\(\int \frac {\sqrt {a+b x} (c+d x)^3 (e+f x)}{x} \, dx\) [26]

Optimal result

Integrand size = 25, antiderivative size = 223 \[ \int \frac {\sqrt {a+b x} (c+d x)^3 (e+f x)}{x} \, dx=2 c^3 e \sqrt {a+b x}-\frac {2 \left (a^3 d^3 f+3 a b^2 c d (d e+c f)-b^3 c^2 (3 d e+c f)-a^2 b d^2 (d e+3 c f)\right ) (a+b x)^{3/2}}{3 b^4}+\frac {2 d \left (3 a^2 d^2 f+3 b^2 c (d e+c f)-2 a b d (d e+3 c f)\right ) (a+b x)^{5/2}}{5 b^4}+\frac {2 d^2 (b d e+3 b c f-3 a d f) (a+b x)^{7/2}}{7 b^4}+\frac {2 d^3 f (a+b x)^{9/2}}{9 b^4}-2 \sqrt {a} c^3 e \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right ) \] Output:

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

Mathematica [A] (verified)

Time = 0.27 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.06 \[ \int \frac {\sqrt {a+b x} (c+d x)^3 (e+f x)}{x} \, dx=\frac {2 \sqrt {a+b x} \left (-16 a^4 d^3 f+8 a^3 b d^2 (3 d e+9 c f+d f x)-6 a^2 b^2 d \left (21 c^2 f+d^2 x (2 e+f x)+3 c d (7 e+2 f x)\right )+a b^3 \left (105 c^3 f+63 c^2 d (5 e+f x)+9 c d^2 x (7 e+3 f x)+d^3 x^2 (9 e+5 f x)\right )+b^4 \left (105 c^3 (3 e+f x)+63 c^2 d x (5 e+3 f x)+27 c d^2 x^2 (7 e+5 f x)+5 d^3 x^3 (9 e+7 f x)\right )\right )}{315 b^4}-2 \sqrt {a} c^3 e \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right ) \] Input:

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

Output:

(2*Sqrt[a + b*x]*(-16*a^4*d^3*f + 8*a^3*b*d^2*(3*d*e + 9*c*f + d*f*x) - 6* 
a^2*b^2*d*(21*c^2*f + d^2*x*(2*e + f*x) + 3*c*d*(7*e + 2*f*x)) + a*b^3*(10 
5*c^3*f + 63*c^2*d*(5*e + f*x) + 9*c*d^2*x*(7*e + 3*f*x) + d^3*x^2*(9*e + 
5*f*x)) + b^4*(105*c^3*(3*e + f*x) + 63*c^2*d*x*(5*e + 3*f*x) + 27*c*d^2*x 
^2*(7*e + 5*f*x) + 5*d^3*x^3*(9*e + 7*f*x))))/(315*b^4) - 2*Sqrt[a]*c^3*e* 
ArcTanh[Sqrt[a + b*x]/Sqrt[a]]
 

Rubi [A] (verified)

Time = 0.42 (sec) , antiderivative size = 241, 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.

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

\(\Big \downarrow \) 170

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 170

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 164

\(\displaystyle \frac {\frac {21 b^2 c^3 e \int \frac {\sqrt {a+b x}}{x}dx-\frac {2 (a+b x)^{3/2} \left (16 a^3 d^3 f-24 a^2 b d^2 (3 c f+d e)-3 b d x \left (4 (b c-a d) (-2 a d f+2 b c f+3 b d e)+21 b^2 c d e\right )+6 a b^2 c d (16 c f+21 d e)-10 b^3 c^2 (4 c f+27 d e)\right )}{15 b^2}}{7 b}+\frac {2 (a+b x)^{3/2} (c+d x)^2 (-2 a d f+2 b c f+3 b d e)}{7 b}}{3 b}+\frac {2 f (a+b x)^{3/2} (c+d x)^3}{9 b}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {\frac {21 b^2 c^3 e \left (a \int \frac {1}{x \sqrt {a+b x}}dx+2 \sqrt {a+b x}\right )-\frac {2 (a+b x)^{3/2} \left (16 a^3 d^3 f-24 a^2 b d^2 (3 c f+d e)-3 b d x \left (4 (b c-a d) (-2 a d f+2 b c f+3 b d e)+21 b^2 c d e\right )+6 a b^2 c d (16 c f+21 d e)-10 b^3 c^2 (4 c f+27 d e)\right )}{15 b^2}}{7 b}+\frac {2 (a+b x)^{3/2} (c+d x)^2 (-2 a d f+2 b c f+3 b d e)}{7 b}}{3 b}+\frac {2 f (a+b x)^{3/2} (c+d x)^3}{9 b}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {\frac {21 b^2 c^3 e \left (\frac {2 a \int \frac {1}{\frac {a+b x}{b}-\frac {a}{b}}d\sqrt {a+b x}}{b}+2 \sqrt {a+b x}\right )-\frac {2 (a+b x)^{3/2} \left (16 a^3 d^3 f-24 a^2 b d^2 (3 c f+d e)-3 b d x \left (4 (b c-a d) (-2 a d f+2 b c f+3 b d e)+21 b^2 c d e\right )+6 a b^2 c d (16 c f+21 d e)-10 b^3 c^2 (4 c f+27 d e)\right )}{15 b^2}}{7 b}+\frac {2 (a+b x)^{3/2} (c+d x)^2 (-2 a d f+2 b c f+3 b d e)}{7 b}}{3 b}+\frac {2 f (a+b x)^{3/2} (c+d x)^3}{9 b}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {\frac {21 b^2 c^3 e \left (2 \sqrt {a+b x}-2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )\right )-\frac {2 (a+b x)^{3/2} \left (16 a^3 d^3 f-24 a^2 b d^2 (3 c f+d e)-3 b d x \left (4 (b c-a d) (-2 a d f+2 b c f+3 b d e)+21 b^2 c d e\right )+6 a b^2 c d (16 c f+21 d e)-10 b^3 c^2 (4 c f+27 d e)\right )}{15 b^2}}{7 b}+\frac {2 (a+b x)^{3/2} (c+d x)^2 (-2 a d f+2 b c f+3 b d e)}{7 b}}{3 b}+\frac {2 f (a+b x)^{3/2} (c+d x)^3}{9 b}\)

Input:

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

Output:

(2*f*(a + b*x)^(3/2)*(c + d*x)^3)/(9*b) + ((2*(3*b*d*e + 2*b*c*f - 2*a*d*f 
)*(a + b*x)^(3/2)*(c + d*x)^2)/(7*b) + ((-2*(a + b*x)^(3/2)*(16*a^3*d^3*f 
- 24*a^2*b*d^2*(d*e + 3*c*f) - 10*b^3*c^2*(27*d*e + 4*c*f) + 6*a*b^2*c*d*( 
21*d*e + 16*c*f) - 3*b*d*(21*b^2*c*d*e + 4*(b*c - a*d)*(3*b*d*e + 2*b*c*f 
- 2*a*d*f))*x))/(15*b^2) + 21*b^2*c^3*e*(2*Sqrt[a + b*x] - 2*Sqrt[a]*ArcTa 
nh[Sqrt[a + b*x]/Sqrt[a]]))/(7*b))/(3*b)
 

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 = 0.34 (sec) , antiderivative size = 217, normalized size of antiderivative = 0.97

method	result	size
pseudoelliptic	\(\frac {-2 \sqrt {a}\, b^{4} c^{3} e \,\operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )-\frac {32 \left (\frac {9 \left (-5 \left (\frac {7 f x}{9}+e \right ) x^{3} d^{3}-21 c \left (\frac {5 f x}{7}+e \right ) x^{2} d^{2}-35 c^{2} x \left (\frac {3 f x}{5}+e \right ) d -35 c^{3} \left (\frac {f x}{3}+e \right )\right ) b^{4}}{16}-\frac {105 a \left (\frac {3 x^{2} \left (\frac {5 f x}{9}+e \right ) d^{3}}{35}+\frac {3 c \left (\frac {3 f x}{7}+e \right ) x \,d^{2}}{5}+3 c^{2} \left (\frac {f x}{5}+e \right ) d +c^{3} f \right ) b^{3}}{16}+\frac {63 a^{2} \left (\frac {2 \left (\frac {f x}{2}+e \right ) x \,d^{2}}{21}+c \left (\frac {2 f x}{7}+e \right ) d +c^{2} f \right ) d \,b^{2}}{8}-\frac {9 a^{3} d^{2} \left (\frac {\left (\frac {f x}{3}+e \right ) d}{3}+c f \right ) b}{2}+a^{4} d^{3} f \right ) \sqrt {b x +a}}{315}}{b^{4}}\)	\(217\)
derivativedivides	\(\frac {\frac {2 d^{3} f \left (b x +a \right )^{\frac {9}{2}}}{9}-\frac {6 a \,d^{3} f \left (b x +a \right )^{\frac {7}{2}}}{7}+\frac {6 b c \,d^{2} f \left (b x +a \right )^{\frac {7}{2}}}{7}+\frac {2 b \,d^{3} e \left (b x +a \right )^{\frac {7}{2}}}{7}+\frac {6 a^{2} d^{3} f \left (b x +a \right )^{\frac {5}{2}}}{5}-\frac {12 a b c \,d^{2} f \left (b x +a \right )^{\frac {5}{2}}}{5}-\frac {4 a b \,d^{3} e \left (b x +a \right )^{\frac {5}{2}}}{5}+\frac {6 b^{2} c^{2} d f \left (b x +a \right )^{\frac {5}{2}}}{5}+\frac {6 b^{2} c \,d^{2} e \left (b x +a \right )^{\frac {5}{2}}}{5}-\frac {2 a^{3} d^{3} f \left (b x +a \right )^{\frac {3}{2}}}{3}+2 a^{2} b c \,d^{2} f \left (b x +a \right )^{\frac {3}{2}}+\frac {2 a^{2} b \,d^{3} e \left (b x +a \right )^{\frac {3}{2}}}{3}-2 a \,b^{2} c^{2} d f \left (b x +a \right )^{\frac {3}{2}}-2 a \,b^{2} c \,d^{2} e \left (b x +a \right )^{\frac {3}{2}}+\frac {2 b^{3} c^{3} f \left (b x +a \right )^{\frac {3}{2}}}{3}+2 b^{3} c^{2} d e \left (b x +a \right )^{\frac {3}{2}}+2 b^{4} c^{3} e \sqrt {b x +a}-2 \sqrt {a}\, b^{4} c^{3} e \,\operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{b^{4}}\)	\(301\)
default	\(\frac {\frac {2 d^{3} f \left (b x +a \right )^{\frac {9}{2}}}{9}-\frac {6 a \,d^{3} f \left (b x +a \right )^{\frac {7}{2}}}{7}+\frac {6 b c \,d^{2} f \left (b x +a \right )^{\frac {7}{2}}}{7}+\frac {2 b \,d^{3} e \left (b x +a \right )^{\frac {7}{2}}}{7}+\frac {6 a^{2} d^{3} f \left (b x +a \right )^{\frac {5}{2}}}{5}-\frac {12 a b c \,d^{2} f \left (b x +a \right )^{\frac {5}{2}}}{5}-\frac {4 a b \,d^{3} e \left (b x +a \right )^{\frac {5}{2}}}{5}+\frac {6 b^{2} c^{2} d f \left (b x +a \right )^{\frac {5}{2}}}{5}+\frac {6 b^{2} c \,d^{2} e \left (b x +a \right )^{\frac {5}{2}}}{5}-\frac {2 a^{3} d^{3} f \left (b x +a \right )^{\frac {3}{2}}}{3}+2 a^{2} b c \,d^{2} f \left (b x +a \right )^{\frac {3}{2}}+\frac {2 a^{2} b \,d^{3} e \left (b x +a \right )^{\frac {3}{2}}}{3}-2 a \,b^{2} c^{2} d f \left (b x +a \right )^{\frac {3}{2}}-2 a \,b^{2} c \,d^{2} e \left (b x +a \right )^{\frac {3}{2}}+\frac {2 b^{3} c^{3} f \left (b x +a \right )^{\frac {3}{2}}}{3}+2 b^{3} c^{2} d e \left (b x +a \right )^{\frac {3}{2}}+2 b^{4} c^{3} e \sqrt {b x +a}-2 \sqrt {a}\, b^{4} c^{3} e \,\operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{b^{4}}\)	\(301\)

Input:

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

Output:

2/315*(-315*a^(1/2)*b^4*c^3*e*arctanh((b*x+a)^(1/2)/a^(1/2))-16*(9/16*(-5* 
(7/9*f*x+e)*x^3*d^3-21*c*(5/7*f*x+e)*x^2*d^2-35*c^2*x*(3/5*f*x+e)*d-35*c^3 
*(1/3*f*x+e))*b^4-105/16*a*(3/35*x^2*(5/9*f*x+e)*d^3+3/5*c*(3/7*f*x+e)*x*d 
^2+3*c^2*(1/5*f*x+e)*d+c^3*f)*b^3+63/8*a^2*(2/21*(1/2*f*x+e)*x*d^2+c*(2/7* 
f*x+e)*d+c^2*f)*d*b^2-9/2*a^3*d^2*(1/3*(1/3*f*x+e)*d+c*f)*b+a^4*d^3*f)*(b* 
x+a)^(1/2))/b^4
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 638, normalized size of antiderivative = 2.86 \[ \int \frac {\sqrt {a+b x} (c+d x)^3 (e+f x)}{x} \, dx=\left [\frac {315 \, \sqrt {a} b^{4} c^{3} e \log \left (\frac {b x - 2 \, \sqrt {b x + a} \sqrt {a} + 2 \, a}{x}\right ) + 2 \, {\left (35 \, b^{4} d^{3} f x^{4} + 5 \, {\left (9 \, b^{4} d^{3} e + {\left (27 \, b^{4} c d^{2} + a b^{3} d^{3}\right )} f\right )} x^{3} + 3 \, {\left (3 \, {\left (21 \, b^{4} c d^{2} + a b^{3} d^{3}\right )} e + {\left (63 \, b^{4} c^{2} d + 9 \, a b^{3} c d^{2} - 2 \, a^{2} b^{2} d^{3}\right )} f\right )} x^{2} + 3 \, {\left (105 \, b^{4} c^{3} + 105 \, a b^{3} c^{2} d - 42 \, a^{2} b^{2} c d^{2} + 8 \, a^{3} b d^{3}\right )} e + {\left (105 \, a b^{3} c^{3} - 126 \, a^{2} b^{2} c^{2} d + 72 \, a^{3} b c d^{2} - 16 \, a^{4} d^{3}\right )} f + {\left (3 \, {\left (105 \, b^{4} c^{2} d + 21 \, a b^{3} c d^{2} - 4 \, a^{2} b^{2} d^{3}\right )} e + {\left (105 \, b^{4} c^{3} + 63 \, a b^{3} c^{2} d - 36 \, a^{2} b^{2} c d^{2} + 8 \, a^{3} b d^{3}\right )} f\right )} x\right )} \sqrt {b x + a}}{315 \, b^{4}}, \frac {2 \, {\left (315 \, \sqrt {-a} b^{4} c^{3} e \arctan \left (\frac {\sqrt {-a}}{\sqrt {b x + a}}\right ) + {\left (35 \, b^{4} d^{3} f x^{4} + 5 \, {\left (9 \, b^{4} d^{3} e + {\left (27 \, b^{4} c d^{2} + a b^{3} d^{3}\right )} f\right )} x^{3} + 3 \, {\left (3 \, {\left (21 \, b^{4} c d^{2} + a b^{3} d^{3}\right )} e + {\left (63 \, b^{4} c^{2} d + 9 \, a b^{3} c d^{2} - 2 \, a^{2} b^{2} d^{3}\right )} f\right )} x^{2} + 3 \, {\left (105 \, b^{4} c^{3} + 105 \, a b^{3} c^{2} d - 42 \, a^{2} b^{2} c d^{2} + 8 \, a^{3} b d^{3}\right )} e + {\left (105 \, a b^{3} c^{3} - 126 \, a^{2} b^{2} c^{2} d + 72 \, a^{3} b c d^{2} - 16 \, a^{4} d^{3}\right )} f + {\left (3 \, {\left (105 \, b^{4} c^{2} d + 21 \, a b^{3} c d^{2} - 4 \, a^{2} b^{2} d^{3}\right )} e + {\left (105 \, b^{4} c^{3} + 63 \, a b^{3} c^{2} d - 36 \, a^{2} b^{2} c d^{2} + 8 \, a^{3} b d^{3}\right )} f\right )} x\right )} \sqrt {b x + a}\right )}}{315 \, b^{4}}\right ] \] Input:

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

Output:

[1/315*(315*sqrt(a)*b^4*c^3*e*log((b*x - 2*sqrt(b*x + a)*sqrt(a) + 2*a)/x) 
 + 2*(35*b^4*d^3*f*x^4 + 5*(9*b^4*d^3*e + (27*b^4*c*d^2 + a*b^3*d^3)*f)*x^ 
3 + 3*(3*(21*b^4*c*d^2 + a*b^3*d^3)*e + (63*b^4*c^2*d + 9*a*b^3*c*d^2 - 2* 
a^2*b^2*d^3)*f)*x^2 + 3*(105*b^4*c^3 + 105*a*b^3*c^2*d - 42*a^2*b^2*c*d^2 
+ 8*a^3*b*d^3)*e + (105*a*b^3*c^3 - 126*a^2*b^2*c^2*d + 72*a^3*b*c*d^2 - 1 
6*a^4*d^3)*f + (3*(105*b^4*c^2*d + 21*a*b^3*c*d^2 - 4*a^2*b^2*d^3)*e + (10 
5*b^4*c^3 + 63*a*b^3*c^2*d - 36*a^2*b^2*c*d^2 + 8*a^3*b*d^3)*f)*x)*sqrt(b* 
x + a))/b^4, 2/315*(315*sqrt(-a)*b^4*c^3*e*arctan(sqrt(-a)/sqrt(b*x + a)) 
+ (35*b^4*d^3*f*x^4 + 5*(9*b^4*d^3*e + (27*b^4*c*d^2 + a*b^3*d^3)*f)*x^3 + 
 3*(3*(21*b^4*c*d^2 + a*b^3*d^3)*e + (63*b^4*c^2*d + 9*a*b^3*c*d^2 - 2*a^2 
*b^2*d^3)*f)*x^2 + 3*(105*b^4*c^3 + 105*a*b^3*c^2*d - 42*a^2*b^2*c*d^2 + 8 
*a^3*b*d^3)*e + (105*a*b^3*c^3 - 126*a^2*b^2*c^2*d + 72*a^3*b*c*d^2 - 16*a 
^4*d^3)*f + (3*(105*b^4*c^2*d + 21*a*b^3*c*d^2 - 4*a^2*b^2*d^3)*e + (105*b 
^4*c^3 + 63*a*b^3*c^2*d - 36*a^2*b^2*c*d^2 + 8*a^3*b*d^3)*f)*x)*sqrt(b*x + 
 a))/b^4]
 

Sympy [A] (verification not implemented)

Time = 11.21 (sec) , antiderivative size = 355, normalized size of antiderivative = 1.59 \[ \int \frac {\sqrt {a+b x} (c+d x)^3 (e+f x)}{x} \, dx=\begin {cases} \frac {2 a c^{3} e \operatorname {atan}{\left (\frac {\sqrt {a + b x}}{\sqrt {- a}} \right )}}{\sqrt {- a}} + 2 c^{3} e \sqrt {a + b x} + \frac {2 d^{3} f \left (a + b x\right )^{\frac {9}{2}}}{9 b^{4}} + \frac {2 \left (a + b x\right )^{\frac {7}{2}} \left (- 3 a d^{3} f + 3 b c d^{2} f + b d^{3} e\right )}{7 b^{4}} + \frac {2 \left (a + b x\right )^{\frac {5}{2}} \cdot \left (3 a^{2} d^{3} f - 6 a b c d^{2} f - 2 a b d^{3} e + 3 b^{2} c^{2} d f + 3 b^{2} c d^{2} e\right )}{5 b^{4}} + \frac {2 \left (a + b x\right )^{\frac {3}{2}} \left (- a^{3} d^{3} f + 3 a^{2} b c d^{2} f + a^{2} b d^{3} e - 3 a b^{2} c^{2} d f - 3 a b^{2} c d^{2} e + b^{3} c^{3} f + 3 b^{3} c^{2} d e\right )}{3 b^{4}} & \text {for}\: b \neq 0 \\\sqrt {a} \left (c^{3} e \log {\left (x \right )} + c^{3} f x + 3 c^{2} d e x + \frac {d^{3} f x^{4}}{4} + \frac {x^{3} \cdot \left (3 c d^{2} f + d^{3} e\right )}{3} + \frac {x^{2} \cdot \left (3 c^{2} d f + 3 c d^{2} e\right )}{2}\right ) & \text {otherwise} \end {cases} \] Input:

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.07 \[ \int \frac {\sqrt {a+b x} (c+d x)^3 (e+f x)}{x} \, dx=\sqrt {a} c^{3} e \log \left (\frac {\sqrt {b x + a} - \sqrt {a}}{\sqrt {b x + a} + \sqrt {a}}\right ) + \frac {2 \, {\left (315 \, \sqrt {b x + a} b^{4} c^{3} e + 35 \, {\left (b x + a\right )}^{\frac {9}{2}} d^{3} f + 45 \, {\left (b d^{3} e + 3 \, {\left (b c d^{2} - a d^{3}\right )} f\right )} {\left (b x + a\right )}^{\frac {7}{2}} + 63 \, {\left ({\left (3 \, b^{2} c d^{2} - 2 \, a b d^{3}\right )} e + 3 \, {\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} f\right )} {\left (b x + a\right )}^{\frac {5}{2}} + 105 \, {\left ({\left (3 \, b^{3} c^{2} d - 3 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} e + {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} f\right )} {\left (b x + a\right )}^{\frac {3}{2}}\right )}}{315 \, b^{4}} \] Input:

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

Output:

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

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 330, normalized size of antiderivative = 1.48 \[ \int \frac {\sqrt {a+b x} (c+d x)^3 (e+f x)}{x} \, dx=\frac {2 \, a c^{3} e \arctan \left (\frac {\sqrt {b x + a}}{\sqrt {-a}}\right )}{\sqrt {-a}} + \frac {2 \, {\left (315 \, \sqrt {b x + a} b^{36} c^{3} e + 315 \, {\left (b x + a\right )}^{\frac {3}{2}} b^{35} c^{2} d e + 189 \, {\left (b x + a\right )}^{\frac {5}{2}} b^{34} c d^{2} e - 315 \, {\left (b x + a\right )}^{\frac {3}{2}} a b^{34} c d^{2} e + 45 \, {\left (b x + a\right )}^{\frac {7}{2}} b^{33} d^{3} e - 126 \, {\left (b x + a\right )}^{\frac {5}{2}} a b^{33} d^{3} e + 105 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{2} b^{33} d^{3} e + 105 \, {\left (b x + a\right )}^{\frac {3}{2}} b^{35} c^{3} f + 189 \, {\left (b x + a\right )}^{\frac {5}{2}} b^{34} c^{2} d f - 315 \, {\left (b x + a\right )}^{\frac {3}{2}} a b^{34} c^{2} d f + 135 \, {\left (b x + a\right )}^{\frac {7}{2}} b^{33} c d^{2} f - 378 \, {\left (b x + a\right )}^{\frac {5}{2}} a b^{33} c d^{2} f + 315 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{2} b^{33} c d^{2} f + 35 \, {\left (b x + a\right )}^{\frac {9}{2}} b^{32} d^{3} f - 135 \, {\left (b x + a\right )}^{\frac {7}{2}} a b^{32} d^{3} f + 189 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{2} b^{32} d^{3} f - 105 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{3} b^{32} d^{3} f\right )}}{315 \, b^{36}} \] Input:

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

Output:

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

Mupad [B] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 413, normalized size of antiderivative = 1.85 \[ \int \frac {\sqrt {a+b x} (c+d x)^3 (e+f x)}{x} \, dx=\left (\frac {2\,b\,d^3\,e-8\,a\,d^3\,f+6\,b\,c\,d^2\,f}{7\,b^4}+\frac {2\,a\,d^3\,f}{7\,b^4}\right )\,{\left (a+b\,x\right )}^{7/2}+\left (\frac {a\,\left (\frac {2\,b\,d^3\,e-8\,a\,d^3\,f+6\,b\,c\,d^2\,f}{b^4}+\frac {2\,a\,d^3\,f}{b^4}\right )}{5}-\frac {6\,d\,\left (a\,d-b\,c\right )\,\left (b\,c\,f-2\,a\,d\,f+b\,d\,e\right )}{5\,b^4}\right )\,{\left (a+b\,x\right )}^{5/2}+\left (a\,\left (a\,\left (a\,\left (\frac {2\,b\,d^3\,e-8\,a\,d^3\,f+6\,b\,c\,d^2\,f}{b^4}+\frac {2\,a\,d^3\,f}{b^4}\right )-\frac {6\,d\,\left (a\,d-b\,c\right )\,\left (b\,c\,f-2\,a\,d\,f+b\,d\,e\right )}{b^4}\right )+\frac {2\,{\left (a\,d-b\,c\right )}^2\,\left (b\,c\,f-4\,a\,d\,f+3\,b\,d\,e\right )}{b^4}\right )+\frac {2\,{\left (a\,d-b\,c\right )}^3\,\left (a\,f-b\,e\right )}{b^4}\right )\,\sqrt {a+b\,x}+\left (\frac {a\,\left (a\,\left (\frac {2\,b\,d^3\,e-8\,a\,d^3\,f+6\,b\,c\,d^2\,f}{b^4}+\frac {2\,a\,d^3\,f}{b^4}\right )-\frac {6\,d\,\left (a\,d-b\,c\right )\,\left (b\,c\,f-2\,a\,d\,f+b\,d\,e\right )}{b^4}\right )}{3}+\frac {2\,{\left (a\,d-b\,c\right )}^2\,\left (b\,c\,f-4\,a\,d\,f+3\,b\,d\,e\right )}{3\,b^4}\right )\,{\left (a+b\,x\right )}^{3/2}+\frac {2\,d^3\,f\,{\left (a+b\,x\right )}^{9/2}}{9\,b^4}+\sqrt {a}\,c^3\,e\,\mathrm {atan}\left (\frac {\sqrt {a+b\,x}\,1{}\mathrm {i}}{\sqrt {a}}\right )\,2{}\mathrm {i} \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 481, normalized size of antiderivative = 2.16 \[ \int \frac {\sqrt {a+b x} (c+d x)^3 (e+f x)}{x} \, dx=\frac {90 \sqrt {b x +a}\, b^{4} d^{3} e \,x^{3}+70 \sqrt {b x +a}\, b^{4} d^{3} f \,x^{4}+315 \sqrt {a}\, \mathrm {log}\left (\sqrt {b x +a}-\sqrt {a}\right ) b^{4} c^{3} e -315 \sqrt {a}\, \mathrm {log}\left (\sqrt {b x +a}+\sqrt {a}\right ) b^{4} c^{3} e +48 \sqrt {b x +a}\, a^{3} b \,d^{3} e +210 \sqrt {b x +a}\, a \,b^{3} c^{3} f +210 \sqrt {b x +a}\, b^{4} c^{3} f x +144 \sqrt {b x +a}\, a^{3} b c \,d^{2} f +16 \sqrt {b x +a}\, a^{3} b \,d^{3} f x -252 \sqrt {b x +a}\, a^{2} b^{2} c^{2} d f -252 \sqrt {b x +a}\, a^{2} b^{2} c \,d^{2} e -24 \sqrt {b x +a}\, a^{2} b^{2} d^{3} e x -12 \sqrt {b x +a}\, a^{2} b^{2} d^{3} f \,x^{2}+630 \sqrt {b x +a}\, a \,b^{3} c^{2} d e +18 \sqrt {b x +a}\, a \,b^{3} d^{3} e \,x^{2}+10 \sqrt {b x +a}\, a \,b^{3} d^{3} f \,x^{3}+630 \sqrt {b x +a}\, b^{4} c^{2} d e x +378 \sqrt {b x +a}\, b^{4} c^{2} d f \,x^{2}+378 \sqrt {b x +a}\, b^{4} c \,d^{2} e \,x^{2}+270 \sqrt {b x +a}\, b^{4} c \,d^{2} f \,x^{3}-32 \sqrt {b x +a}\, a^{4} d^{3} f +630 \sqrt {b x +a}\, b^{4} c^{3} e -72 \sqrt {b x +a}\, a^{2} b^{2} c \,d^{2} f x +126 \sqrt {b x +a}\, a \,b^{3} c^{2} d f x +126 \sqrt {b x +a}\, a \,b^{3} c \,d^{2} e x +54 \sqrt {b x +a}\, a \,b^{3} c \,d^{2} f \,x^{2}}{315 b^{4}} \] Input:

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

Output:

( - 32*sqrt(a + b*x)*a**4*d**3*f + 144*sqrt(a + b*x)*a**3*b*c*d**2*f + 48* 
sqrt(a + b*x)*a**3*b*d**3*e + 16*sqrt(a + b*x)*a**3*b*d**3*f*x - 252*sqrt( 
a + b*x)*a**2*b**2*c**2*d*f - 252*sqrt(a + b*x)*a**2*b**2*c*d**2*e - 72*sq 
rt(a + b*x)*a**2*b**2*c*d**2*f*x - 24*sqrt(a + b*x)*a**2*b**2*d**3*e*x - 1 
2*sqrt(a + b*x)*a**2*b**2*d**3*f*x**2 + 210*sqrt(a + b*x)*a*b**3*c**3*f + 
630*sqrt(a + b*x)*a*b**3*c**2*d*e + 126*sqrt(a + b*x)*a*b**3*c**2*d*f*x + 
126*sqrt(a + b*x)*a*b**3*c*d**2*e*x + 54*sqrt(a + b*x)*a*b**3*c*d**2*f*x** 
2 + 18*sqrt(a + b*x)*a*b**3*d**3*e*x**2 + 10*sqrt(a + b*x)*a*b**3*d**3*f*x 
**3 + 630*sqrt(a + b*x)*b**4*c**3*e + 210*sqrt(a + b*x)*b**4*c**3*f*x + 63 
0*sqrt(a + b*x)*b**4*c**2*d*e*x + 378*sqrt(a + b*x)*b**4*c**2*d*f*x**2 + 3 
78*sqrt(a + b*x)*b**4*c*d**2*e*x**2 + 270*sqrt(a + b*x)*b**4*c*d**2*f*x**3 
 + 90*sqrt(a + b*x)*b**4*d**3*e*x**3 + 70*sqrt(a + b*x)*b**4*d**3*f*x**4 + 
 315*sqrt(a)*log(sqrt(a + b*x) - sqrt(a))*b**4*c**3*e - 315*sqrt(a)*log(sq 
rt(a + b*x) + sqrt(a))*b**4*c**3*e)/(315*b**4)