\(\int \frac {(a+b x)^{3/2} \sqrt {c+d x}}{x^2} \, dx\) [602]

Optimal result

Integrand size = 22, antiderivative size = 144 \[ \int \frac {(a+b x)^{3/2} \sqrt {c+d x}}{x^2} \, dx=2 b \sqrt {a+b x} \sqrt {c+d x}-\frac {(a+b x)^{3/2} \sqrt {c+d x}}{x}-\frac {\sqrt {a} (3 b c+a d) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{\sqrt {c}}+\frac {\sqrt {b} (b c+3 a d) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{\sqrt {d}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.00, number of steps used = 8, 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} \[ \int \frac {(a+b x)^{3/2} \sqrt {c+d x}}{x^2} \, dx=-\frac {\sqrt {a} (a d+3 b c) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{\sqrt {c}}+\frac {\sqrt {b} (3 a d+b c) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{\sqrt {d}}-\frac {(a+b x)^{3/2} \sqrt {c+d x}}{x}+2 b \sqrt {a+b x} \sqrt {c+d x} \]

[In]

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

[Out]

2*b*Sqrt[a + b*x]*Sqrt[c + d*x] - ((a + b*x)^(3/2)*Sqrt[c + d*x])/x - (Sqrt[a]*(3*b*c + a*d)*ArcTanh[(Sqrt[c]*
Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/Sqrt[c] + (Sqrt[b]*(b*c + 3*a*d)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqr
t[b]*Sqrt[c + d*x])])/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} \sqrt {c+d x}}{x}+\int \frac {\sqrt {a+b x} \left (\frac {1}{2} (3 b c+a d)+2 b d x\right )}{x \sqrt {c+d x}} \, dx \\ & = 2 b \sqrt {a+b x} \sqrt {c+d x}-\frac {(a+b x)^{3/2} \sqrt {c+d x}}{x}+\frac {\int \frac {\frac {1}{2} a d (3 b c+a d)+\frac {1}{2} b d (b c+3 a d) x}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{d} \\ & = 2 b \sqrt {a+b x} \sqrt {c+d x}-\frac {(a+b x)^{3/2} \sqrt {c+d x}}{x}+\frac {1}{2} (a (3 b c+a d)) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx+\frac {1}{2} (b (b c+3 a d)) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx \\ & = 2 b \sqrt {a+b x} \sqrt {c+d x}-\frac {(a+b x)^{3/2} \sqrt {c+d x}}{x}+(a (3 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 )+(b c+3 a d) \text {Subst}\left (\int \frac {1}{\sqrt {c-\frac {a d}{b}+\frac {d x^2}{b}}} \, dx,x,\sqrt {a+b x}\right ) \\ & = 2 b \sqrt {a+b x} \sqrt {c+d x}-\frac {(a+b x)^{3/2} \sqrt {c+d x}}{x}-\frac {\sqrt {a} (3 b c+a d) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{\sqrt {c}}+(b c+3 a d) \text {Subst}\left (\int \frac {1}{1-\frac {d x^2}{b}} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right ) \\ & = 2 b \sqrt {a+b x} \sqrt {c+d x}-\frac {(a+b x)^{3/2} \sqrt {c+d x}}{x}-\frac {\sqrt {a} (3 b c+a d) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{\sqrt {c}}+\frac {\sqrt {b} (b c+3 a d) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{\sqrt {d}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.35 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.94 \[ \int \frac {(a+b x)^{3/2} \sqrt {c+d x}}{x^2} \, dx=\frac {\sqrt {c+d x} \left (-a^2+b^2 x^2\right )}{x \sqrt {a+b x}}-\frac {\sqrt {a} (3 b c+a d) \text {arctanh}\left (\frac {\sqrt {a} \sqrt {c+d x}}{\sqrt {c} \sqrt {a+b x}}\right )}{\sqrt {c}}+\frac {\sqrt {b} (b c+3 a d) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {a+b x}}\right )}{\sqrt {d}} \]

[In]

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

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(297\) vs. \(2(112)=224\).

Time = 1.53 (sec) , antiderivative size = 298, normalized size of antiderivative = 2.07

method	result	size
default	\(-\frac {\sqrt {b x +a}\, \sqrt {d x +c}\, \left (\ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a^{2} d x \sqrt {b d}+3 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a b c x \sqrt {b d}-3 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a b d x \sqrt {a c}-\ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) b^{2} c x \sqrt {a c}-2 b x \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\right )}{2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, x \sqrt {b d}\, \sqrt {a c}}\)	\(298\)

[In]

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

[Out]

-1/2*(b*x+a)^(1/2)*(d*x+c)^(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*d*x*(b*d
)^(1/2)+3*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*x*(b*d)^(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*b*d*x*(a*c)^(1/2)-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))*b^2*c*x*(a*c)^(1/2)-2*b*x*(b*d)^(1/2)*(a*c)^(1/2)*((b*x+a)*(d*
x+c))^(1/2)+2*a*(b*d)^(1/2)*(a*c)^(1/2)*((b*x+a)*(d*x+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 = 0.57 (sec) , antiderivative size = 893, normalized size of antiderivative = 6.20 \[ \int \frac {(a+b x)^{3/2} \sqrt {c+d x}}{x^2} \, dx=\left [\frac {{\left (b c + 3 \, a d\right )} x \sqrt {\frac {b}{d}} \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^{2} x + b c d + a d^{2}\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {\frac {b}{d}} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x\right ) + {\left (3 \, b c + a d\right )} x \sqrt {\frac {a}{c}} \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^{2} + {\left (b c^{2} + a c d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {\frac {a}{c}} + 8 \, {\left (a b c^{2} + a^{2} c d\right )} x}{x^{2}}\right ) + 4 \, \sqrt {b x + a} {\left (b x - a\right )} \sqrt {d x + c}}{4 \, x}, -\frac {2 \, {\left (b c + 3 \, a d\right )} x \sqrt {-\frac {b}{d}} \arctan \left (\frac {{\left (2 \, b d x + b c + a d\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {-\frac {b}{d}}}{2 \, {\left (b^{2} d x^{2} + a b c + {\left (b^{2} c + a b d\right )} x\right )}}\right ) - {\left (3 \, b c + a d\right )} x \sqrt {\frac {a}{c}} \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^{2} + {\left (b c^{2} + a c d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {\frac {a}{c}} + 8 \, {\left (a b c^{2} + a^{2} c d\right )} x}{x^{2}}\right ) - 4 \, \sqrt {b x + a} {\left (b x - a\right )} \sqrt {d x + c}}{4 \, x}, \frac {2 \, {\left (3 \, b c + a d\right )} x \sqrt {-\frac {a}{c}} \arctan \left (\frac {{\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {-\frac {a}{c}}}{2 \, {\left (a b d x^{2} + a^{2} c + {\left (a b c + a^{2} d\right )} x\right )}}\right ) + {\left (b c + 3 \, a d\right )} x \sqrt {\frac {b}{d}} \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^{2} x + b c d + a d^{2}\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {\frac {b}{d}} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x\right ) + 4 \, \sqrt {b x + a} {\left (b x - a\right )} \sqrt {d x + c}}{4 \, x}, \frac {{\left (3 \, b c + a d\right )} x \sqrt {-\frac {a}{c}} \arctan \left (\frac {{\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {-\frac {a}{c}}}{2 \, {\left (a b d x^{2} + a^{2} c + {\left (a b c + a^{2} d\right )} x\right )}}\right ) - {\left (b c + 3 \, a d\right )} x \sqrt {-\frac {b}{d}} \arctan \left (\frac {{\left (2 \, b d x + b c + a d\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {-\frac {b}{d}}}{2 \, {\left (b^{2} d x^{2} + a b c + {\left (b^{2} c + a b d\right )} x\right )}}\right ) + 2 \, \sqrt {b x + a} {\left (b x - a\right )} \sqrt {d x + c}}{2 \, x}\right ] \]

[In]

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

[Out]

[1/4*((b*c + 3*a*d)*x*sqrt(b/d)*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^2*x + b*c*d + a*d
^2)*sqrt(b*x + a)*sqrt(d*x + c)*sqrt(b/d) + 8*(b^2*c*d + a*b*d^2)*x) + (3*b*c + a*d)*x*sqrt(a/c)*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^2 + (b*c^2 + a*c*d)*x)*sqrt(b*x + a)*sqrt(d*x + c)*sqrt(a/c
) + 8*(a*b*c^2 + a^2*c*d)*x)/x^2) + 4*sqrt(b*x + a)*(b*x - a)*sqrt(d*x + c))/x, -1/4*(2*(b*c + 3*a*d)*x*sqrt(-
b/d)*arctan(1/2*(2*b*d*x + b*c + a*d)*sqrt(b*x + a)*sqrt(d*x + c)*sqrt(-b/d)/(b^2*d*x^2 + a*b*c + (b^2*c + a*b
*d)*x)) - (3*b*c + a*d)*x*sqrt(a/c)*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^2 + (b*c^2
+ a*c*d)*x)*sqrt(b*x + a)*sqrt(d*x + c)*sqrt(a/c) + 8*(a*b*c^2 + a^2*c*d)*x)/x^2) - 4*sqrt(b*x + a)*(b*x - a)
*sqrt(d*x + c))/x, 1/4*(2*(3*b*c + a*d)*x*sqrt(-a/c)*arctan(1/2*(2*a*c + (b*c + a*d)*x)*sqrt(b*x + a)*sqrt(d*x
+ c)*sqrt(-a/c)/(a*b*d*x^2 + a^2*c + (a*b*c + a^2*d)*x)) + (b*c + 3*a*d)*x*sqrt(b/d)*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^2*x + b*c*d + a*d^2)*sqrt(b*x + a)*sqrt(d*x + c)*sqrt(b/d) + 8*(b^2*c*d +
a*b*d^2)*x) + 4*sqrt(b*x + a)*(b*x - a)*sqrt(d*x + c))/x, 1/2*((3*b*c + a*d)*x*sqrt(-a/c)*arctan(1/2*(2*a*c +
(b*c + a*d)*x)*sqrt(b*x + a)*sqrt(d*x + c)*sqrt(-a/c)/(a*b*d*x^2 + a^2*c + (a*b*c + a^2*d)*x)) - (b*c + 3*a*d
)*x*sqrt(-b/d)*arctan(1/2*(2*b*d*x + b*c + a*d)*sqrt(b*x + a)*sqrt(d*x + c)*sqrt(-b/d)/(b^2*d*x^2 + a*b*c + (b
^2*c + a*b*d)*x)) + 2*sqrt(b*x + a)*(b*x - a)*sqrt(d*x + c))/x]

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F(-2)]

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

[In]

integrate((b*x+a)^(3/2)*(d*x+c)^(1/2)/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. 525 vs. \(2 (112) = 224\).

Time = 0.45 (sec) , antiderivative size = 525, normalized size of antiderivative = 3.65 \[ \int \frac {(a+b x)^{3/2} \sqrt {c+d x}}{x^2} \, dx=\frac {2 \, \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} \sqrt {b x + a} {\left | b \right |} - \frac {{\left (\sqrt {b d} b c {\left | b \right |} + 3 \, \sqrt {b d} a d {\left | b \right |}\right )} \log \left ({\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2}\right )}{d} - \frac {2 \, {\left (3 \, \sqrt {b d} a b^{2} c {\left | b \right |} + \sqrt {b d} a^{2} b d {\left | b \right |}\right )} \arctan \left (-\frac {b^{2} c + a b d - {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2}}{2 \, \sqrt {-a b c d} b}\right )}{\sqrt {-a b c d} b} - \frac {4 \, {\left (\sqrt {b d} a b^{4} c^{2} {\left | b \right |} - 2 \, \sqrt {b d} a^{2} b^{3} c d {\left | b \right |} + \sqrt {b d} a^{3} b^{2} d^{2} {\left | b \right |} - \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a b^{2} c {\left | b \right |} - \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a^{2} b d {\left | b \right |}\right )}}{b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2} - 2 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} b^{2} c - 2 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a b d + {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{4}}}{2 \, b} \]

[In]

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

[Out]

1/2*(2*sqrt(b^2*c + (b*x + a)*b*d - a*b*d)*sqrt(b*x + a)*abs(b) - (sqrt(b*d)*b*c*abs(b) + 3*sqrt(b*d)*a*d*abs(
b))*log((sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2)/d - 2*(3*sqrt(b*d)*a*b^2*c*abs(b) +
sqrt(b*d)*a^2*b*d*abs(b))*arctan(-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) - 4*(sqrt(b*d)*a*b^4*c^2*abs(b) - 2*sqrt(b*d)*a^2*b^3*c*d*
abs(b) + sqrt(b*d)*a^3*b^2*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*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*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. \[ \int \frac {(a+b x)^{3/2} \sqrt {c+d x}}{x^2} \, dx=\int \frac {{\left (a+b\,x\right )}^{3/2}\,\sqrt {c+d\,x}}{x^2} \,d x \]

[In]

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

[Out]

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