\(\int \frac {1}{x^2 \sqrt {a+b x} (c+d x)^{5/2}} \, dx\) [754]

Optimal result

Integrand size = 22, antiderivative size = 189 \[ \int \frac {1}{x^2 \sqrt {a+b x} (c+d x)^{5/2}} \, dx=-\frac {d (3 b c-5 a d) \sqrt {a+b x}}{3 a c^2 (b c-a d) (c+d x)^{3/2}}-\frac {\sqrt {a+b x}}{a c x (c+d x)^{3/2}}-\frac {d \left (3 b^2 c^2-22 a b c d+15 a^2 d^2\right ) \sqrt {a+b x}}{3 a c^3 (b c-a d)^2 \sqrt {c+d x}}+\frac {(b c+5 a d) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{a^{3/2} c^{7/2}} \]

[Out]

(5*a*d+b*c)*arctanh(c^(1/2)*(b*x+a)^(1/2)/a^(1/2)/(d*x+c)^(1/2))/a^(3/2)/c^(7/2)-1/3*d*(-5*a*d+3*b*c)*(b*x+a)^
(1/2)/a/c^2/(-a*d+b*c)/(d*x+c)^(3/2)-(b*x+a)^(1/2)/a/c/x/(d*x+c)^(3/2)-1/3*d*(15*a^2*d^2-22*a*b*c*d+3*b^2*c^2)
*(b*x+a)^(1/2)/a/c^3/(-a*d+b*c)^2/(d*x+c)^(1/2)

Rubi [A] (verified)

Time = 0.13 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {105, 157, 12, 95, 214} \[ \int \frac {1}{x^2 \sqrt {a+b x} (c+d x)^{5/2}} \, dx=\frac {(5 a d+b c) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{a^{3/2} c^{7/2}}-\frac {d \sqrt {a+b x} \left (15 a^2 d^2-22 a b c d+3 b^2 c^2\right )}{3 a c^3 \sqrt {c+d x} (b c-a d)^2}-\frac {d \sqrt {a+b x} (3 b c-5 a d)}{3 a c^2 (c+d x)^{3/2} (b c-a d)}-\frac {\sqrt {a+b x}}{a c x (c+d x)^{3/2}} \]

[In]

Int[1/(x^2*Sqrt[a + b*x]*(c + d*x)^(5/2)),x]

[Out]

-1/3*(d*(3*b*c - 5*a*d)*Sqrt[a + b*x])/(a*c^2*(b*c - a*d)*(c + d*x)^(3/2)) - Sqrt[a + b*x]/(a*c*x*(c + d*x)^(3
/2)) - (d*(3*b^2*c^2 - 22*a*b*c*d + 15*a^2*d^2)*Sqrt[a + b*x])/(3*a*c^3*(b*c - a*d)^2*Sqrt[c + d*x]) + ((b*c +
5*a*d)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(a^(3/2)*c^(7/2))

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, 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 105

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[b*(a +
b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Dist[1/((m + 1)*(b*
c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*(m + 1) - b*(d*e*(m + n + 2) +
c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && ILtQ[m, -1] &
& (IntegerQ[n] || IntegersQ[2*n, 2*p] || ILtQ[m + n + p + 3, 0])

Rule 157

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 + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f
))), x] + Dist[1/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[(a*d*f*
g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a*h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p
+ 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && IntegersQ[2*m, 2*n, 2*p]

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]

Rubi steps \begin{align*} \text {integral}& = -\frac {\sqrt {a+b x}}{a c x (c+d x)^{3/2}}-\frac {\int \frac {\frac {1}{2} (b c+5 a d)+2 b d x}{x \sqrt {a+b x} (c+d x)^{5/2}} \, dx}{a c} \\ & = -\frac {d (3 b c-5 a d) \sqrt {a+b x}}{3 a c^2 (b c-a d) (c+d x)^{3/2}}-\frac {\sqrt {a+b x}}{a c x (c+d x)^{3/2}}+\frac {2 \int \frac {-\frac {3}{4} (b c-a d) (b c+5 a d)-\frac {1}{2} b d (3 b c-5 a d) x}{x \sqrt {a+b x} (c+d x)^{3/2}} \, dx}{3 a c^2 (b c-a d)} \\ & = -\frac {d (3 b c-5 a d) \sqrt {a+b x}}{3 a c^2 (b c-a d) (c+d x)^{3/2}}-\frac {\sqrt {a+b x}}{a c x (c+d x)^{3/2}}-\frac {d \left (3 b^2 c^2-22 a b c d+15 a^2 d^2\right ) \sqrt {a+b x}}{3 a c^3 (b c-a d)^2 \sqrt {c+d x}}-\frac {4 \int \frac {3 (b c-a d)^2 (b c+5 a d)}{8 x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{3 a c^3 (b c-a d)^2} \\ & = -\frac {d (3 b c-5 a d) \sqrt {a+b x}}{3 a c^2 (b c-a d) (c+d x)^{3/2}}-\frac {\sqrt {a+b x}}{a c x (c+d x)^{3/2}}-\frac {d \left (3 b^2 c^2-22 a b c d+15 a^2 d^2\right ) \sqrt {a+b x}}{3 a c^3 (b c-a d)^2 \sqrt {c+d x}}-\frac {(b c+5 a d) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{2 a c^3} \\ & = -\frac {d (3 b c-5 a d) \sqrt {a+b x}}{3 a c^2 (b c-a d) (c+d x)^{3/2}}-\frac {\sqrt {a+b x}}{a c x (c+d x)^{3/2}}-\frac {d \left (3 b^2 c^2-22 a b c d+15 a^2 d^2\right ) \sqrt {a+b x}}{3 a c^3 (b c-a d)^2 \sqrt {c+d x}}-\frac {(b c+5 a d) \text {Subst}\left (\int \frac {1}{-a+c x^2} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )}{a c^3} \\ & = -\frac {d (3 b c-5 a d) \sqrt {a+b x}}{3 a c^2 (b c-a d) (c+d x)^{3/2}}-\frac {\sqrt {a+b x}}{a c x (c+d x)^{3/2}}-\frac {d \left (3 b^2 c^2-22 a b c d+15 a^2 d^2\right ) \sqrt {a+b x}}{3 a c^3 (b c-a d)^2 \sqrt {c+d x}}+\frac {(b c+5 a d) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{a^{3/2} c^{7/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.30 (sec) , antiderivative size = 158, normalized size of antiderivative = 0.84 \[ \int \frac {1}{x^2 \sqrt {a+b x} (c+d x)^{5/2}} \, dx=-\frac {\sqrt {a+b x} \left (3 b^2 c^2 (c+d x)^2-2 a b c d \left (3 c^2+15 c d x+11 d^2 x^2\right )+a^2 d^2 \left (3 c^2+20 c d x+15 d^2 x^2\right )\right )}{3 a c^3 (b c-a d)^2 x (c+d x)^{3/2}}+\frac {(b c+5 a d) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{a^{3/2} c^{7/2}} \]

[In]

Integrate[1/(x^2*Sqrt[a + b*x]*(c + d*x)^(5/2)),x]

[Out]

-1/3*(Sqrt[a + b*x]*(3*b^2*c^2*(c + d*x)^2 - 2*a*b*c*d*(3*c^2 + 15*c*d*x + 11*d^2*x^2) + a^2*d^2*(3*c^2 + 20*c
*d*x + 15*d^2*x^2)))/(a*c^3*(b*c - a*d)^2*x*(c + d*x)^(3/2)) + ((b*c + 5*a*d)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/
(Sqrt[a]*Sqrt[c + d*x])])/(a^(3/2)*c^(7/2))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(918\) vs. \(2(161)=322\).

Time = 0.59 (sec) , antiderivative size = 919, normalized size of antiderivative = 4.86

method	result	size
default	\(\frac {\sqrt {b x +a}\, \left (15 \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} d^{5} x^{3}-27 \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^{2} b c \,d^{4} x^{3}+9 \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 \,b^{2} c^{2} d^{3} x^{3}+3 \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 ) b^{3} c^{3} d^{2} x^{3}+30 \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} c \,d^{4} x^{2}-54 \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^{2} b \,c^{2} d^{3} x^{2}+18 \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 \,b^{2} c^{3} d^{2} x^{2}+6 \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 ) b^{3} c^{4} d \,x^{2}+15 \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} c^{2} d^{3} x -27 \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^{2} b \,c^{3} d^{2} x +9 \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 \,b^{2} c^{4} d x +3 \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 ) b^{3} c^{5} x -30 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a^{2} d^{4} x^{2}+44 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a b c \,d^{3} x^{2}-6 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, b^{2} c^{2} d^{2} x^{2}-40 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a^{2} c \,d^{3} x +60 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a b \,c^{2} d^{2} x -12 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, b^{2} c^{3} d x -6 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a^{2} c^{2} d^{2}+12 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a b \,c^{3} d -6 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, b^{2} c^{4}\right )}{6 a \,c^{3} \left (a d -b c \right )^{2} \sqrt {a c}\, x \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \left (d x +c \right )^{\frac {3}{2}}}\)	\(919\)

[In]

int(1/x^2/(d*x+c)^(5/2)/(b*x+a)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/6*(b*x+a)^(1/2)/a/c^3*(15*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*d^5*x^3-27*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^2*b*c*d^4*x^3+9*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*b^2*c^2*d^3*x^3+3*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a
*c)/x)*b^3*c^3*d^2*x^3+30*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*c*d^4*x^2-54*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^2*b*c^2*d^3*x^2+18*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*b^2*c^3*d^2*x^2+6*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+
2*a*c)/x)*b^3*c^4*d*x^2+15*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*c^2*d^3*x-27*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^2*b*c^3*d^2*x+9*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*b^2*c^4*d*x+3*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)
/x)*b^3*c^5*x-30*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a^2*d^4*x^2+44*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a*b*c*
d^3*x^2-6*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*b^2*c^2*d^2*x^2-40*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a^2*c*d^3
*x+60*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a*b*c^2*d^2*x-12*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*b^2*c^3*d*x-6*(
a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a^2*c^2*d^2+12*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a*b*c^3*d-6*(a*c)^(1/2)*
((b*x+a)*(d*x+c))^(1/2)*b^2*c^4)/(a*d-b*c)^2/(a*c)^(1/2)/x/((b*x+a)*(d*x+c))^(1/2)/(d*x+c)^(3/2)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 454 vs. \(2 (161) = 322\).

Time = 0.68 (sec) , antiderivative size = 928, normalized size of antiderivative = 4.91 \[ \int \frac {1}{x^2 \sqrt {a+b x} (c+d x)^{5/2}} \, dx=\left [\frac {3 \, {\left ({\left (b^{3} c^{3} d^{2} + 3 \, a b^{2} c^{2} d^{3} - 9 \, a^{2} b c d^{4} + 5 \, a^{3} d^{5}\right )} x^{3} + 2 \, {\left (b^{3} c^{4} d + 3 \, a b^{2} c^{3} d^{2} - 9 \, a^{2} b c^{2} d^{3} + 5 \, a^{3} c d^{4}\right )} x^{2} + {\left (b^{3} c^{5} + 3 \, a b^{2} c^{4} d - 9 \, a^{2} b c^{3} d^{2} + 5 \, a^{3} c^{2} d^{3}\right )} x\right )} \sqrt {a c} \log \left (\frac {8 \, a^{2} c^{2} + {\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} + 4 \, {\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {a c} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (a b c^{2} + a^{2} c d\right )} x}{x^{2}}\right ) - 4 \, {\left (3 \, a b^{2} c^{5} - 6 \, a^{2} b c^{4} d + 3 \, a^{3} c^{3} d^{2} + {\left (3 \, a b^{2} c^{3} d^{2} - 22 \, a^{2} b c^{2} d^{3} + 15 \, a^{3} c d^{4}\right )} x^{2} + 2 \, {\left (3 \, a b^{2} c^{4} d - 15 \, a^{2} b c^{3} d^{2} + 10 \, a^{3} c^{2} d^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{12 \, {\left ({\left (a^{2} b^{2} c^{6} d^{2} - 2 \, a^{3} b c^{5} d^{3} + a^{4} c^{4} d^{4}\right )} x^{3} + 2 \, {\left (a^{2} b^{2} c^{7} d - 2 \, a^{3} b c^{6} d^{2} + a^{4} c^{5} d^{3}\right )} x^{2} + {\left (a^{2} b^{2} c^{8} - 2 \, a^{3} b c^{7} d + a^{4} c^{6} d^{2}\right )} x\right )}}, -\frac {3 \, {\left ({\left (b^{3} c^{3} d^{2} + 3 \, a b^{2} c^{2} d^{3} - 9 \, a^{2} b c d^{4} + 5 \, a^{3} d^{5}\right )} x^{3} + 2 \, {\left (b^{3} c^{4} d + 3 \, a b^{2} c^{3} d^{2} - 9 \, a^{2} b c^{2} d^{3} + 5 \, a^{3} c d^{4}\right )} x^{2} + {\left (b^{3} c^{5} + 3 \, a b^{2} c^{4} d - 9 \, a^{2} b c^{3} d^{2} + 5 \, a^{3} c^{2} d^{3}\right )} x\right )} \sqrt {-a c} \arctan \left (\frac {{\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {-a c} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (a b c d x^{2} + a^{2} c^{2} + {\left (a b c^{2} + a^{2} c d\right )} x\right )}}\right ) + 2 \, {\left (3 \, a b^{2} c^{5} - 6 \, a^{2} b c^{4} d + 3 \, a^{3} c^{3} d^{2} + {\left (3 \, a b^{2} c^{3} d^{2} - 22 \, a^{2} b c^{2} d^{3} + 15 \, a^{3} c d^{4}\right )} x^{2} + 2 \, {\left (3 \, a b^{2} c^{4} d - 15 \, a^{2} b c^{3} d^{2} + 10 \, a^{3} c^{2} d^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{6 \, {\left ({\left (a^{2} b^{2} c^{6} d^{2} - 2 \, a^{3} b c^{5} d^{3} + a^{4} c^{4} d^{4}\right )} x^{3} + 2 \, {\left (a^{2} b^{2} c^{7} d - 2 \, a^{3} b c^{6} d^{2} + a^{4} c^{5} d^{3}\right )} x^{2} + {\left (a^{2} b^{2} c^{8} - 2 \, a^{3} b c^{7} d + a^{4} c^{6} d^{2}\right )} x\right )}}\right ] \]

[In]

integrate(1/x^2/(d*x+c)^(5/2)/(b*x+a)^(1/2),x, algorithm="fricas")

[Out]

[1/12*(3*((b^3*c^3*d^2 + 3*a*b^2*c^2*d^3 - 9*a^2*b*c*d^4 + 5*a^3*d^5)*x^3 + 2*(b^3*c^4*d + 3*a*b^2*c^3*d^2 - 9
*a^2*b*c^2*d^3 + 5*a^3*c*d^4)*x^2 + (b^3*c^5 + 3*a*b^2*c^4*d - 9*a^2*b*c^3*d^2 + 5*a^3*c^2*d^3)*x)*sqrt(a*c)*l
og((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*(3*a*b^2*c^5 - 6*a^2*b*c^4*d + 3*a^3*c^3*d^2 + (3*a*b^2*c^3*d^2 -
22*a^2*b*c^2*d^3 + 15*a^3*c*d^4)*x^2 + 2*(3*a*b^2*c^4*d - 15*a^2*b*c^3*d^2 + 10*a^3*c^2*d^3)*x)*sqrt(b*x + a)*
sqrt(d*x + c))/((a^2*b^2*c^6*d^2 - 2*a^3*b*c^5*d^3 + a^4*c^4*d^4)*x^3 + 2*(a^2*b^2*c^7*d - 2*a^3*b*c^6*d^2 + a
^4*c^5*d^3)*x^2 + (a^2*b^2*c^8 - 2*a^3*b*c^7*d + a^4*c^6*d^2)*x), -1/6*(3*((b^3*c^3*d^2 + 3*a*b^2*c^2*d^3 - 9*
a^2*b*c*d^4 + 5*a^3*d^5)*x^3 + 2*(b^3*c^4*d + 3*a*b^2*c^3*d^2 - 9*a^2*b*c^2*d^3 + 5*a^3*c*d^4)*x^2 + (b^3*c^5
+ 3*a*b^2*c^4*d - 9*a^2*b*c^3*d^2 + 5*a^3*c^2*d^3)*x)*sqrt(-a*c)*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*(3*a*b^2*c^5 - 6*a^2*b*c^4*d
+ 3*a^3*c^3*d^2 + (3*a*b^2*c^3*d^2 - 22*a^2*b*c^2*d^3 + 15*a^3*c*d^4)*x^2 + 2*(3*a*b^2*c^4*d - 15*a^2*b*c^3*d
^2 + 10*a^3*c^2*d^3)*x)*sqrt(b*x + a)*sqrt(d*x + c))/((a^2*b^2*c^6*d^2 - 2*a^3*b*c^5*d^3 + a^4*c^4*d^4)*x^3 +
2*(a^2*b^2*c^7*d - 2*a^3*b*c^6*d^2 + a^4*c^5*d^3)*x^2 + (a^2*b^2*c^8 - 2*a^3*b*c^7*d + a^4*c^6*d^2)*x)]

Sympy [F]

\[ \int \frac {1}{x^2 \sqrt {a+b x} (c+d x)^{5/2}} \, dx=\int \frac {1}{x^{2} \sqrt {a + b x} \left (c + d x\right )^{\frac {5}{2}}}\, dx \]

[In]

integrate(1/x**2/(d*x+c)**(5/2)/(b*x+a)**(1/2),x)

[Out]

Integral(1/(x**2*sqrt(a + b*x)*(c + d*x)**(5/2)), x)

Maxima [F]

\[ \int \frac {1}{x^2 \sqrt {a+b x} (c+d x)^{5/2}} \, dx=\int { \frac {1}{\sqrt {b x + a} {\left (d x + c\right )}^{\frac {5}{2}} x^{2}} \,d x } \]

[In]

integrate(1/x^2/(d*x+c)^(5/2)/(b*x+a)^(1/2),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(b*x + a)*(d*x + c)^(5/2)*x^2), x)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 605 vs. \(2 (161) = 322\).

Time = 0.78 (sec) , antiderivative size = 605, normalized size of antiderivative = 3.20 \[ \int \frac {1}{x^2 \sqrt {a+b x} (c+d x)^{5/2}} \, dx=\frac {2 \, \sqrt {b x + a} {\left (\frac {2 \, {\left (4 \, b^{4} c^{4} d^{4} {\left | b \right |} - 3 \, a b^{3} c^{3} d^{5} {\left | b \right |}\right )} {\left (b x + a\right )}}{b^{4} c^{8} d - 2 \, a b^{3} c^{7} d^{2} + a^{2} b^{2} c^{6} d^{3}} + \frac {3 \, {\left (3 \, b^{5} c^{5} d^{3} {\left | b \right |} - 5 \, a b^{4} c^{4} d^{4} {\left | b \right |} + 2 \, a^{2} b^{3} c^{3} d^{5} {\left | b \right |}\right )}}{b^{4} c^{8} d - 2 \, a b^{3} c^{7} d^{2} + a^{2} b^{2} c^{6} d^{3}}\right )}}{3 \, {\left (b^{2} c + {\left (b x + a\right )} b d - a b d\right )}^{\frac {3}{2}}} + \frac {{\left (\sqrt {b d} b^{3} c + 5 \, \sqrt {b d} a b^{2} d\right )} \arctan \left (-\frac {b^{2} c + a b d - {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2}}{2 \, \sqrt {-a b c d} b}\right )}{\sqrt {-a b c d} a b c^{3} {\left | b \right |}} - \frac {2 \, {\left (\sqrt {b d} b^{5} c^{2} - 2 \, \sqrt {b d} a b^{4} c d + \sqrt {b d} a^{2} b^{3} d^{2} - \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} b^{3} c - \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a b^{2} d\right )}}{{\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2} - 2 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} b^{2} c - 2 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a b d + {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{4}\right )} a c^{3} {\left | b \right |}} \]

[In]

integrate(1/x^2/(d*x+c)^(5/2)/(b*x+a)^(1/2),x, algorithm="giac")

[Out]

2/3*sqrt(b*x + a)*(2*(4*b^4*c^4*d^4*abs(b) - 3*a*b^3*c^3*d^5*abs(b))*(b*x + a)/(b^4*c^8*d - 2*a*b^3*c^7*d^2 +
a^2*b^2*c^6*d^3) + 3*(3*b^5*c^5*d^3*abs(b) - 5*a*b^4*c^4*d^4*abs(b) + 2*a^2*b^3*c^3*d^5*abs(b))/(b^4*c^8*d - 2
*a*b^3*c^7*d^2 + a^2*b^2*c^6*d^3))/(b^2*c + (b*x + a)*b*d - a*b*d)^(3/2) + (sqrt(b*d)*b^3*c + 5*sqrt(b*d)*a*b^
2*d)*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^3*abs(b)) - 2*(sqrt(b*d)*b^5*c^2 - 2*sqrt(b*d)*a*b^4*c*d + sqrt(b*d)*a^2*b^3*
d^2 - sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*b^3*c - 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^2*d)/((b^4*c^2 - 2*a*b^3*c*d + a^2*b^2*d^2 - 2*(sq
rt(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)*a*c^3*
abs(b))

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{x^2 \sqrt {a+b x} (c+d x)^{5/2}} \, dx=\int \frac {1}{x^2\,\sqrt {a+b\,x}\,{\left (c+d\,x\right )}^{5/2}} \,d x \]

[In]

int(1/(x^2*(a + b*x)^(1/2)*(c + d*x)^(5/2)),x)

[Out]

int(1/(x^2*(a + b*x)^(1/2)*(c + d*x)^(5/2)), x)