d+e x)^{7/2}}{a+b x} \, dx=\begin {cases} \frac {2 \left (\frac {B \left (d + e x\right )^{\frac {9}{2}}}{9 b} + \frac {\left (d + e x\right )^{\frac {7}{2}} \left (A b e - B a e\right )}{7 b^{2}} + \frac {\left (d + e x\right )^{\frac {5}{2}} \left (- A a b e^{2} + A b^{2} d e + B a^{2} e^{2} - B a b d e\right )}{5 b^{3}} + \frac {\left (d + e x\right )^{\frac {3}{2}} \left (A a^{2} b e^{3} - 2 A a b^{2} d e^{2} + A b^{3} d^{2} e - B a^{3} e^{3} + 2 B a^{2} b d e^{2} - B a b^{2} d^{2} e\right )}{3 b^{4}} + \frac {\sqrt {d + e x} \left (- A a^{3} b e^{4} + 3 A a^{2} b^{2} d e^{3} - 3 A a b^{3} d^{2} e^{2} + A b^{4} d^{3} e + B a^{4} e^{4} - 3 B a^{3} b d e^{3} + 3 B a^{2} b^{2} d^{2} e^{2} - B a b^{3} d^{3} e\right )}{b^{5}} - \frac {e \left (- A b + B a\right ) \left (a e - b d\right )^{4} \operatorname {atan}{\left (\frac {\sqrt {d + e x}}{\sqrt {\frac {a e - b d}{b}}} \right )}}{b^{6} \sqrt {\frac {a e - b d}{b}}}\right )}{e} & \text {for}\: e \neq 0 \\d^{\frac {7}{2}} \left (\frac {B x}{b} - \frac {\left (- A b + B a\right ) \left (\begin {cases} \frac {x}{a} & \text {for}\: b = 0 \\\frac {\log {\left (a + b x \right )}}{b} & \text {otherwise} \end {cases}\right )}{b}\right ) & \text {otherwise} \end {cases} \]

[In]

integrate((B*x+A)*(e*x+d)**(7/2)/(b*x+a),x)

[Out]

Piecewise((2*(B*(d + e*x)**(9/2)/(9*b) + (d + e*x)**(7/2)*(A*b*e - B*a*e)/(7*b**2) + (d + e*x)**(5/2)*(-A*a*b*
e**2 + A*b**2*d*e + B*a**2*e**2 - B*a*b*d*e)/(5*b**3) + (d + e*x)**(3/2)*(A*a**2*b*e**3 - 2*A*a*b**2*d*e**2 +
A*b**3*d**2*e - B*a**3*e**3 + 2*B*a**2*b*d*e**2 - B*a*b**2*d**2*e)/(3*b**4) + sqrt(d + e*x)*(-A*a**3*b*e**4 +
3*A*a**2*b**2*d*e**3 - 3*A*a*b**3*d**2*e**2 + A*b**4*d**3*e + B*a**4*e**4 - 3*B*a**3*b*d*e**3 + 3*B*a**2*b**2*
d**2*e**2 - B*a*b**3*d**3*e)/b**5 - e*(-A*b + B*a)*(a*e - b*d)**4*atan(sqrt(d + e*x)/sqrt((a*e - b*d)/b))/(b**
6*sqrt((a*e - b*d)/b)))/e, Ne(e, 0)), (d**(7/2)*(B*x/b - (-A*b + B*a)*Piecewise((x/a, Eq(b, 0)), (log(a + b*x)
/b, True))/b), True))

Maxima [F(-2)]

Exception generated. \[ \int \frac {(A+B x) (d+e x)^{7/2}}{a+b x} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate((B*x+A)*(e*x+d)^(7/2)/(b*x+a),x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(a*e-b*d>0)', see `assume?` for
 more detail

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 560 vs. \(2 (170) = 340\).

Time = 0.29 (sec) , antiderivative size = 560, normalized size of antiderivative = 2.83 \[ \int \frac {(A+B x) (d+e x)^{7/2}}{a+b x} \, dx=-\frac {2 \, {\left (B a b^{4} d^{4} - A b^{5} d^{4} - 4 \, B a^{2} b^{3} d^{3} e + 4 \, A a b^{4} d^{3} e + 6 \, B a^{3} b^{2} d^{2} e^{2} - 6 \, A a^{2} b^{3} d^{2} e^{2} - 4 \, B a^{4} b d e^{3} + 4 \, A a^{3} b^{2} d e^{3} + B a^{5} e^{4} - A a^{4} b e^{4}\right )} \arctan \left (\frac {\sqrt {e x + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{\sqrt {-b^{2} d + a b e} b^{5}} + \frac {2 \, {\left (35 \, {\left (e x + d\right )}^{\frac {9}{2}} B b^{8} e^{8} - 45 \, {\left (e x + d\right )}^{\frac {7}{2}} B a b^{7} e^{9} + 45 \, {\left (e x + d\right )}^{\frac {7}{2}} A b^{8} e^{9} - 63 \, {\left (e x + d\right )}^{\frac {5}{2}} B a b^{7} d e^{9} + 63 \, {\left (e x + d\right )}^{\frac {5}{2}} A b^{8} d e^{9} - 105 \, {\left (e x + d\right )}^{\frac {3}{2}} B a b^{7} d^{2} e^{9} + 105 \, {\left (e x + d\right )}^{\frac {3}{2}} A b^{8} d^{2} e^{9} - 315 \, \sqrt {e x + d} B a b^{7} d^{3} e^{9} + 315 \, \sqrt {e x + d} A b^{8} d^{3} e^{9} + 63 \, {\left (e x + d\right )}^{\frac {5}{2}} B a^{2} b^{6} e^{10} - 63 \, {\left (e x + d\right )}^{\frac {5}{2}} A a b^{7} e^{10} + 210 \, {\left (e x + d\right )}^{\frac {3}{2}} B a^{2} b^{6} d e^{10} - 210 \, {\left (e x + d\right )}^{\frac {3}{2}} A a b^{7} d e^{10} + 945 \, \sqrt {e x + d} B a^{2} b^{6} d^{2} e^{10} - 945 \, \sqrt {e x + d} A a b^{7} d^{2} e^{10} - 105 \, {\left (e x + d\right )}^{\frac {3}{2}} B a^{3} b^{5} e^{11} + 105 \, {\left (e x + d\right )}^{\frac {3}{2}} A a^{2} b^{6} e^{11} - 945 \, \sqrt {e x + d} B a^{3} b^{5} d e^{11} + 945 \, \sqrt {e x + d} A a^{2} b^{6} d e^{11} + 315 \, \sqrt {e x + d} B a^{4} b^{4} e^{12} - 315 \, \sqrt {e x + d} A a^{3} b^{5} e^{12}\right )}}{315 \, b^{9} e^{9}} \]

[In]

integrate((B*x+A)*(e*x+d)^(7/2)/(b*x+a),x, algorithm="giac")

[Out]

-2*(B*a*b^4*d^4 - A*b^5*d^4 - 4*B*a^2*b^3*d^3*e + 4*A*a*b^4*d^3*e + 6*B*a^3*b^2*d^2*e^2 - 6*A*a^2*b^3*d^2*e^2
- 4*B*a^4*b*d*e^3 + 4*A*a^3*b^2*d*e^3 + B*a^5*e^4 - A*a^4*b*e^4)*arctan(sqrt(e*x + d)*b/sqrt(-b^2*d + a*b*e))/
(sqrt(-b^2*d + a*b*e)*b^5) + 2/315*(35*(e*x + d)^(9/2)*B*b^8*e^8 - 45*(e*x + d)^(7/2)*B*a*b^7*e^9 + 45*(e*x +
d)^(7/2)*A*b^8*e^9 - 63*(e*x + d)^(5/2)*B*a*b^7*d*e^9 + 63*(e*x + d)^(5/2)*A*b^8*d*e^9 - 105*(e*x + d)^(3/2)*B
*a*b^7*d^2*e^9 + 105*(e*x + d)^(3/2)*A*b^8*d^2*e^9 - 315*sqrt(e*x + d)*B*a*b^7*d^3*e^9 + 315*sqrt(e*x + d)*A*b
^8*d^3*e^9 + 63*(e*x + d)^(5/2)*B*a^2*b^6*e^10 - 63*(e*x + d)^(5/2)*A*a*b^7*e^10 + 210*(e*x + d)^(3/2)*B*a^2*b
^6*d*e^10 - 210*(e*x + d)^(3/2)*A*a*b^7*d*e^10 + 945*sqrt(e*x + d)*B*a^2*b^6*d^2*e^10 - 945*sqrt(e*x + d)*A*a*
b^7*d^2*e^10 - 105*(e*x + d)^(3/2)*B*a^3*b^5*e^11 + 105*(e*x + d)^(3/2)*A*a^2*b^6*e^11 - 945*sqrt(e*x + d)*B*a
^3*b^5*d*e^11 + 945*sqrt(e*x + d)*A*a^2*b^6*d*e^11 + 315*sqrt(e*x + d)*B*a^4*b^4*e^12 - 315*sqrt(e*x + d)*A*a^
3*b^5*e^12)/(b^9*e^9)

Mupad [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 426, normalized size of antiderivative = 2.15 \[ \int \frac {(A+B x) (d+e x)^{7/2}}{a+b x} \, dx=\left (\frac {2\,A\,e-2\,B\,d}{7\,b\,e}-\frac {2\,B\,\left (a\,e^2-b\,d\,e\right )}{7\,b^2\,e^2}\right )\,{\left (d+e\,x\right )}^{7/2}+\frac {2\,B\,{\left (d+e\,x\right )}^{9/2}}{9\,b\,e}+\frac {2\,\mathrm {atan}\left (\frac {\sqrt {b}\,\left (A\,b-B\,a\right )\,{\left (a\,e-b\,d\right )}^{7/2}\,\sqrt {d+e\,x}}{-B\,a^5\,e^4+4\,B\,a^4\,b\,d\,e^3+A\,a^4\,b\,e^4-6\,B\,a^3\,b^2\,d^2\,e^2-4\,A\,a^3\,b^2\,d\,e^3+4\,B\,a^2\,b^3\,d^3\,e+6\,A\,a^2\,b^3\,d^2\,e^2-B\,a\,b^4\,d^4-4\,A\,a\,b^4\,d^3\,e+A\,b^5\,d^4}\right )\,\left (A\,b-B\,a\right )\,{\left (a\,e-b\,d\right )}^{7/2}}{b^{11/2}}+\frac {\left (\frac {2\,A\,e-2\,B\,d}{b\,e}-\frac {2\,B\,\left (a\,e^2-b\,d\,e\right )}{b^2\,e^2}\right )\,{\left (a\,e^2-b\,d\,e\right )}^2\,{\left (d+e\,x\right )}^{3/2}}{3\,b^2\,e^2}-\frac {\left (\frac {2\,A\,e-2\,B\,d}{b\,e}-\frac {2\,B\,\left (a\,e^2-b\,d\,e\right )}{b^2\,e^2}\right )\,{\left (a\,e^2-b\,d\,e\right )}^3\,\sqrt {d+e\,x}}{b^3\,e^3}-\frac {\left (\frac {2\,A\,e-2\,B\,d}{b\,e}-\frac {2\,B\,\left (a\,e^2-b\,d\,e\right )}{b^2\,e^2}\right )\,\left (a\,e^2-b\,d\,e\right )\,{\left (d+e\,x\right )}^{5/2}}{5\,b\,e} \]

[In]

int(((A + B*x)*(d + e*x)^(7/2))/(a + b*x),x)

[Out]

((2*A*e - 2*B*d)/(7*b*e) - (2*B*(a*e^2 - b*d*e))/(7*b^2*e^2))*(d + e*x)^(7/2) + (2*B*(d + e*x)^(9/2))/(9*b*e)
+ (2*atan((b^(1/2)*(A*b - B*a)*(a*e - b*d)^(7/2)*(d + e*x)^(1/2))/(A*b^5*d^4 - B*a^5*e^4 + A*a^4*b*e^4 - B*a*b
^4*d^4 - 4*A*a^3*b^2*d*e^3 + 4*B*a^2*b^3*d^3*e + 6*A*a^2*b^3*d^2*e^2 - 6*B*a^3*b^2*d^2*e^2 - 4*A*a*b^4*d^3*e +
 4*B*a^4*b*d*e^3))*(A*b - B*a)*(a*e - b*d)^(7/2))/b^(11/2) + (((2*A*e - 2*B*d)/(b*e) - (2*B*(a*e^2 - b*d*e))/(
b^2*e^2))*(a*e^2 - b*d*e)^2*(d + e*x)^(3/2))/(3*b^2*e^2) - (((2*A*e - 2*B*d)/(b*e) - (2*B*(a*e^2 - b*d*e))/(b^
2*e^2))*(a*e^2 - b*d*e)^3*(d + e*x)^(1/2))/(b^3*e^3) - (((2*A*e - 2*B*d)/(b*e) - (2*B*(a*e^2 - b*d*e))/(b^2*e^
2))*(a*e^2 - b*d*e)*(d + e*x)^(5/2))/(5*b*e)