\(\int \frac {(a+b x)^{3/2}}{x^2 (c+d x)^{3/2}} \, dx\) [638]

Optimal result

Integrand size = 22, antiderivative size = 108 \[ \int \frac {(a+b x)^{3/2}}{x^2 (c+d x)^{3/2}} \, dx=\frac {3 (b c-a d) \sqrt {a+b x}}{c^2 \sqrt {c+d x}}-\frac {(a+b x)^{3/2}}{c x \sqrt {c+d x}}-\frac {3 \sqrt {a} (b c-a d) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{c^{5/2}} \]

[Out]

-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^(5/2)-(b*x+a)^(3/2)/c/x/(d*x+c)^(
1/2)+3*(-a*d+b*c)*(b*x+a)^(1/2)/c^2/(d*x+c)^(1/2)

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {96, 95, 214} \[ \int \frac {(a+b x)^{3/2}}{x^2 (c+d x)^{3/2}} \, dx=-\frac {3 \sqrt {a} (b c-a d) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{c^{5/2}}+\frac {3 \sqrt {a+b x} (b c-a d)}{c^2 \sqrt {c+d x}}-\frac {(a+b x)^{3/2}}{c x \sqrt {c+d x}} \]

[In]

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

[Out]

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

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 96

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 + 1)/((m + 1)*(b*e - a*f))), x] - Dist[n*((d*e - c*f)/((m + 1)*(b*e - a*f
))), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[
m + n + p + 2, 0] && GtQ[n, 0] && (SumSimplerQ[m, 1] ||  !SumSimplerQ[p, 1]) && NeQ[m, -1]

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 {(a+b x)^{3/2}}{c x \sqrt {c+d x}}+\frac {(3 (b c-a d)) \int \frac {\sqrt {a+b x}}{x (c+d x)^{3/2}} \, dx}{2 c} \\ & = \frac {3 (b c-a d) \sqrt {a+b x}}{c^2 \sqrt {c+d x}}-\frac {(a+b x)^{3/2}}{c x \sqrt {c+d x}}+\frac {(3 a (b c-a d)) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{2 c^2} \\ & = \frac {3 (b c-a d) \sqrt {a+b x}}{c^2 \sqrt {c+d x}}-\frac {(a+b x)^{3/2}}{c x \sqrt {c+d x}}+\frac {(3 a (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 )}{c^2} \\ & = \frac {3 (b c-a d) \sqrt {a+b x}}{c^2 \sqrt {c+d x}}-\frac {(a+b x)^{3/2}}{c x \sqrt {c+d x}}-\frac {3 \sqrt {a} (b c-a d) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{c^{5/2}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 10.89 (sec) , antiderivative size = 1648, normalized size of antiderivative = 15.26 \[ \int \frac {(a+b x)^{3/2}}{x^2 (c+d x)^{3/2}} \, dx=\frac {\frac {\sqrt {c} (-2 b c x+a (c+3 d x)) \left (-4 a^2 d+b^2 x (3 c-d x)+b \sqrt {a-\frac {b c}{d}} \sqrt {a+b x} (-c+3 d x)+a \left (3 b c-5 b d x+4 \sqrt {a-\frac {b c}{d}} d \sqrt {a+b x}\right )\right )}{x \sqrt {c+d x} \left (b c \left (\sqrt {a-\frac {b c}{d}}-3 \sqrt {a+b x}\right )+b d x \left (-3 \sqrt {a-\frac {b c}{d}}+\sqrt {a+b x}\right )+a d \left (-4 \sqrt {a-\frac {b c}{d}}+4 \sqrt {a+b x}\right )\right )}+\frac {3 a b c \sqrt {d} \arctan \left (\frac {\sqrt {b c-2 a d-2 i \sqrt {a} \sqrt {d} \sqrt {b c-a d}} \sqrt {c+d x}}{\sqrt {c} \sqrt {d} \left (\sqrt {a-\frac {b c}{d}}-\sqrt {a+b x}\right )}\right )}{\sqrt {b c-2 a d-2 i \sqrt {a} \sqrt {d} \sqrt {b c-a d}}}-\frac {3 a^2 d^{3/2} \arctan \left (\frac {\sqrt {b c-2 a d-2 i \sqrt {a} \sqrt {d} \sqrt {b c-a d}} \sqrt {c+d x}}{\sqrt {c} \sqrt {d} \left (\sqrt {a-\frac {b c}{d}}-\sqrt {a+b x}\right )}\right )}{\sqrt {b c-2 a d-2 i \sqrt {a} \sqrt {d} \sqrt {b c-a d}}}+\frac {3 i \sqrt {a} b^2 c^2 \arctan \left (\frac {\sqrt {b c-2 a d-2 i \sqrt {a} \sqrt {d} \sqrt {b c-a d}} \sqrt {c+d x}}{\sqrt {c} \sqrt {d} \left (\sqrt {a-\frac {b c}{d}}-\sqrt {a+b x}\right )}\right )}{\sqrt {b c-a d} \sqrt {b c-2 a d-2 i \sqrt {a} \sqrt {d} \sqrt {b c-a d}}}-\frac {6 i a^{3/2} b c d \arctan \left (\frac {\sqrt {b c-2 a d-2 i \sqrt {a} \sqrt {d} \sqrt {b c-a d}} \sqrt {c+d x}}{\sqrt {c} \sqrt {d} \left (\sqrt {a-\frac {b c}{d}}-\sqrt {a+b x}\right )}\right )}{\sqrt {b c-a d} \sqrt {b c-2 a d-2 i \sqrt {a} \sqrt {d} \sqrt {b c-a d}}}+\frac {3 i a^{5/2} d^2 \arctan \left (\frac {\sqrt {b c-2 a d-2 i \sqrt {a} \sqrt {d} \sqrt {b c-a d}} \sqrt {c+d x}}{\sqrt {c} \sqrt {d} \left (\sqrt {a-\frac {b c}{d}}-\sqrt {a+b x}\right )}\right )}{\sqrt {b c-a d} \sqrt {b c-2 a d-2 i \sqrt {a} \sqrt {d} \sqrt {b c-a d}}}+\frac {3 a b c \sqrt {d} \arctan \left (\frac {\sqrt {b c-2 a d+2 i \sqrt {a} \sqrt {d} \sqrt {b c-a d}} \sqrt {c+d x}}{\sqrt {c} \sqrt {d} \left (\sqrt {a-\frac {b c}{d}}-\sqrt {a+b x}\right )}\right )}{\sqrt {b c-2 a d+2 i \sqrt {a} \sqrt {d} \sqrt {b c-a d}}}-\frac {3 a^2 d^{3/2} \arctan \left (\frac {\sqrt {b c-2 a d+2 i \sqrt {a} \sqrt {d} \sqrt {b c-a d}} \sqrt {c+d x}}{\sqrt {c} \sqrt {d} \left (\sqrt {a-\frac {b c}{d}}-\sqrt {a+b x}\right )}\right )}{\sqrt {b c-2 a d+2 i \sqrt {a} \sqrt {d} \sqrt {b c-a d}}}-\frac {3 i \sqrt {a} b^2 c^2 \arctan \left (\frac {\sqrt {b c-2 a d+2 i \sqrt {a} \sqrt {d} \sqrt {b c-a d}} \sqrt {c+d x}}{\sqrt {c} \sqrt {d} \left (\sqrt {a-\frac {b c}{d}}-\sqrt {a+b x}\right )}\right )}{\sqrt {b c-a d} \sqrt {b c-2 a d+2 i \sqrt {a} \sqrt {d} \sqrt {b c-a d}}}+\frac {6 i a^{3/2} b c d \arctan \left (\frac {\sqrt {b c-2 a d+2 i \sqrt {a} \sqrt {d} \sqrt {b c-a d}} \sqrt {c+d x}}{\sqrt {c} \sqrt {d} \left (\sqrt {a-\frac {b c}{d}}-\sqrt {a+b x}\right )}\right )}{\sqrt {b c-a d} \sqrt {b c-2 a d+2 i \sqrt {a} \sqrt {d} \sqrt {b c-a d}}}-\frac {3 i a^{5/2} d^2 \arctan \left (\frac {\sqrt {b c-2 a d+2 i \sqrt {a} \sqrt {d} \sqrt {b c-a d}} \sqrt {c+d x}}{\sqrt {c} \sqrt {d} \left (\sqrt {a-\frac {b c}{d}}-\sqrt {a+b x}\right )}\right )}{\sqrt {b c-a d} \sqrt {b c-2 a d+2 i \sqrt {a} \sqrt {d} \sqrt {b c-a d}}}}{c^{5/2}} \]

[In]

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

[Out]

((Sqrt[c]*(-2*b*c*x + a*(c + 3*d*x))*(-4*a^2*d + b^2*x*(3*c - d*x) + b*Sqrt[a - (b*c)/d]*Sqrt[a + b*x]*(-c + 3
*d*x) + a*(3*b*c - 5*b*d*x + 4*Sqrt[a - (b*c)/d]*d*Sqrt[a + b*x])))/(x*Sqrt[c + d*x]*(b*c*(Sqrt[a - (b*c)/d] -
3*Sqrt[a + b*x]) + b*d*x*(-3*Sqrt[a - (b*c)/d] + Sqrt[a + b*x]) + a*d*(-4*Sqrt[a - (b*c)/d] + 4*Sqrt[a + b*x]
))) + (3*a*b*c*Sqrt[d]*ArcTan[(Sqrt[b*c - 2*a*d - (2*I)*Sqrt[a]*Sqrt[d]*Sqrt[b*c - a*d]]*Sqrt[c + d*x])/(Sqrt[
c]*Sqrt[d]*(Sqrt[a - (b*c)/d] - Sqrt[a + b*x]))])/Sqrt[b*c - 2*a*d - (2*I)*Sqrt[a]*Sqrt[d]*Sqrt[b*c - a*d]] -
(3*a^2*d^(3/2)*ArcTan[(Sqrt[b*c - 2*a*d - (2*I)*Sqrt[a]*Sqrt[d]*Sqrt[b*c - a*d]]*Sqrt[c + d*x])/(Sqrt[c]*Sqrt[
d]*(Sqrt[a - (b*c)/d] - Sqrt[a + b*x]))])/Sqrt[b*c - 2*a*d - (2*I)*Sqrt[a]*Sqrt[d]*Sqrt[b*c - a*d]] + ((3*I)*S
qrt[a]*b^2*c^2*ArcTan[(Sqrt[b*c - 2*a*d - (2*I)*Sqrt[a]*Sqrt[d]*Sqrt[b*c - a*d]]*Sqrt[c + d*x])/(Sqrt[c]*Sqrt[
d]*(Sqrt[a - (b*c)/d] - Sqrt[a + b*x]))])/(Sqrt[b*c - a*d]*Sqrt[b*c - 2*a*d - (2*I)*Sqrt[a]*Sqrt[d]*Sqrt[b*c -
a*d]]) - ((6*I)*a^(3/2)*b*c*d*ArcTan[(Sqrt[b*c - 2*a*d - (2*I)*Sqrt[a]*Sqrt[d]*Sqrt[b*c - a*d]]*Sqrt[c + d*x]
)/(Sqrt[c]*Sqrt[d]*(Sqrt[a - (b*c)/d] - Sqrt[a + b*x]))])/(Sqrt[b*c - a*d]*Sqrt[b*c - 2*a*d - (2*I)*Sqrt[a]*Sq
rt[d]*Sqrt[b*c - a*d]]) + ((3*I)*a^(5/2)*d^2*ArcTan[(Sqrt[b*c - 2*a*d - (2*I)*Sqrt[a]*Sqrt[d]*Sqrt[b*c - a*d]]
*Sqrt[c + d*x])/(Sqrt[c]*Sqrt[d]*(Sqrt[a - (b*c)/d] - Sqrt[a + b*x]))])/(Sqrt[b*c - a*d]*Sqrt[b*c - 2*a*d - (2
*I)*Sqrt[a]*Sqrt[d]*Sqrt[b*c - a*d]]) + (3*a*b*c*Sqrt[d]*ArcTan[(Sqrt[b*c - 2*a*d + (2*I)*Sqrt[a]*Sqrt[d]*Sqrt
[b*c - a*d]]*Sqrt[c + d*x])/(Sqrt[c]*Sqrt[d]*(Sqrt[a - (b*c)/d] - Sqrt[a + b*x]))])/Sqrt[b*c - 2*a*d + (2*I)*S
qrt[a]*Sqrt[d]*Sqrt[b*c - a*d]] - (3*a^2*d^(3/2)*ArcTan[(Sqrt[b*c - 2*a*d + (2*I)*Sqrt[a]*Sqrt[d]*Sqrt[b*c - a
*d]]*Sqrt[c + d*x])/(Sqrt[c]*Sqrt[d]*(Sqrt[a - (b*c)/d] - Sqrt[a + b*x]))])/Sqrt[b*c - 2*a*d + (2*I)*Sqrt[a]*S
qrt[d]*Sqrt[b*c - a*d]] - ((3*I)*Sqrt[a]*b^2*c^2*ArcTan[(Sqrt[b*c - 2*a*d + (2*I)*Sqrt[a]*Sqrt[d]*Sqrt[b*c - a
*d]]*Sqrt[c + d*x])/(Sqrt[c]*Sqrt[d]*(Sqrt[a - (b*c)/d] - Sqrt[a + b*x]))])/(Sqrt[b*c - a*d]*Sqrt[b*c - 2*a*d
+ (2*I)*Sqrt[a]*Sqrt[d]*Sqrt[b*c - a*d]]) + ((6*I)*a^(3/2)*b*c*d*ArcTan[(Sqrt[b*c - 2*a*d + (2*I)*Sqrt[a]*Sqrt
[d]*Sqrt[b*c - a*d]]*Sqrt[c + d*x])/(Sqrt[c]*Sqrt[d]*(Sqrt[a - (b*c)/d] - Sqrt[a + b*x]))])/(Sqrt[b*c - a*d]*S
qrt[b*c - 2*a*d + (2*I)*Sqrt[a]*Sqrt[d]*Sqrt[b*c - a*d]]) - ((3*I)*a^(5/2)*d^2*ArcTan[(Sqrt[b*c - 2*a*d + (2*I
)*Sqrt[a]*Sqrt[d]*Sqrt[b*c - a*d]]*Sqrt[c + d*x])/(Sqrt[c]*Sqrt[d]*(Sqrt[a - (b*c)/d] - Sqrt[a + b*x]))])/(Sqr
t[b*c - a*d]*Sqrt[b*c - 2*a*d + (2*I)*Sqrt[a]*Sqrt[d]*Sqrt[b*c - a*d]]))/c^(5/2)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(297\) vs. \(2(88)=176\).

Time = 0.55 (sec) , antiderivative size = 298, normalized size of antiderivative = 2.76

method	result	size
default	\(\frac {\sqrt {b x +a}\, \left (3 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a^{2} d^{2} x^{2}-3 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a b c d \,x^{2}+3 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a^{2} c d x -3 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a b \,c^{2} x -6 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a d x +4 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, b c x -2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a c \sqrt {a c}\right )}{2 c^{2} \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, x \sqrt {a c}\, \sqrt {d x +c}}\)	\(298\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.38 (sec) , antiderivative size = 343, normalized size of antiderivative = 3.18 \[ \int \frac {(a+b x)^{3/2}}{x^2 (c+d x)^{3/2}} \, dx=\left [-\frac {3 \, {\left ({\left (b c d - a d^{2}\right )} x^{2} + {\left (b c^{2} - a c d\right )} x\right )} \sqrt {\frac {a}{c}} \log \left (\frac {8 \, a^{2} c^{2} + {\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} + 4 \, {\left (2 \, a c^{2} + {\left (b c^{2} + a c d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {\frac {a}{c}} + 8 \, {\left (a b c^{2} + a^{2} c d\right )} x}{x^{2}}\right ) + 4 \, {\left (a c - {\left (2 \, b c - 3 \, a d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{4 \, {\left (c^{2} d x^{2} + c^{3} x\right )}}, \frac {3 \, {\left ({\left (b c d - a d^{2}\right )} x^{2} + {\left (b c^{2} - a c d\right )} x\right )} \sqrt {-\frac {a}{c}} \arctan \left (\frac {{\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {-\frac {a}{c}}}{2 \, {\left (a b d x^{2} + a^{2} c + {\left (a b c + a^{2} d\right )} x\right )}}\right ) - 2 \, {\left (a c - {\left (2 \, b c - 3 \, a d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (c^{2} d x^{2} + c^{3} x\right )}}\right ] \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F(-2)]

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

[In]

integrate((b*x+a)^(3/2)/x^2/(d*x+c)^(3/2),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. 475 vs. \(2 (88) = 176\).

Time = 1.02 (sec) , antiderivative size = 475, normalized size of antiderivative = 4.40 \[ \int \frac {(a+b x)^{3/2}}{x^2 (c+d x)^{3/2}} \, dx=\frac {2 \, {\left (b^{3} c - a b^{2} d\right )} \sqrt {b x + a}}{\sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} c^{2} {\left | b \right |}} - \frac {3 \, {\left (\sqrt {b d} a b^{3} c - \sqrt {b d} a^{2} b^{2} d\right )} \arctan \left (-\frac {b^{2} c + a b d - {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2}}{2 \, \sqrt {-a b c d} b}\right )}{\sqrt {-a b c d} b c^{2} {\left | b \right |}} - \frac {2 \, {\left (\sqrt {b d} a b^{5} c^{2} - 2 \, \sqrt {b d} a^{2} b^{4} c d + \sqrt {b d} a^{3} b^{3} d^{2} - \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a b^{3} c - \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a^{2} b^{2} d\right )}}{{\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2} - 2 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} b^{2} c - 2 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a b d + {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{4}\right )} c^{2} {\left | b \right |}} \]

[In]

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

[Out]

2*(b^3*c - a*b^2*d)*sqrt(b*x + a)/(sqrt(b^2*c + (b*x + a)*b*d - a*b*d)*c^2*abs(b)) - 3*(sqrt(b*d)*a*b^3*c - sq
rt(b*d)*a^2*b^2*d)*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*c^2*abs(b)) - 2*(sqrt(b*d)*a*b^5*c^2 - 2*sqrt(b*d)*a^2*b^4*c*d + sq
rt(b*d)*a^3*b^3*d^2 - 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^3*c - sq
rt(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^2*d)/((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)*c^2*abs(b))

Mupad [F(-1)]

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

[In]

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

[Out]

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