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

Optimal. Leaf size=334 \[ -\frac{5 \sqrt{a+b x} (c+d x)^{3/2} \left (-31 a^2 d^2-18 a b c d+b^2 c^2\right )}{96 d}-\frac{5 \sqrt{a+b x} \sqrt{c+d x} \left (-45 a^2 b c d^2-a^3 d^3-19 a b^2 c^2 d+b^3 c^3\right )}{64 b d}-\frac{5 \left (-90 a^2 b^2 c^2 d^2-20 a^3 b c d^3+a^4 d^4-20 a b^3 c^3 d+b^4 c^4\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{64 b^{3/2} d^{3/2}}-5 a^{3/2} c^{3/2} (a d+b c) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )-\frac{(a+b x)^{5/2} (c+d x)^{5/2}}{x}+\frac{5}{4} b (a+b x)^{3/2} (c+d x)^{5/2}+\frac{5 b \sqrt{a+b x} (c+d x)^{5/2} (7 a d+b c)}{24 d} \]

[Out]

(-5*(b^3*c^3 - 19*a*b^2*c^2*d - 45*a^2*b*c*d^2 - a^3*d^3)*Sqrt[a + b*x]*Sqrt[c + d*x])/(64*b*d) - (5*(b^2*c^2
- 18*a*b*c*d - 31*a^2*d^2)*Sqrt[a + b*x]*(c + d*x)^(3/2))/(96*d) + (5*b*(b*c + 7*a*d)*Sqrt[a + b*x]*(c + d*x)^
(5/2))/(24*d) + (5*b*(a + b*x)^(3/2)*(c + d*x)^(5/2))/4 - ((a + b*x)^(5/2)*(c + d*x)^(5/2))/x - 5*a^(3/2)*c^(3
/2)*(b*c + a*d)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])] - (5*(b^4*c^4 - 20*a*b^3*c^3*d - 90*a
^2*b^2*c^2*d^2 - 20*a^3*b*c*d^3 + a^4*d^4)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(64*b^(3/
2)*d^(3/2))

________________________________________________________________________________________

Rubi [A]  time = 0.38849, antiderivative size = 334, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 8, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364, Rules used = {97, 154, 157, 63, 217, 206, 93, 208} \[ -\frac{5 \sqrt{a+b x} (c+d x)^{3/2} \left (-31 a^2 d^2-18 a b c d+b^2 c^2\right )}{96 d}-\frac{5 \sqrt{a+b x} \sqrt{c+d x} \left (-45 a^2 b c d^2-a^3 d^3-19 a b^2 c^2 d+b^3 c^3\right )}{64 b d}-\frac{5 \left (-90 a^2 b^2 c^2 d^2-20 a^3 b c d^3+a^4 d^4-20 a b^3 c^3 d+b^4 c^4\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{64 b^{3/2} d^{3/2}}-5 a^{3/2} c^{3/2} (a d+b c) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )-\frac{(a+b x)^{5/2} (c+d x)^{5/2}}{x}+\frac{5}{4} b (a+b x)^{3/2} (c+d x)^{5/2}+\frac{5 b \sqrt{a+b x} (c+d x)^{5/2} (7 a d+b c)}{24 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-5*(b^3*c^3 - 19*a*b^2*c^2*d - 45*a^2*b*c*d^2 - a^3*d^3)*Sqrt[a + b*x]*Sqrt[c + d*x])/(64*b*d) - (5*(b^2*c^2
- 18*a*b*c*d - 31*a^2*d^2)*Sqrt[a + b*x]*(c + d*x)^(3/2))/(96*d) + (5*b*(b*c + 7*a*d)*Sqrt[a + b*x]*(c + d*x)^
(5/2))/(24*d) + (5*b*(a + b*x)^(3/2)*(c + d*x)^(5/2))/4 - ((a + b*x)^(5/2)*(c + d*x)^(5/2))/x - 5*a^(3/2)*c^(3
/2)*(b*c + a*d)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])] - (5*(b^4*c^4 - 20*a*b^3*c^3*d - 90*a
^2*b^2*c^2*d^2 - 20*a^3*b*c*d^3 + a^4*d^4)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(64*b^(3/
2)*d^(3/2))

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 154

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 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 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 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]

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 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 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]

Rubi steps

\begin{align*} \int \frac{(a+b x)^{5/2} (c+d x)^{5/2}}{x^2} \, dx &=-\frac{(a+b x)^{5/2} (c+d x)^{5/2}}{x}+\int \frac{(a+b x)^{3/2} (c+d x)^{3/2} \left (\frac{5}{2} (b c+a d)+5 b d x\right )}{x} \, dx\\ &=\frac{5}{4} b (a+b x)^{3/2} (c+d x)^{5/2}-\frac{(a+b x)^{5/2} (c+d x)^{5/2}}{x}+\frac{\int \frac{\sqrt{a+b x} (c+d x)^{3/2} \left (10 a d (b c+a d)+\frac{5}{2} b d (b c+7 a d) x\right )}{x} \, dx}{4 d}\\ &=\frac{5 b (b c+7 a d) \sqrt{a+b x} (c+d x)^{5/2}}{24 d}+\frac{5}{4} b (a+b x)^{3/2} (c+d x)^{5/2}-\frac{(a+b x)^{5/2} (c+d x)^{5/2}}{x}+\frac{\int \frac{(c+d x)^{3/2} \left (30 a^2 d^2 (b c+a d)-\frac{5}{4} b d \left (b^2 c^2-18 a b c d-31 a^2 d^2\right ) x\right )}{x \sqrt{a+b x}} \, dx}{12 d^2}\\ &=-\frac{5 \left (b^2 c^2-18 a b c d-31 a^2 d^2\right ) \sqrt{a+b x} (c+d x)^{3/2}}{96 d}+\frac{5 b (b c+7 a d) \sqrt{a+b x} (c+d x)^{5/2}}{24 d}+\frac{5}{4} b (a+b x)^{3/2} (c+d x)^{5/2}-\frac{(a+b x)^{5/2} (c+d x)^{5/2}}{x}+\frac{\int \frac{\sqrt{c+d x} \left (60 a^2 b c d^2 (b c+a d)-\frac{15}{8} b d \left (b^3 c^3-19 a b^2 c^2 d-45 a^2 b c d^2-a^3 d^3\right ) x\right )}{x \sqrt{a+b x}} \, dx}{24 b d^2}\\ &=-\frac{5 \left (b^3 c^3-19 a b^2 c^2 d-45 a^2 b c d^2-a^3 d^3\right ) \sqrt{a+b x} \sqrt{c+d x}}{64 b d}-\frac{5 \left (b^2 c^2-18 a b c d-31 a^2 d^2\right ) \sqrt{a+b x} (c+d x)^{3/2}}{96 d}+\frac{5 b (b c+7 a d) \sqrt{a+b x} (c+d x)^{5/2}}{24 d}+\frac{5}{4} b (a+b x)^{3/2} (c+d x)^{5/2}-\frac{(a+b x)^{5/2} (c+d x)^{5/2}}{x}+\frac{\int \frac{60 a^2 b^2 c^2 d^2 (b c+a d)-\frac{15}{16} b d \left (b^4 c^4-20 a b^3 c^3 d-90 a^2 b^2 c^2 d^2-20 a^3 b c d^3+a^4 d^4\right ) x}{x \sqrt{a+b x} \sqrt{c+d x}} \, dx}{24 b^2 d^2}\\ &=-\frac{5 \left (b^3 c^3-19 a b^2 c^2 d-45 a^2 b c d^2-a^3 d^3\right ) \sqrt{a+b x} \sqrt{c+d x}}{64 b d}-\frac{5 \left (b^2 c^2-18 a b c d-31 a^2 d^2\right ) \sqrt{a+b x} (c+d x)^{3/2}}{96 d}+\frac{5 b (b c+7 a d) \sqrt{a+b x} (c+d x)^{5/2}}{24 d}+\frac{5}{4} b (a+b x)^{3/2} (c+d x)^{5/2}-\frac{(a+b x)^{5/2} (c+d x)^{5/2}}{x}+\frac{1}{2} \left (5 a^2 c^2 (b c+a d)\right ) \int \frac{1}{x \sqrt{a+b x} \sqrt{c+d x}} \, dx-\frac{\left (5 \left (b^4 c^4-20 a b^3 c^3 d-90 a^2 b^2 c^2 d^2-20 a^3 b c d^3+a^4 d^4\right )\right ) \int \frac{1}{\sqrt{a+b x} \sqrt{c+d x}} \, dx}{128 b d}\\ &=-\frac{5 \left (b^3 c^3-19 a b^2 c^2 d-45 a^2 b c d^2-a^3 d^3\right ) \sqrt{a+b x} \sqrt{c+d x}}{64 b d}-\frac{5 \left (b^2 c^2-18 a b c d-31 a^2 d^2\right ) \sqrt{a+b x} (c+d x)^{3/2}}{96 d}+\frac{5 b (b c+7 a d) \sqrt{a+b x} (c+d x)^{5/2}}{24 d}+\frac{5}{4} b (a+b x)^{3/2} (c+d x)^{5/2}-\frac{(a+b x)^{5/2} (c+d x)^{5/2}}{x}+\left (5 a^2 c^2 (b c+a d)\right ) \operatorname{Subst}\left (\int \frac{1}{-a+c x^2} \, dx,x,\frac{\sqrt{a+b x}}{\sqrt{c+d x}}\right )-\frac{\left (5 \left (b^4 c^4-20 a b^3 c^3 d-90 a^2 b^2 c^2 d^2-20 a^3 b c d^3+a^4 d^4\right )\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 )}{64 b^2 d}\\ &=-\frac{5 \left (b^3 c^3-19 a b^2 c^2 d-45 a^2 b c d^2-a^3 d^3\right ) \sqrt{a+b x} \sqrt{c+d x}}{64 b d}-\frac{5 \left (b^2 c^2-18 a b c d-31 a^2 d^2\right ) \sqrt{a+b x} (c+d x)^{3/2}}{96 d}+\frac{5 b (b c+7 a d) \sqrt{a+b x} (c+d x)^{5/2}}{24 d}+\frac{5}{4} b (a+b x)^{3/2} (c+d x)^{5/2}-\frac{(a+b x)^{5/2} (c+d x)^{5/2}}{x}-5 a^{3/2} c^{3/2} (b c+a d) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )-\frac{\left (5 \left (b^4 c^4-20 a b^3 c^3 d-90 a^2 b^2 c^2 d^2-20 a^3 b c d^3+a^4 d^4\right )\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 )}{64 b^2 d}\\ &=-\frac{5 \left (b^3 c^3-19 a b^2 c^2 d-45 a^2 b c d^2-a^3 d^3\right ) \sqrt{a+b x} \sqrt{c+d x}}{64 b d}-\frac{5 \left (b^2 c^2-18 a b c d-31 a^2 d^2\right ) \sqrt{a+b x} (c+d x)^{3/2}}{96 d}+\frac{5 b (b c+7 a d) \sqrt{a+b x} (c+d x)^{5/2}}{24 d}+\frac{5}{4} b (a+b x)^{3/2} (c+d x)^{5/2}-\frac{(a+b x)^{5/2} (c+d x)^{5/2}}{x}-5 a^{3/2} c^{3/2} (b c+a d) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )-\frac{5 \left (b^4 c^4-20 a b^3 c^3 d-90 a^2 b^2 c^2 d^2-20 a^3 b c d^3+a^4 d^4\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{64 b^{3/2} d^{3/2}}\\ \end{align*}

Mathematica [B]  time = 4.76093, size = 1077, normalized size = 3.22 \[ \frac{-15 b^4 x (c+d x)^2 \sinh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b c-a d}}\right ) c^4+15 b \sqrt{d} (b c-a d)^{5/2} x \sqrt{a+b x} \left (\frac{b (c+d x)}{b c-a d}\right )^{5/2} c^3+300 a b^3 d x (c+d x)^2 \sinh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b c-a d}}\right ) c^3-960 a^{3/2} b d^{3/2} (b c-a d)^{3/2} x \sqrt{c+d x} \left (\frac{b (c+d x)}{b c-a d}\right )^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right ) c^{5/2}-\frac{192 a^2 d^{3/2} (b c-a d)^{5/2} \sqrt{a+b x} \left (\frac{b (c+d x)}{b c-a d}\right )^{5/2} c^2}{b}+118 b d^{3/2} (b c-a d)^{5/2} x^2 \sqrt{a+b x} \left (\frac{b (c+d x)}{b c-a d}\right )^{5/2} c^2+601 a d^{3/2} (b c-a d)^{5/2} x \sqrt{a+b x} \left (\frac{b (c+d x)}{b c-a d}\right )^{5/2} c^2+1350 a^2 b^2 d^2 x (c+d x)^2 \sinh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b c-a d}}\right ) c^2-960 a^{5/2} d^{5/2} (b c-a d)^{3/2} x \sqrt{c+d x} \left (\frac{b (c+d x)}{b c-a d}\right )^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right ) c^{3/2}+136 b d^{5/2} (b c-a d)^{5/2} x^3 \sqrt{a+b x} \left (\frac{b (c+d x)}{b c-a d}\right )^{5/2} c+452 a d^{5/2} (b c-a d)^{5/2} x^2 \sqrt{a+b x} \left (\frac{b (c+d x)}{b c-a d}\right )^{5/2} c+\frac{601 a^2 d^{5/2} (b c-a d)^{5/2} x \sqrt{a+b x} \left (\frac{b (c+d x)}{b c-a d}\right )^{5/2} c}{b}+300 a^3 b d^3 x (c+d x)^2 \sinh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b c-a d}}\right ) c+48 b d^{7/2} (b c-a d)^{5/2} x^4 \sqrt{a+b x} \left (\frac{b (c+d x)}{b c-a d}\right )^{5/2}+136 a d^{7/2} (b c-a d)^{5/2} x^3 \sqrt{a+b x} \left (\frac{b (c+d x)}{b c-a d}\right )^{5/2}+\frac{118 a^2 d^{7/2} (b c-a d)^{5/2} x^2 \sqrt{a+b x} \left (\frac{b (c+d x)}{b c-a d}\right )^{5/2}}{b}-15 a^4 d^4 x (c+d x)^2 \sinh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b c-a d}}\right )+15 a^3 d^{7/2} \sqrt{b c-a d} x \sqrt{a+b x} (c+d x)^2 \sqrt{\frac{b (c+d x)}{b c-a d}}}{192 d^{3/2} (b c-a d)^{3/2} x \sqrt{c+d x} \left (\frac{b (c+d x)}{b c-a d}\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(15*a^3*d^(7/2)*Sqrt[b*c - a*d]*x*Sqrt[a + b*x]*(c + d*x)^2*Sqrt[(b*(c + d*x))/(b*c - a*d)] - (192*a^2*c^2*d^(
3/2)*(b*c - a*d)^(5/2)*Sqrt[a + b*x]*((b*(c + d*x))/(b*c - a*d))^(5/2))/b + 15*b*c^3*Sqrt[d]*(b*c - a*d)^(5/2)
*x*Sqrt[a + b*x]*((b*(c + d*x))/(b*c - a*d))^(5/2) + 601*a*c^2*d^(3/2)*(b*c - a*d)^(5/2)*x*Sqrt[a + b*x]*((b*(
c + d*x))/(b*c - a*d))^(5/2) + (601*a^2*c*d^(5/2)*(b*c - a*d)^(5/2)*x*Sqrt[a + b*x]*((b*(c + d*x))/(b*c - a*d)
)^(5/2))/b + 118*b*c^2*d^(3/2)*(b*c - a*d)^(5/2)*x^2*Sqrt[a + b*x]*((b*(c + d*x))/(b*c - a*d))^(5/2) + 452*a*c
*d^(5/2)*(b*c - a*d)^(5/2)*x^2*Sqrt[a + b*x]*((b*(c + d*x))/(b*c - a*d))^(5/2) + (118*a^2*d^(7/2)*(b*c - a*d)^
(5/2)*x^2*Sqrt[a + b*x]*((b*(c + d*x))/(b*c - a*d))^(5/2))/b + 136*b*c*d^(5/2)*(b*c - a*d)^(5/2)*x^3*Sqrt[a +
b*x]*((b*(c + d*x))/(b*c - a*d))^(5/2) + 136*a*d^(7/2)*(b*c - a*d)^(5/2)*x^3*Sqrt[a + b*x]*((b*(c + d*x))/(b*c
- a*d))^(5/2) + 48*b*d^(7/2)*(b*c - a*d)^(5/2)*x^4*Sqrt[a + b*x]*((b*(c + d*x))/(b*c - a*d))^(5/2) - 15*b^4*c
^4*x*(c + d*x)^2*ArcSinh[(Sqrt[d]*Sqrt[a + b*x])/Sqrt[b*c - a*d]] + 300*a*b^3*c^3*d*x*(c + d*x)^2*ArcSinh[(Sqr
t[d]*Sqrt[a + b*x])/Sqrt[b*c - a*d]] + 1350*a^2*b^2*c^2*d^2*x*(c + d*x)^2*ArcSinh[(Sqrt[d]*Sqrt[a + b*x])/Sqrt
[b*c - a*d]] + 300*a^3*b*c*d^3*x*(c + d*x)^2*ArcSinh[(Sqrt[d]*Sqrt[a + b*x])/Sqrt[b*c - a*d]] - 15*a^4*d^4*x*(
c + d*x)^2*ArcSinh[(Sqrt[d]*Sqrt[a + b*x])/Sqrt[b*c - a*d]] - 960*a^(3/2)*b*c^(5/2)*d^(3/2)*(b*c - a*d)^(3/2)*
x*Sqrt[c + d*x]*((b*(c + d*x))/(b*c - a*d))^(3/2)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])] - 9
60*a^(5/2)*c^(3/2)*d^(5/2)*(b*c - a*d)^(3/2)*x*Sqrt[c + d*x]*((b*(c + d*x))/(b*c - a*d))^(3/2)*ArcTanh[(Sqrt[c
]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(192*d^(3/2)*(b*c - a*d)^(3/2)*x*Sqrt[c + d*x]*((b*(c + d*x))/(b*c
- a*d))^(3/2))

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Maple [B]  time = 0.017, size = 950, normalized size = 2.8 \begin{align*} -{\frac{1}{384\,bdx}\sqrt{bx+a}\sqrt{dx+c} \left ( -96\,{x}^{4}{b}^{3}{d}^{3}\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}\sqrt{ac}-272\,{x}^{3}a{b}^{2}{d}^{3}\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}\sqrt{ac}-272\,{x}^{3}{b}^{3}c{d}^{2}\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}\sqrt{ac}+15\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ) \sqrt{ac}x{a}^{4}{d}^{4}-300\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ) \sqrt{ac}x{a}^{3}bc{d}^{3}-1350\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ) \sqrt{ac}x{a}^{2}{b}^{2}{c}^{2}{d}^{2}-300\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ) \sqrt{ac}xa{b}^{3}{c}^{3}d+15\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ) \sqrt{ac}x{b}^{4}{c}^{4}+960\,\sqrt{bd}\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ) x{a}^{3}b{c}^{2}{d}^{2}+960\,\sqrt{bd}\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ) x{a}^{2}{b}^{2}{c}^{3}d-236\,\sqrt{bd}\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{2}{a}^{2}b{d}^{3}-904\,\sqrt{bd}\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{2}a{b}^{2}c{d}^{2}-236\,\sqrt{bd}\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{2}{b}^{3}{c}^{2}d-30\,\sqrt{bd}\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}x{a}^{3}{d}^{3}-1202\,\sqrt{bd}\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}x{a}^{2}bc{d}^{2}-1202\,\sqrt{bd}\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}xa{b}^{2}{c}^{2}d-30\,\sqrt{bd}\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}x{b}^{3}{c}^{3}+384\,{a}^{2}b{c}^{2}d\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}\sqrt{ac} \right ){\frac{1}{\sqrt{d{x}^{2}b+adx+bcx+ac}}}{\frac{1}{\sqrt{bd}}}{\frac{1}{\sqrt{ac}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/384*(b*x+a)^(1/2)*(d*x+c)^(1/2)*(-96*x^4*b^3*d^3*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)*(a*c)^(1/2)-27
2*x^3*a*b^2*d^3*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)*(a*c)^(1/2)-272*x^3*b^3*c*d^2*(b*d*x^2+a*d*x+b*c*x
+a*c)^(1/2)*(b*d)^(1/2)*(a*c)^(1/2)+15*ln(1/2*(2*b*d*x+2*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)+a*d+b*c)/
(b*d)^(1/2))*(a*c)^(1/2)*x*a^4*d^4-300*ln(1/2*(2*b*d*x+2*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)+a*d+b*c)/
(b*d)^(1/2))*(a*c)^(1/2)*x*a^3*b*c*d^3-1350*ln(1/2*(2*b*d*x+2*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)+a*d+
b*c)/(b*d)^(1/2))*(a*c)^(1/2)*x*a^2*b^2*c^2*d^2-300*ln(1/2*(2*b*d*x+2*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1
/2)+a*d+b*c)/(b*d)^(1/2))*(a*c)^(1/2)*x*a*b^3*c^3*d+15*ln(1/2*(2*b*d*x+2*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)
^(1/2)+a*d+b*c)/(b*d)^(1/2))*(a*c)^(1/2)*x*b^4*c^4+960*(b*d)^(1/2)*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*(b*d*x^2+a*d*
x+b*c*x+a*c)^(1/2)+2*a*c)/x)*x*a^3*b*c^2*d^2+960*(b*d)^(1/2)*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*
x+a*c)^(1/2)+2*a*c)/x)*x*a^2*b^2*c^3*d-236*(b*d)^(1/2)*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x^2*a^2*b*d
^3-904*(b*d)^(1/2)*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x^2*a*b^2*c*d^2-236*(b*d)^(1/2)*(a*c)^(1/2)*(b*
d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x^2*b^3*c^2*d-30*(b*d)^(1/2)*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x*a^3*d^
3-1202*(b*d)^(1/2)*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x*a^2*b*c*d^2-1202*(b*d)^(1/2)*(a*c)^(1/2)*(b*d
*x^2+a*d*x+b*c*x+a*c)^(1/2)*x*a*b^2*c^2*d-30*(b*d)^(1/2)*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x*b^3*c^3
+384*a^2*b*c^2*d*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)*(a*c)^(1/2))/b/d/(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)/
(b*d)^(1/2)/(a*c)^(1/2)/x

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 120.139, size = 3625, normalized size = 10.85 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/768*(15*(b^4*c^4 - 20*a*b^3*c^3*d - 90*a^2*b^2*c^2*d^2 - 20*a^3*b*c*d^3 + a^4*d^4)*sqrt(b*d)*x*log(8*b^2*d^
2*x^2 + b^2*c^2 + 6*a*b*c*d + a^2*d^2 - 4*(2*b*d*x + b*c + a*d)*sqrt(b*d)*sqrt(b*x + a)*sqrt(d*x + c) + 8*(b^2
*c*d + a*b*d^2)*x) + 960*(a*b^3*c^2*d^2 + a^2*b^2*c*d^3)*sqrt(a*c)*x*log((8*a^2*c^2 + (b^2*c^2 + 6*a*b*c*d + a
^2*d^2)*x^2 - 4*(2*a*c + (b*c + a*d)*x)*sqrt(a*c)*sqrt(b*x + a)*sqrt(d*x + c) + 8*(a*b*c^2 + a^2*c*d)*x)/x^2)
+ 4*(48*b^4*d^4*x^4 - 192*a^2*b^2*c^2*d^2 + 136*(b^4*c*d^3 + a*b^3*d^4)*x^3 + 2*(59*b^4*c^2*d^2 + 226*a*b^3*c*
d^3 + 59*a^2*b^2*d^4)*x^2 + (15*b^4*c^3*d + 601*a*b^3*c^2*d^2 + 601*a^2*b^2*c*d^3 + 15*a^3*b*d^4)*x)*sqrt(b*x
+ a)*sqrt(d*x + c))/(b^2*d^2*x), 1/384*(15*(b^4*c^4 - 20*a*b^3*c^3*d - 90*a^2*b^2*c^2*d^2 - 20*a^3*b*c*d^3 + a
^4*d^4)*sqrt(-b*d)*x*arctan(1/2*(2*b*d*x + b*c + a*d)*sqrt(-b*d)*sqrt(b*x + a)*sqrt(d*x + c)/(b^2*d^2*x^2 + a*
b*c*d + (b^2*c*d + a*b*d^2)*x)) + 480*(a*b^3*c^2*d^2 + a^2*b^2*c*d^3)*sqrt(a*c)*x*log((8*a^2*c^2 + (b^2*c^2 +
6*a*b*c*d + a^2*d^2)*x^2 - 4*(2*a*c + (b*c + a*d)*x)*sqrt(a*c)*sqrt(b*x + a)*sqrt(d*x + c) + 8*(a*b*c^2 + a^2*
c*d)*x)/x^2) + 2*(48*b^4*d^4*x^4 - 192*a^2*b^2*c^2*d^2 + 136*(b^4*c*d^3 + a*b^3*d^4)*x^3 + 2*(59*b^4*c^2*d^2 +
226*a*b^3*c*d^3 + 59*a^2*b^2*d^4)*x^2 + (15*b^4*c^3*d + 601*a*b^3*c^2*d^2 + 601*a^2*b^2*c*d^3 + 15*a^3*b*d^4)
*x)*sqrt(b*x + a)*sqrt(d*x + c))/(b^2*d^2*x), 1/768*(1920*(a*b^3*c^2*d^2 + a^2*b^2*c*d^3)*sqrt(-a*c)*x*arctan(
1/2*(2*a*c + (b*c + a*d)*x)*sqrt(-a*c)*sqrt(b*x + a)*sqrt(d*x + c)/(a*b*c*d*x^2 + a^2*c^2 + (a*b*c^2 + a^2*c*d
)*x)) + 15*(b^4*c^4 - 20*a*b^3*c^3*d - 90*a^2*b^2*c^2*d^2 - 20*a^3*b*c*d^3 + a^4*d^4)*sqrt(b*d)*x*log(8*b^2*d^
2*x^2 + b^2*c^2 + 6*a*b*c*d + a^2*d^2 - 4*(2*b*d*x + b*c + a*d)*sqrt(b*d)*sqrt(b*x + a)*sqrt(d*x + c) + 8*(b^2
*c*d + a*b*d^2)*x) + 4*(48*b^4*d^4*x^4 - 192*a^2*b^2*c^2*d^2 + 136*(b^4*c*d^3 + a*b^3*d^4)*x^3 + 2*(59*b^4*c^2
*d^2 + 226*a*b^3*c*d^3 + 59*a^2*b^2*d^4)*x^2 + (15*b^4*c^3*d + 601*a*b^3*c^2*d^2 + 601*a^2*b^2*c*d^3 + 15*a^3*
b*d^4)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(b^2*d^2*x), 1/384*(960*(a*b^3*c^2*d^2 + a^2*b^2*c*d^3)*sqrt(-a*c)*x*ar
ctan(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)) + 15*(b^4*c^4 - 20*a*b^3*c^3*d - 90*a^2*b^2*c^2*d^2 - 20*a^3*b*c*d^3 + a^4*d^4)*sqrt(-b*d)*x*arctan
(1/2*(2*b*d*x + b*c + a*d)*sqrt(-b*d)*sqrt(b*x + a)*sqrt(d*x + c)/(b^2*d^2*x^2 + a*b*c*d + (b^2*c*d + a*b*d^2)
*x)) + 2*(48*b^4*d^4*x^4 - 192*a^2*b^2*c^2*d^2 + 136*(b^4*c*d^3 + a*b^3*d^4)*x^3 + 2*(59*b^4*c^2*d^2 + 226*a*b
^3*c*d^3 + 59*a^2*b^2*d^4)*x^2 + (15*b^4*c^3*d + 601*a*b^3*c^2*d^2 + 601*a^2*b^2*c*d^3 + 15*a^3*b*d^4)*x)*sqrt
(b*x + a)*sqrt(d*x + c))/(b^2*d^2*x)]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [B]  time = 5.4762, size = 1052, normalized size = 3.15 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/384*(2*sqrt(b^2*c + (b*x + a)*b*d - a*b*d)*(2*(4*(b*x + a)*(6*(b*x + a)*d^2*abs(b)/b^2 + (17*b^3*c*d^7*abs(b
) - a*b^2*d^8*abs(b))/(b^4*d^6)) + (59*b^4*c^2*d^6*abs(b) + 90*a*b^3*c*d^7*abs(b) - 5*a^2*b^2*d^8*abs(b))/(b^4
*d^6))*(b*x + a) + 3*(5*b^5*c^3*d^5*abs(b) + 161*a*b^4*c^2*d^6*abs(b) + 95*a^2*b^3*c*d^7*abs(b) - 5*a^3*b^2*d^
8*abs(b))/(b^4*d^6))*sqrt(b*x + a) - 1920*(sqrt(b*d)*a^2*b^2*c^3*abs(b) + sqrt(b*d)*a^3*b*c^2*d*abs(b))*arctan
(-1/2*(b^2*c + a*b*d - (sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2)/(sqrt(-a*b*c*d)*b))/
(sqrt(-a*b*c*d)*b) - 768*(sqrt(b*d)*a^2*b^4*c^4*abs(b) - 2*sqrt(b*d)*a^3*b^3*c^3*d*abs(b) + sqrt(b*d)*a^4*b^2*
c^2*d^2*abs(b) - sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^2*b^2*c^3*abs(b
) - sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^3*b*c^2*d*abs(b))/(b^4*c^2 -
2*a*b^3*c*d + a^2*b^2*d^2 - 2*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*b^2*c - 2*(sq
rt(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) + 15*(sqrt(b*d)*b^4*c^4*abs(b) - 20*sqrt(b*d)*a*b^3*c^3*d*abs(b) - 90*sqrt(b*d)*a^
2*b^2*c^2*d^2*abs(b) - 20*sqrt(b*d)*a^3*b*c*d^3*abs(b) + sqrt(b*d)*a^4*d^4*abs(b))*log((sqrt(b*d)*sqrt(b*x + a
) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2)/(b^2*d^2))/b