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

Optimal. Leaf size=313 \[ -\frac{\sqrt{a+b x} \sqrt{c+d x} \left (-17 a^2 b c d^2+3 a^3 d^3+73 a b^2 c^2 d+5 b^3 c^3\right )}{64 a c^2 x}+\frac{\left (-90 a^2 b^2 c^2 d^2+20 a^3 b c d^3-3 a^4 d^4-60 a b^3 c^3 d+5 b^4 c^4\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{64 a^{3/2} c^{5/2}}+2 b^{5/2} d^{3/2} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )-\frac{\sqrt{a+b x} (c+d x)^{3/2} (5 b c-a d) (3 a d+b c)}{32 c^2 x^2}-\frac{(a+b x)^{5/2} (c+d x)^{3/2}}{4 x^4}-\frac{(a+b x)^{3/2} (c+d x)^{3/2} (3 a d+5 b c)}{24 c x^3} \]

[Out]

-((5*b^3*c^3 + 73*a*b^2*c^2*d - 17*a^2*b*c*d^2 + 3*a^3*d^3)*Sqrt[a + b*x]*Sqrt[c + d*x])/(64*a*c^2*x) - ((5*b*
c - a*d)*(b*c + 3*a*d)*Sqrt[a + b*x]*(c + d*x)^(3/2))/(32*c^2*x^2) - ((5*b*c + 3*a*d)*(a + b*x)^(3/2)*(c + d*x
)^(3/2))/(24*c*x^3) - ((a + b*x)^(5/2)*(c + d*x)^(3/2))/(4*x^4) + ((5*b^4*c^4 - 60*a*b^3*c^3*d - 90*a^2*b^2*c^
2*d^2 + 20*a^3*b*c*d^3 - 3*a^4*d^4)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(64*a^(3/2)*c^(5
/2)) + 2*b^(5/2)*d^(3/2)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])]

________________________________________________________________________________________

Rubi [A]  time = 0.316293, antiderivative size = 313, 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} \[ -\frac{\sqrt{a+b x} \sqrt{c+d x} \left (-17 a^2 b c d^2+3 a^3 d^3+73 a b^2 c^2 d+5 b^3 c^3\right )}{64 a c^2 x}+\frac{\left (-90 a^2 b^2 c^2 d^2+20 a^3 b c d^3-3 a^4 d^4-60 a b^3 c^3 d+5 b^4 c^4\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{64 a^{3/2} c^{5/2}}+2 b^{5/2} d^{3/2} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )-\frac{\sqrt{a+b x} (c+d x)^{3/2} (5 b c-a d) (3 a d+b c)}{32 c^2 x^2}-\frac{(a+b x)^{5/2} (c+d x)^{3/2}}{4 x^4}-\frac{(a+b x)^{3/2} (c+d x)^{3/2} (3 a d+5 b c)}{24 c x^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((5*b^3*c^3 + 73*a*b^2*c^2*d - 17*a^2*b*c*d^2 + 3*a^3*d^3)*Sqrt[a + b*x]*Sqrt[c + d*x])/(64*a*c^2*x) - ((5*b*
c - a*d)*(b*c + 3*a*d)*Sqrt[a + b*x]*(c + d*x)^(3/2))/(32*c^2*x^2) - ((5*b*c + 3*a*d)*(a + b*x)^(3/2)*(c + d*x
)^(3/2))/(24*c*x^3) - ((a + b*x)^(5/2)*(c + d*x)^(3/2))/(4*x^4) + ((5*b^4*c^4 - 60*a*b^3*c^3*d - 90*a^2*b^2*c^
2*d^2 + 20*a^3*b*c*d^3 - 3*a^4*d^4)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(64*a^(3/2)*c^(5
/2)) + 2*b^(5/2)*d^(3/2)*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)^{5/2} (c+d x)^{3/2}}{x^5} \, dx &=-\frac{(a+b x)^{5/2} (c+d x)^{3/2}}{4 x^4}+\frac{1}{4} \int \frac{(a+b x)^{3/2} \sqrt{c+d x} \left (\frac{1}{2} (5 b c+3 a d)+4 b d x\right )}{x^4} \, dx\\ &=-\frac{(5 b c+3 a d) (a+b x)^{3/2} (c+d x)^{3/2}}{24 c x^3}-\frac{(a+b x)^{5/2} (c+d x)^{3/2}}{4 x^4}+\frac{\int \frac{\sqrt{a+b x} \sqrt{c+d x} \left (\frac{3}{4} (5 b c-a d) (b c+3 a d)+12 b^2 c d x\right )}{x^3} \, dx}{12 c}\\ &=-\frac{(5 b c-a d) (b c+3 a d) \sqrt{a+b x} (c+d x)^{3/2}}{32 c^2 x^2}-\frac{(5 b c+3 a d) (a+b x)^{3/2} (c+d x)^{3/2}}{24 c x^3}-\frac{(a+b x)^{5/2} (c+d x)^{3/2}}{4 x^4}+\frac{\int \frac{\sqrt{c+d x} \left (\frac{3}{8} \left (5 b^3 c^3+73 a b^2 c^2 d-17 a^2 b c d^2+3 a^3 d^3\right )+24 b^3 c^2 d x\right )}{x^2 \sqrt{a+b x}} \, dx}{24 c^2}\\ &=-\frac{\left (5 b^3 c^3+73 a b^2 c^2 d-17 a^2 b c d^2+3 a^3 d^3\right ) \sqrt{a+b x} \sqrt{c+d x}}{64 a c^2 x}-\frac{(5 b c-a d) (b c+3 a d) \sqrt{a+b x} (c+d x)^{3/2}}{32 c^2 x^2}-\frac{(5 b c+3 a d) (a+b x)^{3/2} (c+d x)^{3/2}}{24 c x^3}-\frac{(a+b x)^{5/2} (c+d x)^{3/2}}{4 x^4}+\frac{\int \frac{-\frac{3}{16} \left (5 b^4 c^4-60 a b^3 c^3 d-90 a^2 b^2 c^2 d^2+20 a^3 b c d^3-3 a^4 d^4\right )+24 a b^3 c^2 d^2 x}{x \sqrt{a+b x} \sqrt{c+d x}} \, dx}{24 a c^2}\\ &=-\frac{\left (5 b^3 c^3+73 a b^2 c^2 d-17 a^2 b c d^2+3 a^3 d^3\right ) \sqrt{a+b x} \sqrt{c+d x}}{64 a c^2 x}-\frac{(5 b c-a d) (b c+3 a d) \sqrt{a+b x} (c+d x)^{3/2}}{32 c^2 x^2}-\frac{(5 b c+3 a d) (a+b x)^{3/2} (c+d x)^{3/2}}{24 c x^3}-\frac{(a+b x)^{5/2} (c+d x)^{3/2}}{4 x^4}+\left (b^3 d^2\right ) \int \frac{1}{\sqrt{a+b x} \sqrt{c+d x}} \, dx-\frac{\left (5 b^4 c^4-60 a b^3 c^3 d-90 a^2 b^2 c^2 d^2+20 a^3 b c d^3-3 a^4 d^4\right ) \int \frac{1}{x \sqrt{a+b x} \sqrt{c+d x}} \, dx}{128 a c^2}\\ &=-\frac{\left (5 b^3 c^3+73 a b^2 c^2 d-17 a^2 b c d^2+3 a^3 d^3\right ) \sqrt{a+b x} \sqrt{c+d x}}{64 a c^2 x}-\frac{(5 b c-a d) (b c+3 a d) \sqrt{a+b x} (c+d x)^{3/2}}{32 c^2 x^2}-\frac{(5 b c+3 a d) (a+b x)^{3/2} (c+d x)^{3/2}}{24 c x^3}-\frac{(a+b x)^{5/2} (c+d x)^{3/2}}{4 x^4}+\left (2 b^2 d^2\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 (5 b^4 c^4-60 a b^3 c^3 d-90 a^2 b^2 c^2 d^2+20 a^3 b c d^3-3 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 c^2}\\ &=-\frac{\left (5 b^3 c^3+73 a b^2 c^2 d-17 a^2 b c d^2+3 a^3 d^3\right ) \sqrt{a+b x} \sqrt{c+d x}}{64 a c^2 x}-\frac{(5 b c-a d) (b c+3 a d) \sqrt{a+b x} (c+d x)^{3/2}}{32 c^2 x^2}-\frac{(5 b c+3 a d) (a+b x)^{3/2} (c+d x)^{3/2}}{24 c x^3}-\frac{(a+b x)^{5/2} (c+d x)^{3/2}}{4 x^4}+\frac{\left (5 b^4 c^4-60 a b^3 c^3 d-90 a^2 b^2 c^2 d^2+20 a^3 b c d^3-3 a^4 d^4\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{64 a^{3/2} c^{5/2}}+\left (2 b^2 d^2\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 (5 b^3 c^3+73 a b^2 c^2 d-17 a^2 b c d^2+3 a^3 d^3\right ) \sqrt{a+b x} \sqrt{c+d x}}{64 a c^2 x}-\frac{(5 b c-a d) (b c+3 a d) \sqrt{a+b x} (c+d x)^{3/2}}{32 c^2 x^2}-\frac{(5 b c+3 a d) (a+b x)^{3/2} (c+d x)^{3/2}}{24 c x^3}-\frac{(a+b x)^{5/2} (c+d x)^{3/2}}{4 x^4}+\frac{\left (5 b^4 c^4-60 a b^3 c^3 d-90 a^2 b^2 c^2 d^2+20 a^3 b c d^3-3 a^4 d^4\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{64 a^{3/2} c^{5/2}}+2 b^{5/2} d^{3/2} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )\\ \end{align*}

Mathematica [A]  time = 3.31724, size = 295, normalized size = 0.94 \[ -\frac{\sqrt{a+b x} \sqrt{c+d x} \left (a^2 b c x \left (136 c^2+244 c d x+57 d^2 x^2\right )+a^3 \left (72 c^2 d x+48 c^3+6 c d^2 x^2-9 d^3 x^3\right )+a b^2 c^2 x^2 (118 c+337 d x)+15 b^3 c^3 x^3\right )}{192 a c^2 x^4}-\frac{\left (90 a^2 b^2 c^2 d^2-20 a^3 b c d^3+3 a^4 d^4+60 a b^3 c^3 d-5 b^4 c^4\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{64 a^{3/2} c^{5/2}}+\frac{2 d^{3/2} (b c-a d)^{5/2} \left (\frac{b (c+d x)}{b c-a d}\right )^{5/2} \sinh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b c-a d}}\right )}{(c+d x)^{5/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(Sqrt[a + b*x]*Sqrt[c + d*x]*(15*b^3*c^3*x^3 + a*b^2*c^2*x^2*(118*c + 337*d*x) + a^2*b*c*x*(136*c^2 + 244*c*d
*x + 57*d^2*x^2) + a^3*(48*c^3 + 72*c^2*d*x + 6*c*d^2*x^2 - 9*d^3*x^3)))/(192*a*c^2*x^4) + (2*d^(3/2)*(b*c - a
*d)^(5/2)*((b*(c + d*x))/(b*c - a*d))^(5/2)*ArcSinh[(Sqrt[d]*Sqrt[a + b*x])/Sqrt[b*c - a*d]])/(c + d*x)^(5/2)
- ((-5*b^4*c^4 + 60*a*b^3*c^3*d + 90*a^2*b^2*c^2*d^2 - 20*a^3*b*c*d^3 + 3*a^4*d^4)*ArcTanh[(Sqrt[c]*Sqrt[a + b
*x])/(Sqrt[a]*Sqrt[c + d*x])])/(64*a^(3/2)*c^(5/2))

________________________________________________________________________________________

Maple [B]  time = 0.017, size = 852, normalized size = 2.7 \begin{align*}{\frac{1}{384\,a{c}^{2}{x}^{4}}\sqrt{bx+a}\sqrt{dx+c} \left ( 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{b}^{3}{c}^{2}{d}^{2}\sqrt{ac}-9\,\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}+60\,\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}-180\,\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}+15\,\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}+18\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}\sqrt{ac}{x}^{3}{a}^{3}{d}^{3}-114\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}\sqrt{ac}{x}^{3}{a}^{2}bc{d}^{2}-674\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}\sqrt{ac}{x}^{3}a{b}^{2}{c}^{2}d-30\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}\sqrt{ac}{x}^{3}{b}^{3}{c}^{3}-12\,\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-236\,\sqrt{bd}\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{2}a{b}^{2}{c}^{3}-144\,\sqrt{bd}\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}x{a}^{3}{c}^{2}d-272\,\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)^(5/2)*(d*x+c)^(3/2)/x^5,x)

[Out]

1/384*(b*x+a)^(1/2)*(d*x+c)^(1/2)/a/c^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*b^3*c^2*d^2*(a*c)^(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*a^4*d^4*(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^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)-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*
b^3*c^3*d*(b*d)^(1/2)+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*b^4*c^4*(
b*d)^(1/2)+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*a^3*d^3-114*(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^2*b*c*d^2-674*(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*b^2*c^2*d-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*b^3*c^3-12*(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^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-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*b^2*c^3-144*(b*d)^(1/2)*(a*c)^
(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x*a^3*c^2*d-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^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)^(5/2)*(d*x+c)^(3/2)/x^5,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 64.639, size = 3437, normalized size = 10.98 \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)^(3/2)/x^5,x, algorithm="fricas")

[Out]

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

[Out]

Timed out

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Giac [B]  time = 5.89676, size = 5247, normalized size = 16.76 \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)^(3/2)/x^5,x, algorithm="giac")

[Out]

-1/192*(192*sqrt(b*d)*b^2*d*abs(b)*log((sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2) - 3*
(5*sqrt(b*d)*b^5*c^4*abs(b) - 60*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) + 20*sqrt(
b*d)*a^3*b^2*c*d^3*abs(b) - 3*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*b*c^2) + 2*(15*sqrt(b*d)*b^1
9*c^11*abs(b) + 217*sqrt(b*d)*a*b^18*c^10*d*abs(b) - 2219*sqrt(b*d)*a^2*b^17*c^9*d^2*abs(b) + 8131*sqrt(b*d)*a
^3*b^16*c^8*d^3*abs(b) - 16154*sqrt(b*d)*a^4*b^15*c^7*d^4*abs(b) + 19306*sqrt(b*d)*a^5*b^14*c^6*d^5*abs(b) - 1
3958*sqrt(b*d)*a^6*b^13*c^5*d^6*abs(b) + 5494*sqrt(b*d)*a^7*b^12*c^4*d^7*abs(b) - 581*sqrt(b*d)*a^8*b^11*c^3*d
^8*abs(b) - 371*sqrt(b*d)*a^9*b^10*c^2*d^9*abs(b) + 129*sqrt(b*d)*a^10*b^9*c*d^10*abs(b) - 9*sqrt(b*d)*a^11*b^
8*d^11*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*b^17*c^10*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*b^16*c^9*d*abs(b) + 95
23*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) - 18024
*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^14*c^7*d^3*abs(b) + 10942*s
qrt(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) - 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^6*b^11*c^4*d^6*abs(b) + 5272*sqrt(b*d)
*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^7*b^10*c^3*d^7*abs(b) + 619*sqrt(b*d)*(sq
rt(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) - 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^9*b^8*c*d^9*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*a^10*b^7*d^10*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*b^15*c^9*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*b^14*c^8*d*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^2*b^13*c^7*d^2*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^3*b^12*c^6*d^3*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^4*b^11*c^5*d^4*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^5*b^10*c^4*d^5*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^6*b^9*c^3*d^6*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^7*b^8*c^2*d^7*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^8*b^7*c*d^8*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*
a^9*b^6*d^9*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*b^13*c^8*
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*b^12*c^7*d*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^2*b^11*c^6*d^2*abs(b) +
5456*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) + 237
8*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) + 8096*sq
rt(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) - 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^6*b^7*c^2*d^6*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^7*b^6*c*d^7*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*a^8*b^5*d^8*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*b^11*c^7*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*b^10*c^6*d*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^2*b^9*c^5*d^2*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^3*b^8*c^4*d^3*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^4*b^7*c^3*d^4*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^5*b^6*c^2*d^5*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^6*b^5*c*d^6*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*a^7*b^4*d^7*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*b^9*c^6*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*b^8*c^
5*d*abs(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^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*
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^4*b^5*c^2*d^4*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^5*b^4*c*d^5*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*a^6*b^3*d^6*abs(b) + 105*sqr
t(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) + 3453*sqrt(b*d)*(sqr
t(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) + 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^2*b^5*c^3*d^2*abs(b) + 4290*sqrt(b*d)*(sqrt(b*d)*sqr
t(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) + 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^4*b^3*c*d^4*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*a^5*b^2*d^5*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*b^5*c^4*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*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) - 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^3*b^2*c*d^3*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*
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*c^2))/b