3.5.78 \(\int \frac {1}{x^2 (a+b x)^2 (c+d x)^{5/2}} \, dx\) [478]

Optimal. Leaf size=277 \[ -\frac {d \left (6 b^2 c^2-6 a b c d+5 a^2 d^2\right )}{3 a^2 c^2 (b c-a d)^2 (c+d x)^{3/2}}-\frac {b (2 b c-a d)}{a^2 c (b c-a d) (a+b x) (c+d x)^{3/2}}-\frac {1}{a c x (a+b x) (c+d x)^{3/2}}-\frac {d (2 b c-a d) \left (b^2 c^2-a b c d+5 a^2 d^2\right )}{a^2 c^3 (b c-a d)^3 \sqrt {c+d x}}+\frac {(4 b c+5 a d) \tanh ^{-1}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{a^3 c^{7/2}}-\frac {b^{7/2} (4 b c-9 a d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{a^3 (b c-a d)^{7/2}} \]

[Out]

-1/3*d*(5*a^2*d^2-6*a*b*c*d+6*b^2*c^2)/a^2/c^2/(-a*d+b*c)^2/(d*x+c)^(3/2)-b*(-a*d+2*b*c)/a^2/c/(-a*d+b*c)/(b*x
+a)/(d*x+c)^(3/2)-1/a/c/x/(b*x+a)/(d*x+c)^(3/2)+(5*a*d+4*b*c)*arctanh((d*x+c)^(1/2)/c^(1/2))/a^3/c^(7/2)-b^(7/
2)*(-9*a*d+4*b*c)*arctanh(b^(1/2)*(d*x+c)^(1/2)/(-a*d+b*c)^(1/2))/a^3/(-a*d+b*c)^(7/2)-d*(-a*d+2*b*c)*(5*a^2*d
^2-a*b*c*d+b^2*c^2)/a^2/c^3/(-a*d+b*c)^3/(d*x+c)^(1/2)

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Rubi [A]
time = 0.30, antiderivative size = 277, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {105, 156, 157, 162, 65, 214} \begin {gather*} -\frac {b^{7/2} (4 b c-9 a d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{a^3 (b c-a d)^{7/2}}+\frac {(5 a d+4 b c) \tanh ^{-1}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{a^3 c^{7/2}}-\frac {d \left (5 a^2 d^2-6 a b c d+6 b^2 c^2\right )}{3 a^2 c^2 (c+d x)^{3/2} (b c-a d)^2}-\frac {d (2 b c-a d) \left (5 a^2 d^2-a b c d+b^2 c^2\right )}{a^2 c^3 \sqrt {c+d x} (b c-a d)^3}-\frac {b (2 b c-a d)}{a^2 c (a+b x) (c+d x)^{3/2} (b c-a d)}-\frac {1}{a c x (a+b x) (c+d x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/3*(d*(6*b^2*c^2 - 6*a*b*c*d + 5*a^2*d^2))/(a^2*c^2*(b*c - a*d)^2*(c + d*x)^(3/2)) - (b*(2*b*c - a*d))/(a^2*
c*(b*c - a*d)*(a + b*x)*(c + d*x)^(3/2)) - 1/(a*c*x*(a + b*x)*(c + d*x)^(3/2)) - (d*(2*b*c - a*d)*(b^2*c^2 - a
*b*c*d + 5*a^2*d^2))/(a^2*c^3*(b*c - a*d)^3*Sqrt[c + d*x]) + ((4*b*c + 5*a*d)*ArcTanh[Sqrt[c + d*x]/Sqrt[c]])/
(a^3*c^(7/2)) - (b^(7/2)*(4*b*c - 9*a*d)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[b*c - a*d]])/(a^3*(b*c - a*d)^(7
/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 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 156

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] && ILtQ[m, -1]

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 162

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

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*} \int \frac {1}{x^2 (a+b x)^2 (c+d x)^{5/2}} \, dx &=-\frac {1}{a c x (a+b x) (c+d x)^{3/2}}-\frac {\int \frac {\frac {1}{2} (4 b c+5 a d)+\frac {7 b d x}{2}}{x (a+b x)^2 (c+d x)^{5/2}} \, dx}{a c}\\ &=-\frac {b (2 b c-a d)}{a^2 c (b c-a d) (a+b x) (c+d x)^{3/2}}-\frac {1}{a c x (a+b x) (c+d x)^{3/2}}-\frac {\int \frac {\frac {1}{2} (b c-a d) (4 b c+5 a d)+\frac {5}{2} b d (2 b c-a d) x}{x (a+b x) (c+d x)^{5/2}} \, dx}{a^2 c (b c-a d)}\\ &=-\frac {d \left (6 b^2 c^2-6 a b c d+5 a^2 d^2\right )}{3 a^2 c^2 (b c-a d)^2 (c+d x)^{3/2}}-\frac {b (2 b c-a d)}{a^2 c (b c-a d) (a+b x) (c+d x)^{3/2}}-\frac {1}{a c x (a+b x) (c+d x)^{3/2}}+\frac {2 \int \frac {-\frac {3}{4} (b c-a d)^2 (4 b c+5 a d)-\frac {3}{4} b d \left (6 b^2 c^2-6 a b c d+5 a^2 d^2\right ) x}{x (a+b x) (c+d x)^{3/2}} \, dx}{3 a^2 c^2 (b c-a d)^2}\\ &=-\frac {d \left (6 b^2 c^2-6 a b c d+5 a^2 d^2\right )}{3 a^2 c^2 (b c-a d)^2 (c+d x)^{3/2}}-\frac {b (2 b c-a d)}{a^2 c (b c-a d) (a+b x) (c+d x)^{3/2}}-\frac {1}{a c x (a+b x) (c+d x)^{3/2}}-\frac {d (2 b c-a d) \left (b^2 c^2-a b c d+5 a^2 d^2\right )}{a^2 c^3 (b c-a d)^3 \sqrt {c+d x}}-\frac {4 \int \frac {\frac {3}{8} (b c-a d)^3 (4 b c+5 a d)+\frac {3}{8} b d (2 b c-a d) \left (b^2 c^2-a b c d+5 a^2 d^2\right ) x}{x (a+b x) \sqrt {c+d x}} \, dx}{3 a^2 c^3 (b c-a d)^3}\\ &=-\frac {d \left (6 b^2 c^2-6 a b c d+5 a^2 d^2\right )}{3 a^2 c^2 (b c-a d)^2 (c+d x)^{3/2}}-\frac {b (2 b c-a d)}{a^2 c (b c-a d) (a+b x) (c+d x)^{3/2}}-\frac {1}{a c x (a+b x) (c+d x)^{3/2}}-\frac {d (2 b c-a d) \left (b^2 c^2-a b c d+5 a^2 d^2\right )}{a^2 c^3 (b c-a d)^3 \sqrt {c+d x}}+\frac {\left (b^4 (4 b c-9 a d)\right ) \int \frac {1}{(a+b x) \sqrt {c+d x}} \, dx}{2 a^3 (b c-a d)^3}-\frac {(4 b c+5 a d) \int \frac {1}{x \sqrt {c+d x}} \, dx}{2 a^3 c^3}\\ &=-\frac {d \left (6 b^2 c^2-6 a b c d+5 a^2 d^2\right )}{3 a^2 c^2 (b c-a d)^2 (c+d x)^{3/2}}-\frac {b (2 b c-a d)}{a^2 c (b c-a d) (a+b x) (c+d x)^{3/2}}-\frac {1}{a c x (a+b x) (c+d x)^{3/2}}-\frac {d (2 b c-a d) \left (b^2 c^2-a b c d+5 a^2 d^2\right )}{a^2 c^3 (b c-a d)^3 \sqrt {c+d x}}+\frac {\left (b^4 (4 b c-9 a d)\right ) \text {Subst}\left (\int \frac {1}{a-\frac {b c}{d}+\frac {b x^2}{d}} \, dx,x,\sqrt {c+d x}\right )}{a^3 d (b c-a d)^3}-\frac {(4 b c+5 a d) \text {Subst}\left (\int \frac {1}{-\frac {c}{d}+\frac {x^2}{d}} \, dx,x,\sqrt {c+d x}\right )}{a^3 c^3 d}\\ &=-\frac {d \left (6 b^2 c^2-6 a b c d+5 a^2 d^2\right )}{3 a^2 c^2 (b c-a d)^2 (c+d x)^{3/2}}-\frac {b (2 b c-a d)}{a^2 c (b c-a d) (a+b x) (c+d x)^{3/2}}-\frac {1}{a c x (a+b x) (c+d x)^{3/2}}-\frac {d (2 b c-a d) \left (b^2 c^2-a b c d+5 a^2 d^2\right )}{a^2 c^3 (b c-a d)^3 \sqrt {c+d x}}+\frac {(4 b c+5 a d) \tanh ^{-1}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{a^3 c^{7/2}}-\frac {b^{7/2} (4 b c-9 a d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{a^3 (b c-a d)^{7/2}}\\ \end {align*}

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Mathematica [A]
time = 1.30, size = 272, normalized size = 0.98 \begin {gather*} \frac {\frac {a \left (-6 b^4 c^3 x (c+d x)^2-3 a b^3 c^2 (c-3 d x) (c+d x)^2+a^4 d^3 \left (3 c^2+20 c d x+15 d^2 x^2\right )+a^2 b^2 c d \left (9 c^3+9 c^2 d x-35 c d^2 x^2-33 d^3 x^3\right )+a^3 b d^2 \left (-9 c^3-41 c^2 d x-13 c d^2 x^2+15 d^3 x^3\right )\right )}{c^3 (b c-a d)^3 x (a+b x) (c+d x)^{3/2}}-\frac {3 b^{7/2} (4 b c-9 a d) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {-b c+a d}}\right )}{(-b c+a d)^{7/2}}+\frac {3 (4 b c+5 a d) \tanh ^{-1}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{c^{7/2}}}{3 a^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((a*(-6*b^4*c^3*x*(c + d*x)^2 - 3*a*b^3*c^2*(c - 3*d*x)*(c + d*x)^2 + a^4*d^3*(3*c^2 + 20*c*d*x + 15*d^2*x^2)
+ a^2*b^2*c*d*(9*c^3 + 9*c^2*d*x - 35*c*d^2*x^2 - 33*d^3*x^3) + a^3*b*d^2*(-9*c^3 - 41*c^2*d*x - 13*c*d^2*x^2
+ 15*d^3*x^3)))/(c^3*(b*c - a*d)^3*x*(a + b*x)*(c + d*x)^(3/2)) - (3*b^(7/2)*(4*b*c - 9*a*d)*ArcTan[(Sqrt[b]*S
qrt[c + d*x])/Sqrt[-(b*c) + a*d]])/(-(b*c) + a*d)^(7/2) + (3*(4*b*c + 5*a*d)*ArcTanh[Sqrt[c + d*x]/Sqrt[c]])/c
^(7/2))/(3*a^3)

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Maple [A]
time = 0.12, size = 204, normalized size = 0.74

method	result	size
derivativedivides	\(2 d^{3} \left (\frac {b^{4} \left (\frac {a d \sqrt {d x +c}}{2 b \left (d x +c \right )+2 a d -2 b c}+\frac {\left (9 a d -4 b c \right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{2 \sqrt {\left (a d -b c \right ) b}}\right )}{a^{3} d^{3} \left (a d -b c \right )^{3}}+\frac {-\frac {a \sqrt {d x +c}}{2 x}+\frac {\left (5 a d +4 b c \right ) \arctanh \left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )}{2 \sqrt {c}}}{a^{3} c^{3} d^{3}}-\frac {1}{3 c^{2} \left (a d -b c \right )^{2} \left (d x +c \right )^{\frac {3}{2}}}-\frac {2 a d -4 b c}{c^{3} \left (a d -b c \right )^{3} \sqrt {d x +c}}\right )\)	\(204\)
default	\(2 d^{3} \left (\frac {b^{4} \left (\frac {a d \sqrt {d x +c}}{2 b \left (d x +c \right )+2 a d -2 b c}+\frac {\left (9 a d -4 b c \right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{2 \sqrt {\left (a d -b c \right ) b}}\right )}{a^{3} d^{3} \left (a d -b c \right )^{3}}+\frac {-\frac {a \sqrt {d x +c}}{2 x}+\frac {\left (5 a d +4 b c \right ) \arctanh \left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )}{2 \sqrt {c}}}{a^{3} c^{3} d^{3}}-\frac {1}{3 c^{2} \left (a d -b c \right )^{2} \left (d x +c \right )^{\frac {3}{2}}}-\frac {2 a d -4 b c}{c^{3} \left (a d -b c \right )^{3} \sqrt {d x +c}}\right )\)	\(204\)
risch	\(-\frac {\sqrt {d x +c}}{c^{3} a^{2} x}-\frac {4 a \,d^{4}}{c^{3} \left (a d -b c \right )^{3} \sqrt {d x +c}}+\frac {8 d^{3} b}{c^{2} \left (a d -b c \right )^{3} \sqrt {d x +c}}-\frac {2 d^{3}}{3 c^{2} \left (a d -b c \right )^{2} \left (d x +c \right )^{\frac {3}{2}}}+\frac {d \,b^{4} \sqrt {d x +c}}{a^{2} \left (a d -b c \right )^{3} \left (b d x +a d \right )}+\frac {9 d \,b^{4} \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{a^{2} \left (a d -b c \right )^{3} \sqrt {\left (a d -b c \right ) b}}-\frac {4 c \,b^{5} \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{a^{3} \left (a d -b c \right )^{3} \sqrt {\left (a d -b c \right ) b}}+\frac {5 d \arctanh \left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )}{a^{2} c^{\frac {7}{2}}}+\frac {4 \arctanh \left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right ) b}{a^{3} c^{\frac {5}{2}}}\)	\(280\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*d^3*(b^4/a^3/d^3/(a*d-b*c)^3*(1/2*a*d*(d*x+c)^(1/2)/(b*(d*x+c)+a*d-b*c)+1/2*(9*a*d-4*b*c)/((a*d-b*c)*b)^(1/2
)*arctan(b*(d*x+c)^(1/2)/((a*d-b*c)*b)^(1/2)))+1/a^3/c^3/d^3*(-1/2*a*(d*x+c)^(1/2)/x+1/2*(5*a*d+4*b*c)/c^(1/2)
*arctanh((d*x+c)^(1/2)/c^(1/2)))-1/3/c^2/(a*d-b*c)^2/(d*x+c)^(3/2)-(2*a*d-4*b*c)/c^3/(a*d-b*c)^3/(d*x+c)^(1/2)
)

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 960 vs. \(2 (251) = 502\).
time = 3.28, size = 3872, normalized size = 13.98 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/6*(3*((4*b^5*c^5*d^2 - 9*a*b^4*c^4*d^3)*x^4 + (8*b^5*c^6*d - 14*a*b^4*c^5*d^2 - 9*a^2*b^3*c^4*d^3)*x^3 + (4
*b^5*c^7 - a*b^4*c^6*d - 18*a^2*b^3*c^5*d^2)*x^2 + (4*a*b^4*c^7 - 9*a^2*b^3*c^6*d)*x)*sqrt(b/(b*c - a*d))*log(
(b*d*x + 2*b*c - a*d - 2*(b*c - a*d)*sqrt(d*x + c)*sqrt(b/(b*c - a*d)))/(b*x + a)) + 3*((4*b^5*c^4*d^2 - 7*a*b
^4*c^3*d^3 - 3*a^2*b^3*c^2*d^4 + 11*a^3*b^2*c*d^5 - 5*a^4*b*d^6)*x^4 + (8*b^5*c^5*d - 10*a*b^4*c^4*d^2 - 13*a^
2*b^3*c^3*d^3 + 19*a^3*b^2*c^2*d^4 + a^4*b*c*d^5 - 5*a^5*d^6)*x^3 + (4*b^5*c^6 + a*b^4*c^5*d - 17*a^2*b^3*c^4*
d^2 + 5*a^3*b^2*c^3*d^3 + 17*a^4*b*c^2*d^4 - 10*a^5*c*d^5)*x^2 + (4*a*b^4*c^6 - 7*a^2*b^3*c^5*d - 3*a^3*b^2*c^
4*d^2 + 11*a^4*b*c^3*d^3 - 5*a^5*c^2*d^4)*x)*sqrt(c)*log((d*x + 2*sqrt(d*x + c)*sqrt(c) + 2*c)/x) - 2*(3*a^2*b
^3*c^6 - 9*a^3*b^2*c^5*d + 9*a^4*b*c^4*d^2 - 3*a^5*c^3*d^3 + 3*(2*a*b^4*c^4*d^2 - 3*a^2*b^3*c^3*d^3 + 11*a^3*b
^2*c^2*d^4 - 5*a^4*b*c*d^5)*x^3 + (12*a*b^4*c^5*d - 15*a^2*b^3*c^4*d^2 + 35*a^3*b^2*c^3*d^3 + 13*a^4*b*c^2*d^4
- 15*a^5*c*d^5)*x^2 + (6*a*b^4*c^6 - 3*a^2*b^3*c^5*d - 9*a^3*b^2*c^4*d^2 + 41*a^4*b*c^3*d^3 - 20*a^5*c^2*d^4)
*x)*sqrt(d*x + c))/((a^3*b^4*c^7*d^2 - 3*a^4*b^3*c^6*d^3 + 3*a^5*b^2*c^5*d^4 - a^6*b*c^4*d^5)*x^4 + (2*a^3*b^4
*c^8*d - 5*a^4*b^3*c^7*d^2 + 3*a^5*b^2*c^6*d^3 + a^6*b*c^5*d^4 - a^7*c^4*d^5)*x^3 + (a^3*b^4*c^9 - a^4*b^3*c^8
*d - 3*a^5*b^2*c^7*d^2 + 5*a^6*b*c^6*d^3 - 2*a^7*c^5*d^4)*x^2 + (a^4*b^3*c^9 - 3*a^5*b^2*c^8*d + 3*a^6*b*c^7*d
^2 - a^7*c^6*d^3)*x), -1/6*(6*((4*b^5*c^5*d^2 - 9*a*b^4*c^4*d^3)*x^4 + (8*b^5*c^6*d - 14*a*b^4*c^5*d^2 - 9*a^2
*b^3*c^4*d^3)*x^3 + (4*b^5*c^7 - a*b^4*c^6*d - 18*a^2*b^3*c^5*d^2)*x^2 + (4*a*b^4*c^7 - 9*a^2*b^3*c^6*d)*x)*sq
rt(-b/(b*c - a*d))*arctan(-(b*c - a*d)*sqrt(d*x + c)*sqrt(-b/(b*c - a*d))/(b*d*x + b*c)) - 3*((4*b^5*c^4*d^2 -
7*a*b^4*c^3*d^3 - 3*a^2*b^3*c^2*d^4 + 11*a^3*b^2*c*d^5 - 5*a^4*b*d^6)*x^4 + (8*b^5*c^5*d - 10*a*b^4*c^4*d^2 -
13*a^2*b^3*c^3*d^3 + 19*a^3*b^2*c^2*d^4 + a^4*b*c*d^5 - 5*a^5*d^6)*x^3 + (4*b^5*c^6 + a*b^4*c^5*d - 17*a^2*b^
3*c^4*d^2 + 5*a^3*b^2*c^3*d^3 + 17*a^4*b*c^2*d^4 - 10*a^5*c*d^5)*x^2 + (4*a*b^4*c^6 - 7*a^2*b^3*c^5*d - 3*a^3*
b^2*c^4*d^2 + 11*a^4*b*c^3*d^3 - 5*a^5*c^2*d^4)*x)*sqrt(c)*log((d*x + 2*sqrt(d*x + c)*sqrt(c) + 2*c)/x) + 2*(3
*a^2*b^3*c^6 - 9*a^3*b^2*c^5*d + 9*a^4*b*c^4*d^2 - 3*a^5*c^3*d^3 + 3*(2*a*b^4*c^4*d^2 - 3*a^2*b^3*c^3*d^3 + 11
*a^3*b^2*c^2*d^4 - 5*a^4*b*c*d^5)*x^3 + (12*a*b^4*c^5*d - 15*a^2*b^3*c^4*d^2 + 35*a^3*b^2*c^3*d^3 + 13*a^4*b*c
^2*d^4 - 15*a^5*c*d^5)*x^2 + (6*a*b^4*c^6 - 3*a^2*b^3*c^5*d - 9*a^3*b^2*c^4*d^2 + 41*a^4*b*c^3*d^3 - 20*a^5*c^
2*d^4)*x)*sqrt(d*x + c))/((a^3*b^4*c^7*d^2 - 3*a^4*b^3*c^6*d^3 + 3*a^5*b^2*c^5*d^4 - a^6*b*c^4*d^5)*x^4 + (2*a
^3*b^4*c^8*d - 5*a^4*b^3*c^7*d^2 + 3*a^5*b^2*c^6*d^3 + a^6*b*c^5*d^4 - a^7*c^4*d^5)*x^3 + (a^3*b^4*c^9 - a^4*b
^3*c^8*d - 3*a^5*b^2*c^7*d^2 + 5*a^6*b*c^6*d^3 - 2*a^7*c^5*d^4)*x^2 + (a^4*b^3*c^9 - 3*a^5*b^2*c^8*d + 3*a^6*b
*c^7*d^2 - a^7*c^6*d^3)*x), -1/6*(6*((4*b^5*c^4*d^2 - 7*a*b^4*c^3*d^3 - 3*a^2*b^3*c^2*d^4 + 11*a^3*b^2*c*d^5 -
5*a^4*b*d^6)*x^4 + (8*b^5*c^5*d - 10*a*b^4*c^4*d^2 - 13*a^2*b^3*c^3*d^3 + 19*a^3*b^2*c^2*d^4 + a^4*b*c*d^5 -
5*a^5*d^6)*x^3 + (4*b^5*c^6 + a*b^4*c^5*d - 17*a^2*b^3*c^4*d^2 + 5*a^3*b^2*c^3*d^3 + 17*a^4*b*c^2*d^4 - 10*a^5
*c*d^5)*x^2 + (4*a*b^4*c^6 - 7*a^2*b^3*c^5*d - 3*a^3*b^2*c^4*d^2 + 11*a^4*b*c^3*d^3 - 5*a^5*c^2*d^4)*x)*sqrt(-
c)*arctan(sqrt(d*x + c)*sqrt(-c)/c) - 3*((4*b^5*c^5*d^2 - 9*a*b^4*c^4*d^3)*x^4 + (8*b^5*c^6*d - 14*a*b^4*c^5*d
^2 - 9*a^2*b^3*c^4*d^3)*x^3 + (4*b^5*c^7 - a*b^4*c^6*d - 18*a^2*b^3*c^5*d^2)*x^2 + (4*a*b^4*c^7 - 9*a^2*b^3*c^
6*d)*x)*sqrt(b/(b*c - a*d))*log((b*d*x + 2*b*c - a*d - 2*(b*c - a*d)*sqrt(d*x + c)*sqrt(b/(b*c - a*d)))/(b*x +
a)) + 2*(3*a^2*b^3*c^6 - 9*a^3*b^2*c^5*d + 9*a^4*b*c^4*d^2 - 3*a^5*c^3*d^3 + 3*(2*a*b^4*c^4*d^2 - 3*a^2*b^3*c
^3*d^3 + 11*a^3*b^2*c^2*d^4 - 5*a^4*b*c*d^5)*x^3 + (12*a*b^4*c^5*d - 15*a^2*b^3*c^4*d^2 + 35*a^3*b^2*c^3*d^3 +
13*a^4*b*c^2*d^4 - 15*a^5*c*d^5)*x^2 + (6*a*b^4*c^6 - 3*a^2*b^3*c^5*d - 9*a^3*b^2*c^4*d^2 + 41*a^4*b*c^3*d^3
- 20*a^5*c^2*d^4)*x)*sqrt(d*x + c))/((a^3*b^4*c^7*d^2 - 3*a^4*b^3*c^6*d^3 + 3*a^5*b^2*c^5*d^4 - a^6*b*c^4*d^5)
*x^4 + (2*a^3*b^4*c^8*d - 5*a^4*b^3*c^7*d^2 + 3*a^5*b^2*c^6*d^3 + a^6*b*c^5*d^4 - a^7*c^4*d^5)*x^3 + (a^3*b^4*
c^9 - a^4*b^3*c^8*d - 3*a^5*b^2*c^7*d^2 + 5*a^6*b*c^6*d^3 - 2*a^7*c^5*d^4)*x^2 + (a^4*b^3*c^9 - 3*a^5*b^2*c^8*
d + 3*a^6*b*c^7*d^2 - a^7*c^6*d^3)*x), -1/3*(3*((4*b^5*c^5*d^2 - 9*a*b^4*c^4*d^3)*x^4 + (8*b^5*c^6*d - 14*a*b^
4*c^5*d^2 - 9*a^2*b^3*c^4*d^3)*x^3 + (4*b^5*c^7 - a*b^4*c^6*d - 18*a^2*b^3*c^5*d^2)*x^2 + (4*a*b^4*c^7 - 9*a^2
*b^3*c^6*d)*x)*sqrt(-b/(b*c - a*d))*arctan(-(b*c - a*d)*sqrt(d*x + c)*sqrt(-b/(b*c - a*d))/(b*d*x + b*c)) + 3*
((4*b^5*c^4*d^2 - 7*a*b^4*c^3*d^3 - 3*a^2*b^3*c^2*d^4 + 11*a^3*b^2*c*d^5 - 5*a^4*b*d^6)*x^4 + (8*b^5*c^5*d - 1
0*a*b^4*c^4*d^2 - 13*a^2*b^3*c^3*d^3 + 19*a^3*b^2*c^2*d^4 + a^4*b*c*d^5 - 5*a^5*d^6)*x^3 + (4*b^5*c^6 + a*b^4*
c^5*d - 17*a^2*b^3*c^4*d^2 + 5*a^3*b^2*c^3*d^3 + 17*a^4*b*c^2*d^4 - 10*a^5*c*d^5)*x^2 + (4*a*b^4*c^6 - 7*a^2*b
^3*c^5*d - 3*a^3*b^2*c^4*d^2 + 11*a^4*b*c^3*d^3...

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x^{2} \left (a + b x\right )^{2} \left (c + d x\right )^{\frac {5}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 0.49, size = 470, normalized size = 1.70 \begin {gather*} \frac {{\left (4 \, b^{5} c - 9 \, a b^{4} d\right )} \arctan \left (\frac {\sqrt {d x + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{{\left (a^{3} b^{3} c^{3} - 3 \, a^{4} b^{2} c^{2} d + 3 \, a^{5} b c d^{2} - a^{6} d^{3}\right )} \sqrt {-b^{2} c + a b d}} - \frac {2 \, {\left (d x + c\right )}^{\frac {3}{2}} b^{4} c^{3} d - 2 \, \sqrt {d x + c} b^{4} c^{4} d - 3 \, {\left (d x + c\right )}^{\frac {3}{2}} a b^{3} c^{2} d^{2} + 4 \, \sqrt {d x + c} a b^{3} c^{3} d^{2} + 3 \, {\left (d x + c\right )}^{\frac {3}{2}} a^{2} b^{2} c d^{3} - 6 \, \sqrt {d x + c} a^{2} b^{2} c^{2} d^{3} - {\left (d x + c\right )}^{\frac {3}{2}} a^{3} b d^{4} + 4 \, \sqrt {d x + c} a^{3} b c d^{4} - \sqrt {d x + c} a^{4} d^{5}}{{\left (a^{2} b^{3} c^{6} - 3 \, a^{3} b^{2} c^{5} d + 3 \, a^{4} b c^{4} d^{2} - a^{5} c^{3} d^{3}\right )} {\left ({\left (d x + c\right )}^{2} b - 2 \, {\left (d x + c\right )} b c + b c^{2} + {\left (d x + c\right )} a d - a c d\right )}} - \frac {2 \, {\left (12 \, {\left (d x + c\right )} b c d^{3} + b c^{2} d^{3} - 6 \, {\left (d x + c\right )} a d^{4} - a c d^{4}\right )}}{3 \, {\left (b^{3} c^{6} - 3 \, a b^{2} c^{5} d + 3 \, a^{2} b c^{4} d^{2} - a^{3} c^{3} d^{3}\right )} {\left (d x + c\right )}^{\frac {3}{2}}} - \frac {{\left (4 \, b c + 5 \, a d\right )} \arctan \left (\frac {\sqrt {d x + c}}{\sqrt {-c}}\right )}{a^{3} \sqrt {-c} c^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(4*b^5*c - 9*a*b^4*d)*arctan(sqrt(d*x + c)*b/sqrt(-b^2*c + a*b*d))/((a^3*b^3*c^3 - 3*a^4*b^2*c^2*d + 3*a^5*b*c
*d^2 - a^6*d^3)*sqrt(-b^2*c + a*b*d)) - (2*(d*x + c)^(3/2)*b^4*c^3*d - 2*sqrt(d*x + c)*b^4*c^4*d - 3*(d*x + c)
^(3/2)*a*b^3*c^2*d^2 + 4*sqrt(d*x + c)*a*b^3*c^3*d^2 + 3*(d*x + c)^(3/2)*a^2*b^2*c*d^3 - 6*sqrt(d*x + c)*a^2*b
^2*c^2*d^3 - (d*x + c)^(3/2)*a^3*b*d^4 + 4*sqrt(d*x + c)*a^3*b*c*d^4 - sqrt(d*x + c)*a^4*d^5)/((a^2*b^3*c^6 -
3*a^3*b^2*c^5*d + 3*a^4*b*c^4*d^2 - a^5*c^3*d^3)*((d*x + c)^2*b - 2*(d*x + c)*b*c + b*c^2 + (d*x + c)*a*d - a*
c*d)) - 2/3*(12*(d*x + c)*b*c*d^3 + b*c^2*d^3 - 6*(d*x + c)*a*d^4 - a*c*d^4)/((b^3*c^6 - 3*a*b^2*c^5*d + 3*a^2
*b*c^4*d^2 - a^3*c^3*d^3)*(d*x + c)^(3/2)) - (4*b*c + 5*a*d)*arctan(sqrt(d*x + c)/sqrt(-c))/(a^3*sqrt(-c)*c^3)

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Mupad [B]
time = 3.96, size = 2500, normalized size = 9.03 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((10*d^3*(a*d - 2*b*c)*(c + d*x))/(3*(b*c^2 - a*c*d)^2) - (2*d^3)/(3*(b*c^2 - a*c*d)) + (d*(c + d*x)^2*(15*a^4
*d^4 + 6*b^4*c^4 + 64*a^2*b^2*c^2*d^2 - 12*a*b^3*c^3*d - 58*a^3*b*c*d^3))/(3*a^2*(b*c^2 - a*c*d)^3) + (d*(a*d
- 2*b*c)*(c + d*x)^3*(b^3*c^2 + 5*a^2*b*d^2 - a*b^2*c*d))/(a^2*(b*c^2 - a*c*d)^3))/(b*(c + d*x)^(7/2) + (b*c^2
- a*c*d)*(c + d*x)^(3/2) + (a*d - 2*b*c)*(c + d*x)^(5/2)) - (atan((a^19*c^15*d^19*(c + d*x)^(1/2)*125i - a^18
*b*c^16*d^18*(c + d*x)^(1/2)*1700i + a^3*b^16*c^31*d^3*(c + d*x)^(1/2)*420i - a^4*b^15*c^30*d^4*(c + d*x)^(1/2
)*4515i + a^5*b^14*c^29*d^5*(c + d*x)^(1/2)*20916i - a^6*b^13*c^28*d^6*(c + d*x)^(1/2)*52836i + a^7*b^12*c^27*
d^7*(c + d*x)^(1/2)*71070i - a^8*b^11*c^26*d^8*(c + d*x)^(1/2)*19530i - a^9*b^10*c^25*d^9*(c + d*x)^(1/2)*1077
40i + a^10*b^9*c^24*d^10*(c + d*x)^(1/2)*212608i - a^11*b^8*c^23*d^11*(c + d*x)^(1/2)*184563i + a^12*b^7*c^22*
d^12*(c + d*x)^(1/2)*40965i + a^13*b^6*c^21*d^13*(c + d*x)^(1/2)*91560i - a^14*b^5*c^20*d^14*(c + d*x)^(1/2)*1
26720i + a^15*b^4*c^19*d^15*(c + d*x)^(1/2)*87276i - a^16*b^3*c^18*d^16*(c + d*x)^(1/2)*37776i + a^17*b^2*c^17
*d^17*(c + d*x)^(1/2)*10440i)/(c^7*(c^7)^(1/2)*(c^7*(c^7*(212608*a^10*b^9*d^10 - 107740*a^9*b^10*c*d^9 + 420*a
^3*b^16*c^7*d^3 - 4515*a^4*b^15*c^6*d^4 + 20916*a^5*b^14*c^5*d^5 - 52836*a^6*b^13*c^4*d^6 + 71070*a^7*b^12*c^3
*d^7 - 19530*a^8*b^11*c^2*d^8) + 10440*a^17*b^2*d^17 - 37776*a^16*b^3*c*d^16 - 184563*a^11*b^8*c^6*d^11 + 4096
5*a^12*b^7*c^5*d^12 + 91560*a^13*b^6*c^4*d^13 - 126720*a^14*b^5*c^3*d^14 + 87276*a^15*b^4*c^2*d^15) + 125*a^19
*c^5*d^19 - 1700*a^18*b*c^6*d^18)))*(5*a*d + 4*b*c)*1i)/(a^3*(c^7)^(1/2)) + (atan((((-b^7*(a*d - b*c)^7)^(1/2)
*(9*a*d - 4*b*c)*((c + d*x)^(1/2)*(64*a^6*b^20*c^26*d^2 - 832*a^7*b^19*c^25*d^3 + 4820*a^8*b^18*c^24*d^4 - 162
40*a^9*b^17*c^23*d^5 + 34490*a^10*b^16*c^22*d^6 - 45430*a^11*b^15*c^21*d^7 + 29414*a^12*b^14*c^20*d^8 + 10670*
a^13*b^13*c^19*d^9 - 39550*a^14*b^12*c^18*d^10 + 25730*a^15*b^11*c^17*d^11 + 19048*a^16*b^10*c^16*d^12 - 53852
*a^17*b^9*c^15*d^13 + 55510*a^18*b^8*c^14*d^14 - 35210*a^19*b^7*c^13*d^15 + 14830*a^20*b^6*c^12*d^16 - 4082*a^
21*b^5*c^11*d^17 + 670*a^22*b^4*c^10*d^18 - 50*a^23*b^3*c^9*d^19) + ((-b^7*(a*d - b*c)^7)^(1/2)*(9*a*d - 4*b*c
)*(8*a^10*b^18*c^28*d^3 - 112*a^11*b^17*c^27*d^4 + 664*a^12*b^16*c^26*d^5 - 2080*a^13*b^15*c^25*d^6 + 2996*a^1
4*b^14*c^24*d^7 + 2528*a^15*b^13*c^23*d^8 - 23056*a^16*b^12*c^22*d^9 + 59312*a^17*b^11*c^21*d^10 - 95700*a^18*
b^10*c^20*d^11 + 109648*a^19*b^9*c^19*d^12 - 92840*a^20*b^8*c^18*d^13 + 58688*a^21*b^7*c^17*d^14 - 27476*a^22*
b^6*c^16*d^15 + 9280*a^23*b^5*c^15*d^16 - 2144*a^24*b^4*c^14*d^17 + 304*a^25*b^3*c^13*d^18 - 20*a^26*b^2*c^12*
d^19 - ((-b^7*(a*d - b*c)^7)^(1/2)*(9*a*d - 4*b*c)*(c + d*x)^(1/2)*(16*a^12*b^18*c^31*d^2 - 248*a^13*b^17*c^30
*d^3 + 1800*a^14*b^16*c^29*d^4 - 8120*a^15*b^15*c^28*d^5 + 25480*a^16*b^14*c^27*d^6 - 58968*a^17*b^13*c^26*d^7
+ 104104*a^18*b^12*c^25*d^8 - 143000*a^19*b^11*c^24*d^9 + 154440*a^20*b^10*c^23*d^10 - 131560*a^21*b^9*c^22*d
^11 + 88088*a^22*b^8*c^21*d^12 - 45864*a^23*b^7*c^20*d^13 + 18200*a^24*b^6*c^19*d^14 - 5320*a^25*b^5*c^18*d^15
+ 1080*a^26*b^4*c^17*d^16 - 136*a^27*b^3*c^16*d^17 + 8*a^28*b^2*c^15*d^18))/(2*(a^10*d^7 - a^3*b^7*c^7 + 7*a^
4*b^6*c^6*d - 21*a^5*b^5*c^5*d^2 + 35*a^6*b^4*c^4*d^3 - 35*a^7*b^3*c^3*d^4 + 21*a^8*b^2*c^2*d^5 - 7*a^9*b*c*d^
6))))/(2*(a^10*d^7 - a^3*b^7*c^7 + 7*a^4*b^6*c^6*d - 21*a^5*b^5*c^5*d^2 + 35*a^6*b^4*c^4*d^3 - 35*a^7*b^3*c^3*
d^4 + 21*a^8*b^2*c^2*d^5 - 7*a^9*b*c*d^6)))*1i)/(2*(a^10*d^7 - a^3*b^7*c^7 + 7*a^4*b^6*c^6*d - 21*a^5*b^5*c^5*
d^2 + 35*a^6*b^4*c^4*d^3 - 35*a^7*b^3*c^3*d^4 + 21*a^8*b^2*c^2*d^5 - 7*a^9*b*c*d^6)) + ((-b^7*(a*d - b*c)^7)^(
1/2)*(9*a*d - 4*b*c)*((c + d*x)^(1/2)*(64*a^6*b^20*c^26*d^2 - 832*a^7*b^19*c^25*d^3 + 4820*a^8*b^18*c^24*d^4 -
16240*a^9*b^17*c^23*d^5 + 34490*a^10*b^16*c^22*d^6 - 45430*a^11*b^15*c^21*d^7 + 29414*a^12*b^14*c^20*d^8 + 10
670*a^13*b^13*c^19*d^9 - 39550*a^14*b^12*c^18*d^10 + 25730*a^15*b^11*c^17*d^11 + 19048*a^16*b^10*c^16*d^12 - 5
3852*a^17*b^9*c^15*d^13 + 55510*a^18*b^8*c^14*d^14 - 35210*a^19*b^7*c^13*d^15 + 14830*a^20*b^6*c^12*d^16 - 408
2*a^21*b^5*c^11*d^17 + 670*a^22*b^4*c^10*d^18 - 50*a^23*b^3*c^9*d^19) - ((-b^7*(a*d - b*c)^7)^(1/2)*(9*a*d - 4
*b*c)*(8*a^10*b^18*c^28*d^3 - 112*a^11*b^17*c^27*d^4 + 664*a^12*b^16*c^26*d^5 - 2080*a^13*b^15*c^25*d^6 + 2996
*a^14*b^14*c^24*d^7 + 2528*a^15*b^13*c^23*d^8 - 23056*a^16*b^12*c^22*d^9 + 59312*a^17*b^11*c^21*d^10 - 95700*a
^18*b^10*c^20*d^11 + 109648*a^19*b^9*c^19*d^12 - 92840*a^20*b^8*c^18*d^13 + 58688*a^21*b^7*c^17*d^14 - 27476*a
^22*b^6*c^16*d^15 + 9280*a^23*b^5*c^15*d^16 - 2144*a^24*b^4*c^14*d^17 + 304*a^25*b^3*c^13*d^18 - 20*a^26*b^2*c
^12*d^19 + ((-b^7*(a*d - b*c)^7)^(1/2)*(9*a*d - 4*b*c)*(c + d*x)^(1/2)*(16*a^12*b^18*c^31*d^2 - 248*a^13*b^17*
c^30*d^3 + 1800*a^14*b^16*c^29*d^4 - 8120*a^15*b^15*c^28*d^5 + 25480*a^16*b^14*c^27*d^6 - 58968*a^17*b^13*c^26
*d^7 + 104104*a^18*b^12*c^25*d^8 - 143000*a^19*b^11*c^24*d^9 + 154440*a^20*b^10*c^23*d^10 - 131560*a^21*b^9*c^
22*d^11 + 88088*a^22*b^8*c^21*d^12 - 45864*a^23...

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