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

Optimal. Leaf size=316 \[ \frac {\sqrt {a+b x} \sqrt {c+d x} \left (5 a^3 d^3-55 a^2 b c d^2-17 a b^2 c^2 d+3 b^3 c^3\right )}{64 a^2 c x}-\frac {\left (-5 a^4 d^4+60 a^3 b c d^3+90 a^2 b^2 c^2 d^2-20 a b^3 c^3 d+3 b^4 c^4\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{64 a^{5/2} c^{3/2}}+2 b^{3/2} d^{5/2} \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )-\frac {\sqrt {a+b x} (c+d x)^{3/2} \left (\frac {3 b^2 c}{a}-\frac {5 a d^2}{c}+50 b d\right )}{96 x^2}-\frac {(a+b x)^{3/2} (c+d x)^{5/2}}{4 x^4}-\frac {\sqrt {a+b x} (c+d x)^{5/2} (5 a d+3 b c)}{24 c x^3} \]

________________________________________________________________________________________

Rubi [A]  time = 0.30, antiderivative size = 316, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {97, 149, 157, 63, 217, 206, 93, 208} \begin {gather*} \frac {\sqrt {a+b x} \sqrt {c+d x} \left (-55 a^2 b c d^2+5 a^3 d^3-17 a b^2 c^2 d+3 b^3 c^3\right )}{64 a^2 c x}-\frac {\left (90 a^2 b^2 c^2 d^2+60 a^3 b c d^3-5 a^4 d^4-20 a b^3 c^3 d+3 b^4 c^4\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{64 a^{5/2} c^{3/2}}-\frac {\sqrt {a+b x} (c+d x)^{3/2} \left (\frac {3 b^2 c}{a}-\frac {5 a d^2}{c}+50 b d\right )}{96 x^2}+2 b^{3/2} d^{5/2} \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )-\frac {(a+b x)^{3/2} (c+d x)^{5/2}}{4 x^4}-\frac {\sqrt {a+b x} (c+d x)^{5/2} (5 a d+3 b c)}{24 c x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((3*b^3*c^3 - 17*a*b^2*c^2*d - 55*a^2*b*c*d^2 + 5*a^3*d^3)*Sqrt[a + b*x]*Sqrt[c + d*x])/(64*a^2*c*x) - (((3*b^
2*c)/a + 50*b*d - (5*a*d^2)/c)*Sqrt[a + b*x]*(c + d*x)^(3/2))/(96*x^2) - ((3*b*c + 5*a*d)*Sqrt[a + b*x]*(c + d
*x)^(5/2))/(24*c*x^3) - ((a + b*x)^(3/2)*(c + d*x)^(5/2))/(4*x^4) - ((3*b^4*c^4 - 20*a*b^3*c^3*d + 90*a^2*b^2*
c^2*d^2 + 60*a^3*b*c*d^3 - 5*a^4*d^4)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(64*a^(5/2)*c^
(3/2)) + 2*b^(3/2)*d^(5/2)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])]

Rule 63

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 93

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 97

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 149

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] && LtQ[m, -1] && GtQ[n, 0] && IntegerQ[m]

Rule 157

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 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rule 217

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*} \int \frac {(a+b x)^{3/2} (c+d x)^{5/2}}{x^5} \, dx &=-\frac {(a+b x)^{3/2} (c+d x)^{5/2}}{4 x^4}+\frac {1}{4} \int \frac {\sqrt {a+b x} (c+d x)^{3/2} \left (\frac {1}{2} (3 b c+5 a d)+4 b d x\right )}{x^4} \, dx\\ &=-\frac {(3 b c+5 a d) \sqrt {a+b x} (c+d x)^{5/2}}{24 c x^3}-\frac {(a+b x)^{3/2} (c+d x)^{5/2}}{4 x^4}+\frac {\int \frac {(c+d x)^{3/2} \left (\frac {1}{4} \left (3 b^2 c^2+50 a b c d-5 a^2 d^2\right )+12 b^2 c d x\right )}{x^3 \sqrt {a+b x}} \, dx}{12 c}\\ &=-\frac {\left (\frac {3 b^2 c}{a}+50 b d-\frac {5 a d^2}{c}\right ) \sqrt {a+b x} (c+d x)^{3/2}}{96 x^2}-\frac {(3 b c+5 a d) \sqrt {a+b x} (c+d x)^{5/2}}{24 c x^3}-\frac {(a+b x)^{3/2} (c+d x)^{5/2}}{4 x^4}+\frac {\int \frac {\sqrt {c+d x} \left (-\frac {3}{8} \left (3 b^3 c^3-17 a b^2 c^2 d-55 a^2 b c d^2+5 a^3 d^3\right )+24 a b^2 c d^2 x\right )}{x^2 \sqrt {a+b x}} \, dx}{24 a c}\\ &=\frac {\left (3 b^3 c^3-17 a b^2 c^2 d-55 a^2 b c d^2+5 a^3 d^3\right ) \sqrt {a+b x} \sqrt {c+d x}}{64 a^2 c x}-\frac {\left (\frac {3 b^2 c}{a}+50 b d-\frac {5 a d^2}{c}\right ) \sqrt {a+b x} (c+d x)^{3/2}}{96 x^2}-\frac {(3 b c+5 a d) \sqrt {a+b x} (c+d x)^{5/2}}{24 c x^3}-\frac {(a+b x)^{3/2} (c+d x)^{5/2}}{4 x^4}+\frac {\int \frac {\frac {3}{16} \left (3 b^4 c^4-20 a b^3 c^3 d+90 a^2 b^2 c^2 d^2+60 a^3 b c d^3-5 a^4 d^4\right )+24 a^2 b^2 c d^3 x}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{24 a^2 c}\\ &=\frac {\left (3 b^3 c^3-17 a b^2 c^2 d-55 a^2 b c d^2+5 a^3 d^3\right ) \sqrt {a+b x} \sqrt {c+d x}}{64 a^2 c x}-\frac {\left (\frac {3 b^2 c}{a}+50 b d-\frac {5 a d^2}{c}\right ) \sqrt {a+b x} (c+d x)^{3/2}}{96 x^2}-\frac {(3 b c+5 a d) \sqrt {a+b x} (c+d x)^{5/2}}{24 c x^3}-\frac {(a+b x)^{3/2} (c+d x)^{5/2}}{4 x^4}+\left (b^2 d^3\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx+\frac {\left (3 b^4 c^4-20 a b^3 c^3 d+90 a^2 b^2 c^2 d^2+60 a^3 b c d^3-5 a^4 d^4\right ) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{128 a^2 c}\\ &=\frac {\left (3 b^3 c^3-17 a b^2 c^2 d-55 a^2 b c d^2+5 a^3 d^3\right ) \sqrt {a+b x} \sqrt {c+d x}}{64 a^2 c x}-\frac {\left (\frac {3 b^2 c}{a}+50 b d-\frac {5 a d^2}{c}\right ) \sqrt {a+b x} (c+d x)^{3/2}}{96 x^2}-\frac {(3 b c+5 a d) \sqrt {a+b x} (c+d x)^{5/2}}{24 c x^3}-\frac {(a+b x)^{3/2} (c+d x)^{5/2}}{4 x^4}+\left (2 b d^3\right ) \operatorname {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 (3 b^4 c^4-20 a b^3 c^3 d+90 a^2 b^2 c^2 d^2+60 a^3 b c d^3-5 a^4 d^4\right ) \operatorname {Subst}\left (\int \frac {1}{-a+c x^2} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )}{64 a^2 c}\\ &=\frac {\left (3 b^3 c^3-17 a b^2 c^2 d-55 a^2 b c d^2+5 a^3 d^3\right ) \sqrt {a+b x} \sqrt {c+d x}}{64 a^2 c x}-\frac {\left (\frac {3 b^2 c}{a}+50 b d-\frac {5 a d^2}{c}\right ) \sqrt {a+b x} (c+d x)^{3/2}}{96 x^2}-\frac {(3 b c+5 a d) \sqrt {a+b x} (c+d x)^{5/2}}{24 c x^3}-\frac {(a+b x)^{3/2} (c+d x)^{5/2}}{4 x^4}-\frac {\left (3 b^4 c^4-20 a b^3 c^3 d+90 a^2 b^2 c^2 d^2+60 a^3 b c d^3-5 a^4 d^4\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{64 a^{5/2} c^{3/2}}+\left (2 b d^3\right ) \operatorname {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 (3 b^3 c^3-17 a b^2 c^2 d-55 a^2 b c d^2+5 a^3 d^3\right ) \sqrt {a+b x} \sqrt {c+d x}}{64 a^2 c x}-\frac {\left (\frac {3 b^2 c}{a}+50 b d-\frac {5 a d^2}{c}\right ) \sqrt {a+b x} (c+d x)^{3/2}}{96 x^2}-\frac {(3 b c+5 a d) \sqrt {a+b x} (c+d x)^{5/2}}{24 c x^3}-\frac {(a+b x)^{3/2} (c+d x)^{5/2}}{4 x^4}-\frac {\left (3 b^4 c^4-20 a b^3 c^3 d+90 a^2 b^2 c^2 d^2+60 a^3 b c d^3-5 a^4 d^4\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{64 a^{5/2} c^{3/2}}+2 b^{3/2} d^{5/2} \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 3.44, size = 296, normalized size = 0.94 \begin {gather*} -\frac {\sqrt {a+b x} \sqrt {c+d x} \left (a^3 \left (48 c^3+136 c^2 d x+118 c d^2 x^2+15 d^3 x^3\right )+a^2 b c x \left (72 c^2+244 c d x+337 d^2 x^2\right )+3 a b^2 c^2 x^2 (2 c+19 d x)-9 b^3 c^3 x^3\right )}{192 a^2 c x^4}+\frac {\left (5 a^4 d^4-60 a^3 b c d^3-90 a^2 b^2 c^2 d^2+20 a b^3 c^3 d-3 b^4 c^4\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{64 a^{5/2} c^{3/2}}+\frac {2 d^{5/2} (b c-a d)^{3/2} \left (\frac {b (c+d x)}{b c-a d}\right )^{3/2} \sinh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b c-a d}}\right )}{(c+d x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/192*(Sqrt[a + b*x]*Sqrt[c + d*x]*(-9*b^3*c^3*x^3 + 3*a*b^2*c^2*x^2*(2*c + 19*d*x) + a^2*b*c*x*(72*c^2 + 244
*c*d*x + 337*d^2*x^2) + a^3*(48*c^3 + 136*c^2*d*x + 118*c*d^2*x^2 + 15*d^3*x^3)))/(a^2*c*x^4) + (2*d^(5/2)*(b*
c - a*d)^(3/2)*((b*(c + d*x))/(b*c - a*d))^(3/2)*ArcSinh[(Sqrt[d]*Sqrt[a + b*x])/Sqrt[b*c - a*d]])/(c + d*x)^(
3/2) + ((-3*b^4*c^4 + 20*a*b^3*c^3*d - 90*a^2*b^2*c^2*d^2 - 60*a^3*b*c*d^3 + 5*a^4*d^4)*ArcTanh[(Sqrt[c]*Sqrt[
a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(64*a^(5/2)*c^(3/2))

________________________________________________________________________________________

IntegrateAlgebraic [B]  time = 0.77, size = 754, normalized size = 2.39 \begin {gather*} \frac {\left (5 a^4 d^4-60 a^3 b c d^3-90 a^2 b^2 c^2 d^2+20 a b^3 c^3 d-3 b^4 c^4\right ) \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {c+d x}}{\sqrt {c} \sqrt {a+b x}}\right )}{64 a^{5/2} c^{3/2}}+\frac {-\frac {15 a^7 d^4 (c+d x)^{7/2}}{(a+b x)^{7/2}}-\frac {73 a^6 c d^4 (c+d x)^{5/2}}{(a+b x)^{5/2}}-\frac {204 a^6 b c d^3 (c+d x)^{7/2}}{(a+b x)^{7/2}}+\frac {270 a^5 b^2 c^2 d^2 (c+d x)^{7/2}}{(a+b x)^{7/2}}+\frac {55 a^5 c^2 d^4 (c+d x)^{3/2}}{(a+b x)^{3/2}}+\frac {876 a^5 b c^2 d^3 (c+d x)^{5/2}}{(a+b x)^{5/2}}-\frac {60 a^4 b^3 c^3 d (c+d x)^{7/2}}{(a+b x)^{7/2}}-\frac {990 a^4 b^2 c^3 d^2 (c+d x)^{5/2}}{(a+b x)^{5/2}}-\frac {15 a^4 c^3 d^4 \sqrt {c+d x}}{\sqrt {a+b x}}-\frac {660 a^4 b c^3 d^3 (c+d x)^{3/2}}{(a+b x)^{3/2}}+\frac {9 a^3 b^4 c^4 (c+d x)^{7/2}}{(a+b x)^{7/2}}+\frac {220 a^3 b^3 c^4 d (c+d x)^{5/2}}{(a+b x)^{5/2}}+\frac {546 a^3 b^2 c^4 d^2 (c+d x)^{3/2}}{(a+b x)^{3/2}}+\frac {180 a^3 b c^4 d^3 \sqrt {c+d x}}{\sqrt {a+b x}}-\frac {33 a^2 b^4 c^5 (c+d x)^{5/2}}{(a+b x)^{5/2}}+\frac {92 a^2 b^3 c^5 d (c+d x)^{3/2}}{(a+b x)^{3/2}}-\frac {114 a^2 b^2 c^5 d^2 \sqrt {c+d x}}{\sqrt {a+b x}}+\frac {9 b^4 c^7 \sqrt {c+d x}}{\sqrt {a+b x}}-\frac {33 a b^4 c^6 (c+d x)^{3/2}}{(a+b x)^{3/2}}-\frac {60 a b^3 c^6 d \sqrt {c+d x}}{\sqrt {a+b x}}}{192 a^2 c \left (c-\frac {a (c+d x)}{a+b x}\right )^4}+2 b^{3/2} d^{5/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {a+b x}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((9*b^4*c^7*Sqrt[c + d*x])/Sqrt[a + b*x] - (60*a*b^3*c^6*d*Sqrt[c + d*x])/Sqrt[a + b*x] - (114*a^2*b^2*c^5*d^2
*Sqrt[c + d*x])/Sqrt[a + b*x] + (180*a^3*b*c^4*d^3*Sqrt[c + d*x])/Sqrt[a + b*x] - (15*a^4*c^3*d^4*Sqrt[c + d*x
])/Sqrt[a + b*x] - (33*a*b^4*c^6*(c + d*x)^(3/2))/(a + b*x)^(3/2) + (92*a^2*b^3*c^5*d*(c + d*x)^(3/2))/(a + b*
x)^(3/2) + (546*a^3*b^2*c^4*d^2*(c + d*x)^(3/2))/(a + b*x)^(3/2) - (660*a^4*b*c^3*d^3*(c + d*x)^(3/2))/(a + b*
x)^(3/2) + (55*a^5*c^2*d^4*(c + d*x)^(3/2))/(a + b*x)^(3/2) - (33*a^2*b^4*c^5*(c + d*x)^(5/2))/(a + b*x)^(5/2)
+ (220*a^3*b^3*c^4*d*(c + d*x)^(5/2))/(a + b*x)^(5/2) - (990*a^4*b^2*c^3*d^2*(c + d*x)^(5/2))/(a + b*x)^(5/2)
+ (876*a^5*b*c^2*d^3*(c + d*x)^(5/2))/(a + b*x)^(5/2) - (73*a^6*c*d^4*(c + d*x)^(5/2))/(a + b*x)^(5/2) + (9*a
^3*b^4*c^4*(c + d*x)^(7/2))/(a + b*x)^(7/2) - (60*a^4*b^3*c^3*d*(c + d*x)^(7/2))/(a + b*x)^(7/2) + (270*a^5*b^
2*c^2*d^2*(c + d*x)^(7/2))/(a + b*x)^(7/2) - (204*a^6*b*c*d^3*(c + d*x)^(7/2))/(a + b*x)^(7/2) - (15*a^7*d^4*(
c + d*x)^(7/2))/(a + b*x)^(7/2))/(192*a^2*c*(c - (a*(c + d*x))/(a + b*x))^4) + ((-3*b^4*c^4 + 20*a*b^3*c^3*d -
90*a^2*b^2*c^2*d^2 - 60*a^3*b*c*d^3 + 5*a^4*d^4)*ArcTanh[(Sqrt[a]*Sqrt[c + d*x])/(Sqrt[c]*Sqrt[a + b*x])])/(6
4*a^(5/2)*c^(3/2)) + 2*b^(3/2)*d^(5/2)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[a + b*x])]

________________________________________________________________________________________

fricas [A]  time = 28.92, size = 1529, normalized size = 4.84

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

giac [B]  time = 11.84, size = 3887, normalized size = 12.30

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/192*(192*sqrt(b*d)*b*d^2*abs(b)*log((sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2) + 3*
(3*sqrt(b*d)*b^5*c^4*abs(b) - 20*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) + 60*sqrt(
b*d)*a^3*b^2*c*d^3*abs(b) - 5*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^2*b*c) - 2*(9*sqrt(b*d)*b^19
*c^11*abs(b) - 129*sqrt(b*d)*a*b^18*c^10*d*abs(b) + 371*sqrt(b*d)*a^2*b^17*c^9*d^2*abs(b) + 581*sqrt(b*d)*a^3*
b^16*c^8*d^3*abs(b) - 5494*sqrt(b*d)*a^4*b^15*c^7*d^4*abs(b) + 13958*sqrt(b*d)*a^5*b^14*c^6*d^5*abs(b) - 19306
*sqrt(b*d)*a^6*b^13*c^5*d^6*abs(b) + 16154*sqrt(b*d)*a^7*b^12*c^4*d^7*abs(b) - 8131*sqrt(b*d)*a^8*b^11*c^3*d^8
*abs(b) + 2219*sqrt(b*d)*a^9*b^10*c^2*d^9*abs(b) - 217*sqrt(b*d)*a^10*b^9*c*d^10*abs(b) - 15*sqrt(b*d)*a^11*b^
8*d^11*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*b^17*c^10*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*b^16*c^9*d*abs(b) - 619*
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) - 5272*sqr
t(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) + 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^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) - 10942*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) + 18024*sqrt(b*d)*(s
qrt(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) - 9523*sqrt(b*d)*(sqrt
(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) + 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^9*b^8*c*d^9*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*a^10*b^7*d^10*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*b^15*c^9*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*b^14*c^8*d*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^2*b^13*c^7*d^2*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^3*b^12*c^6*d^3*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^4*b^11*c^5*d^4*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^5*b^10*c^4*d^5*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^6*b^9*c^3*d^6*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^7*b^8*c^2*d^7*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^8*b^7*c*d^8*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*
a^9*b^6*d^9*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*b^13*c^8*
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*b^12*c^7*d*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^2*b^11*c^6*d^2*abs(b) - 8
096*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) - 2378
*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) - 5456*sqr
t(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) - 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^6*b^7*c^2*d^6*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^7*b^6*c*d^7*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*a^8*b^5*d^8*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*b^11*c^7*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*b^10*c^6*d*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^2*b^9*c^5*d^2*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^3*b^8*c^4*d^3*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^4*b^7*c^3*d^4*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^5*b^6*c^2*d^5*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^6*b^5*c*d^6*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*a^7*b^4*d^7*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*b^9*c^6*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*b^8*c^
5*d*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^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*a
bs(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^4*b^5*c^2*d^4*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^5*b^4*c*d^5*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*a^6*b^3*d^6*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*b^7*c^5*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*b^6*c^4*d*abs(b) - 4290*sqrt(b*d)*(sqrt(b*d)*sq
rt(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) - 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^3*b^4*c^2*d^3*abs(b) - 3453*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) - 105*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) - 9*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) + 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*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) + 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^3*b^2*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))^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^2*c))/b

________________________________________________________________________________________

maple [B]  time = 0.02, size = 852, normalized size = 2.70 \begin {gather*} \frac {\sqrt {b x +a}\, \sqrt {d x +c}\, \left (15 \sqrt {b d}\, a^{4} d^{4} x^{4} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}}{x}\right )-180 \sqrt {b d}\, a^{3} b c \,d^{3} x^{4} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}}{x}\right )-270 \sqrt {b d}\, a^{2} b^{2} c^{2} d^{2} x^{4} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}}{x}\right )+384 \sqrt {a c}\, a^{2} b^{2} c \,d^{3} x^{4} \ln \left (\frac {2 b d x +a d +b c +2 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}}{2 \sqrt {b d}}\right )+60 \sqrt {b d}\, a \,b^{3} c^{3} d \,x^{4} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}}{x}\right )-9 \sqrt {b d}\, b^{4} c^{4} x^{4} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}}{x}\right )-30 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}\, \sqrt {a c}\, a^{3} d^{3} x^{3}-674 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}\, \sqrt {a c}\, a^{2} b c \,d^{2} x^{3}-114 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}\, \sqrt {a c}\, a \,b^{2} c^{2} d \,x^{3}+18 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}\, \sqrt {a c}\, b^{3} c^{3} x^{3}-236 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}\, \sqrt {a c}\, a^{3} c \,d^{2} x^{2}-488 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}\, \sqrt {a c}\, a^{2} b \,c^{2} d \,x^{2}-12 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}\, \sqrt {a c}\, a \,b^{2} c^{3} x^{2}-272 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}\, \sqrt {a c}\, a^{3} c^{2} d x -144 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}\, \sqrt {a c}\, a^{2} b \,c^{3} x -96 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}\, \sqrt {a c}\, a^{3} c^{3}\right )}{384 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}\, \sqrt {a c}\, a^{2} c \,x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(3/2)*(d*x+c)^(5/2)/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 details)Is a*d-b*c zero or nonzero?

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (a+b\,x\right )}^{3/2}\,{\left (c+d\,x\right )}^{5/2}}{x^5} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a + b x\right )^{\frac {3}{2}} \left (c + d x\right )^{\frac {5}{2}}}{x^{5}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________