\(\int \frac {(a+b x)^3 \sqrt {c+d x} (e+f x)}{x} \, dx\) [19]

Optimal result

Integrand size = 25, antiderivative size = 225 \[ \int \frac {(a+b x)^3 \sqrt {c+d x} (e+f x)}{x} \, dx=2 a^3 e \sqrt {c+d x}+\frac {2 \left (a^3 d^3 f+b^3 c^2 (d e-c f)-3 a b^2 c d (d e-c f)+3 a^2 b d^2 (d e-c f)\right ) (c+d x)^{3/2}}{3 d^4}+\frac {2 b \left (3 a^2 d^2 f-b^2 c (2 d e-3 c f)+3 a b d (d e-2 c f)\right ) (c+d x)^{5/2}}{5 d^4}+\frac {2 b^2 (b d e-3 b c f+3 a d f) (c+d x)^{7/2}}{7 d^4}+\frac {2 b^3 f (c+d x)^{9/2}}{9 d^4}-2 a^3 \sqrt {c} e \text {arctanh}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right ) \] Output:

2*a^3*e*(d*x+c)^(1/2)+2/3*(a^3*d^3*f+b^3*c^2*(-c*f+d*e)-3*a*b^2*c*d*(-c*f+ 
d*e)+3*a^2*b*d^2*(-c*f+d*e))*(d*x+c)^(3/2)/d^4+2/5*b*(3*a^2*d^2*f-b^2*c*(- 
3*c*f+2*d*e)+3*a*b*d*(-2*c*f+d*e))*(d*x+c)^(5/2)/d^4+2/7*b^2*(3*a*d*f-3*b* 
c*f+b*d*e)*(d*x+c)^(7/2)/d^4+2/9*b^3*f*(d*x+c)^(9/2)/d^4-2*a^3*c^(1/2)*e*a 
rctanh((d*x+c)^(1/2)/c^(1/2))
 

Mathematica [A] (verified)

Time = 0.28 (sec) , antiderivative size = 197, normalized size of antiderivative = 0.88 \[ \int \frac {(a+b x)^3 \sqrt {c+d x} (e+f x)}{x} \, dx=\frac {2 \sqrt {c+d x} \left (105 a^3 d^3 (3 d e+c f+d f x)+63 a^2 b d^2 (c+d x) (5 d e-2 c f+3 d f x)+9 a b^2 d (c+d x) \left (8 c^2 f+3 d^2 x (7 e+5 f x)-2 c d (7 e+6 f x)\right )-b^3 (c+d x) \left (16 c^3 f-24 c^2 d (e+f x)+6 c d^2 x (6 e+5 f x)-5 d^3 x^2 (9 e+7 f x)\right )\right )}{315 d^4}-2 a^3 \sqrt {c} e \text {arctanh}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right ) \] Input:

Integrate[((a + b*x)^3*Sqrt[c + d*x]*(e + f*x))/x,x]
 

Output:

(2*Sqrt[c + d*x]*(105*a^3*d^3*(3*d*e + c*f + d*f*x) + 63*a^2*b*d^2*(c + d* 
x)*(5*d*e - 2*c*f + 3*d*f*x) + 9*a*b^2*d*(c + d*x)*(8*c^2*f + 3*d^2*x*(7*e 
 + 5*f*x) - 2*c*d*(7*e + 6*f*x)) - b^3*(c + d*x)*(16*c^3*f - 24*c^2*d*(e + 
 f*x) + 6*c*d^2*x*(6*e + 5*f*x) - 5*d^3*x^2*(9*e + 7*f*x))))/(315*d^4) - 2 
*a^3*Sqrt[c]*e*ArcTanh[Sqrt[c + d*x]/Sqrt[c]]
 

Rubi [A] (verified)

Time = 0.46 (sec) , antiderivative size = 242, normalized size of antiderivative = 1.08, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {170, 27, 170, 27, 164, 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.

\(\displaystyle \int \frac {(a+b x)^3 \sqrt {c+d x} (e+f x)}{x} \, dx\)

\(\Big \downarrow \) 170

\(\displaystyle \frac {2 \int \frac {3 (a+b x)^2 \sqrt {c+d x} (3 a d e+(3 b d e-2 b c f+2 a d f) x)}{2 x}dx}{9 d}+\frac {2 f (a+b x)^3 (c+d x)^{3/2}}{9 d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {(a+b x)^2 \sqrt {c+d x} (3 a d e+(3 b d e-2 b c f+2 a d f) x)}{x}dx}{3 d}+\frac {2 f (a+b x)^3 (c+d x)^{3/2}}{9 d}\)

\(\Big \downarrow \) 170

\(\displaystyle \frac {\frac {2 \int \frac {(a+b x) \sqrt {c+d x} \left (21 a^2 e d^2+\left (21 a b d^2 e-4 (b c-a d) (3 b d e-2 b c f+2 a d f)\right ) x\right )}{2 x}dx}{7 d}+\frac {2 (a+b x)^2 (c+d x)^{3/2} (2 a d f-2 b c f+3 b d e)}{7 d}}{3 d}+\frac {2 f (a+b x)^3 (c+d x)^{3/2}}{9 d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\int \frac {(a+b x) \sqrt {c+d x} \left (21 a^2 e d^2+\left (21 a b d^2 e-4 (b c-a d) (3 b d e-2 b c f+2 a d f)\right ) x\right )}{x}dx}{7 d}+\frac {2 (a+b x)^2 (c+d x)^{3/2} (2 a d f-2 b c f+3 b d e)}{7 d}}{3 d}+\frac {2 f (a+b x)^3 (c+d x)^{3/2}}{9 d}\)

\(\Big \downarrow \) 164

\(\displaystyle \frac {\frac {21 a^3 d^2 e \int \frac {\sqrt {c+d x}}{x}dx+\frac {2 (c+d x)^{3/2} \left (40 a^3 d^3 f+6 a^2 b d^2 (45 d e-16 c f)-18 a b^2 c d (7 d e-4 c f)+3 b d x \left (21 a b d^2 e-4 (b c-a d) (2 a d f-2 b c f+3 b d e)\right )+8 b^3 c^2 (3 d e-2 c f)\right )}{15 d^2}}{7 d}+\frac {2 (a+b x)^2 (c+d x)^{3/2} (2 a d f-2 b c f+3 b d e)}{7 d}}{3 d}+\frac {2 f (a+b x)^3 (c+d x)^{3/2}}{9 d}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {\frac {21 a^3 d^2 e \left (c \int \frac {1}{x \sqrt {c+d x}}dx+2 \sqrt {c+d x}\right )+\frac {2 (c+d x)^{3/2} \left (40 a^3 d^3 f+6 a^2 b d^2 (45 d e-16 c f)-18 a b^2 c d (7 d e-4 c f)+3 b d x \left (21 a b d^2 e-4 (b c-a d) (2 a d f-2 b c f+3 b d e)\right )+8 b^3 c^2 (3 d e-2 c f)\right )}{15 d^2}}{7 d}+\frac {2 (a+b x)^2 (c+d x)^{3/2} (2 a d f-2 b c f+3 b d e)}{7 d}}{3 d}+\frac {2 f (a+b x)^3 (c+d x)^{3/2}}{9 d}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {\frac {21 a^3 d^2 e \left (\frac {2 c \int \frac {1}{\frac {c+d x}{d}-\frac {c}{d}}d\sqrt {c+d x}}{d}+2 \sqrt {c+d x}\right )+\frac {2 (c+d x)^{3/2} \left (40 a^3 d^3 f+6 a^2 b d^2 (45 d e-16 c f)-18 a b^2 c d (7 d e-4 c f)+3 b d x \left (21 a b d^2 e-4 (b c-a d) (2 a d f-2 b c f+3 b d e)\right )+8 b^3 c^2 (3 d e-2 c f)\right )}{15 d^2}}{7 d}+\frac {2 (a+b x)^2 (c+d x)^{3/2} (2 a d f-2 b c f+3 b d e)}{7 d}}{3 d}+\frac {2 f (a+b x)^3 (c+d x)^{3/2}}{9 d}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {\frac {21 a^3 d^2 e \left (2 \sqrt {c+d x}-2 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )\right )+\frac {2 (c+d x)^{3/2} \left (40 a^3 d^3 f+6 a^2 b d^2 (45 d e-16 c f)-18 a b^2 c d (7 d e-4 c f)+3 b d x \left (21 a b d^2 e-4 (b c-a d) (2 a d f-2 b c f+3 b d e)\right )+8 b^3 c^2 (3 d e-2 c f)\right )}{15 d^2}}{7 d}+\frac {2 (a+b x)^2 (c+d x)^{3/2} (2 a d f-2 b c f+3 b d e)}{7 d}}{3 d}+\frac {2 f (a+b x)^3 (c+d x)^{3/2}}{9 d}\)

Input:

Int[((a + b*x)^3*Sqrt[c + d*x]*(e + f*x))/x,x]
 

Output:

(2*f*(a + b*x)^3*(c + d*x)^(3/2))/(9*d) + ((2*(3*b*d*e - 2*b*c*f + 2*a*d*f 
)*(a + b*x)^2*(c + d*x)^(3/2))/(7*d) + ((2*(c + d*x)^(3/2)*(40*a^3*d^3*f + 
 6*a^2*b*d^2*(45*d*e - 16*c*f) - 18*a*b^2*c*d*(7*d*e - 4*c*f) + 8*b^3*c^2* 
(3*d*e - 2*c*f) + 3*b*d*(21*a*b*d^2*e - 4*(b*c - a*d)*(3*b*d*e - 2*b*c*f + 
 2*a*d*f))*x))/(15*d^2) + 21*a^3*d^2*e*(2*Sqrt[c + d*x] - 2*Sqrt[c]*ArcTan 
h[Sqrt[c + d*x]/Sqrt[c]]))/(7*d))/(3*d)
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 60

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]
 

rule 73

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]
 

rule 164

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_ 
))*((g_.) + (h_.)*(x_)), x_] :> Simp[(-(a*d*f*h*(n + 2) + b*c*f*h*(m + 2) - 
 b*d*(f*g + e*h)*(m + n + 3) - b*d*f*h*(m + n + 2)*x))*(a + b*x)^(m + 1)*(( 
c + d*x)^(n + 1)/(b^2*d^2*(m + n + 2)*(m + n + 3))), x] + Simp[(a^2*d^2*f*h 
*(n + 1)*(n + 2) + a*b*d*(n + 1)*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(m + n + 
3)) + b^2*(c^2*f*h*(m + 1)*(m + 2) - c*d*(f*g + e*h)*(m + 1)*(m + n + 3) + 
d^2*e*g*(m + n + 2)*(m + n + 3)))/(b^2*d^2*(m + n + 2)*(m + n + 3))   Int[( 
a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] 
&& NeQ[m + n + 2, 0] && NeQ[m + n + 3, 0]
 

rule 170

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[h*(a + b*x)^m*(c + d*x)^(n + 1)*(( 
e + f*x)^(p + 1)/(d*f*(m + n + p + 2))), x] + Simp[1/(d*f*(m + n + p + 2)) 
  Int[(a + b*x)^(m - 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*g*(m + n + p + 2 
) - h*(b*c*e*m + a*(d*e*(n + 1) + c*f*(p + 1))) + (b*d*f*g*(m + n + p + 2) 
+ h*(a*d*f*m - b*(d*e*(m + n + 1) + c*f*(m + p + 1))))*x, x], x], x] /; Fre 
eQ[{a, b, c, d, e, f, g, h, n, p}, x] && GtQ[m, 0] && NeQ[m + n + p + 2, 0] 
 && IntegerQ[m]
 

rule 221

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]
 

Maple [A] (verified)

Time = 0.33 (sec) , antiderivative size = 214, normalized size of antiderivative = 0.95

method	result	size
pseudoelliptic	\(\frac {-2 a^{3} \sqrt {c}\, d^{4} e \,\operatorname {arctanh}\left (\frac {\sqrt {x d +c}}{\sqrt {c}}\right )+\frac {2 \left (3 \left (\frac {\left (\frac {7 f x}{9}+e \right ) x^{3} b^{3}}{7}+\frac {3 a \left (\frac {5 f x}{7}+e \right ) x^{2} b^{2}}{5}+a^{2} x \left (\frac {3 f x}{5}+e \right ) b +a^{3} \left (\frac {f x}{3}+e \right )\right ) d^{4}+\left (\frac {3 x^{2} \left (\frac {5 f x}{9}+e \right ) b^{3}}{35}+\frac {3 a \left (\frac {3 f x}{7}+e \right ) x \,b^{2}}{5}+3 a^{2} \left (\frac {f x}{5}+e \right ) b +f \,a^{3}\right ) c \,d^{3}-\frac {6 c^{2} \left (\frac {2 \left (\frac {f x}{2}+e \right ) x \,b^{2}}{21}+a \left (\frac {2 f x}{7}+e \right ) b +a^{2} f \right ) b \,d^{2}}{5}+\frac {24 c^{3} \left (\frac {\left (\frac {f x}{3}+e \right ) b}{3}+a f \right ) b^{2} d}{35}-\frac {16 b^{3} c^{4} f}{105}\right ) \sqrt {x d +c}}{3}}{d^{4}}\)	\(214\)
derivativedivides	\(\frac {\frac {2 f \,b^{3} \left (x d +c \right )^{\frac {9}{2}}}{9}+\frac {6 a \,b^{2} d f \left (x d +c \right )^{\frac {7}{2}}}{7}-\frac {6 b^{3} c f \left (x d +c \right )^{\frac {7}{2}}}{7}+\frac {2 b^{3} d e \left (x d +c \right )^{\frac {7}{2}}}{7}+\frac {6 a^{2} b \,d^{2} f \left (x d +c \right )^{\frac {5}{2}}}{5}-\frac {12 a \,b^{2} c d f \left (x d +c \right )^{\frac {5}{2}}}{5}+\frac {6 a \,b^{2} d^{2} e \left (x d +c \right )^{\frac {5}{2}}}{5}+\frac {6 b^{3} c^{2} f \left (x d +c \right )^{\frac {5}{2}}}{5}-\frac {4 b^{3} c d e \left (x d +c \right )^{\frac {5}{2}}}{5}+\frac {2 a^{3} d^{3} f \left (x d +c \right )^{\frac {3}{2}}}{3}-2 a^{2} b c \,d^{2} f \left (x d +c \right )^{\frac {3}{2}}+2 a^{2} b \,d^{3} e \left (x d +c \right )^{\frac {3}{2}}+2 a \,b^{2} c^{2} d f \left (x d +c \right )^{\frac {3}{2}}-2 a \,b^{2} c \,d^{2} e \left (x d +c \right )^{\frac {3}{2}}-\frac {2 b^{3} c^{3} f \left (x d +c \right )^{\frac {3}{2}}}{3}+\frac {2 b^{3} c^{2} d e \left (x d +c \right )^{\frac {3}{2}}}{3}+2 a^{3} d^{4} e \sqrt {x d +c}-2 a^{3} \sqrt {c}\, d^{4} e \,\operatorname {arctanh}\left (\frac {\sqrt {x d +c}}{\sqrt {c}}\right )}{d^{4}}\)	\(301\)
default	\(\frac {\frac {2 f \,b^{3} \left (x d +c \right )^{\frac {9}{2}}}{9}+\frac {6 a \,b^{2} d f \left (x d +c \right )^{\frac {7}{2}}}{7}-\frac {6 b^{3} c f \left (x d +c \right )^{\frac {7}{2}}}{7}+\frac {2 b^{3} d e \left (x d +c \right )^{\frac {7}{2}}}{7}+\frac {6 a^{2} b \,d^{2} f \left (x d +c \right )^{\frac {5}{2}}}{5}-\frac {12 a \,b^{2} c d f \left (x d +c \right )^{\frac {5}{2}}}{5}+\frac {6 a \,b^{2} d^{2} e \left (x d +c \right )^{\frac {5}{2}}}{5}+\frac {6 b^{3} c^{2} f \left (x d +c \right )^{\frac {5}{2}}}{5}-\frac {4 b^{3} c d e \left (x d +c \right )^{\frac {5}{2}}}{5}+\frac {2 a^{3} d^{3} f \left (x d +c \right )^{\frac {3}{2}}}{3}-2 a^{2} b c \,d^{2} f \left (x d +c \right )^{\frac {3}{2}}+2 a^{2} b \,d^{3} e \left (x d +c \right )^{\frac {3}{2}}+2 a \,b^{2} c^{2} d f \left (x d +c \right )^{\frac {3}{2}}-2 a \,b^{2} c \,d^{2} e \left (x d +c \right )^{\frac {3}{2}}-\frac {2 b^{3} c^{3} f \left (x d +c \right )^{\frac {3}{2}}}{3}+\frac {2 b^{3} c^{2} d e \left (x d +c \right )^{\frac {3}{2}}}{3}+2 a^{3} d^{4} e \sqrt {x d +c}-2 a^{3} \sqrt {c}\, d^{4} e \,\operatorname {arctanh}\left (\frac {\sqrt {x d +c}}{\sqrt {c}}\right )}{d^{4}}\)	\(301\)

Input:

int((b*x+a)^3*(d*x+c)^(1/2)*(f*x+e)/x,x,method=_RETURNVERBOSE)
 

Output:

2/3*(-3*a^3*c^(1/2)*d^4*e*arctanh((d*x+c)^(1/2)/c^(1/2))+(3*(1/7*(7/9*f*x+ 
e)*x^3*b^3+3/5*a*(5/7*f*x+e)*x^2*b^2+a^2*x*(3/5*f*x+e)*b+a^3*(1/3*f*x+e))* 
d^4+(3/35*x^2*(5/9*f*x+e)*b^3+3/5*a*(3/7*f*x+e)*x*b^2+3*a^2*(1/5*f*x+e)*b+ 
f*a^3)*c*d^3-6/5*c^2*(2/21*(1/2*f*x+e)*x*b^2+a*(2/7*f*x+e)*b+a^2*f)*b*d^2+ 
24/35*c^3*(1/3*(1/3*f*x+e)*b+a*f)*b^2*d-16/105*b^3*c^4*f)*(d*x+c)^(1/2))/d 
^4
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 646, normalized size of antiderivative = 2.87 \[ \int \frac {(a+b x)^3 \sqrt {c+d x} (e+f x)}{x} \, dx=\left [\frac {315 \, a^{3} \sqrt {c} d^{4} e \log \left (\frac {d x - 2 \, \sqrt {d x + c} \sqrt {c} + 2 \, c}{x}\right ) + 2 \, {\left (35 \, b^{3} d^{4} f x^{4} + 5 \, {\left (9 \, b^{3} d^{4} e + {\left (b^{3} c d^{3} + 27 \, a b^{2} d^{4}\right )} f\right )} x^{3} + 3 \, {\left (3 \, {\left (b^{3} c d^{3} + 21 \, a b^{2} d^{4}\right )} e - {\left (2 \, b^{3} c^{2} d^{2} - 9 \, a b^{2} c d^{3} - 63 \, a^{2} b d^{4}\right )} f\right )} x^{2} + 3 \, {\left (8 \, b^{3} c^{3} d - 42 \, a b^{2} c^{2} d^{2} + 105 \, a^{2} b c d^{3} + 105 \, a^{3} d^{4}\right )} e - {\left (16 \, b^{3} c^{4} - 72 \, a b^{2} c^{3} d + 126 \, a^{2} b c^{2} d^{2} - 105 \, a^{3} c d^{3}\right )} f - {\left (3 \, {\left (4 \, b^{3} c^{2} d^{2} - 21 \, a b^{2} c d^{3} - 105 \, a^{2} b d^{4}\right )} e - {\left (8 \, b^{3} c^{3} d - 36 \, a b^{2} c^{2} d^{2} + 63 \, a^{2} b c d^{3} + 105 \, a^{3} d^{4}\right )} f\right )} x\right )} \sqrt {d x + c}}{315 \, d^{4}}, \frac {2 \, {\left (315 \, a^{3} \sqrt {-c} d^{4} e \arctan \left (\frac {\sqrt {-c}}{\sqrt {d x + c}}\right ) + {\left (35 \, b^{3} d^{4} f x^{4} + 5 \, {\left (9 \, b^{3} d^{4} e + {\left (b^{3} c d^{3} + 27 \, a b^{2} d^{4}\right )} f\right )} x^{3} + 3 \, {\left (3 \, {\left (b^{3} c d^{3} + 21 \, a b^{2} d^{4}\right )} e - {\left (2 \, b^{3} c^{2} d^{2} - 9 \, a b^{2} c d^{3} - 63 \, a^{2} b d^{4}\right )} f\right )} x^{2} + 3 \, {\left (8 \, b^{3} c^{3} d - 42 \, a b^{2} c^{2} d^{2} + 105 \, a^{2} b c d^{3} + 105 \, a^{3} d^{4}\right )} e - {\left (16 \, b^{3} c^{4} - 72 \, a b^{2} c^{3} d + 126 \, a^{2} b c^{2} d^{2} - 105 \, a^{3} c d^{3}\right )} f - {\left (3 \, {\left (4 \, b^{3} c^{2} d^{2} - 21 \, a b^{2} c d^{3} - 105 \, a^{2} b d^{4}\right )} e - {\left (8 \, b^{3} c^{3} d - 36 \, a b^{2} c^{2} d^{2} + 63 \, a^{2} b c d^{3} + 105 \, a^{3} d^{4}\right )} f\right )} x\right )} \sqrt {d x + c}\right )}}{315 \, d^{4}}\right ] \] Input:

integrate((b*x+a)^3*(d*x+c)^(1/2)*(f*x+e)/x,x, algorithm="fricas")
 

Output:

[1/315*(315*a^3*sqrt(c)*d^4*e*log((d*x - 2*sqrt(d*x + c)*sqrt(c) + 2*c)/x) 
 + 2*(35*b^3*d^4*f*x^4 + 5*(9*b^3*d^4*e + (b^3*c*d^3 + 27*a*b^2*d^4)*f)*x^ 
3 + 3*(3*(b^3*c*d^3 + 21*a*b^2*d^4)*e - (2*b^3*c^2*d^2 - 9*a*b^2*c*d^3 - 6 
3*a^2*b*d^4)*f)*x^2 + 3*(8*b^3*c^3*d - 42*a*b^2*c^2*d^2 + 105*a^2*b*c*d^3 
+ 105*a^3*d^4)*e - (16*b^3*c^4 - 72*a*b^2*c^3*d + 126*a^2*b*c^2*d^2 - 105* 
a^3*c*d^3)*f - (3*(4*b^3*c^2*d^2 - 21*a*b^2*c*d^3 - 105*a^2*b*d^4)*e - (8* 
b^3*c^3*d - 36*a*b^2*c^2*d^2 + 63*a^2*b*c*d^3 + 105*a^3*d^4)*f)*x)*sqrt(d* 
x + c))/d^4, 2/315*(315*a^3*sqrt(-c)*d^4*e*arctan(sqrt(-c)/sqrt(d*x + c)) 
+ (35*b^3*d^4*f*x^4 + 5*(9*b^3*d^4*e + (b^3*c*d^3 + 27*a*b^2*d^4)*f)*x^3 + 
 3*(3*(b^3*c*d^3 + 21*a*b^2*d^4)*e - (2*b^3*c^2*d^2 - 9*a*b^2*c*d^3 - 63*a 
^2*b*d^4)*f)*x^2 + 3*(8*b^3*c^3*d - 42*a*b^2*c^2*d^2 + 105*a^2*b*c*d^3 + 1 
05*a^3*d^4)*e - (16*b^3*c^4 - 72*a*b^2*c^3*d + 126*a^2*b*c^2*d^2 - 105*a^3 
*c*d^3)*f - (3*(4*b^3*c^2*d^2 - 21*a*b^2*c*d^3 - 105*a^2*b*d^4)*e - (8*b^3 
*c^3*d - 36*a*b^2*c^2*d^2 + 63*a^2*b*c*d^3 + 105*a^3*d^4)*f)*x)*sqrt(d*x + 
 c))/d^4]
 

Sympy [A] (verification not implemented)

Time = 12.53 (sec) , antiderivative size = 355, normalized size of antiderivative = 1.58 \[ \int \frac {(a+b x)^3 \sqrt {c+d x} (e+f x)}{x} \, dx=\begin {cases} \frac {2 a^{3} c e \operatorname {atan}{\left (\frac {\sqrt {c + d x}}{\sqrt {- c}} \right )}}{\sqrt {- c}} + 2 a^{3} e \sqrt {c + d x} + \frac {2 b^{3} f \left (c + d x\right )^{\frac {9}{2}}}{9 d^{4}} + \frac {2 \left (c + d x\right )^{\frac {7}{2}} \cdot \left (3 a b^{2} d f - 3 b^{3} c f + b^{3} d e\right )}{7 d^{4}} + \frac {2 \left (c + d x\right )^{\frac {5}{2}} \cdot \left (3 a^{2} b d^{2} f - 6 a b^{2} c d f + 3 a b^{2} d^{2} e + 3 b^{3} c^{2} f - 2 b^{3} c d e\right )}{5 d^{4}} + \frac {2 \left (c + d x\right )^{\frac {3}{2}} \left (a^{3} d^{3} f - 3 a^{2} b c d^{2} f + 3 a^{2} b d^{3} e + 3 a b^{2} c^{2} d f - 3 a b^{2} c d^{2} e - b^{3} c^{3} f + b^{3} c^{2} d e\right )}{3 d^{4}} & \text {for}\: d \neq 0 \\\sqrt {c} \left (a^{3} e \log {\left (x \right )} + a^{3} f x + 3 a^{2} b e x + \frac {b^{3} f x^{4}}{4} + \frac {x^{3} \cdot \left (3 a b^{2} f + b^{3} e\right )}{3} + \frac {x^{2} \cdot \left (3 a^{2} b f + 3 a b^{2} e\right )}{2}\right ) & \text {otherwise} \end {cases} \] Input:

integrate((b*x+a)**3*(d*x+c)**(1/2)*(f*x+e)/x,x)
 

Output:

Piecewise((2*a**3*c*e*atan(sqrt(c + d*x)/sqrt(-c))/sqrt(-c) + 2*a**3*e*sqr 
t(c + d*x) + 2*b**3*f*(c + d*x)**(9/2)/(9*d**4) + 2*(c + d*x)**(7/2)*(3*a* 
b**2*d*f - 3*b**3*c*f + b**3*d*e)/(7*d**4) + 2*(c + d*x)**(5/2)*(3*a**2*b* 
d**2*f - 6*a*b**2*c*d*f + 3*a*b**2*d**2*e + 3*b**3*c**2*f - 2*b**3*c*d*e)/ 
(5*d**4) + 2*(c + d*x)**(3/2)*(a**3*d**3*f - 3*a**2*b*c*d**2*f + 3*a**2*b* 
d**3*e + 3*a*b**2*c**2*d*f - 3*a*b**2*c*d**2*e - b**3*c**3*f + b**3*c**2*d 
*e)/(3*d**4), Ne(d, 0)), (sqrt(c)*(a**3*e*log(x) + a**3*f*x + 3*a**2*b*e*x 
 + b**3*f*x**4/4 + x**3*(3*a*b**2*f + b**3*e)/3 + x**2*(3*a**2*b*f + 3*a*b 
**2*e)/2), True))
 

Maxima [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 239, normalized size of antiderivative = 1.06 \[ \int \frac {(a+b x)^3 \sqrt {c+d x} (e+f x)}{x} \, dx=a^{3} \sqrt {c} e \log \left (\frac {\sqrt {d x + c} - \sqrt {c}}{\sqrt {d x + c} + \sqrt {c}}\right ) + \frac {2 \, {\left (315 \, \sqrt {d x + c} a^{3} d^{4} e + 35 \, {\left (d x + c\right )}^{\frac {9}{2}} b^{3} f + 45 \, {\left (b^{3} d e - 3 \, {\left (b^{3} c - a b^{2} d\right )} f\right )} {\left (d x + c\right )}^{\frac {7}{2}} - 63 \, {\left ({\left (2 \, b^{3} c d - 3 \, a b^{2} d^{2}\right )} e - 3 \, {\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} f\right )} {\left (d x + c\right )}^{\frac {5}{2}} + 105 \, {\left ({\left (b^{3} c^{2} d - 3 \, a b^{2} c d^{2} + 3 \, a^{2} b d^{3}\right )} e - {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} f\right )} {\left (d x + c\right )}^{\frac {3}{2}}\right )}}{315 \, d^{4}} \] Input:

integrate((b*x+a)^3*(d*x+c)^(1/2)*(f*x+e)/x,x, algorithm="maxima")
 

Output:

a^3*sqrt(c)*e*log((sqrt(d*x + c) - sqrt(c))/(sqrt(d*x + c) + sqrt(c))) + 2 
/315*(315*sqrt(d*x + c)*a^3*d^4*e + 35*(d*x + c)^(9/2)*b^3*f + 45*(b^3*d*e 
 - 3*(b^3*c - a*b^2*d)*f)*(d*x + c)^(7/2) - 63*((2*b^3*c*d - 3*a*b^2*d^2)* 
e - 3*(b^3*c^2 - 2*a*b^2*c*d + a^2*b*d^2)*f)*(d*x + c)^(5/2) + 105*((b^3*c 
^2*d - 3*a*b^2*c*d^2 + 3*a^2*b*d^3)*e - (b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b 
*c*d^2 - a^3*d^3)*f)*(d*x + c)^(3/2))/d^4
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 330, normalized size of antiderivative = 1.47 \[ \int \frac {(a+b x)^3 \sqrt {c+d x} (e+f x)}{x} \, dx=\frac {2 \, a^{3} c e \arctan \left (\frac {\sqrt {d x + c}}{\sqrt {-c}}\right )}{\sqrt {-c}} + \frac {2 \, {\left (45 \, {\left (d x + c\right )}^{\frac {7}{2}} b^{3} d^{33} e - 126 \, {\left (d x + c\right )}^{\frac {5}{2}} b^{3} c d^{33} e + 105 \, {\left (d x + c\right )}^{\frac {3}{2}} b^{3} c^{2} d^{33} e + 189 \, {\left (d x + c\right )}^{\frac {5}{2}} a b^{2} d^{34} e - 315 \, {\left (d x + c\right )}^{\frac {3}{2}} a b^{2} c d^{34} e + 315 \, {\left (d x + c\right )}^{\frac {3}{2}} a^{2} b d^{35} e + 315 \, \sqrt {d x + c} a^{3} d^{36} e + 35 \, {\left (d x + c\right )}^{\frac {9}{2}} b^{3} d^{32} f - 135 \, {\left (d x + c\right )}^{\frac {7}{2}} b^{3} c d^{32} f + 189 \, {\left (d x + c\right )}^{\frac {5}{2}} b^{3} c^{2} d^{32} f - 105 \, {\left (d x + c\right )}^{\frac {3}{2}} b^{3} c^{3} d^{32} f + 135 \, {\left (d x + c\right )}^{\frac {7}{2}} a b^{2} d^{33} f - 378 \, {\left (d x + c\right )}^{\frac {5}{2}} a b^{2} c d^{33} f + 315 \, {\left (d x + c\right )}^{\frac {3}{2}} a b^{2} c^{2} d^{33} f + 189 \, {\left (d x + c\right )}^{\frac {5}{2}} a^{2} b d^{34} f - 315 \, {\left (d x + c\right )}^{\frac {3}{2}} a^{2} b c d^{34} f + 105 \, {\left (d x + c\right )}^{\frac {3}{2}} a^{3} d^{35} f\right )}}{315 \, d^{36}} \] Input:

integrate((b*x+a)^3*(d*x+c)^(1/2)*(f*x+e)/x,x, algorithm="giac")
 

Output:

2*a^3*c*e*arctan(sqrt(d*x + c)/sqrt(-c))/sqrt(-c) + 2/315*(45*(d*x + c)^(7 
/2)*b^3*d^33*e - 126*(d*x + c)^(5/2)*b^3*c*d^33*e + 105*(d*x + c)^(3/2)*b^ 
3*c^2*d^33*e + 189*(d*x + c)^(5/2)*a*b^2*d^34*e - 315*(d*x + c)^(3/2)*a*b^ 
2*c*d^34*e + 315*(d*x + c)^(3/2)*a^2*b*d^35*e + 315*sqrt(d*x + c)*a^3*d^36 
*e + 35*(d*x + c)^(9/2)*b^3*d^32*f - 135*(d*x + c)^(7/2)*b^3*c*d^32*f + 18 
9*(d*x + c)^(5/2)*b^3*c^2*d^32*f - 105*(d*x + c)^(3/2)*b^3*c^3*d^32*f + 13 
5*(d*x + c)^(7/2)*a*b^2*d^33*f - 378*(d*x + c)^(5/2)*a*b^2*c*d^33*f + 315* 
(d*x + c)^(3/2)*a*b^2*c^2*d^33*f + 189*(d*x + c)^(5/2)*a^2*b*d^34*f - 315* 
(d*x + c)^(3/2)*a^2*b*c*d^34*f + 105*(d*x + c)^(3/2)*a^3*d^35*f)/d^36
 

Mupad [B] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 413, normalized size of antiderivative = 1.84 \[ \int \frac {(a+b x)^3 \sqrt {c+d x} (e+f x)}{x} \, dx=\left (c\,\left (c\,\left (c\,\left (\frac {2\,b^3\,d\,e-8\,b^3\,c\,f+6\,a\,b^2\,d\,f}{d^4}+\frac {2\,b^3\,c\,f}{d^4}\right )+\frac {6\,b\,\left (a\,d-b\,c\right )\,\left (a\,d\,f-2\,b\,c\,f+b\,d\,e\right )}{d^4}\right )+\frac {2\,{\left (a\,d-b\,c\right )}^2\,\left (a\,d\,f-4\,b\,c\,f+3\,b\,d\,e\right )}{d^4}\right )-\frac {2\,{\left (a\,d-b\,c\right )}^3\,\left (c\,f-d\,e\right )}{d^4}\right )\,\sqrt {c+d\,x}+\left (\frac {c\,\left (c\,\left (\frac {2\,b^3\,d\,e-8\,b^3\,c\,f+6\,a\,b^2\,d\,f}{d^4}+\frac {2\,b^3\,c\,f}{d^4}\right )+\frac {6\,b\,\left (a\,d-b\,c\right )\,\left (a\,d\,f-2\,b\,c\,f+b\,d\,e\right )}{d^4}\right )}{3}+\frac {2\,{\left (a\,d-b\,c\right )}^2\,\left (a\,d\,f-4\,b\,c\,f+3\,b\,d\,e\right )}{3\,d^4}\right )\,{\left (c+d\,x\right )}^{3/2}+\left (\frac {2\,b^3\,d\,e-8\,b^3\,c\,f+6\,a\,b^2\,d\,f}{7\,d^4}+\frac {2\,b^3\,c\,f}{7\,d^4}\right )\,{\left (c+d\,x\right )}^{7/2}+\left (\frac {c\,\left (\frac {2\,b^3\,d\,e-8\,b^3\,c\,f+6\,a\,b^2\,d\,f}{d^4}+\frac {2\,b^3\,c\,f}{d^4}\right )}{5}+\frac {6\,b\,\left (a\,d-b\,c\right )\,\left (a\,d\,f-2\,b\,c\,f+b\,d\,e\right )}{5\,d^4}\right )\,{\left (c+d\,x\right )}^{5/2}+\frac {2\,b^3\,f\,{\left (c+d\,x\right )}^{9/2}}{9\,d^4}+a^3\,\sqrt {c}\,e\,\mathrm {atan}\left (\frac {\sqrt {c+d\,x}\,1{}\mathrm {i}}{\sqrt {c}}\right )\,2{}\mathrm {i} \] Input:

int(((e + f*x)*(a + b*x)^3*(c + d*x)^(1/2))/x,x)
 

Output:

(c*(c*(c*((2*b^3*d*e - 8*b^3*c*f + 6*a*b^2*d*f)/d^4 + (2*b^3*c*f)/d^4) + ( 
6*b*(a*d - b*c)*(a*d*f - 2*b*c*f + b*d*e))/d^4) + (2*(a*d - b*c)^2*(a*d*f 
- 4*b*c*f + 3*b*d*e))/d^4) - (2*(a*d - b*c)^3*(c*f - d*e))/d^4)*(c + d*x)^ 
(1/2) + ((c*(c*((2*b^3*d*e - 8*b^3*c*f + 6*a*b^2*d*f)/d^4 + (2*b^3*c*f)/d^ 
4) + (6*b*(a*d - b*c)*(a*d*f - 2*b*c*f + b*d*e))/d^4))/3 + (2*(a*d - b*c)^ 
2*(a*d*f - 4*b*c*f + 3*b*d*e))/(3*d^4))*(c + d*x)^(3/2) + ((2*b^3*d*e - 8* 
b^3*c*f + 6*a*b^2*d*f)/(7*d^4) + (2*b^3*c*f)/(7*d^4))*(c + d*x)^(7/2) + (( 
c*((2*b^3*d*e - 8*b^3*c*f + 6*a*b^2*d*f)/d^4 + (2*b^3*c*f)/d^4))/5 + (6*b* 
(a*d - b*c)*(a*d*f - 2*b*c*f + b*d*e))/(5*d^4))*(c + d*x)^(5/2) + a^3*c^(1 
/2)*e*atan(((c + d*x)^(1/2)*1i)/c^(1/2))*2i + (2*b^3*f*(c + d*x)^(9/2))/(9 
*d^4)
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 481, normalized size of antiderivative = 2.14 \[ \int \frac {(a+b x)^3 \sqrt {c+d x} (e+f x)}{x} \, dx=\frac {630 \sqrt {d x +c}\, a^{3} d^{4} e -32 \sqrt {d x +c}\, b^{3} c^{4} f +210 \sqrt {d x +c}\, a^{3} c \,d^{3} f +210 \sqrt {d x +c}\, a^{3} d^{4} f x +48 \sqrt {d x +c}\, b^{3} c^{3} d e +90 \sqrt {d x +c}\, b^{3} d^{4} e \,x^{3}+70 \sqrt {d x +c}\, b^{3} d^{4} f \,x^{4}+315 \sqrt {c}\, \mathrm {log}\left (\sqrt {d x +c}-\sqrt {c}\right ) a^{3} d^{4} e -315 \sqrt {c}\, \mathrm {log}\left (\sqrt {d x +c}+\sqrt {c}\right ) a^{3} d^{4} e -252 \sqrt {d x +c}\, a^{2} b \,c^{2} d^{2} f +630 \sqrt {d x +c}\, a^{2} b c \,d^{3} e +630 \sqrt {d x +c}\, a^{2} b \,d^{4} e x +378 \sqrt {d x +c}\, a^{2} b \,d^{4} f \,x^{2}+144 \sqrt {d x +c}\, a \,b^{2} c^{3} d f -252 \sqrt {d x +c}\, a \,b^{2} c^{2} d^{2} e +378 \sqrt {d x +c}\, a \,b^{2} d^{4} e \,x^{2}+270 \sqrt {d x +c}\, a \,b^{2} d^{4} f \,x^{3}+16 \sqrt {d x +c}\, b^{3} c^{3} d f x -24 \sqrt {d x +c}\, b^{3} c^{2} d^{2} e x -12 \sqrt {d x +c}\, b^{3} c^{2} d^{2} f \,x^{2}+18 \sqrt {d x +c}\, b^{3} c \,d^{3} e \,x^{2}+10 \sqrt {d x +c}\, b^{3} c \,d^{3} f \,x^{3}+126 \sqrt {d x +c}\, a^{2} b c \,d^{3} f x -72 \sqrt {d x +c}\, a \,b^{2} c^{2} d^{2} f x +126 \sqrt {d x +c}\, a \,b^{2} c \,d^{3} e x +54 \sqrt {d x +c}\, a \,b^{2} c \,d^{3} f \,x^{2}}{315 d^{4}} \] Input:

int((b*x+a)^3*(d*x+c)^(1/2)*(f*x+e)/x,x)
 

Output:

(210*sqrt(c + d*x)*a**3*c*d**3*f + 630*sqrt(c + d*x)*a**3*d**4*e + 210*sqr 
t(c + d*x)*a**3*d**4*f*x - 252*sqrt(c + d*x)*a**2*b*c**2*d**2*f + 630*sqrt 
(c + d*x)*a**2*b*c*d**3*e + 126*sqrt(c + d*x)*a**2*b*c*d**3*f*x + 630*sqrt 
(c + d*x)*a**2*b*d**4*e*x + 378*sqrt(c + d*x)*a**2*b*d**4*f*x**2 + 144*sqr 
t(c + d*x)*a*b**2*c**3*d*f - 252*sqrt(c + d*x)*a*b**2*c**2*d**2*e - 72*sqr 
t(c + d*x)*a*b**2*c**2*d**2*f*x + 126*sqrt(c + d*x)*a*b**2*c*d**3*e*x + 54 
*sqrt(c + d*x)*a*b**2*c*d**3*f*x**2 + 378*sqrt(c + d*x)*a*b**2*d**4*e*x**2 
 + 270*sqrt(c + d*x)*a*b**2*d**4*f*x**3 - 32*sqrt(c + d*x)*b**3*c**4*f + 4 
8*sqrt(c + d*x)*b**3*c**3*d*e + 16*sqrt(c + d*x)*b**3*c**3*d*f*x - 24*sqrt 
(c + d*x)*b**3*c**2*d**2*e*x - 12*sqrt(c + d*x)*b**3*c**2*d**2*f*x**2 + 18 
*sqrt(c + d*x)*b**3*c*d**3*e*x**2 + 10*sqrt(c + d*x)*b**3*c*d**3*f*x**3 + 
90*sqrt(c + d*x)*b**3*d**4*e*x**3 + 70*sqrt(c + d*x)*b**3*d**4*f*x**4 + 31 
5*sqrt(c)*log(sqrt(c + d*x) - sqrt(c))*a**3*d**4*e - 315*sqrt(c)*log(sqrt( 
c + d*x) + sqrt(c))*a**3*d**4*e)/(315*d**4)