\(\int \frac {(a+b x)^{5/2} (c+d x)^{5/2}}{x} \, dx\) [667]

Optimal result

Integrand size = 22, antiderivative size = 391 \[ \int \frac {(a+b x)^{5/2} (c+d x)^{5/2}}{x} \, dx=\frac {\left (3 b^4 c^4-22 a b^3 c^3 d+128 a^2 b^2 c^2 d^2+22 a^3 b c d^3-3 a^4 d^4\right ) \sqrt {a+b x} \sqrt {c+d x}}{128 b^2 d^2}+\frac {\left (3 b^3 c^3-19 a b^2 c^2 d+109 a^2 b c d^2+3 a^3 d^3\right ) \sqrt {a+b x} (c+d x)^{3/2}}{192 b d^2}-\frac {\left (3 b^2 c^2-16 a b c d-3 a^2 d^2\right ) \sqrt {a+b x} (c+d x)^{5/2}}{48 d^2}+\frac {(b c+a d) (a+b x)^{3/2} (c+d x)^{5/2}}{8 d}+\frac {1}{5} (a+b x)^{5/2} (c+d x)^{5/2}-2 a^{5/2} c^{5/2} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )+\frac {(b c+a d) \left (3 b^4 c^4-28 a b^3 c^3 d+178 a^2 b^2 c^2 d^2-28 a^3 b c d^3+3 a^4 d^4\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{128 b^{5/2} d^{5/2}} \]

[Out]

1/8*(a*d+b*c)*(b*x+a)^(3/2)*(d*x+c)^(5/2)/d+1/5*(b*x+a)^(5/2)*(d*x+c)^(5/2)-2*a^(5/2)*c^(5/2)*arctanh(c^(1/2)*
(b*x+a)^(1/2)/a^(1/2)/(d*x+c)^(1/2))+1/128*(a*d+b*c)*(3*a^4*d^4-28*a^3*b*c*d^3+178*a^2*b^2*c^2*d^2-28*a*b^3*c^
3*d+3*b^4*c^4)*arctanh(d^(1/2)*(b*x+a)^(1/2)/b^(1/2)/(d*x+c)^(1/2))/b^(5/2)/d^(5/2)+1/192*(3*a^3*d^3+109*a^2*b
*c*d^2-19*a*b^2*c^2*d+3*b^3*c^3)*(d*x+c)^(3/2)*(b*x+a)^(1/2)/b/d^2-1/48*(-3*a^2*d^2-16*a*b*c*d+3*b^2*c^2)*(d*x
+c)^(5/2)*(b*x+a)^(1/2)/d^2+1/128*(-3*a^4*d^4+22*a^3*b*c*d^3+128*a^2*b^2*c^2*d^2-22*a*b^3*c^3*d+3*b^4*c^4)*(b*
x+a)^(1/2)*(d*x+c)^(1/2)/b^2/d^2

Rubi [A] (verified)

Time = 0.31 (sec) , antiderivative size = 391, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {103, 159, 163, 65, 223, 212, 95, 214} \[ \int \frac {(a+b x)^{5/2} (c+d x)^{5/2}}{x} \, dx=-2 a^{5/2} c^{5/2} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )-\frac {\sqrt {a+b x} (c+d x)^{5/2} \left (-3 a^2 d^2-16 a b c d+3 b^2 c^2\right )}{48 d^2}+\frac {\sqrt {a+b x} (c+d x)^{3/2} \left (3 a^3 d^3+109 a^2 b c d^2-19 a b^2 c^2 d+3 b^3 c^3\right )}{192 b d^2}+\frac {(a d+b c) \left (3 a^4 d^4-28 a^3 b c d^3+178 a^2 b^2 c^2 d^2-28 a b^3 c^3 d+3 b^4 c^4\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{128 b^{5/2} d^{5/2}}+\frac {\sqrt {a+b x} \sqrt {c+d x} \left (-3 a^4 d^4+22 a^3 b c d^3+128 a^2 b^2 c^2 d^2-22 a b^3 c^3 d+3 b^4 c^4\right )}{128 b^2 d^2}+\frac {1}{5} (a+b x)^{5/2} (c+d x)^{5/2}+\frac {(a+b x)^{3/2} (c+d x)^{5/2} (a d+b c)}{8 d} \]

[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 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 95

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> With[{q = Denomin
ator[m]}, Dist[q, Subst[Int[x^(q*(m + 1) - 1)/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^
(1/q)], x]] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && LtQ[-1, m, 0] && SimplerQ[
a + b*x, c + d*x]

Rule 103

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 159

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 163

Int[(((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/((a_.) + (b_.)*(x_)), x_Symbol]
:> Dist[h/b, Int[(c + d*x)^n*(e + f*x)^p, x], x] + Dist[(b*g - a*h)/b, Int[(c + d*x)^n*((e + f*x)^p/(a + b*x)
), x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x]

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 214

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x/Rt[-a/b, 2]], x] /; FreeQ[{a, b},
x] && NegQ[a/b]

Rule 223

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {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 ) \text {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 ) \text {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 ) \text {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 [A] (verified)

Time = 1.04 (sec) , antiderivative size = 317, normalized size of antiderivative = 0.81 \[ \int \frac {(a+b x)^{5/2} (c+d x)^{5/2}}{x} \, dx=\frac {\sqrt {a+b x} \sqrt {c+d x} \left (-45 a^4 d^4+30 a^3 b d^3 (12 c+d x)+2 a^2 b^2 d^2 \left (1877 c^2+1289 c d x+372 d^2 x^2\right )+2 a b^3 d \left (180 c^3+1289 c^2 d x+1448 c d^2 x^2+504 d^3 x^3\right )+b^4 \left (-45 c^4+30 c^3 d x+744 c^2 d^2 x^2+1008 c d^3 x^3+384 d^4 x^4\right )\right )}{1920 b^2 d^2}-2 a^{5/2} c^{5/2} \text {arctanh}\left (\frac {\sqrt {a} \sqrt {c+d x}}{\sqrt {c} \sqrt {a+b x}}\right )+\frac {\left (3 b^5 c^5-25 a b^4 c^4 d+150 a^2 b^3 c^3 d^2+150 a^3 b^2 c^2 d^3-25 a^4 b c d^4+3 a^5 d^5\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {a+b x}}\right )}{128 b^{5/2} d^{5/2}} \]

[In]

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

[Out]

(Sqrt[a + b*x]*Sqrt[c + d*x]*(-45*a^4*d^4 + 30*a^3*b*d^3*(12*c + d*x) + 2*a^2*b^2*d^2*(1877*c^2 + 1289*c*d*x +
372*d^2*x^2) + 2*a*b^3*d*(180*c^3 + 1289*c^2*d*x + 1448*c*d^2*x^2 + 504*d^3*x^3) + b^4*(-45*c^4 + 30*c^3*d*x
+ 744*c^2*d^2*x^2 + 1008*c*d^3*x^3 + 384*d^4*x^4)))/(1920*b^2*d^2) - 2*a^(5/2)*c^(5/2)*ArcTanh[(Sqrt[a]*Sqrt[c
+ d*x])/(Sqrt[c]*Sqrt[a + b*x])] + ((3*b^5*c^5 - 25*a*b^4*c^4*d + 150*a^2*b^3*c^3*d^2 + 150*a^3*b^2*c^2*d^3 -
25*a^4*b*c*d^4 + 3*a^5*d^5)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[a + b*x])])/(128*b^(5/2)*d^(5/2))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(954\) vs. \(2(335)=670\).

Time = 0.56 (sec) , antiderivative size = 955, normalized size of antiderivative = 2.44

method	result	size
default	\(-\frac {\sqrt {b x +a}\, \sqrt {d x +c}\, \left (-768 b^{4} d^{4} x^{4} \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, \sqrt {a c}-2016 a \,b^{3} d^{4} x^{3} \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, \sqrt {a c}-2016 b^{4} c \,d^{3} x^{3} \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, \sqrt {a c}-1488 a^{2} b^{2} d^{4} x^{2} \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, \sqrt {a c}-5792 a \,b^{3} c \,d^{3} x^{2} \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, \sqrt {a c}-1488 b^{4} c^{2} d^{2} x^{2} \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, \sqrt {a c}+3840 \sqrt {b d}\, \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a^{3} b^{2} c^{3} d^{2}-45 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) \sqrt {a c}\, a^{5} d^{5}+375 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) \sqrt {a c}\, a^{4} b c \,d^{4}-2250 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) \sqrt {a c}\, a^{3} b^{2} c^{2} d^{3}-2250 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) \sqrt {a c}\, a^{2} b^{3} c^{3} d^{2}+375 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) \sqrt {a c}\, a \,b^{4} c^{4} d -45 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) \sqrt {a c}\, b^{5} c^{5}-60 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, \sqrt {a c}\, a^{3} b \,d^{4} x -5156 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, \sqrt {a c}\, a^{2} b^{2} c \,d^{3} x -5156 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, \sqrt {a c}\, a \,b^{3} c^{2} d^{2} x -60 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, \sqrt {a c}\, b^{4} c^{3} d x +90 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, \sqrt {a c}\, a^{4} d^{4}-720 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, \sqrt {a c}\, a^{3} b c \,d^{3}-7508 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, \sqrt {a c}\, a^{2} b^{2} c^{2} d^{2}-720 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, \sqrt {a c}\, a \,b^{3} c^{3} d +90 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, \sqrt {a c}\, b^{4} c^{4}\right )}{3840 b^{2} d^{2} \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, \sqrt {a c}}\)	\(955\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 29.52 (sec) , antiderivative size = 1801, normalized size of antiderivative = 4.61 \[ \int \frac {(a+b x)^{5/2} (c+d x)^{5/2}}{x} \, dx=\text {Too large to display} \]

[In]

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

[Out]

[1/7680*(3840*sqrt(a*c)*a^2*b^3*c^2*d^3*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) + 15*(3*b^5*c^5 - 25*a*b^4*c^
4*d + 150*a^2*b^3*c^3*d^2 + 150*a^3*b^2*c^2*d^3 - 25*a^4*b*c*d^4 + 3*a^5*d^5)*sqrt(b*d)*log(8*b^2*d^2*x^2 + b^
2*c^2 + 6*a*b*c*d + a^2*d^2 + 4*(2*b*d*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*(384*b^5*d^5*x^4 - 45*b^5*c^4*d + 360*a*b^4*c^3*d^2 + 3754*a^2*b^3*c^2*d^3 + 360*a^3*b^2*c*d^4 -
45*a^4*b*d^5 + 1008*(b^5*c*d^4 + a*b^4*d^5)*x^3 + 8*(93*b^5*c^2*d^3 + 362*a*b^4*c*d^4 + 93*a^2*b^3*d^5)*x^2 +
2*(15*b^5*c^3*d^2 + 1289*a*b^4*c^2*d^3 + 1289*a^2*b^3*c*d^4 + 15*a^3*b^2*d^5)*x)*sqrt(b*x + a)*sqrt(d*x + c))/
(b^3*d^3), 1/3840*(1920*sqrt(a*c)*a^2*b^3*c^2*d^3*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) - 15*(3*b^5*c^5 - 2
5*a*b^4*c^4*d + 150*a^2*b^3*c^3*d^2 + 150*a^3*b^2*c^2*d^3 - 25*a^4*b*c*d^4 + 3*a^5*d^5)*sqrt(-b*d)*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*(384*b^5*d^5*x^4 - 45*b^5*c^4*d + 360*a*b^4*c^3*d^2 + 3754*a^2*b^3*c^2*d^3 + 360*a^3*b^2*c*d^4 - 45*a^4*b*
d^5 + 1008*(b^5*c*d^4 + a*b^4*d^5)*x^3 + 8*(93*b^5*c^2*d^3 + 362*a*b^4*c*d^4 + 93*a^2*b^3*d^5)*x^2 + 2*(15*b^5
*c^3*d^2 + 1289*a*b^4*c^2*d^3 + 1289*a^2*b^3*c*d^4 + 15*a^3*b^2*d^5)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(b^3*d^3)
, 1/7680*(7680*sqrt(-a*c)*a^2*b^3*c^2*d^3*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*(3*b^5*c^5 - 25*a*b^4*c^4*d + 150*a^2*b^3*c^3*d^2
+ 150*a^3*b^2*c^2*d^3 - 25*a^4*b*c*d^4 + 3*a^5*d^5)*sqrt(b*d)*log(8*b^2*d^2*x^2 + b^2*c^2 + 6*a*b*c*d + a^2*d^
2 + 4*(2*b*d*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*(384*b^5*d^5*
x^4 - 45*b^5*c^4*d + 360*a*b^4*c^3*d^2 + 3754*a^2*b^3*c^2*d^3 + 360*a^3*b^2*c*d^4 - 45*a^4*b*d^5 + 1008*(b^5*c
*d^4 + a*b^4*d^5)*x^3 + 8*(93*b^5*c^2*d^3 + 362*a*b^4*c*d^4 + 93*a^2*b^3*d^5)*x^2 + 2*(15*b^5*c^3*d^2 + 1289*a
*b^4*c^2*d^3 + 1289*a^2*b^3*c*d^4 + 15*a^3*b^2*d^5)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(b^3*d^3), 1/3840*(3840*sq
rt(-a*c)*a^2*b^3*c^2*d^3*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*(3*b^5*c^5 - 25*a*b^4*c^4*d + 150*a^2*b^3*c^3*d^2 + 150*a^3*b^2*c^2
*d^3 - 25*a^4*b*c*d^4 + 3*a^5*d^5)*sqrt(-b*d)*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*(384*b^5*d^5*x^4 - 45*b^5*c^4*d + 360*a*b^4*c^3*d
^2 + 3754*a^2*b^3*c^2*d^3 + 360*a^3*b^2*c*d^4 - 45*a^4*b*d^5 + 1008*(b^5*c*d^4 + a*b^4*d^5)*x^3 + 8*(93*b^5*c^
2*d^3 + 362*a*b^4*c*d^4 + 93*a^2*b^3*d^5)*x^2 + 2*(15*b^5*c^3*d^2 + 1289*a*b^4*c^2*d^3 + 1289*a^2*b^3*c*d^4 +
15*a^3*b^2*d^5)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(b^3*d^3)]

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F(-2)]

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

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(a*d-b*c>0)', see `assume?` for
more detail

Giac [F(-2)]

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

[In]

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

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:index.cc index_m i_lex_is_greater Error: Bad Argument Value

Mupad [F(-1)]

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

[In]

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

[Out]

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