\(\int \frac {\sqrt {a+b x} (e+f x)}{x (c+d x)^3} \, dx\) [21]

Optimal result

Integrand size = 25, antiderivative size = 205 \[ \int \frac {\sqrt {a+b x} (e+f x)}{x (c+d x)^3} \, dx=\frac {(d e-c f) \sqrt {a+b x}}{2 c d (c+d x)^2}-\frac {\left (4 a d^2 e-b c (3 d e+c f)\right ) \sqrt {a+b x}}{4 c^2 d (b c-a d) (c+d x)}-\frac {\left (12 a b c d^2 e-8 a^2 d^3 e-b^2 c^2 (3 d e+c f)\right ) \arctan \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b c-a d}}\right )}{4 c^3 d^{3/2} (b c-a d)^{3/2}}-\frac {2 \sqrt {a} e \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{c^3} \]

[Out]

-1/4*(12*a*b*c*d^2*e-8*a^2*d^3*e-b^2*c^2*(c*f+3*d*e))*arctan(d^(1/2)*(b*x+a)^(1/2)/(-a*d+b*c)^(1/2))/c^3/d^(3/
2)/(-a*d+b*c)^(3/2)-2*e*arctanh((b*x+a)^(1/2)/a^(1/2))*a^(1/2)/c^3+1/2*(-c*f+d*e)*(b*x+a)^(1/2)/c/d/(d*x+c)^2-
1/4*(4*a*d^2*e-b*c*(c*f+3*d*e))*(b*x+a)^(1/2)/c^2/d/(-a*d+b*c)/(d*x+c)

Rubi [A] (verified)

Time = 0.19 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {154, 156, 162, 65, 214, 211} \[ \int \frac {\sqrt {a+b x} (e+f x)}{x (c+d x)^3} \, dx=-\frac {\arctan \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b c-a d}}\right ) \left (-8 a^2 d^3 e+12 a b c d^2 e-b^2 c^2 (c f+3 d e)\right )}{4 c^3 d^{3/2} (b c-a d)^{3/2}}-\frac {2 \sqrt {a} e \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{c^3}-\frac {\sqrt {a+b x} \left (4 a d^2 e-b c (c f+3 d e)\right )}{4 c^2 d (c+d x) (b c-a d)}+\frac {\sqrt {a+b x} (d e-c f)}{2 c d (c+d x)^2} \]

[In]

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

[Out]

((d*e - c*f)*Sqrt[a + b*x])/(2*c*d*(c + d*x)^2) - ((4*a*d^2*e - b*c*(3*d*e + c*f))*Sqrt[a + b*x])/(4*c^2*d*(b*
c - a*d)*(c + d*x)) - ((12*a*b*c*d^2*e - 8*a^2*d^3*e - b^2*c^2*(3*d*e + c*f))*ArcTan[(Sqrt[d]*Sqrt[a + b*x])/S
qrt[b*c - a*d]])/(4*c^3*d^(3/2)*(b*c - a*d)^(3/2)) - (2*Sqrt[a]*e*ArcTanh[Sqrt[a + b*x]/Sqrt[a]])/c^3

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 154

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

Rule 156

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

Rule 162

Int[(((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :>
Dist[(b*g - a*h)/(b*c - a*d), Int[(e + f*x)^p/(a + b*x), x], x] - Dist[(d*g - c*h)/(b*c - a*d), Int[(e + f*x)
^p/(c + d*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x]

Rule 211

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/Rt[a/b, 2]], x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

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 {(d e-c f) \sqrt {a+b x}}{2 c d (c+d x)^2}-\frac {\int \frac {-2 a d e-\frac {1}{2} b (3 d e+c f) x}{x \sqrt {a+b x} (c+d x)^2} \, dx}{2 c d} \\ & = \frac {(d e-c f) \sqrt {a+b x}}{2 c d (c+d x)^2}-\frac {\left (4 a d^2 e-b c (3 d e+c f)\right ) \sqrt {a+b x}}{4 c^2 d (b c-a d) (c+d x)}+\frac {\int \frac {2 a d (b c-a d) e-\frac {1}{4} b \left (4 a d^2 e-b c (3 d e+c f)\right ) x}{x \sqrt {a+b x} (c+d x)} \, dx}{2 c^2 d (b c-a d)} \\ & = \frac {(d e-c f) \sqrt {a+b x}}{2 c d (c+d x)^2}-\frac {\left (4 a d^2 e-b c (3 d e+c f)\right ) \sqrt {a+b x}}{4 c^2 d (b c-a d) (c+d x)}+\frac {(a e) \int \frac {1}{x \sqrt {a+b x}} \, dx}{c^3}-\frac {\left (12 a b c d^2 e-8 a^2 d^3 e-b^2 c^2 (3 d e+c f)\right ) \int \frac {1}{\sqrt {a+b x} (c+d x)} \, dx}{8 c^3 d (b c-a d)} \\ & = \frac {(d e-c f) \sqrt {a+b x}}{2 c d (c+d x)^2}-\frac {\left (4 a d^2 e-b c (3 d e+c f)\right ) \sqrt {a+b x}}{4 c^2 d (b c-a d) (c+d x)}+\frac {(2 a e) \text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x}\right )}{b c^3}-\frac {\left (12 a b c d^2 e-8 a^2 d^3 e-b^2 c^2 (3 d e+c f)\right ) \text {Subst}\left (\int \frac {1}{c-\frac {a d}{b}+\frac {d x^2}{b}} \, dx,x,\sqrt {a+b x}\right )}{4 b c^3 d (b c-a d)} \\ & = \frac {(d e-c f) \sqrt {a+b x}}{2 c d (c+d x)^2}-\frac {\left (4 a d^2 e-b c (3 d e+c f)\right ) \sqrt {a+b x}}{4 c^2 d (b c-a d) (c+d x)}-\frac {\left (12 a b c d^2 e-8 a^2 d^3 e-b^2 c^2 (3 d e+c f)\right ) \tan ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b c-a d}}\right )}{4 c^3 d^{3/2} (b c-a d)^{3/2}}-\frac {2 \sqrt {a} e \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{c^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.91 (sec) , antiderivative size = 194, normalized size of antiderivative = 0.95 \[ \int \frac {\sqrt {a+b x} (e+f x)}{x (c+d x)^3} \, dx=\frac {\frac {c \sqrt {a+b x} \left (2 a d \left (3 c d e-c^2 f+2 d^2 e x\right )+b c \left (c^2 f-3 d^2 e x-c d (5 e+f x)\right )\right )}{d (-b c+a d) (c+d x)^2}+\frac {\left (-12 a b c d^2 e+8 a^2 d^3 e+b^2 c^2 (3 d e+c f)\right ) \arctan \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b c-a d}}\right )}{d^{3/2} (b c-a d)^{3/2}}-8 \sqrt {a} e \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{4 c^3} \]

[In]

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

[Out]

((c*Sqrt[a + b*x]*(2*a*d*(3*c*d*e - c^2*f + 2*d^2*e*x) + b*c*(c^2*f - 3*d^2*e*x - c*d*(5*e + f*x))))/(d*(-(b*c
) + a*d)*(c + d*x)^2) + ((-12*a*b*c*d^2*e + 8*a^2*d^3*e + b^2*c^2*(3*d*e + c*f))*ArcTan[(Sqrt[d]*Sqrt[a + b*x]
)/Sqrt[b*c - a*d]])/(d^(3/2)*(b*c - a*d)^(3/2)) - 8*Sqrt[a]*e*ArcTanh[Sqrt[a + b*x]/Sqrt[a]])/(4*c^3)

Maple [A] (verified)

Time = 1.64 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.04

method	result	size
pseudoelliptic	\(-\frac {2 \left (-\left (d x +c \right )^{2} \left (a^{2} d^{3} e -\frac {3}{2} a b c \,d^{2} e +\frac {1}{8} b^{2} c^{3} f +\frac {3}{8} b^{2} c^{2} d e \right ) \operatorname {arctanh}\left (\frac {d \sqrt {b x +a}}{\sqrt {\left (a d -b c \right ) d}}\right )+\sqrt {\left (a d -b c \right ) d}\, \left (\left (d x +c \right )^{2} e \left (a^{\frac {3}{2}} d -b c \sqrt {a}\right ) d \,\operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )+\frac {\sqrt {b x +a}\, c \left (-2 a \,d^{3} e x -3 e \left (-\frac {b x}{2}+a \right ) c \,d^{2}+\left (\frac {\left (f x +5 e \right ) b}{2}+a f \right ) c^{2} d -\frac {b \,c^{3} f}{2}\right )}{4}\right )\right )}{\sqrt {\left (a d -b c \right ) d}\, \left (a d -b c \right ) d \left (d x +c \right )^{2} c^{3}}\)	\(214\)
derivativedivides	\(2 b^{2} \left (-\frac {e \sqrt {a}\, \operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{b^{2} c^{3}}+\frac {\frac {\frac {b c \left (4 a e \,d^{2}-c^{2} b f -3 b c d e \right ) \left (b x +a \right )^{\frac {3}{2}}}{8 a d -8 b c}-\frac {\left (4 a e \,d^{2}+c^{2} b f -5 b c d e \right ) b c \sqrt {b x +a}}{8 d}}{\left (-d \left (b x +a \right )+a d -b c \right )^{2}}+\frac {\left (8 a^{2} d^{3} e -12 a b c \,d^{2} e +b^{2} c^{3} f +3 b^{2} c^{2} d e \right ) \operatorname {arctanh}\left (\frac {d \sqrt {b x +a}}{\sqrt {\left (a d -b c \right ) d}}\right )}{8 \left (a d -b c \right ) d \sqrt {\left (a d -b c \right ) d}}}{c^{3} b^{2}}\right )\)	\(221\)
default	\(2 b^{2} \left (-\frac {e \sqrt {a}\, \operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{b^{2} c^{3}}+\frac {\frac {\frac {b c \left (4 a e \,d^{2}-c^{2} b f -3 b c d e \right ) \left (b x +a \right )^{\frac {3}{2}}}{8 a d -8 b c}-\frac {\left (4 a e \,d^{2}+c^{2} b f -5 b c d e \right ) b c \sqrt {b x +a}}{8 d}}{\left (-d \left (b x +a \right )+a d -b c \right )^{2}}+\frac {\left (8 a^{2} d^{3} e -12 a b c \,d^{2} e +b^{2} c^{3} f +3 b^{2} c^{2} d e \right ) \operatorname {arctanh}\left (\frac {d \sqrt {b x +a}}{\sqrt {\left (a d -b c \right ) d}}\right )}{8 \left (a d -b c \right ) d \sqrt {\left (a d -b c \right ) d}}}{c^{3} b^{2}}\right )\)	\(221\)

[In]

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

[Out]

-2/((a*d-b*c)*d)^(1/2)*(-(d*x+c)^2*(a^2*d^3*e-3/2*a*b*c*d^2*e+1/8*b^2*c^3*f+3/8*b^2*c^2*d*e)*arctanh(d*(b*x+a)
^(1/2)/((a*d-b*c)*d)^(1/2))+((a*d-b*c)*d)^(1/2)*((d*x+c)^2*e*(a^(3/2)*d-b*c*a^(1/2))*d*arctanh((b*x+a)^(1/2)/a
^(1/2))+1/4*(b*x+a)^(1/2)*c*(-2*a*d^3*e*x-3*e*(-1/2*b*x+a)*c*d^2+(1/2*(f*x+5*e)*b+a*f)*c^2*d-1/2*b*c^3*f)))/(a
*d-b*c)/d/(d*x+c)^2/c^3

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 544 vs. \(2 (179) = 358\).

Time = 0.71 (sec) , antiderivative size = 2211, normalized size of antiderivative = 10.79 \[ \int \frac {\sqrt {a+b x} (e+f x)}{x (c+d x)^3} \, dx=\text {Too large to display} \]

[In]

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

[Out]

[1/8*((b^2*c^5*f + (b^2*c^3*d^2*f + (3*b^2*c^2*d^3 - 12*a*b*c*d^4 + 8*a^2*d^5)*e)*x^2 + (3*b^2*c^4*d - 12*a*b*
c^3*d^2 + 8*a^2*c^2*d^3)*e + 2*(b^2*c^4*d*f + (3*b^2*c^3*d^2 - 12*a*b*c^2*d^3 + 8*a^2*c*d^4)*e)*x)*sqrt(-b*c*d
+ a*d^2)*log((b*d*x - b*c + 2*a*d + 2*sqrt(-b*c*d + a*d^2)*sqrt(b*x + a))/(d*x + c)) + 8*((b^2*c^2*d^4 - 2*a*
b*c*d^5 + a^2*d^6)*e*x^2 + 2*(b^2*c^3*d^3 - 2*a*b*c^2*d^4 + a^2*c*d^5)*e*x + (b^2*c^4*d^2 - 2*a*b*c^3*d^3 + a^
2*c^2*d^4)*e)*sqrt(a)*log((b*x - 2*sqrt(b*x + a)*sqrt(a) + 2*a)/x) + 2*((5*b^2*c^4*d^2 - 11*a*b*c^3*d^3 + 6*a^
2*c^2*d^4)*e - (b^2*c^5*d - 3*a*b*c^4*d^2 + 2*a^2*c^3*d^3)*f + ((3*b^2*c^3*d^3 - 7*a*b*c^2*d^4 + 4*a^2*c*d^5)*
e + (b^2*c^4*d^2 - a*b*c^3*d^3)*f)*x)*sqrt(b*x + a))/(b^2*c^7*d^2 - 2*a*b*c^6*d^3 + a^2*c^5*d^4 + (b^2*c^5*d^4
- 2*a*b*c^4*d^5 + a^2*c^3*d^6)*x^2 + 2*(b^2*c^6*d^3 - 2*a*b*c^5*d^4 + a^2*c^4*d^5)*x), 1/8*(16*((b^2*c^2*d^4
- 2*a*b*c*d^5 + a^2*d^6)*e*x^2 + 2*(b^2*c^3*d^3 - 2*a*b*c^2*d^4 + a^2*c*d^5)*e*x + (b^2*c^4*d^2 - 2*a*b*c^3*d^
3 + a^2*c^2*d^4)*e)*sqrt(-a)*arctan(sqrt(b*x + a)*sqrt(-a)/a) + (b^2*c^5*f + (b^2*c^3*d^2*f + (3*b^2*c^2*d^3 -
12*a*b*c*d^4 + 8*a^2*d^5)*e)*x^2 + (3*b^2*c^4*d - 12*a*b*c^3*d^2 + 8*a^2*c^2*d^3)*e + 2*(b^2*c^4*d*f + (3*b^2
*c^3*d^2 - 12*a*b*c^2*d^3 + 8*a^2*c*d^4)*e)*x)*sqrt(-b*c*d + a*d^2)*log((b*d*x - b*c + 2*a*d + 2*sqrt(-b*c*d +
a*d^2)*sqrt(b*x + a))/(d*x + c)) + 2*((5*b^2*c^4*d^2 - 11*a*b*c^3*d^3 + 6*a^2*c^2*d^4)*e - (b^2*c^5*d - 3*a*b
*c^4*d^2 + 2*a^2*c^3*d^3)*f + ((3*b^2*c^3*d^3 - 7*a*b*c^2*d^4 + 4*a^2*c*d^5)*e + (b^2*c^4*d^2 - a*b*c^3*d^3)*f
)*x)*sqrt(b*x + a))/(b^2*c^7*d^2 - 2*a*b*c^6*d^3 + a^2*c^5*d^4 + (b^2*c^5*d^4 - 2*a*b*c^4*d^5 + a^2*c^3*d^6)*x
^2 + 2*(b^2*c^6*d^3 - 2*a*b*c^5*d^4 + a^2*c^4*d^5)*x), -1/4*((b^2*c^5*f + (b^2*c^3*d^2*f + (3*b^2*c^2*d^3 - 12
*a*b*c*d^4 + 8*a^2*d^5)*e)*x^2 + (3*b^2*c^4*d - 12*a*b*c^3*d^2 + 8*a^2*c^2*d^3)*e + 2*(b^2*c^4*d*f + (3*b^2*c^
3*d^2 - 12*a*b*c^2*d^3 + 8*a^2*c*d^4)*e)*x)*sqrt(b*c*d - a*d^2)*arctan(sqrt(b*c*d - a*d^2)*sqrt(b*x + a)/(b*d*
x + a*d)) - 4*((b^2*c^2*d^4 - 2*a*b*c*d^5 + a^2*d^6)*e*x^2 + 2*(b^2*c^3*d^3 - 2*a*b*c^2*d^4 + a^2*c*d^5)*e*x +
(b^2*c^4*d^2 - 2*a*b*c^3*d^3 + a^2*c^2*d^4)*e)*sqrt(a)*log((b*x - 2*sqrt(b*x + a)*sqrt(a) + 2*a)/x) - ((5*b^2
*c^4*d^2 - 11*a*b*c^3*d^3 + 6*a^2*c^2*d^4)*e - (b^2*c^5*d - 3*a*b*c^4*d^2 + 2*a^2*c^3*d^3)*f + ((3*b^2*c^3*d^3
- 7*a*b*c^2*d^4 + 4*a^2*c*d^5)*e + (b^2*c^4*d^2 - a*b*c^3*d^3)*f)*x)*sqrt(b*x + a))/(b^2*c^7*d^2 - 2*a*b*c^6*
d^3 + a^2*c^5*d^4 + (b^2*c^5*d^4 - 2*a*b*c^4*d^5 + a^2*c^3*d^6)*x^2 + 2*(b^2*c^6*d^3 - 2*a*b*c^5*d^4 + a^2*c^4
*d^5)*x), -1/4*((b^2*c^5*f + (b^2*c^3*d^2*f + (3*b^2*c^2*d^3 - 12*a*b*c*d^4 + 8*a^2*d^5)*e)*x^2 + (3*b^2*c^4*d
- 12*a*b*c^3*d^2 + 8*a^2*c^2*d^3)*e + 2*(b^2*c^4*d*f + (3*b^2*c^3*d^2 - 12*a*b*c^2*d^3 + 8*a^2*c*d^4)*e)*x)*s
qrt(b*c*d - a*d^2)*arctan(sqrt(b*c*d - a*d^2)*sqrt(b*x + a)/(b*d*x + a*d)) - 8*((b^2*c^2*d^4 - 2*a*b*c*d^5 + a
^2*d^6)*e*x^2 + 2*(b^2*c^3*d^3 - 2*a*b*c^2*d^4 + a^2*c*d^5)*e*x + (b^2*c^4*d^2 - 2*a*b*c^3*d^3 + a^2*c^2*d^4)*
e)*sqrt(-a)*arctan(sqrt(b*x + a)*sqrt(-a)/a) - ((5*b^2*c^4*d^2 - 11*a*b*c^3*d^3 + 6*a^2*c^2*d^4)*e - (b^2*c^5*
d - 3*a*b*c^4*d^2 + 2*a^2*c^3*d^3)*f + ((3*b^2*c^3*d^3 - 7*a*b*c^2*d^4 + 4*a^2*c*d^5)*e + (b^2*c^4*d^2 - a*b*c
^3*d^3)*f)*x)*sqrt(b*x + a))/(b^2*c^7*d^2 - 2*a*b*c^6*d^3 + a^2*c^5*d^4 + (b^2*c^5*d^4 - 2*a*b*c^4*d^5 + a^2*c
^3*d^6)*x^2 + 2*(b^2*c^6*d^3 - 2*a*b*c^5*d^4 + a^2*c^4*d^5)*x)]

Sympy [F(-1)]

Timed out. \[ \int \frac {\sqrt {a+b x} (e+f x)}{x (c+d x)^3} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F(-2)]

Exception generated. \[ \int \frac {\sqrt {a+b x} (e+f x)}{x (c+d x)^3} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate((f*x+e)*(b*x+a)^(1/2)/x/(d*x+c)^3,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*d-b*c>0)', see `assume?` for
more detail

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 291, normalized size of antiderivative = 1.42 \[ \int \frac {\sqrt {a+b x} (e+f x)}{x (c+d x)^3} \, dx=\frac {{\left (3 \, b^{2} c^{2} d e - 12 \, a b c d^{2} e + 8 \, a^{2} d^{3} e + b^{2} c^{3} f\right )} \arctan \left (\frac {\sqrt {b x + a} d}{\sqrt {b c d - a d^{2}}}\right )}{4 \, {\left (b c^{4} d - a c^{3} d^{2}\right )} \sqrt {b c d - a d^{2}}} + \frac {2 \, a e \arctan \left (\frac {\sqrt {b x + a}}{\sqrt {-a}}\right )}{\sqrt {-a} c^{3}} + \frac {5 \, \sqrt {b x + a} b^{3} c^{2} d e + 3 \, {\left (b x + a\right )}^{\frac {3}{2}} b^{2} c d^{2} e - 9 \, \sqrt {b x + a} a b^{2} c d^{2} e - 4 \, {\left (b x + a\right )}^{\frac {3}{2}} a b d^{3} e + 4 \, \sqrt {b x + a} a^{2} b d^{3} e - \sqrt {b x + a} b^{3} c^{3} f + {\left (b x + a\right )}^{\frac {3}{2}} b^{2} c^{2} d f + \sqrt {b x + a} a b^{2} c^{2} d f}{4 \, {\left (b c^{3} d - a c^{2} d^{2}\right )} {\left (b c + {\left (b x + a\right )} d - a d\right )}^{2}} \]

[In]

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

[Out]

1/4*(3*b^2*c^2*d*e - 12*a*b*c*d^2*e + 8*a^2*d^3*e + b^2*c^3*f)*arctan(sqrt(b*x + a)*d/sqrt(b*c*d - a*d^2))/((b
*c^4*d - a*c^3*d^2)*sqrt(b*c*d - a*d^2)) + 2*a*e*arctan(sqrt(b*x + a)/sqrt(-a))/(sqrt(-a)*c^3) + 1/4*(5*sqrt(b
*x + a)*b^3*c^2*d*e + 3*(b*x + a)^(3/2)*b^2*c*d^2*e - 9*sqrt(b*x + a)*a*b^2*c*d^2*e - 4*(b*x + a)^(3/2)*a*b*d^
3*e + 4*sqrt(b*x + a)*a^2*b*d^3*e - sqrt(b*x + a)*b^3*c^3*f + (b*x + a)^(3/2)*b^2*c^2*d*f + sqrt(b*x + a)*a*b^
2*c^2*d*f)/((b*c^3*d - a*c^2*d^2)*(b*c + (b*x + a)*d - a*d)^2)

Mupad [B] (verification not implemented)

Time = 4.91 (sec) , antiderivative size = 4839, normalized size of antiderivative = 23.60 \[ \int \frac {\sqrt {a+b x} (e+f x)}{x (c+d x)^3} \, dx=\text {Too large to display} \]

[In]

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

[Out]

(atan((((d^3*(a*d - b*c)^3)^(1/2)*(((a + b*x)^(1/2)*(b^6*c^6*f^2 + 128*a^4*b^2*d^6*e^2 + 9*b^6*c^4*d^2*e^2 + 6
*b^6*c^5*d*e*f + 256*a^2*b^4*c^2*d^4*e^2 - 72*a*b^5*c^3*d^3*e^2 - 320*a^3*b^3*c*d^5*e^2 + 16*a^2*b^4*c^3*d^3*e
*f - 24*a*b^5*c^4*d^2*e*f))/(8*(b^2*c^6*d + a^2*c^4*d^3 - 2*a*b*c^5*d^2)) - ((d^3*(a*d - b*c)^3)^(1/2)*((5*a*b
^5*c^8*d^3*e - a*b^5*c^9*d^2*f - 9*a^2*b^4*c^7*d^4*e + 4*a^3*b^3*c^6*d^5*e + a^2*b^4*c^8*d^3*f)/(b^2*c^8*d + a
^2*c^6*d^3 - 2*a*b*c^7*d^2) - ((d^3*(a*d - b*c)^3)^(1/2)*(a + b*x)^(1/2)*(8*a^2*d^3*e + b^2*c^3*f + 3*b^2*c^2*
d*e - 12*a*b*c*d^2*e)*(64*b^5*c^9*d^3 - 256*a*b^4*c^8*d^4 + 320*a^2*b^3*c^7*d^5 - 128*a^3*b^2*c^6*d^6))/(64*(b
^2*c^6*d + a^2*c^4*d^3 - 2*a*b*c^5*d^2)*(a^3*c^3*d^6 - b^3*c^6*d^3 + 3*a*b^2*c^5*d^4 - 3*a^2*b*c^4*d^5)))*(8*a
^2*d^3*e + b^2*c^3*f + 3*b^2*c^2*d*e - 12*a*b*c*d^2*e))/(8*(a^3*c^3*d^6 - b^3*c^6*d^3 + 3*a*b^2*c^5*d^4 - 3*a^
2*b*c^4*d^5)))*(8*a^2*d^3*e + b^2*c^3*f + 3*b^2*c^2*d*e - 12*a*b*c*d^2*e)*1i)/(8*(a^3*c^3*d^6 - b^3*c^6*d^3 +
3*a*b^2*c^5*d^4 - 3*a^2*b*c^4*d^5)) + ((d^3*(a*d - b*c)^3)^(1/2)*(((a + b*x)^(1/2)*(b^6*c^6*f^2 + 128*a^4*b^2*
d^6*e^2 + 9*b^6*c^4*d^2*e^2 + 6*b^6*c^5*d*e*f + 256*a^2*b^4*c^2*d^4*e^2 - 72*a*b^5*c^3*d^3*e^2 - 320*a^3*b^3*c
*d^5*e^2 + 16*a^2*b^4*c^3*d^3*e*f - 24*a*b^5*c^4*d^2*e*f))/(8*(b^2*c^6*d + a^2*c^4*d^3 - 2*a*b*c^5*d^2)) + ((d
^3*(a*d - b*c)^3)^(1/2)*((5*a*b^5*c^8*d^3*e - a*b^5*c^9*d^2*f - 9*a^2*b^4*c^7*d^4*e + 4*a^3*b^3*c^6*d^5*e + a^
2*b^4*c^8*d^3*f)/(b^2*c^8*d + a^2*c^6*d^3 - 2*a*b*c^7*d^2) + ((d^3*(a*d - b*c)^3)^(1/2)*(a + b*x)^(1/2)*(8*a^2
*d^3*e + b^2*c^3*f + 3*b^2*c^2*d*e - 12*a*b*c*d^2*e)*(64*b^5*c^9*d^3 - 256*a*b^4*c^8*d^4 + 320*a^2*b^3*c^7*d^5
- 128*a^3*b^2*c^6*d^6))/(64*(b^2*c^6*d + a^2*c^4*d^3 - 2*a*b*c^5*d^2)*(a^3*c^3*d^6 - b^3*c^6*d^3 + 3*a*b^2*c^
5*d^4 - 3*a^2*b*c^4*d^5)))*(8*a^2*d^3*e + b^2*c^3*f + 3*b^2*c^2*d*e - 12*a*b*c*d^2*e))/(8*(a^3*c^3*d^6 - b^3*c
^6*d^3 + 3*a*b^2*c^5*d^4 - 3*a^2*b*c^4*d^5)))*(8*a^2*d^3*e + b^2*c^3*f + 3*b^2*c^2*d*e - 12*a*b*c*d^2*e)*1i)/(
8*(a^3*c^3*d^6 - b^3*c^6*d^3 + 3*a*b^2*c^5*d^4 - 3*a^2*b*c^4*d^5)))/(((a*b^6*c^5*e*f^2)/4 - 12*a^2*b^5*c^2*d^3
*e^3 - 8*a^4*b^3*d^5*e^3 + (9*a*b^6*c^3*d^2*e^3)/4 + 18*a^3*b^4*c*d^4*e^3 - 4*a^2*b^5*c^3*d^2*e^2*f + 2*a^3*b^
4*c^2*d^3*e^2*f + (3*a*b^6*c^4*d*e^2*f)/2)/(b^2*c^8*d + a^2*c^6*d^3 - 2*a*b*c^7*d^2) - ((d^3*(a*d - b*c)^3)^(1
/2)*(((a + b*x)^(1/2)*(b^6*c^6*f^2 + 128*a^4*b^2*d^6*e^2 + 9*b^6*c^4*d^2*e^2 + 6*b^6*c^5*d*e*f + 256*a^2*b^4*c
^2*d^4*e^2 - 72*a*b^5*c^3*d^3*e^2 - 320*a^3*b^3*c*d^5*e^2 + 16*a^2*b^4*c^3*d^3*e*f - 24*a*b^5*c^4*d^2*e*f))/(8
*(b^2*c^6*d + a^2*c^4*d^3 - 2*a*b*c^5*d^2)) - ((d^3*(a*d - b*c)^3)^(1/2)*((5*a*b^5*c^8*d^3*e - a*b^5*c^9*d^2*f
- 9*a^2*b^4*c^7*d^4*e + 4*a^3*b^3*c^6*d^5*e + a^2*b^4*c^8*d^3*f)/(b^2*c^8*d + a^2*c^6*d^3 - 2*a*b*c^7*d^2) -
((d^3*(a*d - b*c)^3)^(1/2)*(a + b*x)^(1/2)*(8*a^2*d^3*e + b^2*c^3*f + 3*b^2*c^2*d*e - 12*a*b*c*d^2*e)*(64*b^5*
c^9*d^3 - 256*a*b^4*c^8*d^4 + 320*a^2*b^3*c^7*d^5 - 128*a^3*b^2*c^6*d^6))/(64*(b^2*c^6*d + a^2*c^4*d^3 - 2*a*b
*c^5*d^2)*(a^3*c^3*d^6 - b^3*c^6*d^3 + 3*a*b^2*c^5*d^4 - 3*a^2*b*c^4*d^5)))*(8*a^2*d^3*e + b^2*c^3*f + 3*b^2*c
^2*d*e - 12*a*b*c*d^2*e))/(8*(a^3*c^3*d^6 - b^3*c^6*d^3 + 3*a*b^2*c^5*d^4 - 3*a^2*b*c^4*d^5)))*(8*a^2*d^3*e +
b^2*c^3*f + 3*b^2*c^2*d*e - 12*a*b*c*d^2*e))/(8*(a^3*c^3*d^6 - b^3*c^6*d^3 + 3*a*b^2*c^5*d^4 - 3*a^2*b*c^4*d^5
)) + ((d^3*(a*d - b*c)^3)^(1/2)*(((a + b*x)^(1/2)*(b^6*c^6*f^2 + 128*a^4*b^2*d^6*e^2 + 9*b^6*c^4*d^2*e^2 + 6*b
^6*c^5*d*e*f + 256*a^2*b^4*c^2*d^4*e^2 - 72*a*b^5*c^3*d^3*e^2 - 320*a^3*b^3*c*d^5*e^2 + 16*a^2*b^4*c^3*d^3*e*f
- 24*a*b^5*c^4*d^2*e*f))/(8*(b^2*c^6*d + a^2*c^4*d^3 - 2*a*b*c^5*d^2)) + ((d^3*(a*d - b*c)^3)^(1/2)*((5*a*b^5
*c^8*d^3*e - a*b^5*c^9*d^2*f - 9*a^2*b^4*c^7*d^4*e + 4*a^3*b^3*c^6*d^5*e + a^2*b^4*c^8*d^3*f)/(b^2*c^8*d + a^2
*c^6*d^3 - 2*a*b*c^7*d^2) + ((d^3*(a*d - b*c)^3)^(1/2)*(a + b*x)^(1/2)*(8*a^2*d^3*e + b^2*c^3*f + 3*b^2*c^2*d*
e - 12*a*b*c*d^2*e)*(64*b^5*c^9*d^3 - 256*a*b^4*c^8*d^4 + 320*a^2*b^3*c^7*d^5 - 128*a^3*b^2*c^6*d^6))/(64*(b^2
*c^6*d + a^2*c^4*d^3 - 2*a*b*c^5*d^2)*(a^3*c^3*d^6 - b^3*c^6*d^3 + 3*a*b^2*c^5*d^4 - 3*a^2*b*c^4*d^5)))*(8*a^2
*d^3*e + b^2*c^3*f + 3*b^2*c^2*d*e - 12*a*b*c*d^2*e))/(8*(a^3*c^3*d^6 - b^3*c^6*d^3 + 3*a*b^2*c^5*d^4 - 3*a^2*
b*c^4*d^5)))*(8*a^2*d^3*e + b^2*c^3*f + 3*b^2*c^2*d*e - 12*a*b*c*d^2*e))/(8*(a^3*c^3*d^6 - b^3*c^6*d^3 + 3*a*b
^2*c^5*d^4 - 3*a^2*b*c^4*d^5))))*(d^3*(a*d - b*c)^3)^(1/2)*(8*a^2*d^3*e + b^2*c^3*f + 3*b^2*c^2*d*e - 12*a*b*c
*d^2*e)*1i)/(4*(a^3*c^3*d^6 - b^3*c^6*d^3 + 3*a*b^2*c^5*d^4 - 3*a^2*b*c^4*d^5)) - (((a + b*x)^(1/2)*(b^2*c^2*f
+ 4*a*b*d^2*e - 5*b^2*c*d*e))/(4*c^2*d) + ((a + b*x)^(3/2)*(b^2*c^2*f - 4*a*b*d^2*e + 3*b^2*c*d*e))/(4*c^2*(a
*d - b*c)))/(d^2*(a + b*x)^2 - (2*a*d^2 - 2*b*c*d)*(a + b*x) + a^2*d^2 + b^2*c^2 - 2*a*b*c*d) + (a^(1/2)*e*ata
n(((a^(1/2)*e*(((a + b*x)^(1/2)*(b^6*c^6*f^2 + 128*a^4*b^2*d^6*e^2 + 9*b^6*c^4*d^2*e^2 + 6*b^6*c^5*d*e*f + 256
*a^2*b^4*c^2*d^4*e^2 - 72*a*b^5*c^3*d^3*e^2 - 320*a^3*b^3*c*d^5*e^2 + 16*a^2*b^4*c^3*d^3*e*f - 24*a*b^5*c^4*d^
2*e*f))/(8*(b^2*c^6*d + a^2*c^4*d^3 - 2*a*b*c^5*d^2)) + (a^(1/2)*e*((5*a*b^5*c^8*d^3*e - a*b^5*c^9*d^2*f - 9*a
^2*b^4*c^7*d^4*e + 4*a^3*b^3*c^6*d^5*e + a^2*b^4*c^8*d^3*f)/(b^2*c^8*d + a^2*c^6*d^3 - 2*a*b*c^7*d^2) + (a^(1/
2)*e*(a + b*x)^(1/2)*(64*b^5*c^9*d^3 - 256*a*b^4*c^8*d^4 + 320*a^2*b^3*c^7*d^5 - 128*a^3*b^2*c^6*d^6))/(8*c^3*
(b^2*c^6*d + a^2*c^4*d^3 - 2*a*b*c^5*d^2))))/c^3)*1i)/c^3 + (a^(1/2)*e*(((a + b*x)^(1/2)*(b^6*c^6*f^2 + 128*a^
4*b^2*d^6*e^2 + 9*b^6*c^4*d^2*e^2 + 6*b^6*c^5*d*e*f + 256*a^2*b^4*c^2*d^4*e^2 - 72*a*b^5*c^3*d^3*e^2 - 320*a^3
*b^3*c*d^5*e^2 + 16*a^2*b^4*c^3*d^3*e*f - 24*a*b^5*c^4*d^2*e*f))/(8*(b^2*c^6*d + a^2*c^4*d^3 - 2*a*b*c^5*d^2))
- (a^(1/2)*e*((5*a*b^5*c^8*d^3*e - a*b^5*c^9*d^2*f - 9*a^2*b^4*c^7*d^4*e + 4*a^3*b^3*c^6*d^5*e + a^2*b^4*c^8*
d^3*f)/(b^2*c^8*d + a^2*c^6*d^3 - 2*a*b*c^7*d^2) - (a^(1/2)*e*(a + b*x)^(1/2)*(64*b^5*c^9*d^3 - 256*a*b^4*c^8*
d^4 + 320*a^2*b^3*c^7*d^5 - 128*a^3*b^2*c^6*d^6))/(8*c^3*(b^2*c^6*d + a^2*c^4*d^3 - 2*a*b*c^5*d^2))))/c^3)*1i)
/c^3)/(((a*b^6*c^5*e*f^2)/4 - 12*a^2*b^5*c^2*d^3*e^3 - 8*a^4*b^3*d^5*e^3 + (9*a*b^6*c^3*d^2*e^3)/4 + 18*a^3*b^
4*c*d^4*e^3 - 4*a^2*b^5*c^3*d^2*e^2*f + 2*a^3*b^4*c^2*d^3*e^2*f + (3*a*b^6*c^4*d*e^2*f)/2)/(b^2*c^8*d + a^2*c^
6*d^3 - 2*a*b*c^7*d^2) + (a^(1/2)*e*(((a + b*x)^(1/2)*(b^6*c^6*f^2 + 128*a^4*b^2*d^6*e^2 + 9*b^6*c^4*d^2*e^2 +
6*b^6*c^5*d*e*f + 256*a^2*b^4*c^2*d^4*e^2 - 72*a*b^5*c^3*d^3*e^2 - 320*a^3*b^3*c*d^5*e^2 + 16*a^2*b^4*c^3*d^3
*e*f - 24*a*b^5*c^4*d^2*e*f))/(8*(b^2*c^6*d + a^2*c^4*d^3 - 2*a*b*c^5*d^2)) + (a^(1/2)*e*((5*a*b^5*c^8*d^3*e -
a*b^5*c^9*d^2*f - 9*a^2*b^4*c^7*d^4*e + 4*a^3*b^3*c^6*d^5*e + a^2*b^4*c^8*d^3*f)/(b^2*c^8*d + a^2*c^6*d^3 - 2
*a*b*c^7*d^2) + (a^(1/2)*e*(a + b*x)^(1/2)*(64*b^5*c^9*d^3 - 256*a*b^4*c^8*d^4 + 320*a^2*b^3*c^7*d^5 - 128*a^3
*b^2*c^6*d^6))/(8*c^3*(b^2*c^6*d + a^2*c^4*d^3 - 2*a*b*c^5*d^2))))/c^3))/c^3 - (a^(1/2)*e*(((a + b*x)^(1/2)*(b
^6*c^6*f^2 + 128*a^4*b^2*d^6*e^2 + 9*b^6*c^4*d^2*e^2 + 6*b^6*c^5*d*e*f + 256*a^2*b^4*c^2*d^4*e^2 - 72*a*b^5*c^
3*d^3*e^2 - 320*a^3*b^3*c*d^5*e^2 + 16*a^2*b^4*c^3*d^3*e*f - 24*a*b^5*c^4*d^2*e*f))/(8*(b^2*c^6*d + a^2*c^4*d^
3 - 2*a*b*c^5*d^2)) - (a^(1/2)*e*((5*a*b^5*c^8*d^3*e - a*b^5*c^9*d^2*f - 9*a^2*b^4*c^7*d^4*e + 4*a^3*b^3*c^6*d
^5*e + a^2*b^4*c^8*d^3*f)/(b^2*c^8*d + a^2*c^6*d^3 - 2*a*b*c^7*d^2) - (a^(1/2)*e*(a + b*x)^(1/2)*(64*b^5*c^9*d
^3 - 256*a*b^4*c^8*d^4 + 320*a^2*b^3*c^7*d^5 - 128*a^3*b^2*c^6*d^6))/(8*c^3*(b^2*c^6*d + a^2*c^4*d^3 - 2*a*b*c
^5*d^2))))/c^3))/c^3))*2i)/c^3