\(\int \frac {(c+d x)^{3/2}}{x^2 \sqrt {a+b x}} \, dx\) [710]

Optimal result

Integrand size = 22, antiderivative size = 119 \[ \int \frac {(c+d x)^{3/2}}{x^2 \sqrt {a+b x}} \, dx=-\frac {c \sqrt {a+b x} \sqrt {c+d x}}{a x}+\frac {\sqrt {c} (b c-3 a d) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{a^{3/2}}+\frac {2 d^{3/2} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{\sqrt {b}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {100, 163, 65, 223, 212, 95, 214} \[ \int \frac {(c+d x)^{3/2}}{x^2 \sqrt {a+b x}} \, dx=\frac {\sqrt {c} (b c-3 a d) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{a^{3/2}}+\frac {2 d^{3/2} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{\sqrt {b}}-\frac {c \sqrt {a+b x} \sqrt {c+d x}}{a x} \]

[In]

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

[Out]

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

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 100

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

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

Mathematica [A] (verified)

Time = 0.33 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.00 \[ \int \frac {(c+d x)^{3/2}}{x^2 \sqrt {a+b x}} \, dx=-\frac {c \sqrt {a+b x} \sqrt {c+d x}}{a x}+\frac {\sqrt {c} (b c-3 a d) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{a^{3/2}}+\frac {2 d^{3/2} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{\sqrt {b}} \]

[In]

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

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(222\) vs. \(2(91)=182\).

Time = 0.57 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.87

method	result	size
default	\(\frac {\sqrt {d x +c}\, \sqrt {b x +a}\, \left (2 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a \,d^{2} x \sqrt {a c}-3 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a c d x \sqrt {b d}+\ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) b \,c^{2} x \sqrt {b d}-2 c \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\right )}{2 a \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, x \sqrt {b d}\, \sqrt {a c}}\)	\(223\)

[In]

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

[Out]

1/2*(d*x+c)^(1/2)*(b*x+a)^(1/2)/a*(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
))*a*d^2*x*(a*c)^(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*c*d*x*(b*d)^(1/2)+l
n((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*b*c^2*x*(b*d)^(1/2)-2*c*(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 [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 187 vs. \(2 (91) = 182\).

Time = 0.47 (sec) , antiderivative size = 858, normalized size of antiderivative = 7.21 \[ \int \frac {(c+d x)^{3/2}}{x^2 \sqrt {a+b x}} \, dx=\left [\frac {2 \, a d x \sqrt {\frac {d}{b}} \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^{2} d x + b^{2} c + a b d\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {\frac {d}{b}} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x\right ) - {\left (b c - 3 \, a d\right )} x \sqrt {\frac {c}{a}} \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^{2} c + {\left (a b c + a^{2} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {\frac {c}{a}} + 8 \, {\left (a b c^{2} + a^{2} c d\right )} x}{x^{2}}\right ) - 4 \, \sqrt {b x + a} \sqrt {d x + c} c}{4 \, a x}, -\frac {4 \, a d x \sqrt {-\frac {d}{b}} \arctan \left (\frac {{\left (2 \, b d x + b c + a d\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {-\frac {d}{b}}}{2 \, {\left (b d^{2} x^{2} + a c d + {\left (b c d + a d^{2}\right )} x\right )}}\right ) + {\left (b c - 3 \, a d\right )} x \sqrt {\frac {c}{a}} \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^{2} c + {\left (a b c + a^{2} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {\frac {c}{a}} + 8 \, {\left (a b c^{2} + a^{2} c d\right )} x}{x^{2}}\right ) + 4 \, \sqrt {b x + a} \sqrt {d x + c} c}{4 \, a x}, \frac {a d x \sqrt {\frac {d}{b}} \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^{2} d x + b^{2} c + a b d\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {\frac {d}{b}} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x\right ) - {\left (b c - 3 \, a d\right )} x \sqrt {-\frac {c}{a}} \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 {c}{a}}}{2 \, {\left (b c d x^{2} + a c^{2} + {\left (b c^{2} + a c d\right )} x\right )}}\right ) - 2 \, \sqrt {b x + a} \sqrt {d x + c} c}{2 \, a x}, -\frac {2 \, a d x \sqrt {-\frac {d}{b}} \arctan \left (\frac {{\left (2 \, b d x + b c + a d\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {-\frac {d}{b}}}{2 \, {\left (b d^{2} x^{2} + a c d + {\left (b c d + a d^{2}\right )} x\right )}}\right ) + {\left (b c - 3 \, a d\right )} x \sqrt {-\frac {c}{a}} \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 {c}{a}}}{2 \, {\left (b c d x^{2} + a c^{2} + {\left (b c^{2} + a c d\right )} x\right )}}\right ) + 2 \, \sqrt {b x + a} \sqrt {d x + c} c}{2 \, a x}\right ] \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F(-2)]

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

[In]

integrate((d*x+c)^(3/2)/x^2/(b*x+a)^(1/2),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. 484 vs. \(2 (91) = 182\).

Time = 0.42 (sec) , antiderivative size = 484, normalized size of antiderivative = 4.07 \[ \int \frac {(c+d x)^{3/2}}{x^2 \sqrt {a+b x}} \, dx=-\frac {\frac {\sqrt {b d} d {\left | b \right |} \log \left ({\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2}\right )}{b} - \frac {{\left (\sqrt {b d} b^{2} c^{2} {\left | b \right |} - 3 \, \sqrt {b d} a b c d {\left | b \right |}\right )} \arctan \left (-\frac {b^{2} c + a b d - {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2}}{2 \, \sqrt {-a b c d} b}\right )}{\sqrt {-a b c d} a b} + \frac {2 \, {\left (\sqrt {b d} b^{4} c^{3} {\left | b \right |} - 2 \, \sqrt {b d} a b^{3} c^{2} d {\left | b \right |} + \sqrt {b d} a^{2} b^{2} c d^{2} {\left | b \right |} - \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} b^{2} c^{2} {\left | b \right |} - \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a b c d {\left | b \right |}\right )}}{{\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2} - 2 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} b^{2} c - 2 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a b d + {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{4}\right )} a}}{b} \]

[In]

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

[Out]

-(sqrt(b*d)*d*abs(b)*log((sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2)/b - (sqrt(b*d)*b^2
*c^2*abs(b) - 3*sqrt(b*d)*a*b*c*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)*a*b) + 2*(sqrt(b*d)*b^4*c^3*abs(b) - 2*sqrt(b*
d)*a*b^3*c^2*d*abs(b) + sqrt(b*d)*a^2*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*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*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)*a))/b

Mupad [F(-1)]

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

[In]

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

[Out]

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