\(\int \frac {(a+b x)^3 \sqrt {c+d x} (e+f x)}{x} \, dx\) [8]

Optimal result

Integrand size = 25, antiderivative size = 227 \[ \int \frac {(a+b x)^3 \sqrt {c+d x} (e+f x)}{x} \, dx=2 a^3 e \sqrt {c+d x}+\frac {2 (3 b d e-2 b c f+2 a d f) (a+b x)^2 (c+d x)^{3/2}}{21 d^2}+\frac {2 f (a+b x)^3 (c+d x)^{3/2}}{9 d}+\frac {2 (c+d x)^{3/2} \left (2 \left (20 a^3 d^3 f+3 a^2 b d^2 (45 d e-16 c f)-9 a b^2 c d (7 d e-4 c f)+4 b^3 c^2 (3 d e-2 c f)\right )+3 b d \left (21 a b d^2 e-4 (b c-a d) (3 b d e-2 b c f+2 a d f)\right ) x\right )}{315 d^4}-2 a^3 \sqrt {c} e \text {arctanh}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right ) \]

[Out]

2/21*(2*a*d*f-2*b*c*f+3*b*d*e)*(b*x+a)^2*(d*x+c)^(3/2)/d^2+2/9*f*(b*x+a)^3*(d*x+c)^(3/2)/d+2/315*(d*x+c)^(3/2)
*(40*a^3*d^3*f+6*a^2*b*d^2*(-16*c*f+45*d*e)-18*a*b^2*c*d*(-4*c*f+7*d*e)+8*b^3*c^2*(-2*c*f+3*d*e)+3*b*d*(21*a*b
*d^2*e-4*(-a*d+b*c)*(2*a*d*f-2*b*c*f+3*b*d*e))*x)/d^4-2*a^3*e*arctanh((d*x+c)^(1/2)/c^(1/2))*c^(1/2)+2*a^3*e*(
d*x+c)^(1/2)

Rubi [A] (verified)

Time = 0.18 (sec) , antiderivative size = 227, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {158, 152, 52, 65, 214} \[ \int \frac {(a+b x)^3 \sqrt {c+d x} (e+f x)}{x} \, dx=-2 a^3 \sqrt {c} e \text {arctanh}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )+2 a^3 e \sqrt {c+d x}+\frac {2 (c+d x)^{3/2} \left (2 \left (20 a^3 d^3 f+3 a^2 b d^2 (45 d e-16 c f)-9 a b^2 c d (7 d e-4 c f)+4 b^3 c^2 (3 d e-2 c f)\right )+3 b d x \left (21 a b d^2 e-4 (b c-a d) (2 a d f-2 b c f+3 b d e)\right )\right )}{315 d^4}+\frac {2 (a+b x)^2 (c+d x)^{3/2} (2 a d f-2 b c f+3 b d e)}{21 d^2}+\frac {2 f (a+b x)^3 (c+d x)^{3/2}}{9 d} \]

[In]

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

[Out]

2*a^3*e*Sqrt[c + d*x] + (2*(3*b*d*e - 2*b*c*f + 2*a*d*f)*(a + b*x)^2*(c + d*x)^(3/2))/(21*d^2) + (2*f*(a + b*x
)^3*(c + d*x)^(3/2))/(9*d) + (2*(c + d*x)^(3/2)*(2*(20*a^3*d^3*f + 3*a^2*b*d^2*(45*d*e - 16*c*f) - 9*a*b^2*c*d
*(7*d*e - 4*c*f) + 4*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))/(315*d^4) - 2*a^3*Sqrt[c]*e*ArcTanh[Sqrt[c + d*x]/Sqrt[c]]

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 152

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_))*((g_.) + (h_.)*(x_)), x_Symbol]
:> 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] + Dist[(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 158

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

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 f (a+b x)^3 (c+d x)^{3/2}}{9 d}+\frac {2 \int \frac {(a+b x)^2 \sqrt {c+d x} \left (\frac {9 a d e}{2}+\frac {3}{2} (3 b d e-2 b c f+2 a d f) x\right )}{x} \, dx}{9 d} \\ & = \frac {2 (3 b d e-2 b c f+2 a d f) (a+b x)^2 (c+d x)^{3/2}}{21 d^2}+\frac {2 f (a+b x)^3 (c+d x)^{3/2}}{9 d}+\frac {4 \int \frac {(a+b x) \sqrt {c+d x} \left (\frac {63}{4} a^2 d^2 e+\frac {3}{4} \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}{63 d^2} \\ & = \frac {2 (3 b d e-2 b c f+2 a d f) (a+b x)^2 (c+d x)^{3/2}}{21 d^2}+\frac {2 f (a+b x)^3 (c+d x)^{3/2}}{9 d}+\frac {2 (c+d x)^{3/2} \left (2 \left (20 a^3 d^3 f+3 a^2 b d^2 (45 d e-16 c f)-9 a b^2 c d (7 d e-4 c f)+4 b^3 c^2 (3 d e-2 c f)\right )+3 b d \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 )}{315 d^4}+\left (a^3 e\right ) \int \frac {\sqrt {c+d x}}{x} \, dx \\ & = 2 a^3 e \sqrt {c+d x}+\frac {2 (3 b d e-2 b c f+2 a d f) (a+b x)^2 (c+d x)^{3/2}}{21 d^2}+\frac {2 f (a+b x)^3 (c+d x)^{3/2}}{9 d}+\frac {2 (c+d x)^{3/2} \left (2 \left (20 a^3 d^3 f+3 a^2 b d^2 (45 d e-16 c f)-9 a b^2 c d (7 d e-4 c f)+4 b^3 c^2 (3 d e-2 c f)\right )+3 b d \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 )}{315 d^4}+\left (a^3 c e\right ) \int \frac {1}{x \sqrt {c+d x}} \, dx \\ & = 2 a^3 e \sqrt {c+d x}+\frac {2 (3 b d e-2 b c f+2 a d f) (a+b x)^2 (c+d x)^{3/2}}{21 d^2}+\frac {2 f (a+b x)^3 (c+d x)^{3/2}}{9 d}+\frac {2 (c+d x)^{3/2} \left (2 \left (20 a^3 d^3 f+3 a^2 b d^2 (45 d e-16 c f)-9 a b^2 c d (7 d e-4 c f)+4 b^3 c^2 (3 d e-2 c f)\right )+3 b d \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 )}{315 d^4}+\frac {\left (2 a^3 c e\right ) \text {Subst}\left (\int \frac {1}{-\frac {c}{d}+\frac {x^2}{d}} \, dx,x,\sqrt {c+d x}\right )}{d} \\ & = 2 a^3 e \sqrt {c+d x}+\frac {2 (3 b d e-2 b c f+2 a d f) (a+b x)^2 (c+d x)^{3/2}}{21 d^2}+\frac {2 f (a+b x)^3 (c+d x)^{3/2}}{9 d}+\frac {2 (c+d x)^{3/2} \left (2 \left (20 a^3 d^3 f+3 a^2 b d^2 (45 d e-16 c f)-9 a b^2 c d (7 d e-4 c f)+4 b^3 c^2 (3 d e-2 c f)\right )+3 b d \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 )}{315 d^4}-2 a^3 \sqrt {c} e \tanh ^{-1}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.29 (sec) , antiderivative size = 197, normalized size of antiderivative = 0.87 \[ \int \frac {(a+b x)^3 \sqrt {c+d x} (e+f x)}{x} \, dx=\frac {2 \sqrt {c+d x} \left (105 a^3 d^3 (3 d e+c f+d f x)+63 a^2 b d^2 (c+d x) (5 d e-2 c f+3 d f x)+9 a b^2 d (c+d x) \left (8 c^2 f+3 d^2 x (7 e+5 f x)-2 c d (7 e+6 f x)\right )-b^3 (c+d x) \left (16 c^3 f-24 c^2 d (e+f x)+6 c d^2 x (6 e+5 f x)-5 d^3 x^2 (9 e+7 f x)\right )\right )}{315 d^4}-2 a^3 \sqrt {c} e \text {arctanh}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right ) \]

[In]

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

[Out]

(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]]

Maple [A] (verified)

Time = 1.66 (sec) , antiderivative size = 214, normalized size of antiderivative = 0.94

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

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 649, normalized size of antiderivative = 2.86 \[ \int \frac {(a+b x)^3 \sqrt {c+d x} (e+f x)}{x} \, dx=\left [\frac {315 \, a^{3} \sqrt {c} d^{4} e \log \left (\frac {d x - 2 \, \sqrt {d x + c} \sqrt {c} + 2 \, c}{x}\right ) + 2 \, {\left (35 \, b^{3} d^{4} f x^{4} + 5 \, {\left (9 \, b^{3} d^{4} e + {\left (b^{3} c d^{3} + 27 \, a b^{2} d^{4}\right )} f\right )} x^{3} + 3 \, {\left (3 \, {\left (b^{3} c d^{3} + 21 \, a b^{2} d^{4}\right )} e - {\left (2 \, b^{3} c^{2} d^{2} - 9 \, a b^{2} c d^{3} - 63 \, a^{2} b d^{4}\right )} f\right )} x^{2} + 3 \, {\left (8 \, b^{3} c^{3} d - 42 \, a b^{2} c^{2} d^{2} + 105 \, a^{2} b c d^{3} + 105 \, a^{3} d^{4}\right )} e - {\left (16 \, b^{3} c^{4} - 72 \, a b^{2} c^{3} d + 126 \, a^{2} b c^{2} d^{2} - 105 \, a^{3} c d^{3}\right )} f - {\left (3 \, {\left (4 \, b^{3} c^{2} d^{2} - 21 \, a b^{2} c d^{3} - 105 \, a^{2} b d^{4}\right )} e - {\left (8 \, b^{3} c^{3} d - 36 \, a b^{2} c^{2} d^{2} + 63 \, a^{2} b c d^{3} + 105 \, a^{3} d^{4}\right )} f\right )} x\right )} \sqrt {d x + c}}{315 \, d^{4}}, \frac {2 \, {\left (315 \, a^{3} \sqrt {-c} d^{4} e \arctan \left (\frac {\sqrt {d x + c} \sqrt {-c}}{c}\right ) + {\left (35 \, b^{3} d^{4} f x^{4} + 5 \, {\left (9 \, b^{3} d^{4} e + {\left (b^{3} c d^{3} + 27 \, a b^{2} d^{4}\right )} f\right )} x^{3} + 3 \, {\left (3 \, {\left (b^{3} c d^{3} + 21 \, a b^{2} d^{4}\right )} e - {\left (2 \, b^{3} c^{2} d^{2} - 9 \, a b^{2} c d^{3} - 63 \, a^{2} b d^{4}\right )} f\right )} x^{2} + 3 \, {\left (8 \, b^{3} c^{3} d - 42 \, a b^{2} c^{2} d^{2} + 105 \, a^{2} b c d^{3} + 105 \, a^{3} d^{4}\right )} e - {\left (16 \, b^{3} c^{4} - 72 \, a b^{2} c^{3} d + 126 \, a^{2} b c^{2} d^{2} - 105 \, a^{3} c d^{3}\right )} f - {\left (3 \, {\left (4 \, b^{3} c^{2} d^{2} - 21 \, a b^{2} c d^{3} - 105 \, a^{2} b d^{4}\right )} e - {\left (8 \, b^{3} c^{3} d - 36 \, a b^{2} c^{2} d^{2} + 63 \, a^{2} b c d^{3} + 105 \, a^{3} d^{4}\right )} f\right )} x\right )} \sqrt {d x + c}\right )}}{315 \, d^{4}}\right ] \]

[In]

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

[Out]

[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 -
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 + 105*a^3*d^4)*e - (16*b^3*c^4 - 7
2*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(d*x + c)*sqrt(-c)/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 + 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]

Sympy [A] (verification not implemented)

Time = 12.08 (sec) , antiderivative size = 355, normalized size of antiderivative = 1.56 \[ \int \frac {(a+b x)^3 \sqrt {c+d x} (e+f x)}{x} \, dx=\begin {cases} \frac {2 a^{3} c e \operatorname {atan}{\left (\frac {\sqrt {c + d x}}{\sqrt {- c}} \right )}}{\sqrt {- c}} + 2 a^{3} e \sqrt {c + d x} + \frac {2 b^{3} f \left (c + d x\right )^{\frac {9}{2}}}{9 d^{4}} + \frac {2 \left (c + d x\right )^{\frac {7}{2}} \cdot \left (3 a b^{2} d f - 3 b^{3} c f + b^{3} d e\right )}{7 d^{4}} + \frac {2 \left (c + d x\right )^{\frac {5}{2}} \cdot \left (3 a^{2} b d^{2} f - 6 a b^{2} c d f + 3 a b^{2} d^{2} e + 3 b^{3} c^{2} f - 2 b^{3} c d e\right )}{5 d^{4}} + \frac {2 \left (c + d x\right )^{\frac {3}{2}} \left (a^{3} d^{3} f - 3 a^{2} b c d^{2} f + 3 a^{2} b d^{3} e + 3 a b^{2} c^{2} d f - 3 a b^{2} c d^{2} e - b^{3} c^{3} f + b^{3} c^{2} d e\right )}{3 d^{4}} & \text {for}\: d \neq 0 \\\sqrt {c} \left (a^{3} e \log {\left (x \right )} + a^{3} f x + 3 a^{2} b e x + \frac {b^{3} f x^{4}}{4} + \frac {x^{3} \cdot \left (3 a b^{2} f + b^{3} e\right )}{3} + \frac {x^{2} \cdot \left (3 a^{2} b f + 3 a b^{2} e\right )}{2}\right ) & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((2*a**3*c*e*atan(sqrt(c + d*x)/sqrt(-c))/sqrt(-c) + 2*a**3*e*sqrt(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)

none

Time = 0.28 (sec) , antiderivative size = 239, normalized size of antiderivative = 1.05 \[ \int \frac {(a+b x)^3 \sqrt {c+d x} (e+f x)}{x} \, dx=a^{3} \sqrt {c} e \log \left (\frac {\sqrt {d x + c} - \sqrt {c}}{\sqrt {d x + c} + \sqrt {c}}\right ) + \frac {2 \, {\left (315 \, \sqrt {d x + c} a^{3} d^{4} e + 35 \, {\left (d x + c\right )}^{\frac {9}{2}} b^{3} f + 45 \, {\left (b^{3} d e - 3 \, {\left (b^{3} c - a b^{2} d\right )} f\right )} {\left (d x + c\right )}^{\frac {7}{2}} - 63 \, {\left ({\left (2 \, b^{3} c d - 3 \, a b^{2} d^{2}\right )} e - 3 \, {\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} f\right )} {\left (d x + c\right )}^{\frac {5}{2}} + 105 \, {\left ({\left (b^{3} c^{2} d - 3 \, a b^{2} c d^{2} + 3 \, a^{2} b d^{3}\right )} e - {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} f\right )} {\left (d x + c\right )}^{\frac {3}{2}}\right )}}{315 \, d^{4}} \]

[In]

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

[Out]

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)

none

Time = 0.30 (sec) , antiderivative size = 330, normalized size of antiderivative = 1.45 \[ \int \frac {(a+b x)^3 \sqrt {c+d x} (e+f x)}{x} \, dx=\frac {2 \, a^{3} c e \arctan \left (\frac {\sqrt {d x + c}}{\sqrt {-c}}\right )}{\sqrt {-c}} + \frac {2 \, {\left (45 \, {\left (d x + c\right )}^{\frac {7}{2}} b^{3} d^{33} e - 126 \, {\left (d x + c\right )}^{\frac {5}{2}} b^{3} c d^{33} e + 105 \, {\left (d x + c\right )}^{\frac {3}{2}} b^{3} c^{2} d^{33} e + 189 \, {\left (d x + c\right )}^{\frac {5}{2}} a b^{2} d^{34} e - 315 \, {\left (d x + c\right )}^{\frac {3}{2}} a b^{2} c d^{34} e + 315 \, {\left (d x + c\right )}^{\frac {3}{2}} a^{2} b d^{35} e + 315 \, \sqrt {d x + c} a^{3} d^{36} e + 35 \, {\left (d x + c\right )}^{\frac {9}{2}} b^{3} d^{32} f - 135 \, {\left (d x + c\right )}^{\frac {7}{2}} b^{3} c d^{32} f + 189 \, {\left (d x + c\right )}^{\frac {5}{2}} b^{3} c^{2} d^{32} f - 105 \, {\left (d x + c\right )}^{\frac {3}{2}} b^{3} c^{3} d^{32} f + 135 \, {\left (d x + c\right )}^{\frac {7}{2}} a b^{2} d^{33} f - 378 \, {\left (d x + c\right )}^{\frac {5}{2}} a b^{2} c d^{33} f + 315 \, {\left (d x + c\right )}^{\frac {3}{2}} a b^{2} c^{2} d^{33} f + 189 \, {\left (d x + c\right )}^{\frac {5}{2}} a^{2} b d^{34} f - 315 \, {\left (d x + c\right )}^{\frac {3}{2}} a^{2} b c d^{34} f + 105 \, {\left (d x + c\right )}^{\frac {3}{2}} a^{3} d^{35} f\right )}}{315 \, d^{36}} \]

[In]

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

[Out]

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 + 189*(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
+ 135*(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^3
6

Mupad [B] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 413, normalized size of antiderivative = 1.82 \[ \int \frac {(a+b x)^3 \sqrt {c+d x} (e+f x)}{x} \, dx=\left (c\,\left (c\,\left (c\,\left (\frac {2\,b^3\,d\,e-8\,b^3\,c\,f+6\,a\,b^2\,d\,f}{d^4}+\frac {2\,b^3\,c\,f}{d^4}\right )+\frac {6\,b\,\left (a\,d-b\,c\right )\,\left (a\,d\,f-2\,b\,c\,f+b\,d\,e\right )}{d^4}\right )+\frac {2\,{\left (a\,d-b\,c\right )}^2\,\left (a\,d\,f-4\,b\,c\,f+3\,b\,d\,e\right )}{d^4}\right )-\frac {2\,{\left (a\,d-b\,c\right )}^3\,\left (c\,f-d\,e\right )}{d^4}\right )\,\sqrt {c+d\,x}+\left (\frac {c\,\left (c\,\left (\frac {2\,b^3\,d\,e-8\,b^3\,c\,f+6\,a\,b^2\,d\,f}{d^4}+\frac {2\,b^3\,c\,f}{d^4}\right )+\frac {6\,b\,\left (a\,d-b\,c\right )\,\left (a\,d\,f-2\,b\,c\,f+b\,d\,e\right )}{d^4}\right )}{3}+\frac {2\,{\left (a\,d-b\,c\right )}^2\,\left (a\,d\,f-4\,b\,c\,f+3\,b\,d\,e\right )}{3\,d^4}\right )\,{\left (c+d\,x\right )}^{3/2}+\left (\frac {2\,b^3\,d\,e-8\,b^3\,c\,f+6\,a\,b^2\,d\,f}{7\,d^4}+\frac {2\,b^3\,c\,f}{7\,d^4}\right )\,{\left (c+d\,x\right )}^{7/2}+\left (\frac {c\,\left (\frac {2\,b^3\,d\,e-8\,b^3\,c\,f+6\,a\,b^2\,d\,f}{d^4}+\frac {2\,b^3\,c\,f}{d^4}\right )}{5}+\frac {6\,b\,\left (a\,d-b\,c\right )\,\left (a\,d\,f-2\,b\,c\,f+b\,d\,e\right )}{5\,d^4}\right )\,{\left (c+d\,x\right )}^{5/2}+\frac {2\,b^3\,f\,{\left (c+d\,x\right )}^{9/2}}{9\,d^4}+a^3\,\sqrt {c}\,e\,\mathrm {atan}\left (\frac {\sqrt {c+d\,x}\,1{}\mathrm {i}}{\sqrt {c}}\right )\,2{}\mathrm {i} \]

[In]

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

[Out]

(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)