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

Optimal. Leaf size=391 \[ -\frac{\sqrt{a+b x} (c+d x)^{5/2} \left (-3 a^2 d^2-16 a b c d+3 b^2 c^2\right )}{48 d^2}+\frac{\sqrt{a+b x} (c+d x)^{3/2} \left (109 a^2 b c d^2+3 a^3 d^3-19 a b^2 c^2 d+3 b^3 c^3\right )}{192 b d^2}+\frac{\sqrt{a+b x} \sqrt{c+d x} \left (128 a^2 b^2 c^2 d^2+22 a^3 b c d^3-3 a^4 d^4-22 a b^3 c^3 d+3 b^4 c^4\right )}{128 b^2 d^2}+\frac{(a d+b c) \left (178 a^2 b^2 c^2 d^2-28 a^3 b c d^3+3 a^4 d^4-28 a b^3 c^3 d+3 b^4 c^4\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{128 b^{5/2} d^{5/2}}-2 a^{5/2} c^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )+\frac{1}{5} (a+b x)^{5/2} (c+d x)^{5/2}+\frac{(a+b x)^{3/2} (c+d x)^{5/2} (a d+b c)}{8 d} \]

[Out]

((3*b^4*c^4 - 22*a*b^3*c^3*d + 128*a^2*b^2*c^2*d^2 + 22*a^3*b*c*d^3 - 3*a^4*d^4)*Sqrt[a + b*x]*Sqrt[c + d*x])/
(128*b^2*d^2) + ((3*b^3*c^3 - 19*a*b^2*c^2*d + 109*a^2*b*c*d^2 + 3*a^3*d^3)*Sqrt[a + b*x]*(c + d*x)^(3/2))/(19
2*b*d^2) - ((3*b^2*c^2 - 16*a*b*c*d - 3*a^2*d^2)*Sqrt[a + b*x]*(c + d*x)^(5/2))/(48*d^2) + ((b*c + a*d)*(a + b
*x)^(3/2)*(c + d*x)^(5/2))/(8*d) + ((a + b*x)^(5/2)*(c + d*x)^(5/2))/5 - 2*a^(5/2)*c^(5/2)*ArcTanh[(Sqrt[c]*Sq
rt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])] + ((b*c + a*d)*(3*b^4*c^4 - 28*a*b^3*c^3*d + 178*a^2*b^2*c^2*d^2 - 28*a^
3*b*c*d^3 + 3*a^4*d^4)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(128*b^(5/2)*d^(5/2))

________________________________________________________________________________________

Rubi [A]  time = 0.429556, antiderivative size = 391, 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 = {101, 154, 157, 63, 217, 206, 93, 208} \[ -\frac{\sqrt{a+b x} (c+d x)^{5/2} \left (-3 a^2 d^2-16 a b c d+3 b^2 c^2\right )}{48 d^2}+\frac{\sqrt{a+b x} (c+d x)^{3/2} \left (109 a^2 b c d^2+3 a^3 d^3-19 a b^2 c^2 d+3 b^3 c^3\right )}{192 b d^2}+\frac{\sqrt{a+b x} \sqrt{c+d x} \left (128 a^2 b^2 c^2 d^2+22 a^3 b c d^3-3 a^4 d^4-22 a b^3 c^3 d+3 b^4 c^4\right )}{128 b^2 d^2}+\frac{(a d+b c) \left (178 a^2 b^2 c^2 d^2-28 a^3 b c d^3+3 a^4 d^4-28 a b^3 c^3 d+3 b^4 c^4\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{128 b^{5/2} d^{5/2}}-2 a^{5/2} c^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )+\frac{1}{5} (a+b x)^{5/2} (c+d x)^{5/2}+\frac{(a+b x)^{3/2} (c+d x)^{5/2} (a d+b c)}{8 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((3*b^4*c^4 - 22*a*b^3*c^3*d + 128*a^2*b^2*c^2*d^2 + 22*a^3*b*c*d^3 - 3*a^4*d^4)*Sqrt[a + b*x]*Sqrt[c + d*x])/
(128*b^2*d^2) + ((3*b^3*c^3 - 19*a*b^2*c^2*d + 109*a^2*b*c*d^2 + 3*a^3*d^3)*Sqrt[a + b*x]*(c + d*x)^(3/2))/(19
2*b*d^2) - ((3*b^2*c^2 - 16*a*b*c*d - 3*a^2*d^2)*Sqrt[a + b*x]*(c + d*x)^(5/2))/(48*d^2) + ((b*c + a*d)*(a + b
*x)^(3/2)*(c + d*x)^(5/2))/(8*d) + ((a + b*x)^(5/2)*(c + d*x)^(5/2))/5 - 2*a^(5/2)*c^(5/2)*ArcTanh[(Sqrt[c]*Sq
rt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])] + ((b*c + a*d)*(3*b^4*c^4 - 28*a*b^3*c^3*d + 178*a^2*b^2*c^2*d^2 - 28*a^
3*b*c*d^3 + 3*a^4*d^4)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(128*b^(5/2)*d^(5/2))

Rule 101

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[((a +
b*x)^m*(c + d*x)^n*(e + f*x)^(p + 1))/(f*(m + n + p + 1)), x] - Dist[1/(f*(m + n + p + 1)), Int[(a + b*x)^(m -
1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[c*m*(b*e - a*f) + a*n*(d*e - c*f) + (d*m*(b*e - a*f) + b*n*(d*e - c*f))
*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && GtQ[m, 0] && GtQ[n, 0] && NeQ[m + n + p + 1, 0] && (Integ
ersQ[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} \, dx &=\frac{1}{5} (a+b x)^{5/2} (c+d x)^{5/2}-\frac{1}{5} \int \frac{(a+b x)^{3/2} (c+d x)^{3/2} \left (-5 a c-\frac{5}{2} (b c+a d) x\right )}{x} \, dx\\ &=\frac{(b c+a d) (a+b x)^{3/2} (c+d x)^{5/2}}{8 d}+\frac{1}{5} (a+b x)^{5/2} (c+d x)^{5/2}-\frac{\int \frac{\sqrt{a+b x} (c+d x)^{3/2} \left (-20 a^2 c d+\frac{5}{4} \left (3 b^2 c^2-16 a b c d-3 a^2 d^2\right ) x\right )}{x} \, dx}{20 d}\\ &=-\frac{\left (3 b^2 c^2-16 a b c d-3 a^2 d^2\right ) \sqrt{a+b x} (c+d x)^{5/2}}{48 d^2}+\frac{(b c+a d) (a+b x)^{3/2} (c+d x)^{5/2}}{8 d}+\frac{1}{5} (a+b x)^{5/2} (c+d x)^{5/2}-\frac{\int \frac{(c+d x)^{3/2} \left (-60 a^3 c d^2-\frac{5}{8} \left (3 b^3 c^3-19 a b^2 c^2 d+109 a^2 b c d^2+3 a^3 d^3\right ) x\right )}{x \sqrt{a+b x}} \, dx}{60 d^2}\\ &=\frac{\left (3 b^3 c^3-19 a b^2 c^2 d+109 a^2 b c d^2+3 a^3 d^3\right ) \sqrt{a+b x} (c+d x)^{3/2}}{192 b d^2}-\frac{\left (3 b^2 c^2-16 a b c d-3 a^2 d^2\right ) \sqrt{a+b x} (c+d x)^{5/2}}{48 d^2}+\frac{(b c+a d) (a+b x)^{3/2} (c+d x)^{5/2}}{8 d}+\frac{1}{5} (a+b x)^{5/2} (c+d x)^{5/2}-\frac{\int \frac{\sqrt{c+d x} \left (-120 a^3 b c^2 d^2-\frac{15}{16} \left (3 b^4 c^4-22 a b^3 c^3 d+128 a^2 b^2 c^2 d^2+22 a^3 b c d^3-3 a^4 d^4\right ) x\right )}{x \sqrt{a+b x}} \, dx}{120 b d^2}\\ &=\frac{\left (3 b^4 c^4-22 a b^3 c^3 d+128 a^2 b^2 c^2 d^2+22 a^3 b c d^3-3 a^4 d^4\right ) \sqrt{a+b x} \sqrt{c+d x}}{128 b^2 d^2}+\frac{\left (3 b^3 c^3-19 a b^2 c^2 d+109 a^2 b c d^2+3 a^3 d^3\right ) \sqrt{a+b x} (c+d x)^{3/2}}{192 b d^2}-\frac{\left (3 b^2 c^2-16 a b c d-3 a^2 d^2\right ) \sqrt{a+b x} (c+d x)^{5/2}}{48 d^2}+\frac{(b c+a d) (a+b x)^{3/2} (c+d x)^{5/2}}{8 d}+\frac{1}{5} (a+b x)^{5/2} (c+d x)^{5/2}-\frac{\int \frac{-120 a^3 b^2 c^3 d^2-\frac{15}{32} (b c+a d) \left (3 b^4 c^4-28 a b^3 c^3 d+178 a^2 b^2 c^2 d^2-28 a^3 b c d^3+3 a^4 d^4\right ) x}{x \sqrt{a+b x} \sqrt{c+d x}} \, dx}{120 b^2 d^2}\\ &=\frac{\left (3 b^4 c^4-22 a b^3 c^3 d+128 a^2 b^2 c^2 d^2+22 a^3 b c d^3-3 a^4 d^4\right ) \sqrt{a+b x} \sqrt{c+d x}}{128 b^2 d^2}+\frac{\left (3 b^3 c^3-19 a b^2 c^2 d+109 a^2 b c d^2+3 a^3 d^3\right ) \sqrt{a+b x} (c+d x)^{3/2}}{192 b d^2}-\frac{\left (3 b^2 c^2-16 a b c d-3 a^2 d^2\right ) \sqrt{a+b x} (c+d x)^{5/2}}{48 d^2}+\frac{(b c+a d) (a+b x)^{3/2} (c+d x)^{5/2}}{8 d}+\frac{1}{5} (a+b x)^{5/2} (c+d x)^{5/2}+\left (a^3 c^3\right ) \int \frac{1}{x \sqrt{a+b x} \sqrt{c+d x}} \, dx+\frac{\left ((b c+a d) \left (3 b^4 c^4-28 a b^3 c^3 d+178 a^2 b^2 c^2 d^2-28 a^3 b c d^3+3 a^4 d^4\right )\right ) \int \frac{1}{\sqrt{a+b x} \sqrt{c+d x}} \, dx}{256 b^2 d^2}\\ &=\frac{\left (3 b^4 c^4-22 a b^3 c^3 d+128 a^2 b^2 c^2 d^2+22 a^3 b c d^3-3 a^4 d^4\right ) \sqrt{a+b x} \sqrt{c+d x}}{128 b^2 d^2}+\frac{\left (3 b^3 c^3-19 a b^2 c^2 d+109 a^2 b c d^2+3 a^3 d^3\right ) \sqrt{a+b x} (c+d x)^{3/2}}{192 b d^2}-\frac{\left (3 b^2 c^2-16 a b c d-3 a^2 d^2\right ) \sqrt{a+b x} (c+d x)^{5/2}}{48 d^2}+\frac{(b c+a d) (a+b x)^{3/2} (c+d x)^{5/2}}{8 d}+\frac{1}{5} (a+b x)^{5/2} (c+d x)^{5/2}+\left (2 a^3 c^3\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 ((b c+a d) \left (3 b^4 c^4-28 a b^3 c^3 d+178 a^2 b^2 c^2 d^2-28 a^3 b c d^3+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 )}{128 b^3 d^2}\\ &=\frac{\left (3 b^4 c^4-22 a b^3 c^3 d+128 a^2 b^2 c^2 d^2+22 a^3 b c d^3-3 a^4 d^4\right ) \sqrt{a+b x} \sqrt{c+d x}}{128 b^2 d^2}+\frac{\left (3 b^3 c^3-19 a b^2 c^2 d+109 a^2 b c d^2+3 a^3 d^3\right ) \sqrt{a+b x} (c+d x)^{3/2}}{192 b d^2}-\frac{\left (3 b^2 c^2-16 a b c d-3 a^2 d^2\right ) \sqrt{a+b x} (c+d x)^{5/2}}{48 d^2}+\frac{(b c+a d) (a+b x)^{3/2} (c+d x)^{5/2}}{8 d}+\frac{1}{5} (a+b x)^{5/2} (c+d x)^{5/2}-2 a^{5/2} c^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )+\frac{\left ((b c+a d) \left (3 b^4 c^4-28 a b^3 c^3 d+178 a^2 b^2 c^2 d^2-28 a^3 b c d^3+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 )}{128 b^3 d^2}\\ &=\frac{\left (3 b^4 c^4-22 a b^3 c^3 d+128 a^2 b^2 c^2 d^2+22 a^3 b c d^3-3 a^4 d^4\right ) \sqrt{a+b x} \sqrt{c+d x}}{128 b^2 d^2}+\frac{\left (3 b^3 c^3-19 a b^2 c^2 d+109 a^2 b c d^2+3 a^3 d^3\right ) \sqrt{a+b x} (c+d x)^{3/2}}{192 b d^2}-\frac{\left (3 b^2 c^2-16 a b c d-3 a^2 d^2\right ) \sqrt{a+b x} (c+d x)^{5/2}}{48 d^2}+\frac{(b c+a d) (a+b x)^{3/2} (c+d x)^{5/2}}{8 d}+\frac{1}{5} (a+b x)^{5/2} (c+d x)^{5/2}-2 a^{5/2} c^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )+\frac{(b c+a d) \left (3 b^4 c^4-28 a b^3 c^3 d+178 a^2 b^2 c^2 d^2-28 a^3 b c d^3+3 a^4 d^4\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{128 b^{5/2} d^{5/2}}\\ \end{align*}

Mathematica [B]  time = 3.54928, size = 1249, normalized size = 3.19 \[ \frac{\sqrt{c+d x} \left (45 b^5 (c+d x)^2 \sinh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b c-a d}}\right ) c^5-45 b^2 \sqrt{d} (b c-a d)^{5/2} \sqrt{a+b x} \left (\frac{b (c+d x)}{b c-a d}\right )^{5/2} c^4-375 a b^4 d (c+d x)^2 \sinh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b c-a d}}\right ) c^4+360 a b 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^3+30 b^2 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^3+2250 a^2 b^3 d^2 (c+d x)^2 \sinh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b c-a d}}\right ) c^3-3840 a^{5/2} b d^{5/2} (b c-a d)^{3/2} \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}+3754 a^2 d^{5/2} (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+744 b^2 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^2+2578 a b 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^2+2250 a^3 b^2 d^3 (c+d x)^2 \sinh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b c-a d}}\right ) c^2+1008 b^2 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} c+\frac{360 a^3 d^{7/2} (b c-a d)^{5/2} \sqrt{a+b x} \left (\frac{b (c+d x)}{b c-a d}\right )^{5/2} c}{b}+2896 a b 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} c+2578 a^2 d^{7/2} (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-375 a^4 b d^4 (c+d x)^2 \sinh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b c-a d}}\right ) c+384 b^2 d^{9/2} (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}+1008 a b d^{9/2} (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}+744 a^2 d^{9/2} (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}+\frac{30 a^3 d^{9/2} (b c-a d)^{5/2} x \sqrt{a+b x} \left (\frac{b (c+d x)}{b c-a d}\right )^{5/2}}{b}+45 a^5 d^5 (c+d x)^2 \sinh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b c-a d}}\right )-45 a^4 d^{9/2} \sqrt{b c-a d} \sqrt{a+b x} (c+d x)^2 \sqrt{\frac{b (c+d x)}{b c-a d}}\right )}{1920 d^{5/2} (b c-a d)^{5/2} \left (\frac{b (c+d x)}{b c-a d}\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[c + d*x]*(-45*a^4*d^(9/2)*Sqrt[b*c - a*d]*Sqrt[a + b*x]*(c + d*x)^2*Sqrt[(b*(c + d*x))/(b*c - a*d)] - 45
*b^2*c^4*Sqrt[d]*(b*c - a*d)^(5/2)*Sqrt[a + b*x]*((b*(c + d*x))/(b*c - a*d))^(5/2) + 360*a*b*c^3*d^(3/2)*(b*c
- a*d)^(5/2)*Sqrt[a + b*x]*((b*(c + d*x))/(b*c - a*d))^(5/2) + 3754*a^2*c^2*d^(5/2)*(b*c - a*d)^(5/2)*Sqrt[a +
b*x]*((b*(c + d*x))/(b*c - a*d))^(5/2) + (360*a^3*c*d^(7/2)*(b*c - a*d)^(5/2)*Sqrt[a + b*x]*((b*(c + d*x))/(b
*c - a*d))^(5/2))/b + 30*b^2*c^3*d^(3/2)*(b*c - a*d)^(5/2)*x*Sqrt[a + b*x]*((b*(c + d*x))/(b*c - a*d))^(5/2) +
2578*a*b*c^2*d^(5/2)*(b*c - a*d)^(5/2)*x*Sqrt[a + b*x]*((b*(c + d*x))/(b*c - a*d))^(5/2) + 2578*a^2*c*d^(7/2)
*(b*c - a*d)^(5/2)*x*Sqrt[a + b*x]*((b*(c + d*x))/(b*c - a*d))^(5/2) + (30*a^3*d^(9/2)*(b*c - a*d)^(5/2)*x*Sqr
t[a + b*x]*((b*(c + d*x))/(b*c - a*d))^(5/2))/b + 744*b^2*c^2*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) + 2896*a*b*c*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) + 744*a^2*d^(9/2)*(b*c - a*d)^(5/2)*x^2*Sqrt[a + b*x]*((b*(c + d*x))/(b*c - a*d))^(5/2) + 1008*b^2*
c*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) + 1008*a*b*d^(9/2)*(b*c - a*d)
^(5/2)*x^3*Sqrt[a + b*x]*((b*(c + d*x))/(b*c - a*d))^(5/2) + 384*b^2*d^(9/2)*(b*c - a*d)^(5/2)*x^4*Sqrt[a + b*
x]*((b*(c + d*x))/(b*c - a*d))^(5/2) + 45*b^5*c^5*(c + d*x)^2*ArcSinh[(Sqrt[d]*Sqrt[a + b*x])/Sqrt[b*c - a*d]]
- 375*a*b^4*c^4*d*(c + d*x)^2*ArcSinh[(Sqrt[d]*Sqrt[a + b*x])/Sqrt[b*c - a*d]] + 2250*a^2*b^3*c^3*d^2*(c + d*
x)^2*ArcSinh[(Sqrt[d]*Sqrt[a + b*x])/Sqrt[b*c - a*d]] + 2250*a^3*b^2*c^2*d^3*(c + d*x)^2*ArcSinh[(Sqrt[d]*Sqrt
[a + b*x])/Sqrt[b*c - a*d]] - 375*a^4*b*c*d^4*(c + d*x)^2*ArcSinh[(Sqrt[d]*Sqrt[a + b*x])/Sqrt[b*c - a*d]] + 4
5*a^5*d^5*(c + d*x)^2*ArcSinh[(Sqrt[d]*Sqrt[a + b*x])/Sqrt[b*c - a*d]] - 3840*a^(5/2)*b*c^(5/2)*d^(5/2)*(b*c -
a*d)^(3/2)*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])]))/(1920*d^(5/2)*(b*c - a*d)^(5/2)*((b*(c + d*x))/(b*c - a*d))^(5/2))

________________________________________________________________________________________

Maple [B]  time = 0.016, size = 1116, normalized size = 2.9 \begin{align*} \text{result too large to display} \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,x)

[Out]

1/3840*(b*x+a)^(1/2)*(d*x+c)^(1/2)*(768*x^4*b^4*d^4*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)*(a*c)^(1/2)+20
16*x^3*a*b^3*d^4*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)*(a*c)^(1/2)+2016*x^3*b^4*c*d^3*(b*d*x^2+a*d*x+b*c
*x+a*c)^(1/2)*(b*d)^(1/2)*(a*c)^(1/2)+1488*x^2*a^2*b^2*d^4*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)*(a*c)^(
1/2)+5792*x^2*a*b^3*c*d^3*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)*(a*c)^(1/2)+1488*x^2*b^4*c^2*d^2*(b*d*x^
2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)*(a*c)^(1/2)+45*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)*a^5*d^5-375*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)*a^4*b*c*d^4+2250*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)*a^3*b^2*c^2*d^3+2250*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)*a^2*b^3*c^3*d^2-375*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)*a*b^4*c^4*d+45*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)*b^5*c^5-3840*(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)*a^3*b^2*c^3*d^2+60*(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*b*d^4+5156*(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^2*c*d^3+5156*(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^3*c^2*d^2+60*(b*d)^(1/2)*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a
*c)^(1/2)*x*b^4*c^3*d-90*(b*d)^(1/2)*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*a^4*d^4+720*(b*d)^(1/2)*(a*c)
^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*a^3*b*c*d^3+7508*(b*d)^(1/2)*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2
)*a^2*b^2*c^2*d^2+720*(b*d)^(1/2)*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*a*b^3*c^3*d-90*(b*d)^(1/2)*(a*c)
^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*b^4*c^4)/b^2/d^2/(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)/(b*d)^(1/2)/(a*c)^(1/2
)

________________________________________________________________________________________

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,x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [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,x, algorithm="fricas")

[Out]

Timed out

________________________________________________________________________________________

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,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [A]  time = 1.98975, size = 722, normalized size = 1.85 \begin{align*} -\frac{2 \, \sqrt{b d} a^{3} c^{3}{\left | b \right |} \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} b} + \frac{1}{1920} \, \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}{\left (2 \,{\left (4 \,{\left (b x + a\right )}{\left (6 \,{\left (b x + a\right )}{\left (\frac{8 \,{\left (b x + a\right )} d^{2}{\left | b \right |}}{b^{4}} + \frac{21 \, b^{11} c d^{9}{\left | b \right |} - 11 \, a b^{10} d^{10}{\left | b \right |}}{b^{14} d^{8}}\right )} + \frac{93 \, b^{12} c^{2} d^{8}{\left | b \right |} - 16 \, a b^{11} c d^{9}{\left | b \right |} + 3 \, a^{2} b^{10} d^{10}{\left | b \right |}}{b^{14} d^{8}}\right )} + \frac{5 \,{\left (3 \, b^{13} c^{3} d^{7}{\left | b \right |} + 109 \, a b^{12} c^{2} d^{8}{\left | b \right |} - 19 \, a^{2} b^{11} c d^{9}{\left | b \right |} + 3 \, a^{3} b^{10} d^{10}{\left | b \right |}\right )}}{b^{14} d^{8}}\right )}{\left (b x + a\right )} - \frac{15 \,{\left (3 \, b^{14} c^{4} d^{6}{\left | b \right |} - 22 \, a b^{13} c^{3} d^{7}{\left | b \right |} - 128 \, a^{2} b^{12} c^{2} d^{8}{\left | b \right |} + 22 \, a^{3} b^{11} c d^{9}{\left | b \right |} - 3 \, a^{4} b^{10} d^{10}{\left | b \right |}\right )}}{b^{14} d^{8}}\right )} \sqrt{b x + a} - \frac{{\left (3 \, \sqrt{b d} b^{5} c^{5}{\left | b \right |} - 25 \, \sqrt{b d} a b^{4} c^{4} d{\left | b \right |} + 150 \, \sqrt{b d} a^{2} b^{3} c^{3} d^{2}{\left | b \right |} + 150 \, \sqrt{b d} a^{3} b^{2} c^{2} d^{3}{\left | b \right |} - 25 \, \sqrt{b d} a^{4} b c d^{4}{\left | b \right |} + 3 \, \sqrt{b d} a^{5} d^{5}{\left | b \right |}\right )} \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 )}{256 \, b^{4} d^{3}} \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,x, algorithm="giac")

[Out]

-2*sqrt(b*d)*a^3*c^3*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) + 1/1920*sqrt(b^2*c + (b*x + a)*b*d - a*b*d)*(2*(4*(b*x +
a)*(6*(b*x + a)*(8*(b*x + a)*d^2*abs(b)/b^4 + (21*b^11*c*d^9*abs(b) - 11*a*b^10*d^10*abs(b))/(b^14*d^8)) + (9
3*b^12*c^2*d^8*abs(b) - 16*a*b^11*c*d^9*abs(b) + 3*a^2*b^10*d^10*abs(b))/(b^14*d^8)) + 5*(3*b^13*c^3*d^7*abs(b
) + 109*a*b^12*c^2*d^8*abs(b) - 19*a^2*b^11*c*d^9*abs(b) + 3*a^3*b^10*d^10*abs(b))/(b^14*d^8))*(b*x + a) - 15*
(3*b^14*c^4*d^6*abs(b) - 22*a*b^13*c^3*d^7*abs(b) - 128*a^2*b^12*c^2*d^8*abs(b) + 22*a^3*b^11*c*d^9*abs(b) - 3
*a^4*b^10*d^10*abs(b))/(b^14*d^8))*sqrt(b*x + a) - 1/256*(3*sqrt(b*d)*b^5*c^5*abs(b) - 25*sqrt(b*d)*a*b^4*c^4*
d*abs(b) + 150*sqrt(b*d)*a^2*b^3*c^3*d^2*abs(b) + 150*sqrt(b*d)*a^3*b^2*c^2*d^3*abs(b) - 25*sqrt(b*d)*a^4*b*c*
d^4*abs(b) + 3*sqrt(b*d)*a^5*d^5*abs(b))*log((sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2
)/(b^4*d^3)