\(\int \frac {(a+b x)^{5/2} (c+d x)^{3/2}}{x^4} \, dx\) [662]

Optimal result

Integrand size = 22, antiderivative size = 294 \[ \int \frac {(a+b x)^{5/2} (c+d x)^{3/2}}{x^4} \, dx=\frac {d \left (19 b^2 c^2+14 a b c d-a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{8 c^2}-\frac {\left (5 b^2 c^2+12 a b c d-a^2 d^2\right ) \sqrt {a+b x} (c+d x)^{3/2}}{8 c^2 x}-\frac {(5 b c+3 a d) (a+b x)^{3/2} (c+d x)^{3/2}}{12 c x^2}-\frac {(a+b x)^{5/2} (c+d x)^{3/2}}{3 x^3}-\frac {\left (5 b^3 c^3+45 a b^2 c^2 d+15 a^2 b c d^2-a^3 d^3\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{8 \sqrt {a} c^{3/2}}+b^{3/2} \sqrt {d} (3 b c+5 a d) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right ) \]

[Out]

-1/12*(3*a*d+5*b*c)*(b*x+a)^(3/2)*(d*x+c)^(3/2)/c/x^2-1/3*(b*x+a)^(5/2)*(d*x+c)^(3/2)/x^3-1/8*(-a^3*d^3+15*a^2
*b*c*d^2+45*a*b^2*c^2*d+5*b^3*c^3)*arctanh(c^(1/2)*(b*x+a)^(1/2)/a^(1/2)/(d*x+c)^(1/2))/c^(3/2)/a^(1/2)+b^(3/2
)*(5*a*d+3*b*c)*arctanh(d^(1/2)*(b*x+a)^(1/2)/b^(1/2)/(d*x+c)^(1/2))*d^(1/2)-1/8*(-a^2*d^2+12*a*b*c*d+5*b^2*c^
2)*(d*x+c)^(3/2)*(b*x+a)^(1/2)/c^2/x+1/8*d*(-a^2*d^2+14*a*b*c*d+19*b^2*c^2)*(b*x+a)^(1/2)*(d*x+c)^(1/2)/c^2

Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 294, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.409, Rules used = {99, 154, 159, 163, 65, 223, 212, 95, 214} \[ \int \frac {(a+b x)^{5/2} (c+d x)^{3/2}}{x^4} \, dx=-\frac {\sqrt {a+b x} (c+d x)^{3/2} \left (-a^2 d^2+12 a b c d+5 b^2 c^2\right )}{8 c^2 x}+\frac {d \sqrt {a+b x} \sqrt {c+d x} \left (-a^2 d^2+14 a b c d+19 b^2 c^2\right )}{8 c^2}-\frac {\left (-a^3 d^3+15 a^2 b c d^2+45 a b^2 c^2 d+5 b^3 c^3\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{8 \sqrt {a} c^{3/2}}+b^{3/2} \sqrt {d} (5 a d+3 b c) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )-\frac {(a+b x)^{5/2} (c+d x)^{3/2}}{3 x^3}-\frac {(a+b x)^{3/2} (c+d x)^{3/2} (3 a d+5 b c)}{12 c x^2} \]

[In]

Int[((a + b*x)^(5/2)*(c + d*x)^(3/2))/x^4,x]

[Out]

(d*(19*b^2*c^2 + 14*a*b*c*d - a^2*d^2)*Sqrt[a + b*x]*Sqrt[c + d*x])/(8*c^2) - ((5*b^2*c^2 + 12*a*b*c*d - a^2*d
^2)*Sqrt[a + b*x]*(c + d*x)^(3/2))/(8*c^2*x) - ((5*b*c + 3*a*d)*(a + b*x)^(3/2)*(c + d*x)^(3/2))/(12*c*x^2) -
((a + b*x)^(5/2)*(c + d*x)^(3/2))/(3*x^3) - ((5*b^3*c^3 + 45*a*b^2*c^2*d + 15*a^2*b*c*d^2 - a^3*d^3)*ArcTanh[(
Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(8*Sqrt[a]*c^(3/2)) + b^(3/2)*Sqrt[d]*(3*b*c + 5*a*d)*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 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}}{3 x^3}+\frac {1}{3} \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^3} \, dx \\ & = -\frac {(5 b c+3 a d) (a+b x)^{3/2} (c+d x)^{3/2}}{12 c x^2}-\frac {(a+b x)^{5/2} (c+d x)^{3/2}}{3 x^3}+\frac {\int \frac {\sqrt {a+b x} \sqrt {c+d x} \left (\frac {3}{4} \left (5 b^2 c^2+12 a b c d-a^2 d^2\right )+\frac {3}{2} b d (7 b c+a d) x\right )}{x^2} \, dx}{6 c} \\ & = -\frac {\left (5 b^2 c^2+12 a b c d-a^2 d^2\right ) \sqrt {a+b x} (c+d x)^{3/2}}{8 c^2 x}-\frac {(5 b c+3 a d) (a+b x)^{3/2} (c+d x)^{3/2}}{12 c x^2}-\frac {(a+b x)^{5/2} (c+d x)^{3/2}}{3 x^3}+\frac {\int \frac {\sqrt {c+d x} \left (\frac {3}{8} \left (5 b^3 c^3+45 a b^2 c^2 d+15 a^2 b c d^2-a^3 d^3\right )+\frac {3}{4} b d \left (19 b^2 c^2+14 a b c d-a^2 d^2\right ) x\right )}{x \sqrt {a+b x}} \, dx}{6 c^2} \\ & = \frac {d \left (19 b^2 c^2+14 a b c d-a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{8 c^2}-\frac {\left (5 b^2 c^2+12 a b c d-a^2 d^2\right ) \sqrt {a+b x} (c+d x)^{3/2}}{8 c^2 x}-\frac {(5 b c+3 a d) (a+b x)^{3/2} (c+d x)^{3/2}}{12 c x^2}-\frac {(a+b x)^{5/2} (c+d x)^{3/2}}{3 x^3}+\frac {\int \frac {\frac {3}{8} b c \left (5 b^3 c^3+45 a b^2 c^2 d+15 a^2 b c d^2-a^3 d^3\right )+3 b^3 c^2 d (3 b c+5 a d) x}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{6 b c^2} \\ & = \frac {d \left (19 b^2 c^2+14 a b c d-a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{8 c^2}-\frac {\left (5 b^2 c^2+12 a b c d-a^2 d^2\right ) \sqrt {a+b x} (c+d x)^{3/2}}{8 c^2 x}-\frac {(5 b c+3 a d) (a+b x)^{3/2} (c+d x)^{3/2}}{12 c x^2}-\frac {(a+b x)^{5/2} (c+d x)^{3/2}}{3 x^3}+\frac {1}{2} \left (b^2 d (3 b c+5 a d)\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx+\frac {\left (5 b^3 c^3+45 a b^2 c^2 d+15 a^2 b c d^2-a^3 d^3\right ) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{16 c} \\ & = \frac {d \left (19 b^2 c^2+14 a b c d-a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{8 c^2}-\frac {\left (5 b^2 c^2+12 a b c d-a^2 d^2\right ) \sqrt {a+b x} (c+d x)^{3/2}}{8 c^2 x}-\frac {(5 b c+3 a d) (a+b x)^{3/2} (c+d x)^{3/2}}{12 c x^2}-\frac {(a+b x)^{5/2} (c+d x)^{3/2}}{3 x^3}+(b d (3 b c+5 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 )+\frac {\left (5 b^3 c^3+45 a b^2 c^2 d+15 a^2 b c d^2-a^3 d^3\right ) \text {Subst}\left (\int \frac {1}{-a+c x^2} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )}{8 c} \\ & = \frac {d \left (19 b^2 c^2+14 a b c d-a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{8 c^2}-\frac {\left (5 b^2 c^2+12 a b c d-a^2 d^2\right ) \sqrt {a+b x} (c+d x)^{3/2}}{8 c^2 x}-\frac {(5 b c+3 a d) (a+b x)^{3/2} (c+d x)^{3/2}}{12 c x^2}-\frac {(a+b x)^{5/2} (c+d x)^{3/2}}{3 x^3}-\frac {\left (5 b^3 c^3+45 a b^2 c^2 d+15 a^2 b c d^2-a^3 d^3\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{8 \sqrt {a} c^{3/2}}+(b d (3 b c+5 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 ) \\ & = \frac {d \left (19 b^2 c^2+14 a b c d-a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{8 c^2}-\frac {\left (5 b^2 c^2+12 a b c d-a^2 d^2\right ) \sqrt {a+b x} (c+d x)^{3/2}}{8 c^2 x}-\frac {(5 b c+3 a d) (a+b x)^{3/2} (c+d x)^{3/2}}{12 c x^2}-\frac {(a+b x)^{5/2} (c+d x)^{3/2}}{3 x^3}-\frac {\left (5 b^3 c^3+45 a b^2 c^2 d+15 a^2 b c d^2-a^3 d^3\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{8 \sqrt {a} c^{3/2}}+b^{3/2} \sqrt {d} (3 b c+5 a d) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.75 (sec) , antiderivative size = 214, normalized size of antiderivative = 0.73 \[ \int \frac {(a+b x)^{5/2} (c+d x)^{3/2}}{x^4} \, dx=-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (3 b^2 c x^2 (11 c-8 d x)+2 a b c x (13 c+34 d x)+a^2 \left (8 c^2+14 c d x+3 d^2 x^2\right )\right )}{24 c x^3}+\frac {\left (-5 b^3 c^3-45 a b^2 c^2 d-15 a^2 b c d^2+a^3 d^3\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{8 \sqrt {a} c^{3/2}}+b^{3/2} \sqrt {d} (3 b c+5 a d) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right ) \]

[In]

Integrate[((a + b*x)^(5/2)*(c + d*x)^(3/2))/x^4,x]

[Out]

-1/24*(Sqrt[a + b*x]*Sqrt[c + d*x]*(3*b^2*c*x^2*(11*c - 8*d*x) + 2*a*b*c*x*(13*c + 34*d*x) + a^2*(8*c^2 + 14*c
*d*x + 3*d^2*x^2)))/(c*x^3) + ((-5*b^3*c^3 - 45*a*b^2*c^2*d - 15*a^2*b*c*d^2 + a^3*d^3)*ArcTanh[(Sqrt[c]*Sqrt[
a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(8*Sqrt[a]*c^(3/2)) + b^(3/2)*Sqrt[d]*(3*b*c + 5*a*d)*ArcTanh[(Sqrt[d]*Sqr
t[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. \(607\) vs. \(2(244)=488\).

Time = 1.60 (sec) , antiderivative size = 608, normalized size of antiderivative = 2.07

method	result	size
default	\(\frac {\sqrt {b x +a}\, \sqrt {d x +c}\, \left (120 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a \,b^{2} c \,d^{2} x^{3} \sqrt {a c}+72 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) b^{3} c^{2} d \,x^{3} \sqrt {a c}+3 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a^{3} d^{3} x^{3} \sqrt {b d}-45 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a^{2} b c \,d^{2} x^{3} \sqrt {b d}-135 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a \,b^{2} c^{2} d \,x^{3} \sqrt {b d}-15 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) b^{3} c^{3} x^{3} \sqrt {b d}+48 \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, b^{2} c d \,x^{3}-6 \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a^{2} d^{2} x^{2}-136 \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a b c d \,x^{2}-66 \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, b^{2} c^{2} x^{2}-28 \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a^{2} c d x -52 \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a b \,c^{2} x -16 \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a^{2} c^{2}\right )}{48 c \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, x^{3} \sqrt {b d}\, \sqrt {a c}}\)	\(608\)

[In]

int((b*x+a)^(5/2)*(d*x+c)^(3/2)/x^4,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 1.94 (sec) , antiderivative size = 1357, normalized size of antiderivative = 4.62 \[ \int \frac {(a+b x)^{5/2} (c+d x)^{3/2}}{x^4} \, dx=\text {Too large to display} \]

[In]

integrate((b*x+a)^(5/2)*(d*x+c)^(3/2)/x^4,x, algorithm="fricas")

[Out]

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

Sympy [F]

\[ \int \frac {(a+b x)^{5/2} (c+d x)^{3/2}}{x^4} \, dx=\int \frac {\left (a + b x\right )^{\frac {5}{2}} \left (c + d x\right )^{\frac {3}{2}}}{x^{4}}\, dx \]

[In]

integrate((b*x+a)**(5/2)*(d*x+c)**(3/2)/x**4,x)

[Out]

Integral((a + b*x)**(5/2)*(c + d*x)**(3/2)/x**4, x)

Maxima [F(-2)]

Exception generated. \[ \int \frac {(a+b x)^{5/2} (c+d x)^{3/2}}{x^4} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate((b*x+a)^(5/2)*(d*x+c)^(3/2)/x^4,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. 2316 vs. \(2 (244) = 488\).

Time = 3.71 (sec) , antiderivative size = 2316, normalized size of antiderivative = 7.88 \[ \int \frac {(a+b x)^{5/2} (c+d x)^{3/2}}{x^4} \, dx=\text {Too large to display} \]

[In]

integrate((b*x+a)^(5/2)*(d*x+c)^(3/2)/x^4,x, algorithm="giac")

[Out]

1/24*(24*sqrt(b^2*c + (b*x + a)*b*d - a*b*d)*sqrt(b*x + a)*b*d*abs(b) - 12*(3*sqrt(b*d)*b^2*c*abs(b) + 5*sqrt(
b*d)*a*b*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^4
*c^3*abs(b) + 45*sqrt(b*d)*a*b^3*c^2*d*abs(b) + 15*sqrt(b*d)*a^2*b^2*c*d^2*abs(b) - sqrt(b*d)*a^3*b*d^3*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*c) - 2*(33*sqrt(b*d)*b^14*c^8*abs(b) - 130*sqrt(b*d)*a*b^13*c^7*d*abs(b) + 90*sqrt(b
*d)*a^2*b^12*c^6*d^2*abs(b) + 342*sqrt(b*d)*a^3*b^11*c^5*d^3*abs(b) - 820*sqrt(b*d)*a^4*b^10*c^4*d^4*abs(b) +
762*sqrt(b*d)*a^5*b^9*c^3*d^5*abs(b) - 330*sqrt(b*d)*a^6*b^8*c^2*d^6*abs(b) + 50*sqrt(b*d)*a^7*b^7*c*d^7*abs(b
) + 3*sqrt(b*d)*a^8*b^6*d^8*abs(b) - 165*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b
*d))^2*b^12*c^7*abs(b) + 207*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^1
1*c^6*d*abs(b) + 495*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^10*c^5*
d^2*abs(b) - 765*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^9*c^4*d^3*a
bs(b) - 255*sqrt(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^8*c^3*d^4*abs(b)
+ 765*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^7*c^2*d^5*abs(b) - 26
7*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^6*c*d^6*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))^2*a^7*b^5*d^7*abs(b) + 330*sqrt(b*d)*(sqrt
(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*b^10*c^6*abs(b) + 312*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^9*c^5*d*abs(b) - 534*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^8*c^4*d^2*abs(b) - 528*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^7*c^3*d^3*abs(b) - 210*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^6*c^2*d^4*abs(b) + 600*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^5*c*d^5*abs(b) + 30*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^4*d^6*abs(b) - 330*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*
b*d))^6*b^8*c^5*abs(b) - 878*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^7
*c^4*d*abs(b) - 636*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^6*c^3*d^
2*abs(b) - 756*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^5*c^2*d^3*abs
(b) - 698*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^4*c*d^4*abs(b) - 3
0*sqrt(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^3*d^5*abs(b) + 165*sqrt(b*
d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^8*b^6*c^4*abs(b) + 642*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^5*c^3*d*abs(b) + 684*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^4*c^2*d^2*abs(b) + 414*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^3*c*d^3*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))^8*a^4*b^2*d^4*abs(b) - 33*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b
*x + a)*b*d - a*b*d))^10*b^4*c^3*abs(b) - 153*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^3*c^2*d*abs(b) - 99*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^2*c*d^2*abs(b) - 3*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*d^3*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)*sqr
t(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4)^3*c))/b

Mupad [F(-1)]

Timed out. \[ \int \frac {(a+b x)^{5/2} (c+d x)^{3/2}}{x^4} \, dx=\int \frac {{\left (a+b\,x\right )}^{5/2}\,{\left (c+d\,x\right )}^{3/2}}{x^4} \,d x \]

[In]

int(((a + b*x)^(5/2)*(c + d*x)^(3/2))/x^4,x)

[Out]

int(((a + b*x)^(5/2)*(c + d*x)^(3/2))/x^4, x)