\(\int \frac {(a+b x)^{3/2} (c+d x)^{3/2}}{x^2} \, dx\) [611]

Optimal result

Integrand size = 22, antiderivative size = 191 \[ \int \frac {(a+b x)^{3/2} (c+d x)^{3/2}}{x^2} \, dx=\frac {3}{4} (b c+3 a d) \sqrt {a+b x} \sqrt {c+d x}+\frac {3}{2} b \sqrt {a+b x} (c+d x)^{3/2}-\frac {(a+b x)^{3/2} (c+d x)^{3/2}}{x}-3 \sqrt {a} \sqrt {c} (b c+a d) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )+\frac {3 \left (b^2 c^2+6 a b c d+a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{4 \sqrt {b} \sqrt {d}} \]

[Out]

-(b*x+a)^(3/2)*(d*x+c)^(3/2)/x-3*(a*d+b*c)*arctanh(c^(1/2)*(b*x+a)^(1/2)/a^(1/2)/(d*x+c)^(1/2))*a^(1/2)*c^(1/2
)+3/4*(a^2*d^2+6*a*b*c*d+b^2*c^2)*arctanh(d^(1/2)*(b*x+a)^(1/2)/b^(1/2)/(d*x+c)^(1/2))/b^(1/2)/d^(1/2)+3/2*b*(
d*x+c)^(3/2)*(b*x+a)^(1/2)+3/4*(3*a*d+b*c)*(b*x+a)^(1/2)*(d*x+c)^(1/2)

Rubi [A] (verified)

Time = 0.13 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {99, 159, 163, 65, 223, 212, 95, 214} \[ \int \frac {(a+b x)^{3/2} (c+d x)^{3/2}}{x^2} \, dx=\frac {3 \left (a^2 d^2+6 a b c d+b^2 c^2\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{4 \sqrt {b} \sqrt {d}}-3 \sqrt {a} \sqrt {c} (a d+b c) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )-\frac {(a+b x)^{3/2} (c+d x)^{3/2}}{x}+\frac {3}{2} b \sqrt {a+b x} (c+d x)^{3/2}+\frac {3}{4} \sqrt {a+b x} \sqrt {c+d x} (3 a d+b c) \]

[In]

Int[((a + b*x)^(3/2)*(c + d*x)^(3/2))/x^2,x]

[Out]

(3*(b*c + 3*a*d)*Sqrt[a + b*x]*Sqrt[c + d*x])/4 + (3*b*Sqrt[a + b*x]*(c + d*x)^(3/2))/2 - ((a + b*x)^(3/2)*(c
+ d*x)^(3/2))/x - 3*Sqrt[a]*Sqrt[c]*(b*c + a*d)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])] + (3*
(b^2*c^2 + 6*a*b*c*d + a^2*d^2)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(4*Sqrt[b]*Sqrt[d])

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 95

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> With[{q = Denomin
ator[m]}, Dist[q, Subst[Int[x^(q*(m + 1) - 1)/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^
(1/q)], x]] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && LtQ[-1, m, 0] && SimplerQ[
a + b*x, c + d*x]

Rule 99

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

Rule 159

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] && IntegersQ[2*m, 2
*n, 2*p]

Rule 163

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

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 214

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

Rule 223

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rubi steps \begin{align*} \text {integral}& = -\frac {(a+b x)^{3/2} (c+d x)^{3/2}}{x}+\int \frac {\sqrt {a+b x} \sqrt {c+d x} \left (\frac {3}{2} (b c+a d)+3 b d x\right )}{x} \, dx \\ & = \frac {3}{2} b \sqrt {a+b x} (c+d x)^{3/2}-\frac {(a+b x)^{3/2} (c+d x)^{3/2}}{x}+\frac {\int \frac {\sqrt {c+d x} \left (3 a d (b c+a d)+\frac {3}{2} b d (b c+3 a d) x\right )}{x \sqrt {a+b x}} \, dx}{2 d} \\ & = \frac {3}{4} (b c+3 a d) \sqrt {a+b x} \sqrt {c+d x}+\frac {3}{2} b \sqrt {a+b x} (c+d x)^{3/2}-\frac {(a+b x)^{3/2} (c+d x)^{3/2}}{x}+\frac {\int \frac {3 a b c d (b c+a d)+\frac {3}{4} b d \left (b^2 c^2+6 a b c d+a^2 d^2\right ) x}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{2 b d} \\ & = \frac {3}{4} (b c+3 a d) \sqrt {a+b x} \sqrt {c+d x}+\frac {3}{2} b \sqrt {a+b x} (c+d x)^{3/2}-\frac {(a+b x)^{3/2} (c+d x)^{3/2}}{x}+\frac {1}{2} (3 a c (b c+a d)) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx+\frac {1}{8} \left (3 \left (b^2 c^2+6 a b c d+a^2 d^2\right )\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx \\ & = \frac {3}{4} (b c+3 a d) \sqrt {a+b x} \sqrt {c+d x}+\frac {3}{2} b \sqrt {a+b x} (c+d x)^{3/2}-\frac {(a+b x)^{3/2} (c+d x)^{3/2}}{x}+(3 a c (b c+a d)) \text {Subst}\left (\int \frac {1}{-a+c x^2} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )+\frac {\left (3 \left (b^2 c^2+6 a b c d+a^2 d^2\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {c-\frac {a d}{b}+\frac {d x^2}{b}}} \, dx,x,\sqrt {a+b x}\right )}{4 b} \\ & = \frac {3}{4} (b c+3 a d) \sqrt {a+b x} \sqrt {c+d x}+\frac {3}{2} b \sqrt {a+b x} (c+d x)^{3/2}-\frac {(a+b x)^{3/2} (c+d x)^{3/2}}{x}-3 \sqrt {a} \sqrt {c} (b c+a d) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )+\frac {\left (3 \left (b^2 c^2+6 a b c d+a^2 d^2\right )\right ) \text {Subst}\left (\int \frac {1}{1-\frac {d x^2}{b}} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )}{4 b} \\ & = \frac {3}{4} (b c+3 a d) \sqrt {a+b x} \sqrt {c+d x}+\frac {3}{2} b \sqrt {a+b x} (c+d x)^{3/2}-\frac {(a+b x)^{3/2} (c+d x)^{3/2}}{x}-3 \sqrt {a} \sqrt {c} (b c+a d) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )+\frac {3 \left (b^2 c^2+6 a b c d+a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{4 \sqrt {b} \sqrt {d}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.69 (sec) , antiderivative size = 162, normalized size of antiderivative = 0.85 \[ \int \frac {(a+b x)^{3/2} (c+d x)^{3/2}}{x^2} \, dx=-3 \sqrt {a} \sqrt {c} (b c+a d) \text {arctanh}\left (\frac {\sqrt {a} \sqrt {c+d x}}{\sqrt {c} \sqrt {a+b x}}\right )+\frac {1}{4} \left (\frac {\sqrt {a+b x} \sqrt {c+d x} (b x (5 c+2 d x)+a (-4 c+5 d x))}{x}+\frac {3 \left (b^2 c^2+6 a b c d+a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {a+b x}}\right )}{\sqrt {b} \sqrt {d}}\right ) \]

[In]

Integrate[((a + b*x)^(3/2)*(c + d*x)^(3/2))/x^2,x]

[Out]

-3*Sqrt[a]*Sqrt[c]*(b*c + a*d)*ArcTanh[(Sqrt[a]*Sqrt[c + d*x])/(Sqrt[c]*Sqrt[a + b*x])] + ((Sqrt[a + b*x]*Sqrt
[c + d*x]*(b*x*(5*c + 2*d*x) + a*(-4*c + 5*d*x)))/x + (3*(b^2*c^2 + 6*a*b*c*d + a^2*d^2)*ArcTanh[(Sqrt[b]*Sqrt
[c + d*x])/(Sqrt[d]*Sqrt[a + b*x])])/(Sqrt[b]*Sqrt[d]))/4

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(418\) vs. \(2(149)=298\).

Time = 0.54 (sec) , antiderivative size = 419, normalized size of antiderivative = 2.19

method	result	size
default	\(\frac {\sqrt {b x +a}\, \sqrt {d x +c}\, \left (3 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) \sqrt {a c}\, a^{2} d^{2} x +18 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) \sqrt {a c}\, a b c d x +3 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) \sqrt {a c}\, b^{2} c^{2} x -12 \sqrt {b d}\, \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a^{2} c d x -12 \sqrt {b d}\, \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a b \,c^{2} x +4 \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, b d \,x^{2}+10 \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a d x +10 \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, b c x -8 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a c \sqrt {b d}\, \sqrt {a c}\right )}{8 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, x \sqrt {b d}\, \sqrt {a c}}\)	\(419\)

[In]

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

[Out]

1/8*(b*x+a)^(1/2)*(d*x+c)^(1/2)*(3*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))
*(a*c)^(1/2)*a^2*d^2*x+18*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*(a*c)^(1
/2)*a*b*c*d*x+3*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*(a*c)^(1/2)*b^2*c^
2*x-12*(b*d)^(1/2)*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*a^2*c*d*x-12*(b*d)^(1/2)*ln
((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*a*b*c^2*x+4*(b*d)^(1/2)*(a*c)^(1/2)*((b*x+a)*(d*
x+c))^(1/2)*b*d*x^2+10*(b*d)^(1/2)*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a*d*x+10*(b*d)^(1/2)*(a*c)^(1/2)*((b*x+
a)*(d*x+c))^(1/2)*b*c*x-8*((b*x+a)*(d*x+c))^(1/2)*a*c*(b*d)^(1/2)*(a*c)^(1/2))/((b*x+a)*(d*x+c))^(1/2)/x/(b*d)
^(1/2)/(a*c)^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.89 (sec) , antiderivative size = 1073, normalized size of antiderivative = 5.62 \[ \int \frac {(a+b x)^{3/2} (c+d x)^{3/2}}{x^2} \, dx=\left [\frac {3 \, {\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} \sqrt {b d} x \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 4 \, {\left (2 \, b d x + b c + a d\right )} \sqrt {b d} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x\right ) + 12 \, {\left (b^{2} c d + a b d^{2}\right )} \sqrt {a c} x \log \left (\frac {8 \, a^{2} c^{2} + {\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} - 4 \, {\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {a c} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (a b c^{2} + a^{2} c d\right )} x}{x^{2}}\right ) + 4 \, {\left (2 \, b^{2} d^{2} x^{2} - 4 \, a b c d + 5 \, {\left (b^{2} c d + a b d^{2}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{16 \, b d x}, -\frac {3 \, {\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} \sqrt {-b d} x \arctan \left (\frac {{\left (2 \, b d x + b c + a d\right )} \sqrt {-b d} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (b^{2} d^{2} x^{2} + a b c d + {\left (b^{2} c d + a b d^{2}\right )} x\right )}}\right ) - 6 \, {\left (b^{2} c d + a b d^{2}\right )} \sqrt {a c} x \log \left (\frac {8 \, a^{2} c^{2} + {\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} - 4 \, {\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {a c} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (a b c^{2} + a^{2} c d\right )} x}{x^{2}}\right ) - 2 \, {\left (2 \, b^{2} d^{2} x^{2} - 4 \, a b c d + 5 \, {\left (b^{2} c d + a b d^{2}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{8 \, b d x}, \frac {24 \, {\left (b^{2} c d + a b d^{2}\right )} \sqrt {-a c} x \arctan \left (\frac {{\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {-a c} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (a b c d x^{2} + a^{2} c^{2} + {\left (a b c^{2} + a^{2} c d\right )} x\right )}}\right ) + 3 \, {\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} \sqrt {b d} x \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 4 \, {\left (2 \, b d x + b c + a d\right )} \sqrt {b d} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x\right ) + 4 \, {\left (2 \, b^{2} d^{2} x^{2} - 4 \, a b c d + 5 \, {\left (b^{2} c d + a b d^{2}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{16 \, b d x}, \frac {12 \, {\left (b^{2} c d + a b d^{2}\right )} \sqrt {-a c} x \arctan \left (\frac {{\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {-a c} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (a b c d x^{2} + a^{2} c^{2} + {\left (a b c^{2} + a^{2} c d\right )} x\right )}}\right ) - 3 \, {\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} \sqrt {-b d} x \arctan \left (\frac {{\left (2 \, b d x + b c + a d\right )} \sqrt {-b d} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (b^{2} d^{2} x^{2} + a b c d + {\left (b^{2} c d + a b d^{2}\right )} x\right )}}\right ) + 2 \, {\left (2 \, b^{2} d^{2} x^{2} - 4 \, a b c d + 5 \, {\left (b^{2} c d + a b d^{2}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{8 \, b d x}\right ] \]

[In]

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

[Out]

[1/16*(3*(b^2*c^2 + 6*a*b*c*d + a^2*d^2)*sqrt(b*d)*x*log(8*b^2*d^2*x^2 + b^2*c^2 + 6*a*b*c*d + a^2*d^2 + 4*(2*
b*d*x + b*c + a*d)*sqrt(b*d)*sqrt(b*x + a)*sqrt(d*x + c) + 8*(b^2*c*d + a*b*d^2)*x) + 12*(b^2*c*d + a*b*d^2)*s
qrt(a*c)*x*log((8*a^2*c^2 + (b^2*c^2 + 6*a*b*c*d + a^2*d^2)*x^2 - 4*(2*a*c + (b*c + a*d)*x)*sqrt(a*c)*sqrt(b*x
+ a)*sqrt(d*x + c) + 8*(a*b*c^2 + a^2*c*d)*x)/x^2) + 4*(2*b^2*d^2*x^2 - 4*a*b*c*d + 5*(b^2*c*d + a*b*d^2)*x)*
sqrt(b*x + a)*sqrt(d*x + c))/(b*d*x), -1/8*(3*(b^2*c^2 + 6*a*b*c*d + a^2*d^2)*sqrt(-b*d)*x*arctan(1/2*(2*b*d*x
+ b*c + a*d)*sqrt(-b*d)*sqrt(b*x + a)*sqrt(d*x + c)/(b^2*d^2*x^2 + a*b*c*d + (b^2*c*d + a*b*d^2)*x)) - 6*(b^2
*c*d + a*b*d^2)*sqrt(a*c)*x*log((8*a^2*c^2 + (b^2*c^2 + 6*a*b*c*d + a^2*d^2)*x^2 - 4*(2*a*c + (b*c + a*d)*x)*s
qrt(a*c)*sqrt(b*x + a)*sqrt(d*x + c) + 8*(a*b*c^2 + a^2*c*d)*x)/x^2) - 2*(2*b^2*d^2*x^2 - 4*a*b*c*d + 5*(b^2*c
*d + a*b*d^2)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(b*d*x), 1/16*(24*(b^2*c*d + a*b*d^2)*sqrt(-a*c)*x*arctan(1/2*(2
*a*c + (b*c + a*d)*x)*sqrt(-a*c)*sqrt(b*x + a)*sqrt(d*x + c)/(a*b*c*d*x^2 + a^2*c^2 + (a*b*c^2 + a^2*c*d)*x))
+ 3*(b^2*c^2 + 6*a*b*c*d + a^2*d^2)*sqrt(b*d)*x*log(8*b^2*d^2*x^2 + b^2*c^2 + 6*a*b*c*d + a^2*d^2 + 4*(2*b*d*x
+ b*c + a*d)*sqrt(b*d)*sqrt(b*x + a)*sqrt(d*x + c) + 8*(b^2*c*d + a*b*d^2)*x) + 4*(2*b^2*d^2*x^2 - 4*a*b*c*d
+ 5*(b^2*c*d + a*b*d^2)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(b*d*x), 1/8*(12*(b^2*c*d + a*b*d^2)*sqrt(-a*c)*x*arct
an(1/2*(2*a*c + (b*c + a*d)*x)*sqrt(-a*c)*sqrt(b*x + a)*sqrt(d*x + c)/(a*b*c*d*x^2 + a^2*c^2 + (a*b*c^2 + a^2*
c*d)*x)) - 3*(b^2*c^2 + 6*a*b*c*d + a^2*d^2)*sqrt(-b*d)*x*arctan(1/2*(2*b*d*x + b*c + a*d)*sqrt(-b*d)*sqrt(b*x
+ a)*sqrt(d*x + c)/(b^2*d^2*x^2 + a*b*c*d + (b^2*c*d + a*b*d^2)*x)) + 2*(2*b^2*d^2*x^2 - 4*a*b*c*d + 5*(b^2*c
*d + a*b*d^2)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(b*d*x)]

Sympy [F]

\[ \int \frac {(a+b x)^{3/2} (c+d x)^{3/2}}{x^2} \, dx=\int \frac {\left (a + b x\right )^{\frac {3}{2}} \left (c + d x\right )^{\frac {3}{2}}}{x^{2}}\, dx \]

[In]

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

[Out]

Integral((a + b*x)**(3/2)*(c + d*x)**(3/2)/x**2, x)

Maxima [F(-2)]

Exception generated. \[ \int \frac {(a+b x)^{3/2} (c+d x)^{3/2}}{x^2} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate((b*x+a)^(3/2)*(d*x+c)^(3/2)/x^2,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 [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 577 vs. \(2 (149) = 298\).

Time = 0.59 (sec) , antiderivative size = 577, normalized size of antiderivative = 3.02 \[ \int \frac {(a+b x)^{3/2} (c+d x)^{3/2}}{x^2} \, dx=\frac {2 \, \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} \sqrt {b x + a} {\left (\frac {2 \, {\left (b x + a\right )} d {\left | b \right |}}{b} + \frac {5 \, b c d^{2} {\left | b \right |} + 3 \, a d^{3} {\left | b \right |}}{b d^{2}}\right )} - \frac {3 \, {\left (b^{2} c^{2} {\left | b \right |} + 6 \, a b c d {\left | b \right |} + a^{2} d^{2} {\left | b \right |}\right )} \log \left ({\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2}\right )}{\sqrt {b d}} - \frac {24 \, {\left (\sqrt {b d} a b^{2} c^{2} {\left | b \right |} + \sqrt {b d} a^{2} b c d {\left | b \right |}\right )} \arctan \left (-\frac {b^{2} c + a b d - {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2}}{2 \, \sqrt {-a b c d} b}\right )}{\sqrt {-a b c d} b} - \frac {16 \, {\left (\sqrt {b d} a b^{4} c^{3} {\left | b \right |} - 2 \, \sqrt {b d} a^{2} b^{3} c^{2} d {\left | b \right |} + \sqrt {b d} a^{3} b^{2} c d^{2} {\left | b \right |} - \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a b^{2} c^{2} {\left | b \right |} - \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a^{2} b c d {\left | b \right |}\right )}}{b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2} - 2 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} b^{2} c - 2 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a b d + {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{4}}}{8 \, b} \]

[In]

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

[Out]

1/8*(2*sqrt(b^2*c + (b*x + a)*b*d - a*b*d)*sqrt(b*x + a)*(2*(b*x + a)*d*abs(b)/b + (5*b*c*d^2*abs(b) + 3*a*d^3
*abs(b))/(b*d^2)) - 3*(b^2*c^2*abs(b) + 6*a*b*c*d*abs(b) + a^2*d^2*abs(b))*log((sqrt(b*d)*sqrt(b*x + a) - sqrt
(b^2*c + (b*x + a)*b*d - a*b*d))^2)/sqrt(b*d) - 24*(sqrt(b*d)*a*b^2*c^2*abs(b) + sqrt(b*d)*a^2*b*c*d*abs(b))*a
rctan(-1/2*(b^2*c + a*b*d - (sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2)/(sqrt(-a*b*c*d)
*b))/(sqrt(-a*b*c*d)*b) - 16*(sqrt(b*d)*a*b^4*c^3*abs(b) - 2*sqrt(b*d)*a^2*b^3*c^2*d*abs(b) + sqrt(b*d)*a^3*b^
2*c*d^2*abs(b) - sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a*b^2*c^2*abs(b)
- sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^2*b*c*d*abs(b))/(b^4*c^2 - 2*a
*b^3*c*d + a^2*b^2*d^2 - 2*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*b^2*c - 2*(sqrt(b
*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a*b*d + (sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*
x + a)*b*d - a*b*d))^4))/b

Mupad [F(-1)]

Timed out. \[ \int \frac {(a+b x)^{3/2} (c+d x)^{3/2}}{x^2} \, dx=\int \frac {{\left (a+b\,x\right )}^{3/2}\,{\left (c+d\,x\right )}^{3/2}}{x^2} \,d x \]

[In]

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

[Out]

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