\(\int \frac {(a+b x)^{5/2} (c+d x)^{3/2}}{x^2} \, dx\) [660]

Optimal result

Integrand size = 22, antiderivative size = 259 \[ \int \frac {(a+b x)^{5/2} (c+d x)^{3/2}}{x^2} \, dx=-\frac {\left (b^2 c^2-14 a b c d-19 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{8 d}+\frac {b (b c+7 a d) \sqrt {a+b x} (c+d x)^{3/2}}{4 d}+\frac {4}{3} b (a+b x)^{3/2} (c+d x)^{3/2}-\frac {(a+b x)^{5/2} (c+d x)^{3/2}}{x}-a^{3/2} \sqrt {c} (5 b c+3 a d) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )-\frac {\left (b^3 c^3-15 a b^2 c^2 d-45 a^2 b c d^2-5 a^3 d^3\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{8 \sqrt {b} d^{3/2}} \]

[Out]

4/3*b*(b*x+a)^(3/2)*(d*x+c)^(3/2)-(b*x+a)^(5/2)*(d*x+c)^(3/2)/x-1/8*(-5*a^3*d^3-45*a^2*b*c*d^2-15*a*b^2*c^2*d+
b^3*c^3)*arctanh(d^(1/2)*(b*x+a)^(1/2)/b^(1/2)/(d*x+c)^(1/2))/d^(3/2)/b^(1/2)-a^(3/2)*(3*a*d+5*b*c)*arctanh(c^
(1/2)*(b*x+a)^(1/2)/a^(1/2)/(d*x+c)^(1/2))*c^(1/2)+1/4*b*(7*a*d+b*c)*(d*x+c)^(3/2)*(b*x+a)^(1/2)/d-1/8*(-19*a^
2*d^2-14*a*b*c*d+b^2*c^2)*(b*x+a)^(1/2)*(d*x+c)^(1/2)/d

Rubi [A] (verified)

Time = 0.20 (sec) , antiderivative size = 259, normalized size of antiderivative = 1.00, number of steps used = 10, 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} \[ \int \frac {(a+b x)^{5/2} (c+d x)^{3/2}}{x^2} \, dx=-a^{3/2} \sqrt {c} (3 a d+5 b c) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (-19 a^2 d^2-14 a b c d+b^2 c^2\right )}{8 d}-\frac {\left (-5 a^3 d^3-45 a^2 b c d^2-15 a b^2 c^2 d+b^3 c^3\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{8 \sqrt {b} d^{3/2}}-\frac {(a+b x)^{5/2} (c+d x)^{3/2}}{x}+\frac {4}{3} b (a+b x)^{3/2} (c+d x)^{3/2}+\frac {b \sqrt {a+b x} (c+d x)^{3/2} (7 a d+b c)}{4 d} \]

[In]

Int[((a + b*x)^(5/2)*(c + d*x)^(3/2))/x^2,x]

[Out]

-1/8*((b^2*c^2 - 14*a*b*c*d - 19*a^2*d^2)*Sqrt[a + b*x]*Sqrt[c + d*x])/d + (b*(b*c + 7*a*d)*Sqrt[a + b*x]*(c +
d*x)^(3/2))/(4*d) + (4*b*(a + b*x)^(3/2)*(c + d*x)^(3/2))/3 - ((a + b*x)^(5/2)*(c + d*x)^(3/2))/x - a^(3/2)*S
qrt[c]*(5*b*c + 3*a*d)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])] - ((b^3*c^3 - 15*a*b^2*c^2*d -
45*a^2*b*c*d^2 - 5*a^3*d^3)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(8*Sqrt[b]*d^(3/2))

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)^{5/2} (c+d x)^{3/2}}{x}+\int \frac {(a+b x)^{3/2} \sqrt {c+d x} \left (\frac {1}{2} (5 b c+3 a d)+4 b d x\right )}{x} \, dx \\ & = \frac {4}{3} b (a+b x)^{3/2} (c+d x)^{3/2}-\frac {(a+b x)^{5/2} (c+d x)^{3/2}}{x}+\frac {\int \frac {\sqrt {a+b x} \sqrt {c+d x} \left (\frac {3}{2} a d (5 b c+3 a d)+\frac {3}{2} b d (b c+7 a d) x\right )}{x} \, dx}{3 d} \\ & = \frac {b (b c+7 a d) \sqrt {a+b x} (c+d x)^{3/2}}{4 d}+\frac {4}{3} b (a+b x)^{3/2} (c+d x)^{3/2}-\frac {(a+b x)^{5/2} (c+d x)^{3/2}}{x}+\frac {\int \frac {\sqrt {c+d x} \left (3 a^2 d^2 (5 b c+3 a d)-\frac {3}{4} b d \left (b^2 c^2-14 a b c d-19 a^2 d^2\right ) x\right )}{x \sqrt {a+b x}} \, dx}{6 d^2} \\ & = -\frac {\left (b^2 c^2-14 a b c d-19 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{8 d}+\frac {b (b c+7 a d) \sqrt {a+b x} (c+d x)^{3/2}}{4 d}+\frac {4}{3} b (a+b x)^{3/2} (c+d x)^{3/2}-\frac {(a+b x)^{5/2} (c+d x)^{3/2}}{x}+\frac {\int \frac {3 a^2 b c d^2 (5 b c+3 a d)-\frac {3}{8} b d \left (b^3 c^3-15 a b^2 c^2 d-45 a^2 b c d^2-5 a^3 d^3\right ) x}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{6 b d^2} \\ & = -\frac {\left (b^2 c^2-14 a b c d-19 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{8 d}+\frac {b (b c+7 a d) \sqrt {a+b x} (c+d x)^{3/2}}{4 d}+\frac {4}{3} b (a+b x)^{3/2} (c+d x)^{3/2}-\frac {(a+b x)^{5/2} (c+d x)^{3/2}}{x}+\frac {1}{2} \left (a^2 c (5 b c+3 a d)\right ) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx-\frac {\left (b^3 c^3-15 a b^2 c^2 d-45 a^2 b c d^2-5 a^3 d^3\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{16 d} \\ & = -\frac {\left (b^2 c^2-14 a b c d-19 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{8 d}+\frac {b (b c+7 a d) \sqrt {a+b x} (c+d x)^{3/2}}{4 d}+\frac {4}{3} b (a+b x)^{3/2} (c+d x)^{3/2}-\frac {(a+b x)^{5/2} (c+d x)^{3/2}}{x}+\left (a^2 c (5 b c+3 a d)\right ) \text {Subst}\left (\int \frac {1}{-a+c x^2} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )-\frac {\left (b^3 c^3-15 a b^2 c^2 d-45 a^2 b c d^2-5 a^3 d^3\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 )}{8 b d} \\ & = -\frac {\left (b^2 c^2-14 a b c d-19 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{8 d}+\frac {b (b c+7 a d) \sqrt {a+b x} (c+d x)^{3/2}}{4 d}+\frac {4}{3} b (a+b x)^{3/2} (c+d x)^{3/2}-\frac {(a+b x)^{5/2} (c+d x)^{3/2}}{x}-a^{3/2} \sqrt {c} (5 b c+3 a d) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )-\frac {\left (b^3 c^3-15 a b^2 c^2 d-45 a^2 b c d^2-5 a^3 d^3\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 )}{8 b d} \\ & = -\frac {\left (b^2 c^2-14 a b c d-19 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{8 d}+\frac {b (b c+7 a d) \sqrt {a+b x} (c+d x)^{3/2}}{4 d}+\frac {4}{3} b (a+b x)^{3/2} (c+d x)^{3/2}-\frac {(a+b x)^{5/2} (c+d x)^{3/2}}{x}-a^{3/2} \sqrt {c} (5 b c+3 a d) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )-\frac {\left (b^3 c^3-15 a b^2 c^2 d-45 a^2 b c d^2-5 a^3 d^3\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{8 \sqrt {b} d^{3/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.73 (sec) , antiderivative size = 213, normalized size of antiderivative = 0.82 \[ \int \frac {(a+b x)^{5/2} (c+d x)^{3/2}}{x^2} \, dx=\frac {\sqrt {a+b x} \sqrt {c+d x} \left (3 a^2 d (-8 c+11 d x)+2 a b d x (34 c+13 d x)+b^2 x \left (3 c^2+14 c d x+8 d^2 x^2\right )\right )}{24 d x}-a^{3/2} \sqrt {c} (5 b c+3 a d) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )-\frac {\left (b^3 c^3-15 a b^2 c^2 d-45 a^2 b c d^2-5 a^3 d^3\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{8 \sqrt {b} d^{3/2}} \]

[In]

Integrate[((a + b*x)^(5/2)*(c + d*x)^(3/2))/x^2,x]

[Out]

(Sqrt[a + b*x]*Sqrt[c + d*x]*(3*a^2*d*(-8*c + 11*d*x) + 2*a*b*d*x*(34*c + 13*d*x) + b^2*x*(3*c^2 + 14*c*d*x +
8*d^2*x^2)))/(24*d*x) - a^(3/2)*Sqrt[c]*(5*b*c + 3*a*d)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x]
)] - ((b^3*c^3 - 15*a*b^2*c^2*d - 45*a^2*b*c*d^2 - 5*a^3*d^3)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c
+ d*x])])/(8*Sqrt[b]*d^(3/2))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(597\) vs. \(2(211)=422\).

Time = 0.56 (sec) , antiderivative size = 598, normalized size of antiderivative = 2.31

method	result	size
default	\(\frac {\sqrt {b x +a}\, \sqrt {d x +c}\, \left (16 b^{2} d^{2} x^{3} \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+15 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) \sqrt {a c}\, a^{3} d^{3} x +135 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) \sqrt {a c}\, a^{2} b c \,d^{2} x +45 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) \sqrt {a c}\, a \,b^{2} c^{2} d x -3 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) \sqrt {a c}\, b^{3} c^{3} x -72 \sqrt {b d}\, \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a^{3} c \,d^{2} x -120 \sqrt {b d}\, \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a^{2} b \,c^{2} d x +52 \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a b \,d^{2} x^{2}+28 \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, b^{2} c d \,x^{2}+66 \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a^{2} d^{2} x +136 \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a b c d x +6 \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, b^{2} c^{2} x -48 \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a^{2} c d \right )}{48 d \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, x \sqrt {a c}}\)	\(598\)

[In]

int((b*x+a)^(5/2)*(d*x+c)^(3/2)/x^2,x,method=_RETURNVERBOSE)

[Out]

1/48*(b*x+a)^(1/2)*(d*x+c)^(1/2)*(16*b^2*d^2*x^3*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+15*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^3*d^3*x+135*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*b*c*d^2*x+45*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^2*c^2*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^3*c^3*x-72*(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^3*c*d^2*x-120*(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*b*c^2*d*x+52*(b*d)^(1/2)*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a*b*d^2*x^2+28*(b*d)^(1/2
)*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*b^2*c*d*x^2+66*(b*d)^(1/2)*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a^2*d^2*x
+136*(b*d)^(1/2)*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a*b*c*d*x+6*(b*d)^(1/2)*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/
2)*b^2*c^2*x-48*(b*d)^(1/2)*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a^2*c*d)/d/((b*x+a)*(d*x+c))^(1/2)/(b*d)^(1/2)
/x/(a*c)^(1/2)

Fricas [A] (verification not implemented)

none

Time = 3.22 (sec) , antiderivative size = 1333, normalized size of antiderivative = 5.15 \[ \int \frac {(a+b x)^{5/2} (c+d x)^{3/2}}{x^2} \, dx=\text {Too large to display} \]

[In]

integrate((b*x+a)^(5/2)*(d*x+c)^(3/2)/x^2,x, algorithm="fricas")

[Out]

[-1/96*(3*(b^3*c^3 - 15*a*b^2*c^2*d - 45*a^2*b*c*d^2 - 5*a^3*d^3)*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) -
24*(5*a*b^2*c*d^2 + 3*a^2*b*d^3)*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)*sqrt(a*c)*sqrt(b*x + a)*sqrt(d*x + c) + 8*(a*b*c^2 + a^2*c*d)*x)/x^2) - 4*(8*b^3*d^3*x^3 - 24
*a^2*b*c*d^2 + 2*(7*b^3*c*d^2 + 13*a*b^2*d^3)*x^2 + (3*b^3*c^2*d + 68*a*b^2*c*d^2 + 33*a^2*b*d^3)*x)*sqrt(b*x
+ a)*sqrt(d*x + c))/(b*d^2*x), 1/48*(3*(b^3*c^3 - 15*a*b^2*c^2*d - 45*a^2*b*c*d^2 - 5*a^3*d^3)*sqrt(-b*d)*x*ar
ctan(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)) + 12*(5*a*b^2*c*d^2 + 3*a^2*b*d^3)*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)*sqrt(a*c)*sqrt(b*x + a)*sqrt(d*x + c) + 8*(a*b*c^2 + a^2*c*d)*x)/x^2) + 2*(8*b^3*d^
3*x^3 - 24*a^2*b*c*d^2 + 2*(7*b^3*c*d^2 + 13*a*b^2*d^3)*x^2 + (3*b^3*c^2*d + 68*a*b^2*c*d^2 + 33*a^2*b*d^3)*x)
*sqrt(b*x + a)*sqrt(d*x + c))/(b*d^2*x), 1/96*(48*(5*a*b^2*c*d^2 + 3*a^2*b*d^3)*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^3*c^3 - 15*a*b^2*c^2*d - 45*a^2*b*c*d^2 - 5*a^3*d^3)*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*(8*b^3*
d^3*x^3 - 24*a^2*b*c*d^2 + 2*(7*b^3*c*d^2 + 13*a*b^2*d^3)*x^2 + (3*b^3*c^2*d + 68*a*b^2*c*d^2 + 33*a^2*b*d^3)*
x)*sqrt(b*x + a)*sqrt(d*x + c))/(b*d^2*x), 1/48*(24*(5*a*b^2*c*d^2 + 3*a^2*b*d^3)*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^3*c^3 - 15*a*b^2*c^2*d - 45*a^2*b*c*d^2 - 5*a^3*d^3)*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*(8*b^3*d^3*x^3 - 24*a^2*
b*c*d^2 + 2*(7*b^3*c*d^2 + 13*a*b^2*d^3)*x^2 + (3*b^3*c^2*d + 68*a*b^2*c*d^2 + 33*a^2*b*d^3)*x)*sqrt(b*x + a)*
sqrt(d*x + c))/(b*d^2*x)]

Sympy [F]

\[ \int \frac {(a+b x)^{5/2} (c+d x)^{3/2}}{x^2} \, dx=\int \frac {\left (a + b x\right )^{\frac {5}{2}} \left (c + d x\right )^{\frac {3}{2}}}{x^{2}}\, dx \]

[In]

integrate((b*x+a)**(5/2)*(d*x+c)**(3/2)/x**2,x)

[Out]

Integral((a + b*x)**(5/2)*(c + d*x)**(3/2)/x**2, x)

Maxima [F(-2)]

Exception generated. \[ \int \frac {(a+b x)^{5/2} (c+d x)^{3/2}}{x^2} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate((b*x+a)^(5/2)*(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. 654 vs. \(2 (211) = 422\).

Time = 0.75 (sec) , antiderivative size = 654, normalized size of antiderivative = 2.53 \[ \int \frac {(a+b x)^{5/2} (c+d x)^{3/2}}{x^2} \, dx=\frac {2 \, \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} {\left (2 \, {\left (b x + a\right )} {\left (\frac {4 \, {\left (b x + a\right )} d {\left | b \right |}}{b} + \frac {7 \, b c d^{4} {\left | b \right |} + 5 \, a d^{5} {\left | b \right |}}{b d^{4}}\right )} + \frac {3 \, {\left (b^{2} c^{2} d^{3} {\left | b \right |} + 18 \, a b c d^{4} {\left | b \right |} + 5 \, a^{2} d^{5} {\left | b \right |}\right )}}{b d^{4}}\right )} \sqrt {b x + a} - \frac {48 \, {\left (5 \, \sqrt {b d} a^{2} b^{2} c^{2} {\left | b \right |} + 3 \, \sqrt {b d} a^{3} 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 {3 \, {\left (b^{3} c^{3} {\left | b \right |} - 15 \, a b^{2} c^{2} d {\left | b \right |} - 45 \, a^{2} b c d^{2} {\left | b \right |} - 5 \, a^{3} d^{3} {\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} d} - \frac {96 \, {\left (\sqrt {b d} a^{2} b^{4} c^{3} {\left | b \right |} - 2 \, \sqrt {b d} a^{3} b^{3} c^{2} d {\left | b \right |} + \sqrt {b d} a^{4} 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^{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^{3} 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}}}{48 \, b} \]

[In]

integrate((b*x+a)^(5/2)*(d*x+c)^(3/2)/x^2,x, algorithm="giac")

[Out]

1/48*(2*sqrt(b^2*c + (b*x + a)*b*d - a*b*d)*(2*(b*x + a)*(4*(b*x + a)*d*abs(b)/b + (7*b*c*d^4*abs(b) + 5*a*d^5
*abs(b))/(b*d^4)) + 3*(b^2*c^2*d^3*abs(b) + 18*a*b*c*d^4*abs(b) + 5*a^2*d^5*abs(b))/(b*d^4))*sqrt(b*x + a) - 4
8*(5*sqrt(b*d)*a^2*b^2*c^2*abs(b) + 3*sqrt(b*d)*a^3*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)*b) + 3*(b^3*c^3*abs(b)
- 15*a*b^2*c^2*d*abs(b) - 45*a^2*b*c*d^2*abs(b) - 5*a^3*d^3*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)*d) - 96*(sqrt(b*d)*a^2*b^4*c^3*abs(b) - 2*sqrt(b*d)*a^3*b^3*c^2*d*abs
(b) + sqrt(b*d)*a^4*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^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^3*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. \[ \int \frac {(a+b x)^{5/2} (c+d x)^{3/2}}{x^2} \, dx=\int \frac {{\left (a+b\,x\right )}^{5/2}\,{\left (c+d\,x\right )}^{3/2}}{x^2} \,d x \]

[In]

int(((a + b*x)^(5/2)*(c + d*x)^(3/2))/x^2,x)

[Out]

int(((a + b*x)^(5/2)*(c + d*x)^(3/2))/x^2, x)