\(\int \frac {(a+b x)^{5/2}}{x (c+d x)^{5/2}} \, dx\) [694]

Optimal result

Integrand size = 22, antiderivative size = 157 \[ \int \frac {(a+b x)^{5/2}}{x (c+d x)^{5/2}} \, dx=-\frac {2 (b c-a d) (a+b x)^{3/2}}{3 c d (c+d x)^{3/2}}+\frac {2 \left (\frac {a^2}{c^2}-\frac {b^2}{d^2}\right ) \sqrt {a+b x}}{\sqrt {c+d x}}-\frac {2 a^{5/2} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{c^{5/2}}+\frac {2 b^{5/2} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{d^{5/2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {100, 155, 163, 65, 223, 212, 95, 214} \[ \int \frac {(a+b x)^{5/2}}{x (c+d x)^{5/2}} \, dx=-\frac {2 a^{5/2} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{c^{5/2}}+\frac {2 \sqrt {a+b x} \left (\frac {a^2}{c^2}-\frac {b^2}{d^2}\right )}{\sqrt {c+d x}}+\frac {2 b^{5/2} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{d^{5/2}}-\frac {2 (a+b x)^{3/2} (b c-a d)}{3 c d (c+d x)^{3/2}} \]

[In]

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

[Out]

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

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 100

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*c -
a*d)*(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*((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 - 2)*(e + f*x)^p*Simp[a*d*(d*e*(n - 1) + c*f*(p + 1)) + b*c*(d
*e*(m - n + 2) - c*f*(m + p + 2)) + d*(a*d*f*(n + p) + b*(d*e*(m + 1) - c*f*(m + n + p + 1)))*x, x], x], x] /;
FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 1] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p
] || IntegersQ[p, m + n])

Rule 155

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] && 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 {2 (b c-a d) (a+b x)^{3/2}}{3 c d (c+d x)^{3/2}}+\frac {2 \int \frac {\sqrt {a+b x} \left (\frac {3 a^2 d}{2}+\frac {3}{2} b^2 c x\right )}{x (c+d x)^{3/2}} \, dx}{3 c d} \\ & = -\frac {2 (b c-a d) (a+b x)^{3/2}}{3 c d (c+d x)^{3/2}}+\frac {2 \left (\frac {a^2}{c^2}-\frac {b^2}{d^2}\right ) \sqrt {a+b x}}{\sqrt {c+d x}}-\frac {4 \int \frac {-\frac {3}{4} a^3 d^2-\frac {3}{4} b^3 c^2 x}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{3 c^2 d^2} \\ & = -\frac {2 (b c-a d) (a+b x)^{3/2}}{3 c d (c+d x)^{3/2}}+\frac {2 \left (\frac {a^2}{c^2}-\frac {b^2}{d^2}\right ) \sqrt {a+b x}}{\sqrt {c+d x}}+\frac {a^3 \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{c^2}+\frac {b^3 \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{d^2} \\ & = -\frac {2 (b c-a d) (a+b x)^{3/2}}{3 c d (c+d x)^{3/2}}+\frac {2 \left (\frac {a^2}{c^2}-\frac {b^2}{d^2}\right ) \sqrt {a+b x}}{\sqrt {c+d x}}+\frac {\left (2 a^3\right ) \text {Subst}\left (\int \frac {1}{-a+c x^2} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )}{c^2}+\frac {\left (2 b^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 )}{d^2} \\ & = -\frac {2 (b c-a d) (a+b x)^{3/2}}{3 c d (c+d x)^{3/2}}+\frac {2 \left (\frac {a^2}{c^2}-\frac {b^2}{d^2}\right ) \sqrt {a+b x}}{\sqrt {c+d x}}-\frac {2 a^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{c^{5/2}}+\frac {\left (2 b^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 )}{d^2} \\ & = -\frac {2 (b c-a d) (a+b x)^{3/2}}{3 c d (c+d x)^{3/2}}+\frac {2 \left (\frac {a^2}{c^2}-\frac {b^2}{d^2}\right ) \sqrt {a+b x}}{\sqrt {c+d x}}-\frac {2 a^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{c^{5/2}}+\frac {2 b^{5/2} \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{d^{5/2}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 7.17 (sec) , antiderivative size = 1440, normalized size of antiderivative = 9.17 \[ \int \frac {(a+b x)^{5/2}}{x (c+d x)^{5/2}} \, dx=\frac {8 a^3 b d^2 \left (2 c^2+9 c d x+3 d^2 x^2\right )+8 b^3 c^2 (3 c+4 d x) \left (b x (3 c-d x)+\sqrt {a-\frac {b c}{d}} \sqrt {a+b x} (-c+3 d x)\right )+2 a^2 b d \left (-4 \sqrt {a-\frac {b c}{d}} d \sqrt {a+b x} \left (2 c^2+9 c d x+3 d^2 x^2\right )+b \left (-51 c^3-84 c^2 d x+21 c d^2 x^2+6 d^3 x^3\right )\right )+a b^2 c \left (2 \sqrt {a-\frac {b c}{d}} d \sqrt {a+b x} \left (47 c^2+70 c d x-9 d^2 x^2\right )+b \left (73 c^3-15 c^2 d x-177 c d^2 x^2+7 d^3 x^3\right )\right )}{12 c^2 d^2 (c+d x)^{3/2} \left (b c \left (\sqrt {a-\frac {b c}{d}}-3 \sqrt {a+b x}\right )+b d x \left (-3 \sqrt {a-\frac {b c}{d}}+\sqrt {a+b x}\right )+a d \left (-4 \sqrt {a-\frac {b c}{d}}+4 \sqrt {a+b x}\right )\right )}+\frac {2 a^3 \sqrt {d} \arctan \left (\frac {\sqrt {b c-2 a d-2 i \sqrt {a} \sqrt {d} \sqrt {b c-a d}} \sqrt {c+d x}}{\sqrt {c} \sqrt {d} \left (\sqrt {a-\frac {b c}{d}}-\sqrt {a+b x}\right )}\right )}{c^{5/2} \sqrt {b c-2 a d-2 i \sqrt {a} \sqrt {d} \sqrt {b c-a d}}}-\frac {2 i a^{7/2} d \arctan \left (\frac {\sqrt {b c-2 a d-2 i \sqrt {a} \sqrt {d} \sqrt {b c-a d}} \sqrt {c+d x}}{\sqrt {c} \sqrt {d} \left (\sqrt {a-\frac {b c}{d}}-\sqrt {a+b x}\right )}\right )}{c^{5/2} \sqrt {b c-a d} \sqrt {b c-2 a d-2 i \sqrt {a} \sqrt {d} \sqrt {b c-a d}}}+\frac {2 a^3 \sqrt {d} \arctan \left (\frac {\sqrt {b c-2 a d+2 i \sqrt {a} \sqrt {d} \sqrt {b c-a d}} \sqrt {c+d x}}{\sqrt {c} \sqrt {d} \left (\sqrt {a-\frac {b c}{d}}-\sqrt {a+b x}\right )}\right )}{c^{5/2} \sqrt {b c-2 a d+2 i \sqrt {a} \sqrt {d} \sqrt {b c-a d}}}+\frac {2 i a^{7/2} d \arctan \left (\frac {\sqrt {b c-2 a d+2 i \sqrt {a} \sqrt {d} \sqrt {b c-a d}} \sqrt {c+d x}}{\sqrt {c} \sqrt {d} \left (\sqrt {a-\frac {b c}{d}}-\sqrt {a+b x}\right )}\right )}{c^{5/2} \sqrt {b c-a d} \sqrt {b c-2 a d+2 i \sqrt {a} \sqrt {d} \sqrt {b c-a d}}}+\frac {a \left (\frac {b \left (\sqrt {a-\frac {b c}{d}}-3 \sqrt {a+b x}\right )}{d}-\frac {4 a (4 c+3 d x) \left (\sqrt {a-\frac {b c}{d}}-\sqrt {a+b x}\right )}{c^2}+\frac {b x \left (-3 \sqrt {a-\frac {b c}{d}}+\sqrt {a+b x}\right )}{c}+\frac {24 i a^{3/2} b (c+d x)^{3/2} \arctan \left (\frac {\sqrt {b c-2 a d-2 i \sqrt {a} \sqrt {d} \sqrt {b c-a d}} \sqrt {c+d x}}{\sqrt {c} \sqrt {d} \left (\sqrt {a-\frac {b c}{d}}-\sqrt {a+b x}\right )}\right )}{c^{3/2} \sqrt {b c-a d} \sqrt {b c-2 a d-2 i \sqrt {a} \sqrt {d} \sqrt {b c-a d}}}-\frac {24 i a^{3/2} b (c+d x)^{3/2} \arctan \left (\frac {\sqrt {b c-2 a d+2 i \sqrt {a} \sqrt {d} \sqrt {b c-a d}} \sqrt {c+d x}}{\sqrt {c} \sqrt {d} \left (\sqrt {a-\frac {b c}{d}}-\sqrt {a+b x}\right )}\right )}{c^{3/2} \sqrt {b c-a d} \sqrt {b c-2 a d+2 i \sqrt {a} \sqrt {d} \sqrt {b c-a d}}}\right )}{12 (c+d x)^{3/2}}-\frac {4 b^{5/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d} \left (\sqrt {a-\frac {b c}{d}}-\sqrt {a+b x}\right )}\right )}{d^{5/2}} \]

[In]

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

[Out]

(8*a^3*b*d^2*(2*c^2 + 9*c*d*x + 3*d^2*x^2) + 8*b^3*c^2*(3*c + 4*d*x)*(b*x*(3*c - d*x) + Sqrt[a - (b*c)/d]*Sqrt
[a + b*x]*(-c + 3*d*x)) + 2*a^2*b*d*(-4*Sqrt[a - (b*c)/d]*d*Sqrt[a + b*x]*(2*c^2 + 9*c*d*x + 3*d^2*x^2) + b*(-
51*c^3 - 84*c^2*d*x + 21*c*d^2*x^2 + 6*d^3*x^3)) + a*b^2*c*(2*Sqrt[a - (b*c)/d]*d*Sqrt[a + b*x]*(47*c^2 + 70*c
*d*x - 9*d^2*x^2) + b*(73*c^3 - 15*c^2*d*x - 177*c*d^2*x^2 + 7*d^3*x^3)))/(12*c^2*d^2*(c + d*x)^(3/2)*(b*c*(Sq
rt[a - (b*c)/d] - 3*Sqrt[a + b*x]) + b*d*x*(-3*Sqrt[a - (b*c)/d] + Sqrt[a + b*x]) + a*d*(-4*Sqrt[a - (b*c)/d]
+ 4*Sqrt[a + b*x]))) + (2*a^3*Sqrt[d]*ArcTan[(Sqrt[b*c - 2*a*d - (2*I)*Sqrt[a]*Sqrt[d]*Sqrt[b*c - a*d]]*Sqrt[c
+ d*x])/(Sqrt[c]*Sqrt[d]*(Sqrt[a - (b*c)/d] - Sqrt[a + b*x]))])/(c^(5/2)*Sqrt[b*c - 2*a*d - (2*I)*Sqrt[a]*Sqr
t[d]*Sqrt[b*c - a*d]]) - ((2*I)*a^(7/2)*d*ArcTan[(Sqrt[b*c - 2*a*d - (2*I)*Sqrt[a]*Sqrt[d]*Sqrt[b*c - a*d]]*Sq
rt[c + d*x])/(Sqrt[c]*Sqrt[d]*(Sqrt[a - (b*c)/d] - Sqrt[a + b*x]))])/(c^(5/2)*Sqrt[b*c - a*d]*Sqrt[b*c - 2*a*d
- (2*I)*Sqrt[a]*Sqrt[d]*Sqrt[b*c - a*d]]) + (2*a^3*Sqrt[d]*ArcTan[(Sqrt[b*c - 2*a*d + (2*I)*Sqrt[a]*Sqrt[d]*S
qrt[b*c - a*d]]*Sqrt[c + d*x])/(Sqrt[c]*Sqrt[d]*(Sqrt[a - (b*c)/d] - Sqrt[a + b*x]))])/(c^(5/2)*Sqrt[b*c - 2*a
*d + (2*I)*Sqrt[a]*Sqrt[d]*Sqrt[b*c - a*d]]) + ((2*I)*a^(7/2)*d*ArcTan[(Sqrt[b*c - 2*a*d + (2*I)*Sqrt[a]*Sqrt[
d]*Sqrt[b*c - a*d]]*Sqrt[c + d*x])/(Sqrt[c]*Sqrt[d]*(Sqrt[a - (b*c)/d] - Sqrt[a + b*x]))])/(c^(5/2)*Sqrt[b*c -
a*d]*Sqrt[b*c - 2*a*d + (2*I)*Sqrt[a]*Sqrt[d]*Sqrt[b*c - a*d]]) + (a*((b*(Sqrt[a - (b*c)/d] - 3*Sqrt[a + b*x]
))/d - (4*a*(4*c + 3*d*x)*(Sqrt[a - (b*c)/d] - Sqrt[a + b*x]))/c^2 + (b*x*(-3*Sqrt[a - (b*c)/d] + Sqrt[a + b*x
]))/c + ((24*I)*a^(3/2)*b*(c + d*x)^(3/2)*ArcTan[(Sqrt[b*c - 2*a*d - (2*I)*Sqrt[a]*Sqrt[d]*Sqrt[b*c - a*d]]*Sq
rt[c + d*x])/(Sqrt[c]*Sqrt[d]*(Sqrt[a - (b*c)/d] - Sqrt[a + b*x]))])/(c^(3/2)*Sqrt[b*c - a*d]*Sqrt[b*c - 2*a*d
- (2*I)*Sqrt[a]*Sqrt[d]*Sqrt[b*c - a*d]]) - ((24*I)*a^(3/2)*b*(c + d*x)^(3/2)*ArcTan[(Sqrt[b*c - 2*a*d + (2*I
)*Sqrt[a]*Sqrt[d]*Sqrt[b*c - a*d]]*Sqrt[c + d*x])/(Sqrt[c]*Sqrt[d]*(Sqrt[a - (b*c)/d] - Sqrt[a + b*x]))])/(c^(
3/2)*Sqrt[b*c - a*d]*Sqrt[b*c - 2*a*d + (2*I)*Sqrt[a]*Sqrt[d]*Sqrt[b*c - a*d]])))/(12*(c + d*x)^(3/2)) - (4*b^
(5/2)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/(Sqrt[d]*(Sqrt[a - (b*c)/d] - Sqrt[a + b*x]))])/d^(5/2)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(565\) vs. \(2(123)=246\).

Time = 0.57 (sec) , antiderivative size = 566, normalized size of antiderivative = 3.61

method	result	size
default	\(-\frac {\sqrt {b x +a}\, \left (3 \sqrt {b d}\, \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a^{3} d^{4} x^{2}-3 \sqrt {a c}\, \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) b^{3} c^{2} d^{2} x^{2}+6 \sqrt {b d}\, \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a^{3} c \,d^{3} x -6 \sqrt {a c}\, \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) b^{3} c^{3} d x +3 \sqrt {b d}\, \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a^{3} c^{2} d^{2}-3 \sqrt {a c}\, \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) b^{3} c^{4}-6 a^{2} d^{3} x \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}-2 \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a b c \,d^{2} x +8 \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, b^{2} c^{2} d x -8 a^{2} c \,d^{2} \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+2 \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a b \,c^{2} d +6 \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, b^{2} c^{3}\right )}{3 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, \sqrt {a c}\, \left (d x +c \right )^{\frac {3}{2}} d^{2} c^{2}}\)	\(566\)

[In]

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

[Out]

-1/3*(b*x+a)^(1/2)*(3*(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*d^4*x^2-
3*(a*c)^(1/2)*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^2*x^2+6*(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^3*x-6*(a*c)^(1/2)*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^3*d*x+3*(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^2*d^2-3*(a*c)^(1/2)*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^4-6*a^2*d^3*x*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-2*(b
*d)^(1/2)*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a*b*c*d^2*x+8*(b*d)^(1/2)*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*b^
2*c^2*d*x-8*a^2*c*d^2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+2*(b*d)^(1/2)*(a*c)^(1/2)*((b*x+a)*(d*x+
c))^(1/2)*a*b*c^2*d+6*(b*d)^(1/2)*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*b^2*c^3)/((b*x+a)*(d*x+c))^(1/2)/(b*d)^(
1/2)/(a*c)^(1/2)/(d*x+c)^(3/2)/d^2/c^2

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 313 vs. \(2 (123) = 246\).

Time = 1.08 (sec) , antiderivative size = 1361, normalized size of antiderivative = 8.67 \[ \int \frac {(a+b x)^{5/2}}{x (c+d x)^{5/2}} \, dx=\text {Too large to display} \]

[In]

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

[Out]

[1/6*(3*(b^2*c^2*d^2*x^2 + 2*b^2*c^3*d*x + b^2*c^4)*sqrt(b/d)*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^2*x + b*c*d + a*d^2)*sqrt(b*x + a)*sqrt(d*x + c)*sqrt(b/d) + 8*(b^2*c*d + a*b*d^2)*x) + 3*(a^2*d^
4*x^2 + 2*a^2*c*d^3*x + a^2*c^2*d^2)*sqrt(a/c)*log((8*a^2*c^2 + (b^2*c^2 + 6*a*b*c*d + a^2*d^2)*x^2 - 4*(2*a*c
^2 + (b*c^2 + a*c*d)*x)*sqrt(b*x + a)*sqrt(d*x + c)*sqrt(a/c) + 8*(a*b*c^2 + a^2*c*d)*x)/x^2) - 4*(3*b^2*c^3 +
a*b*c^2*d - 4*a^2*c*d^2 + (4*b^2*c^2*d - a*b*c*d^2 - 3*a^2*d^3)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(c^2*d^4*x^2
+ 2*c^3*d^3*x + c^4*d^2), -1/6*(6*(b^2*c^2*d^2*x^2 + 2*b^2*c^3*d*x + b^2*c^4)*sqrt(-b/d)*arctan(1/2*(2*b*d*x +
b*c + a*d)*sqrt(b*x + a)*sqrt(d*x + c)*sqrt(-b/d)/(b^2*d*x^2 + a*b*c + (b^2*c + a*b*d)*x)) - 3*(a^2*d^4*x^2 +
2*a^2*c*d^3*x + a^2*c^2*d^2)*sqrt(a/c)*log((8*a^2*c^2 + (b^2*c^2 + 6*a*b*c*d + a^2*d^2)*x^2 - 4*(2*a*c^2 + (b
*c^2 + a*c*d)*x)*sqrt(b*x + a)*sqrt(d*x + c)*sqrt(a/c) + 8*(a*b*c^2 + a^2*c*d)*x)/x^2) + 4*(3*b^2*c^3 + a*b*c^
2*d - 4*a^2*c*d^2 + (4*b^2*c^2*d - a*b*c*d^2 - 3*a^2*d^3)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(c^2*d^4*x^2 + 2*c^3
*d^3*x + c^4*d^2), 1/6*(6*(a^2*d^4*x^2 + 2*a^2*c*d^3*x + a^2*c^2*d^2)*sqrt(-a/c)*arctan(1/2*(2*a*c + (b*c + a*
d)*x)*sqrt(b*x + a)*sqrt(d*x + c)*sqrt(-a/c)/(a*b*d*x^2 + a^2*c + (a*b*c + a^2*d)*x)) + 3*(b^2*c^2*d^2*x^2 + 2
*b^2*c^3*d*x + b^2*c^4)*sqrt(b/d)*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^2*x + b*c*d + a
*d^2)*sqrt(b*x + a)*sqrt(d*x + c)*sqrt(b/d) + 8*(b^2*c*d + a*b*d^2)*x) - 4*(3*b^2*c^3 + a*b*c^2*d - 4*a^2*c*d^
2 + (4*b^2*c^2*d - a*b*c*d^2 - 3*a^2*d^3)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(c^2*d^4*x^2 + 2*c^3*d^3*x + c^4*d^2
), 1/3*(3*(a^2*d^4*x^2 + 2*a^2*c*d^3*x + a^2*c^2*d^2)*sqrt(-a/c)*arctan(1/2*(2*a*c + (b*c + a*d)*x)*sqrt(b*x +
a)*sqrt(d*x + c)*sqrt(-a/c)/(a*b*d*x^2 + a^2*c + (a*b*c + a^2*d)*x)) - 3*(b^2*c^2*d^2*x^2 + 2*b^2*c^3*d*x + b
^2*c^4)*sqrt(-b/d)*arctan(1/2*(2*b*d*x + b*c + a*d)*sqrt(b*x + a)*sqrt(d*x + c)*sqrt(-b/d)/(b^2*d*x^2 + a*b*c
+ (b^2*c + a*b*d)*x)) - 2*(3*b^2*c^3 + a*b*c^2*d - 4*a^2*c*d^2 + (4*b^2*c^2*d - a*b*c*d^2 - 3*a^2*d^3)*x)*sqrt
(b*x + a)*sqrt(d*x + c))/(c^2*d^4*x^2 + 2*c^3*d^3*x + c^4*d^2)]

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F(-2)]

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

[In]

integrate((b*x+a)^(5/2)/x/(d*x+c)^(5/2),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. 347 vs. \(2 (123) = 246\).

Time = 0.43 (sec) , antiderivative size = 347, normalized size of antiderivative = 2.21 \[ \int \frac {(a+b x)^{5/2}}{x (c+d x)^{5/2}} \, dx=-\frac {2 \, \sqrt {b d} a^{3} b \arctan \left (-\frac {b^{2} c + a b d - {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2}}{2 \, \sqrt {-a b c d} b}\right )}{\sqrt {-a b c d} c^{2} {\left | b \right |}} - \frac {\sqrt {b d} b^{3} \log \left ({\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2}\right )}{d^{3} {\left | b \right |}} - \frac {2 \, \sqrt {b x + a} {\left (\frac {{\left (4 \, b^{8} c^{5} d^{2} - 5 \, a b^{7} c^{4} d^{3} - 2 \, a^{2} b^{6} c^{3} d^{4} + 3 \, a^{3} b^{5} c^{2} d^{5}\right )} {\left (b x + a\right )}}{b^{3} c^{5} d^{3} {\left | b \right |} - a b^{2} c^{4} d^{4} {\left | b \right |}} + \frac {3 \, {\left (b^{9} c^{6} d - 2 \, a b^{8} c^{5} d^{2} + 2 \, a^{3} b^{6} c^{3} d^{4} - a^{4} b^{5} c^{2} d^{5}\right )}}{b^{3} c^{5} d^{3} {\left | b \right |} - a b^{2} c^{4} d^{4} {\left | b \right |}}\right )}}{3 \, {\left (b^{2} c + {\left (b x + a\right )} b d - a b d\right )}^{\frac {3}{2}}} \]

[In]

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

[Out]

-2*sqrt(b*d)*a^3*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)*c^2*abs(b)) - sqrt(b*d)*b^3*log((sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c
+ (b*x + a)*b*d - a*b*d))^2)/(d^3*abs(b)) - 2/3*sqrt(b*x + a)*((4*b^8*c^5*d^2 - 5*a*b^7*c^4*d^3 - 2*a^2*b^6*c
^3*d^4 + 3*a^3*b^5*c^2*d^5)*(b*x + a)/(b^3*c^5*d^3*abs(b) - a*b^2*c^4*d^4*abs(b)) + 3*(b^9*c^6*d - 2*a*b^8*c^5
*d^2 + 2*a^3*b^6*c^3*d^4 - a^4*b^5*c^2*d^5)/(b^3*c^5*d^3*abs(b) - a*b^2*c^4*d^4*abs(b)))/(b^2*c + (b*x + a)*b*
d - a*b*d)^(3/2)

Mupad [F(-1)]

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

[In]

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

[Out]

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