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

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

[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])]

________________________________________________________________________________________

Rubi [A]  time = 0.300439, antiderivative size = 316, normalized size of antiderivative = 1., 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} \[ \frac{\sqrt{a+b x} \sqrt{c+d x} \left (-55 a^2 b c d^2+5 a^3 d^3-17 a b^2 c^2 d+3 b^3 c^3\right )}{64 a^2 c x}-\frac{\left (90 a^2 b^2 c^2 d^2+60 a^3 b c d^3-5 a^4 d^4-20 a b^3 c^3 d+3 b^4 c^4\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{64 a^{5/2} c^{3/2}}-\frac{\sqrt{a+b x} (c+d x)^{3/2} \left (\frac{3 b^2 c}{a}-\frac{5 a d^2}{c}+50 b d\right )}{96 x^2}+2 b^{3/2} d^{5/2} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )-\frac{(a+b x)^{3/2} (c+d x)^{5/2}}{4 x^4}-\frac{\sqrt{a+b x} (c+d x)^{5/2} (5 a d+3 b c)}{24 c x^3} \]

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 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 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)^{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 = 2.92679, size = 296, normalized size = 0.94 \[ -\frac{\sqrt{a+b x} \sqrt{c+d x} \left (a^2 b c x \left (72 c^2+244 c d x+337 d^2 x^2\right )+a^3 \left (136 c^2 d x+48 c^3+118 c d^2 x^2+15 d^3 x^3\right )+3 a b^2 c^2 x^2 (2 c+19 d x)-9 b^3 c^3 x^3\right )}{192 a^2 c x^4}+\frac{\left (-90 a^2 b^2 c^2 d^2-60 a^3 b c d^3+5 a^4 d^4+20 a b^3 c^3 d-3 b^4 c^4\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{64 a^{5/2} c^{3/2}}+\frac{2 d^{5/2} (b c-a d)^{3/2} \left (\frac{b (c+d x)}{b c-a d}\right )^{3/2} \sinh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b c-a d}}\right )}{(c+d x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(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)))/(192*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))

________________________________________________________________________________________

Maple [B]  time = 0.017, size = 852, normalized size = 2.7 \begin{align*}{\frac{1}{384\,{a}^{2}c{x}^{4}}\sqrt{bx+a}\sqrt{dx+c} \left ( 15\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){x}^{4}{a}^{4}{d}^{4}\sqrt{bd}-180\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){x}^{4}{a}^{3}bc{d}^{3}\sqrt{bd}-270\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){x}^{4}{a}^{2}{b}^{2}{c}^{2}{d}^{2}\sqrt{bd}+60\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){x}^{4}a{b}^{3}{c}^{3}d\sqrt{bd}-9\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){x}^{4}{b}^{4}{c}^{4}\sqrt{bd}+384\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){x}^{4}{a}^{2}{b}^{2}c{d}^{3}\sqrt{ac}-30\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}\sqrt{ac}{x}^{3}{a}^{3}{d}^{3}-674\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}\sqrt{ac}{x}^{3}{a}^{2}bc{d}^{2}-114\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}\sqrt{ac}{x}^{3}a{b}^{2}{c}^{2}d+18\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}\sqrt{ac}{x}^{3}{b}^{3}{c}^{3}-236\,\sqrt{bd}\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{2}{a}^{3}c{d}^{2}-488\,\sqrt{bd}\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{2}{a}^{2}b{c}^{2}d-12\,\sqrt{bd}\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{2}a{b}^{2}{c}^{3}-272\,\sqrt{bd}\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}x{a}^{3}{c}^{2}d-144\,\sqrt{bd}\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}x{a}^{2}b{c}^{3}-96\,\sqrt{bd}\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{a}^{3}{c}^{3} \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)^(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)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)+2*a*
c)/x)*x^4*a^4*d^4*(b*d)^(1/2)-180*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^4*
a^3*b*c*d^3*(b*d)^(1/2)-270*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^4*a^2*b^
2*c^2*d^2*(b*d)^(1/2)+60*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^4*a*b^3*c^3
*d*(b*d)^(1/2)-9*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^4*b^4*c^4*(b*d)^(1/
2)+384*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))*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)^(1/2)*(a*c)^(1/
2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x^2*a^3*c*d^2-488*(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*c^2*d-12*(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^3-272*(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*c^2*d-144*(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^3-96*(b*d)^(1/2)*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+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)

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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)^(3/2)*(d*x+c)^(5/2)/x^5,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 70.3979, size = 3437, normalized size = 10.88 \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)^(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)]

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

[Out]

Timed out

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Giac [B]  time = 5.51985, size = 5247, normalized size = 16.6 \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)^(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