\(\int \frac {(a+b x)^{5/2} (c+d x)^{3/2}}{x^5} \, dx\) [663]

Optimal result

Integrand size = 22, antiderivative size = 313 \[ \int \frac {(a+b x)^{5/2} (c+d x)^{3/2}}{x^5} \, dx=-\frac {\left (5 b^3 c^3+73 a b^2 c^2 d-17 a^2 b c d^2+3 a^3 d^3\right ) \sqrt {a+b x} \sqrt {c+d x}}{64 a c^2 x}-\frac {(5 b c-a d) (b c+3 a d) \sqrt {a+b x} (c+d x)^{3/2}}{32 c^2 x^2}-\frac {(5 b c+3 a d) (a+b x)^{3/2} (c+d x)^{3/2}}{24 c x^3}-\frac {(a+b x)^{5/2} (c+d x)^{3/2}}{4 x^4}+\frac {\left (5 b^4 c^4-60 a b^3 c^3 d-90 a^2 b^2 c^2 d^2+20 a^3 b c d^3-3 a^4 d^4\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{64 a^{3/2} c^{5/2}}+2 b^{5/2} d^{3/2} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right ) \]

[Out]

-1/24*(3*a*d+5*b*c)*(b*x+a)^(3/2)*(d*x+c)^(3/2)/c/x^3-1/4*(b*x+a)^(5/2)*(d*x+c)^(3/2)/x^4+1/64*(-3*a^4*d^4+20*
a^3*b*c*d^3-90*a^2*b^2*c^2*d^2-60*a*b^3*c^3*d+5*b^4*c^4)*arctanh(c^(1/2)*(b*x+a)^(1/2)/a^(1/2)/(d*x+c)^(1/2))/
a^(3/2)/c^(5/2)+2*b^(5/2)*d^(3/2)*arctanh(d^(1/2)*(b*x+a)^(1/2)/b^(1/2)/(d*x+c)^(1/2))-1/32*(-a*d+5*b*c)*(3*a*
d+b*c)*(d*x+c)^(3/2)*(b*x+a)^(1/2)/c^2/x^2-1/64*(3*a^3*d^3-17*a^2*b*c*d^2+73*a*b^2*c^2*d+5*b^3*c^3)*(b*x+a)^(1
/2)*(d*x+c)^(1/2)/a/c^2/x

Rubi [A] (verified)

Time = 0.23 (sec) , antiderivative size = 313, 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, 154, 163, 65, 223, 212, 95, 214} \[ \int \frac {(a+b x)^{5/2} (c+d x)^{3/2}}{x^5} \, dx=-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (3 a^3 d^3-17 a^2 b c d^2+73 a b^2 c^2 d+5 b^3 c^3\right )}{64 a c^2 x}+\frac {\left (-3 a^4 d^4+20 a^3 b c d^3-90 a^2 b^2 c^2 d^2-60 a b^3 c^3 d+5 b^4 c^4\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{64 a^{3/2} c^{5/2}}+2 b^{5/2} d^{3/2} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )-\frac {\sqrt {a+b x} (c+d x)^{3/2} (5 b c-a d) (3 a d+b c)}{32 c^2 x^2}-\frac {(a+b x)^{5/2} (c+d x)^{3/2}}{4 x^4}-\frac {(a+b x)^{3/2} (c+d x)^{3/2} (3 a d+5 b c)}{24 c x^3} \]

[In]

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

[Out]

-1/64*((5*b^3*c^3 + 73*a*b^2*c^2*d - 17*a^2*b*c*d^2 + 3*a^3*d^3)*Sqrt[a + b*x]*Sqrt[c + d*x])/(a*c^2*x) - ((5*
b*c - a*d)*(b*c + 3*a*d)*Sqrt[a + b*x]*(c + d*x)^(3/2))/(32*c^2*x^2) - ((5*b*c + 3*a*d)*(a + b*x)^(3/2)*(c + d
*x)^(3/2))/(24*c*x^3) - ((a + b*x)^(5/2)*(c + d*x)^(3/2))/(4*x^4) + ((5*b^4*c^4 - 60*a*b^3*c^3*d - 90*a^2*b^2*
c^2*d^2 + 20*a^3*b*c*d^3 - 3*a^4*d^4)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(64*a^(3/2)*c^
(5/2)) + 2*b^(5/2)*d^(3/2)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])]

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 154

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^n*((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 - 1)*(e + f*x)^p*Simp[b*c*(f*g - e*h)*(m + 1) + (
b*g - a*h)*(d*e*n + c*f*(p + 1)) + d*(b*(f*g - e*h)*(m + 1) + f*(b*g - a*h)*(n + p + 1))*x, x], x], x] /; Free
Q[{a, b, c, d, e, f, g, h, p}, x] && ILtQ[m, -1] && GtQ[n, 0]

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}}{4 x^4}+\frac {1}{4} \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^4} \, dx \\ & = -\frac {(5 b c+3 a d) (a+b x)^{3/2} (c+d x)^{3/2}}{24 c x^3}-\frac {(a+b x)^{5/2} (c+d x)^{3/2}}{4 x^4}+\frac {\int \frac {\sqrt {a+b x} \sqrt {c+d x} \left (\frac {3}{4} (5 b c-a d) (b c+3 a d)+12 b^2 c d x\right )}{x^3} \, dx}{12 c} \\ & = -\frac {(5 b c-a d) (b c+3 a d) \sqrt {a+b x} (c+d x)^{3/2}}{32 c^2 x^2}-\frac {(5 b c+3 a d) (a+b x)^{3/2} (c+d x)^{3/2}}{24 c x^3}-\frac {(a+b x)^{5/2} (c+d x)^{3/2}}{4 x^4}+\frac {\int \frac {\sqrt {c+d x} \left (\frac {3}{8} \left (5 b^3 c^3+73 a b^2 c^2 d-17 a^2 b c d^2+3 a^3 d^3\right )+24 b^3 c^2 d x\right )}{x^2 \sqrt {a+b x}} \, dx}{24 c^2} \\ & = -\frac {\left (5 b^3 c^3+73 a b^2 c^2 d-17 a^2 b c d^2+3 a^3 d^3\right ) \sqrt {a+b x} \sqrt {c+d x}}{64 a c^2 x}-\frac {(5 b c-a d) (b c+3 a d) \sqrt {a+b x} (c+d x)^{3/2}}{32 c^2 x^2}-\frac {(5 b c+3 a d) (a+b x)^{3/2} (c+d x)^{3/2}}{24 c x^3}-\frac {(a+b x)^{5/2} (c+d x)^{3/2}}{4 x^4}+\frac {\int \frac {-\frac {3}{16} \left (5 b^4 c^4-60 a b^3 c^3 d-90 a^2 b^2 c^2 d^2+20 a^3 b c d^3-3 a^4 d^4\right )+24 a b^3 c^2 d^2 x}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{24 a c^2} \\ & = -\frac {\left (5 b^3 c^3+73 a b^2 c^2 d-17 a^2 b c d^2+3 a^3 d^3\right ) \sqrt {a+b x} \sqrt {c+d x}}{64 a c^2 x}-\frac {(5 b c-a d) (b c+3 a d) \sqrt {a+b x} (c+d x)^{3/2}}{32 c^2 x^2}-\frac {(5 b c+3 a d) (a+b x)^{3/2} (c+d x)^{3/2}}{24 c x^3}-\frac {(a+b x)^{5/2} (c+d x)^{3/2}}{4 x^4}+\left (b^3 d^2\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx-\frac {\left (5 b^4 c^4-60 a b^3 c^3 d-90 a^2 b^2 c^2 d^2+20 a^3 b c d^3-3 a^4 d^4\right ) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{128 a c^2} \\ & = -\frac {\left (5 b^3 c^3+73 a b^2 c^2 d-17 a^2 b c d^2+3 a^3 d^3\right ) \sqrt {a+b x} \sqrt {c+d x}}{64 a c^2 x}-\frac {(5 b c-a d) (b c+3 a d) \sqrt {a+b x} (c+d x)^{3/2}}{32 c^2 x^2}-\frac {(5 b c+3 a d) (a+b x)^{3/2} (c+d x)^{3/2}}{24 c x^3}-\frac {(a+b x)^{5/2} (c+d x)^{3/2}}{4 x^4}+\left (2 b^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 )-\frac {\left (5 b^4 c^4-60 a b^3 c^3 d-90 a^2 b^2 c^2 d^2+20 a^3 b c d^3-3 a^4 d^4\right ) \text {Subst}\left (\int \frac {1}{-a+c x^2} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )}{64 a c^2} \\ & = -\frac {\left (5 b^3 c^3+73 a b^2 c^2 d-17 a^2 b c d^2+3 a^3 d^3\right ) \sqrt {a+b x} \sqrt {c+d x}}{64 a c^2 x}-\frac {(5 b c-a d) (b c+3 a d) \sqrt {a+b x} (c+d x)^{3/2}}{32 c^2 x^2}-\frac {(5 b c+3 a d) (a+b x)^{3/2} (c+d x)^{3/2}}{24 c x^3}-\frac {(a+b x)^{5/2} (c+d x)^{3/2}}{4 x^4}+\frac {\left (5 b^4 c^4-60 a b^3 c^3 d-90 a^2 b^2 c^2 d^2+20 a^3 b c d^3-3 a^4 d^4\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{64 a^{3/2} c^{5/2}}+\left (2 b^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 ) \\ & = -\frac {\left (5 b^3 c^3+73 a b^2 c^2 d-17 a^2 b c d^2+3 a^3 d^3\right ) \sqrt {a+b x} \sqrt {c+d x}}{64 a c^2 x}-\frac {(5 b c-a d) (b c+3 a d) \sqrt {a+b x} (c+d x)^{3/2}}{32 c^2 x^2}-\frac {(5 b c+3 a d) (a+b x)^{3/2} (c+d x)^{3/2}}{24 c x^3}-\frac {(a+b x)^{5/2} (c+d x)^{3/2}}{4 x^4}+\frac {\left (5 b^4 c^4-60 a b^3 c^3 d-90 a^2 b^2 c^2 d^2+20 a^3 b c d^3-3 a^4 d^4\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{64 a^{3/2} c^{5/2}}+2 b^{5/2} d^{3/2} \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.76 (sec) , antiderivative size = 260, normalized size of antiderivative = 0.83 \[ \int \frac {(a+b x)^{5/2} (c+d x)^{3/2}}{x^5} \, dx=-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (15 b^3 c^3 x^3+a b^2 c^2 x^2 (118 c+337 d x)+a^2 b c x \left (136 c^2+244 c d x+57 d^2 x^2\right )+a^3 \left (48 c^3+72 c^2 d x+6 c d^2 x^2-9 d^3 x^3\right )\right )}{192 a c^2 x^4}-\frac {\left (-5 b^4 c^4+60 a b^3 c^3 d+90 a^2 b^2 c^2 d^2-20 a^3 b c d^3+3 a^4 d^4\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{64 a^{3/2} c^{5/2}}+2 b^{5/2} d^{3/2} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right ) \]

[In]

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

[Out]

-1/192*(Sqrt[a + b*x]*Sqrt[c + d*x]*(15*b^3*c^3*x^3 + a*b^2*c^2*x^2*(118*c + 337*d*x) + a^2*b*c*x*(136*c^2 + 2
44*c*d*x + 57*d^2*x^2) + a^3*(48*c^3 + 72*c^2*d*x + 6*c*d^2*x^2 - 9*d^3*x^3)))/(a*c^2*x^4) - ((-5*b^4*c^4 + 60
*a*b^3*c^3*d + 90*a^2*b^2*c^2*d^2 - 20*a^3*b*c*d^3 + 3*a^4*d^4)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[
c + d*x])])/(64*a^(3/2)*c^(5/2)) + 2*b^(5/2)*d^(3/2)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])]

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(732\) vs. \(2(263)=526\).

Time = 0.57 (sec) , antiderivative size = 733, normalized size of antiderivative = 2.34

method	result	size
default	\(\frac {\sqrt {b x +a}\, \sqrt {d x +c}\, \left (384 \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^{3} c^{2} d^{2} x^{4} \sqrt {a c}-9 \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^{4} d^{4} x^{4} \sqrt {b d}+60 \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} b c \,d^{3} x^{4} \sqrt {b d}-270 \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^{2} c^{2} d^{2} x^{4} \sqrt {b d}-180 \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^{3} c^{3} d \,x^{4} \sqrt {b d}+15 \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^{4} c^{4} x^{4} \sqrt {b d}+18 \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a^{3} d^{3} x^{3}-114 \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a^{2} b c \,d^{2} x^{3}-674 \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a \,b^{2} c^{2} d \,x^{3}-30 \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, b^{3} c^{3} x^{3}-12 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, \sqrt {a c}\, a^{3} c \,d^{2} x^{2}-488 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, \sqrt {a c}\, a^{2} b \,c^{2} d \,x^{2}-236 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, \sqrt {a c}\, a \,b^{2} c^{3} x^{2}-144 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, \sqrt {a c}\, a^{3} c^{2} d x -272 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, \sqrt {a c}\, a^{2} b \,c^{3} x -96 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, \sqrt {a c}\, a^{3} c^{3}\right )}{384 a \,c^{2} \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, x^{4} \sqrt {b d}\, \sqrt {a c}}\)	\(733\)

[In]

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

[Out]

1/384*(b*x+a)^(1/2)*(d*x+c)^(1/2)/a/c^2*(384*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^3*c^2*d^2*x^4*(a*c)^(1/2)-9*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^4
*d^4*x^4*(b*d)^(1/2)+60*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*b*c*d^3*x^4*(b*d)^
(1/2)-270*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^2*c^2*d^2*x^4*(b*d)^(1/2)-180*
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^3*c^3*d*x^4*(b*d)^(1/2)+15*ln((a*d*x+b*c*x
+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*b^4*c^4*x^4*(b*d)^(1/2)+18*(b*d)^(1/2)*(a*c)^(1/2)*((b*x+a)*(
d*x+c))^(1/2)*a^3*d^3*x^3-114*(b*d)^(1/2)*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a^2*b*c*d^2*x^3-674*(b*d)^(1/2)*
(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a*b^2*c^2*d*x^3-30*(b*d)^(1/2)*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*b^3*c^3
*x^3-12*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)*(a*c)^(1/2)*a^3*c*d^2*x^2-488*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)*
(a*c)^(1/2)*a^2*b*c^2*d*x^2-236*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)*(a*c)^(1/2)*a*b^2*c^3*x^2-144*((b*x+a)*(d*
x+c))^(1/2)*(b*d)^(1/2)*(a*c)^(1/2)*a^3*c^2*d*x-272*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)*(a*c)^(1/2)*a^2*b*c^3*
x-96*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)*(a*c)^(1/2)*a^3*c^3)/((b*x+a)*(d*x+c))^(1/2)/x^4/(b*d)^(1/2)/(a*c)^(1
/2)

Fricas [A] (verification not implemented)

none

Time = 4.42 (sec) , antiderivative size = 1525, normalized size of antiderivative = 4.87 \[ \int \frac {(a+b x)^{5/2} (c+d x)^{3/2}}{x^5} \, dx=\text {Too large to display} \]

[In]

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

[Out]

[1/768*(384*sqrt(b*d)*a^2*b^2*c^3*d*x^4*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) - 3*(5*b^4*c^4 - 60*a*b^3*c^3*d - 90*a^
2*b^2*c^2*d^2 + 20*a^3*b*c*d^3 - 3*a^4*d^4)*sqrt(a*c)*x^4*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*(48*a^4
*c^4 + (15*a*b^3*c^4 + 337*a^2*b^2*c^3*d + 57*a^3*b*c^2*d^2 - 9*a^4*c*d^3)*x^3 + 2*(59*a^2*b^2*c^4 + 122*a^3*b
*c^3*d + 3*a^4*c^2*d^2)*x^2 + 8*(17*a^3*b*c^4 + 9*a^4*c^3*d)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(a^2*c^3*x^4), -1
/768*(768*sqrt(-b*d)*a^2*b^2*c^3*d*x^4*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)) + 3*(5*b^4*c^4 - 60*a*b^3*c^3*d - 90*a^2*b^2*c^2*d^2 + 20*a^
3*b*c*d^3 - 3*a^4*d^4)*sqrt(a*c)*x^4*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*(48*a^4*c^4 + (15*a*b^3*c^4
+ 337*a^2*b^2*c^3*d + 57*a^3*b*c^2*d^2 - 9*a^4*c*d^3)*x^3 + 2*(59*a^2*b^2*c^4 + 122*a^3*b*c^3*d + 3*a^4*c^2*d^
2)*x^2 + 8*(17*a^3*b*c^4 + 9*a^4*c^3*d)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(a^2*c^3*x^4), 1/384*(192*sqrt(b*d)*a^
2*b^2*c^3*d*x^4*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) - 3*(5*b^4*c^4 - 60*a*b^3*c^3*d - 90*a^2*b^2*c^2*d^2 + 20*a^3*b
*c*d^3 - 3*a^4*d^4)*sqrt(-a*c)*x^4*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)) - 2*(48*a^4*c^4 + (15*a*b^3*c^4 + 337*a^2*b^2*c^3*d + 57*a^3*b
*c^2*d^2 - 9*a^4*c*d^3)*x^3 + 2*(59*a^2*b^2*c^4 + 122*a^3*b*c^3*d + 3*a^4*c^2*d^2)*x^2 + 8*(17*a^3*b*c^4 + 9*a
^4*c^3*d)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(a^2*c^3*x^4), -1/384*(384*sqrt(-b*d)*a^2*b^2*c^3*d*x^4*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)) +
3*(5*b^4*c^4 - 60*a*b^3*c^3*d - 90*a^2*b^2*c^2*d^2 + 20*a^3*b*c*d^3 - 3*a^4*d^4)*sqrt(-a*c)*x^4*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))
+ 2*(48*a^4*c^4 + (15*a*b^3*c^4 + 337*a^2*b^2*c^3*d + 57*a^3*b*c^2*d^2 - 9*a^4*c*d^3)*x^3 + 2*(59*a^2*b^2*c^4
+ 122*a^3*b*c^3*d + 3*a^4*c^2*d^2)*x^2 + 8*(17*a^3*b*c^4 + 9*a^4*c^3*d)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(a^2*c
^3*x^4)]

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F(-2)]

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

[In]

integrate((b*x+a)^(5/2)*(d*x+c)^(3/2)/x^5,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. 3887 vs. \(2 (263) = 526\).

Time = 1.32 (sec) , antiderivative size = 3887, normalized size of antiderivative = 12.42 \[ \int \frac {(a+b x)^{5/2} (c+d x)^{3/2}}{x^5} \, dx=\text {Too large to display} \]

[In]

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

[Out]

-1/192*(192*sqrt(b*d)*b^2*d*abs(b)*log((sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2) - 3*
(5*sqrt(b*d)*b^5*c^4*abs(b) - 60*sqrt(b*d)*a*b^4*c^3*d*abs(b) - 90*sqrt(b*d)*a^2*b^3*c^2*d^2*abs(b) + 20*sqrt(
b*d)*a^3*b^2*c*d^3*abs(b) - 3*sqrt(b*d)*a^4*b*d^4*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*c^2) + 2*(15*sqrt(b*d)*b^1
9*c^11*abs(b) + 217*sqrt(b*d)*a*b^18*c^10*d*abs(b) - 2219*sqrt(b*d)*a^2*b^17*c^9*d^2*abs(b) + 8131*sqrt(b*d)*a
^3*b^16*c^8*d^3*abs(b) - 16154*sqrt(b*d)*a^4*b^15*c^7*d^4*abs(b) + 19306*sqrt(b*d)*a^5*b^14*c^6*d^5*abs(b) - 1
3958*sqrt(b*d)*a^6*b^13*c^5*d^6*abs(b) + 5494*sqrt(b*d)*a^7*b^12*c^4*d^7*abs(b) - 581*sqrt(b*d)*a^8*b^11*c^3*d
^8*abs(b) - 371*sqrt(b*d)*a^9*b^10*c^2*d^9*abs(b) + 129*sqrt(b*d)*a^10*b^9*c*d^10*abs(b) - 9*sqrt(b*d)*a^11*b^
8*d^11*abs(b) - 105*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*b^17*c^10*abs(
b) - 1598*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^16*c^9*d*abs(b) + 95
23*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^15*c^8*d^2*abs(b) - 18024
*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^14*c^7*d^3*abs(b) + 10942*s
qrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^4*b^13*c^6*d^4*abs(b) + 7372*sqrt
(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^5*b^12*c^5*d^5*abs(b) - 13362*sqrt(b
*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^6*b^11*c^4*d^6*abs(b) + 5272*sqrt(b*d)
*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^7*b^10*c^3*d^7*abs(b) + 619*sqrt(b*d)*(sq
rt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^8*b^9*c^2*d^8*abs(b) - 702*sqrt(b*d)*(sqrt(b*
d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^9*b^8*c*d^9*abs(b) + 63*sqrt(b*d)*(sqrt(b*d)*sqrt(
b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^10*b^7*d^10*abs(b) + 315*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a
) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*b^15*c^9*abs(b) + 5323*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^
2*c + (b*x + a)*b*d - a*b*d))^4*a*b^14*c^8*d*abs(b) - 14628*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c +
(b*x + a)*b*d - a*b*d))^4*a^2*b^13*c^7*d^2*abs(b) + 6156*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*
x + a)*b*d - a*b*d))^4*a^3*b^12*c^6*d^3*abs(b) + 5050*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x +
a)*b*d - a*b*d))^4*a^4*b^11*c^5*d^4*abs(b) + 6666*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)
*b*d - a*b*d))^4*a^5*b^10*c^4*d^5*abs(b) - 12852*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b
*d - a*b*d))^4*a^6*b^9*c^3*d^6*abs(b) + 2524*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d -
a*b*d))^4*a^7*b^8*c^2*d^7*abs(b) + 1635*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b
*d))^4*a^8*b^7*c*d^8*abs(b) - 189*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*
a^9*b^6*d^9*abs(b) - 525*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*b^13*c^8*
abs(b) - 10240*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*a*b^12*c^7*d*abs(b)
+ 5900*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*a^2*b^11*c^6*d^2*abs(b) +
5456*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*a^3*b^10*c^5*d^3*abs(b) + 237
8*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*a^4*b^9*c^4*d^4*abs(b) + 8096*sq
rt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*a^5*b^8*c^3*d^5*abs(b) - 9220*sqrt(b
*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*a^6*b^7*c^2*d^6*abs(b) - 2160*sqrt(b*d)*
(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*a^7*b^6*c*d^7*abs(b) + 315*sqrt(b*d)*(sqrt(b
*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*a^8*b^5*d^8*abs(b) + 525*sqrt(b*d)*(sqrt(b*d)*sqrt(
b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^8*b^11*c^7*abs(b) + 12095*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) -
sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^8*a*b^10*c^6*d*abs(b) + 9005*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b
^2*c + (b*x + a)*b*d - a*b*d))^8*a^2*b^9*c^5*d^2*abs(b) + 7191*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c
+ (b*x + a)*b*d - a*b*d))^8*a^3*b^8*c^4*d^3*abs(b) + 8111*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (
b*x + a)*b*d - a*b*d))^8*a^4*b^7*c^3*d^4*abs(b) + 12773*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x
+ a)*b*d - a*b*d))^8*a^5*b^6*c^2*d^5*abs(b) + 1815*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a
)*b*d - a*b*d))^8*a^6*b^5*c*d^6*abs(b) - 315*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d -
a*b*d))^8*a^7*b^4*d^7*abs(b) - 315*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^
10*b^9*c^6*abs(b) - 8662*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^10*a*b^8*c^
5*d*abs(b) - 12381*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^10*a^2*b^7*c^4*d^
2*abs(b) - 13140*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^10*a^3*b^6*c^3*d^3*
abs(b) - 9733*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^10*a^4*b^5*c^2*d^4*abs
(b) - 1014*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^10*a^5*b^4*c*d^5*abs(b) +
189*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^10*a^6*b^3*d^6*abs(b) + 105*sqr
t(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^12*b^7*c^5*abs(b) + 3453*sqrt(b*d)*(sqr
t(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^12*a*b^6*c^4*d*abs(b) + 5682*sqrt(b*d)*(sqrt(b*d)*
sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^12*a^2*b^5*c^3*d^2*abs(b) + 4290*sqrt(b*d)*(sqrt(b*d)*sqr
t(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^12*a^3*b^4*c^2*d^3*abs(b) + 357*sqrt(b*d)*(sqrt(b*d)*sqrt(b*
x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^12*a^4*b^3*c*d^4*abs(b) - 63*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a)
- sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^12*a^5*b^2*d^5*abs(b) - 15*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^
2*c + (b*x + a)*b*d - a*b*d))^14*b^5*c^4*abs(b) - 588*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x +
a)*b*d - a*b*d))^14*a*b^4*c^3*d*abs(b) - 882*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d
- a*b*d))^14*a^2*b^3*c^2*d^2*abs(b) - 60*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b
*d))^14*a^3*b^2*c*d^3*abs(b) + 9*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^14*
a^4*b*d^4*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)^4*a*c^2))/b

Mupad [F(-1)]

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

[In]

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

[Out]

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