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

Optimal. Leaf size=252 \[ -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{a^{5/2} c^{5/2}}+\frac {2 d \sqrt {a+b x} (a d+b c) \left (3 a^2 d^2-14 a b c d+3 b^2 c^2\right )}{3 a^2 c^2 \sqrt {c+d x} (b c-a d)^4}+\frac {2 d \sqrt {a+b x} \left (-a^2 d^2-10 a b c d+3 b^2 c^2\right )}{3 a^2 c (c+d x)^{3/2} (b c-a d)^3}+\frac {2 b (b c-3 a d)}{a^2 \sqrt {a+b x} (c+d x)^{3/2} (b c-a d)^2}+\frac {2 b}{3 a (a+b x)^{3/2} (c+d x)^{3/2} (b c-a d)} \]

[Out]

2/3*b/a/(-a*d+b*c)/(b*x+a)^(3/2)/(d*x+c)^(3/2)-2*arctanh(c^(1/2)*(b*x+a)^(1/2)/a^(1/2)/(d*x+c)^(1/2))/a^(5/2)/
c^(5/2)+2*b*(-3*a*d+b*c)/a^2/(-a*d+b*c)^2/(d*x+c)^(3/2)/(b*x+a)^(1/2)+2/3*d*(-a^2*d^2-10*a*b*c*d+3*b^2*c^2)*(b
*x+a)^(1/2)/a^2/c/(-a*d+b*c)^3/(d*x+c)^(3/2)+2/3*d*(a*d+b*c)*(3*a^2*d^2-14*a*b*c*d+3*b^2*c^2)*(b*x+a)^(1/2)/a^
2/c^2/(-a*d+b*c)^4/(d*x+c)^(1/2)

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Rubi [A]  time = 0.26, antiderivative size = 252, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {104, 152, 12, 93, 208} \[ \frac {2 d \sqrt {a+b x} (a d+b c) \left (3 a^2 d^2-14 a b c d+3 b^2 c^2\right )}{3 a^2 c^2 \sqrt {c+d x} (b c-a d)^4}+\frac {2 d \sqrt {a+b x} \left (-a^2 d^2-10 a b c d+3 b^2 c^2\right )}{3 a^2 c (c+d x)^{3/2} (b c-a d)^3}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{a^{5/2} c^{5/2}}+\frac {2 b (b c-3 a d)}{a^2 \sqrt {a+b x} (c+d x)^{3/2} (b c-a d)^2}+\frac {2 b}{3 a (a+b x)^{3/2} (c+d x)^{3/2} (b c-a d)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*b)/(3*a*(b*c - a*d)*(a + b*x)^(3/2)*(c + d*x)^(3/2)) + (2*b*(b*c - 3*a*d))/(a^2*(b*c - a*d)^2*Sqrt[a + b*x]
*(c + d*x)^(3/2)) + (2*d*(3*b^2*c^2 - 10*a*b*c*d - a^2*d^2)*Sqrt[a + b*x])/(3*a^2*c*(b*c - a*d)^3*(c + d*x)^(3
/2)) + (2*d*(b*c + a*d)*(3*b^2*c^2 - 14*a*b*c*d + 3*a^2*d^2)*Sqrt[a + b*x])/(3*a^2*c^2*(b*c - a*d)^4*Sqrt[c +
d*x]) - (2*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(a^(5/2)*c^(5/2))

Rule 12

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

Rule 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 104

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] && LtQ[m, -1] &&
IntegersQ[2*m, 2*n, 2*p]

Rule 152

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 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 {1}{x (a+b x)^{5/2} (c+d x)^{5/2}} \, dx &=\frac {2 b}{3 a (b c-a d) (a+b x)^{3/2} (c+d x)^{3/2}}+\frac {2 \int \frac {\frac {3}{2} (b c-a d)+3 b d x}{x (a+b x)^{3/2} (c+d x)^{5/2}} \, dx}{3 a (b c-a d)}\\ &=\frac {2 b}{3 a (b c-a d) (a+b x)^{3/2} (c+d x)^{3/2}}+\frac {2 b (b c-3 a d)}{a^2 (b c-a d)^2 \sqrt {a+b x} (c+d x)^{3/2}}+\frac {4 \int \frac {\frac {3}{4} (b c-a d)^2+3 b d (b c-3 a d) x}{x \sqrt {a+b x} (c+d x)^{5/2}} \, dx}{3 a^2 (b c-a d)^2}\\ &=\frac {2 b}{3 a (b c-a d) (a+b x)^{3/2} (c+d x)^{3/2}}+\frac {2 b (b c-3 a d)}{a^2 (b c-a d)^2 \sqrt {a+b x} (c+d x)^{3/2}}+\frac {2 d \left (3 b^2 c^2-10 a b c d-a^2 d^2\right ) \sqrt {a+b x}}{3 a^2 c (b c-a d)^3 (c+d x)^{3/2}}-\frac {8 \int \frac {-\frac {9}{8} (b c-a d)^3-\frac {3}{4} b d \left (3 b^2 c^2-10 a b c d-a^2 d^2\right ) x}{x \sqrt {a+b x} (c+d x)^{3/2}} \, dx}{9 a^2 c (b c-a d)^3}\\ &=\frac {2 b}{3 a (b c-a d) (a+b x)^{3/2} (c+d x)^{3/2}}+\frac {2 b (b c-3 a d)}{a^2 (b c-a d)^2 \sqrt {a+b x} (c+d x)^{3/2}}+\frac {2 d \left (3 b^2 c^2-10 a b c d-a^2 d^2\right ) \sqrt {a+b x}}{3 a^2 c (b c-a d)^3 (c+d x)^{3/2}}+\frac {2 d (b c+a d) \left (3 b^2 c^2-14 a b c d+3 a^2 d^2\right ) \sqrt {a+b x}}{3 a^2 c^2 (b c-a d)^4 \sqrt {c+d x}}+\frac {16 \int \frac {9 (b c-a d)^4}{16 x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{9 a^2 c^2 (b c-a d)^4}\\ &=\frac {2 b}{3 a (b c-a d) (a+b x)^{3/2} (c+d x)^{3/2}}+\frac {2 b (b c-3 a d)}{a^2 (b c-a d)^2 \sqrt {a+b x} (c+d x)^{3/2}}+\frac {2 d \left (3 b^2 c^2-10 a b c d-a^2 d^2\right ) \sqrt {a+b x}}{3 a^2 c (b c-a d)^3 (c+d x)^{3/2}}+\frac {2 d (b c+a d) \left (3 b^2 c^2-14 a b c d+3 a^2 d^2\right ) \sqrt {a+b x}}{3 a^2 c^2 (b c-a d)^4 \sqrt {c+d x}}+\frac {\int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{a^2 c^2}\\ &=\frac {2 b}{3 a (b c-a d) (a+b x)^{3/2} (c+d x)^{3/2}}+\frac {2 b (b c-3 a d)}{a^2 (b c-a d)^2 \sqrt {a+b x} (c+d x)^{3/2}}+\frac {2 d \left (3 b^2 c^2-10 a b c d-a^2 d^2\right ) \sqrt {a+b x}}{3 a^2 c (b c-a d)^3 (c+d x)^{3/2}}+\frac {2 d (b c+a d) \left (3 b^2 c^2-14 a b c d+3 a^2 d^2\right ) \sqrt {a+b x}}{3 a^2 c^2 (b c-a d)^4 \sqrt {c+d x}}+\frac {2 \operatorname {Subst}\left (\int \frac {1}{-a+c x^2} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )}{a^2 c^2}\\ &=\frac {2 b}{3 a (b c-a d) (a+b x)^{3/2} (c+d x)^{3/2}}+\frac {2 b (b c-3 a d)}{a^2 (b c-a d)^2 \sqrt {a+b x} (c+d x)^{3/2}}+\frac {2 d \left (3 b^2 c^2-10 a b c d-a^2 d^2\right ) \sqrt {a+b x}}{3 a^2 c (b c-a d)^3 (c+d x)^{3/2}}+\frac {2 d (b c+a d) \left (3 b^2 c^2-14 a b c d+3 a^2 d^2\right ) \sqrt {a+b x}}{3 a^2 c^2 (b c-a d)^4 \sqrt {c+d x}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{a^{5/2} c^{5/2}}\\ \end {align*}

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Mathematica [A]  time = 0.58, size = 265, normalized size = 1.05 \[ \frac {2 \left (-\frac {3 d \sqrt {a+b x} \left (a^2 d^2+10 a b c d-3 b^2 c^2\right )}{a c (b c-a d)^2}-\frac {(c+d x) \left (9 \sqrt {c+d x} (b c-a d)^4 \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )-3 \sqrt {a} \sqrt {c} d \sqrt {a+b x} \left (3 a^3 d^3-11 a^2 b c d^2-11 a b^2 c^2 d+3 b^3 c^3\right )\right )}{a^{3/2} c^{5/2} (b c-a d)^3}-\frac {18 b d}{\sqrt {a+b x} (b c-a d)}+\frac {9 b}{a \sqrt {a+b x}}+\frac {3 b}{(a+b x)^{3/2}}\right )}{9 a (c+d x)^{3/2} (b c-a d)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*((3*b)/(a + b*x)^(3/2) + (9*b)/(a*Sqrt[a + b*x]) - (18*b*d)/((b*c - a*d)*Sqrt[a + b*x]) - (3*d*(-3*b^2*c^2
+ 10*a*b*c*d + a^2*d^2)*Sqrt[a + b*x])/(a*c*(b*c - a*d)^2) - ((c + d*x)*(-3*Sqrt[a]*Sqrt[c]*d*(3*b^3*c^3 - 11*
a*b^2*c^2*d - 11*a^2*b*c*d^2 + 3*a^3*d^3)*Sqrt[a + b*x] + 9*(b*c - a*d)^4*Sqrt[c + d*x]*ArcTanh[(Sqrt[c]*Sqrt[
a + b*x])/(Sqrt[a]*Sqrt[c + d*x])]))/(a^(3/2)*c^(5/2)*(b*c - a*d)^3)))/(9*a*(b*c - a*d)*(c + d*x)^(3/2))

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fricas [B]  time = 9.18, size = 2078, normalized size = 8.25 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/6*(3*(a^2*b^4*c^6 - 4*a^3*b^3*c^5*d + 6*a^4*b^2*c^4*d^2 - 4*a^5*b*c^3*d^3 + a^6*c^2*d^4 + (b^6*c^4*d^2 - 4*
a*b^5*c^3*d^3 + 6*a^2*b^4*c^2*d^4 - 4*a^3*b^3*c*d^5 + a^4*b^2*d^6)*x^4 + 2*(b^6*c^5*d - 3*a*b^5*c^4*d^2 + 2*a^
2*b^4*c^3*d^3 + 2*a^3*b^3*c^2*d^4 - 3*a^4*b^2*c*d^5 + a^5*b*d^6)*x^3 + (b^6*c^6 - 9*a^2*b^4*c^4*d^2 + 16*a^3*b
^3*c^3*d^3 - 9*a^4*b^2*c^2*d^4 + a^6*d^6)*x^2 + 2*(a*b^5*c^6 - 3*a^2*b^4*c^5*d + 2*a^3*b^3*c^4*d^2 + 2*a^4*b^2
*c^3*d^3 - 3*a^5*b*c^2*d^4 + a^6*c*d^5)*x)*sqrt(a*c)*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*(4*a^2*b^4*c
^6 - 12*a^3*b^3*c^5*d - 12*a^5*b*c^3*d^3 + 4*a^6*c^2*d^4 + (3*a*b^5*c^4*d^2 - 11*a^2*b^4*c^3*d^3 - 11*a^3*b^3*
c^2*d^4 + 3*a^4*b^2*c*d^5)*x^3 + 6*(a*b^5*c^5*d - 3*a^2*b^4*c^4*d^2 - 4*a^3*b^3*c^3*d^3 - 3*a^4*b^2*c^2*d^4 +
a^5*b*c*d^5)*x^2 + 3*(a*b^5*c^6 - a^2*b^4*c^5*d - 8*a^3*b^3*c^4*d^2 - 8*a^4*b^2*c^3*d^3 - a^5*b*c^2*d^4 + a^6*
c*d^5)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(a^5*b^4*c^9 - 4*a^6*b^3*c^8*d + 6*a^7*b^2*c^7*d^2 - 4*a^8*b*c^6*d^3 +
a^9*c^5*d^4 + (a^3*b^6*c^7*d^2 - 4*a^4*b^5*c^6*d^3 + 6*a^5*b^4*c^5*d^4 - 4*a^6*b^3*c^4*d^5 + a^7*b^2*c^3*d^6)*
x^4 + 2*(a^3*b^6*c^8*d - 3*a^4*b^5*c^7*d^2 + 2*a^5*b^4*c^6*d^3 + 2*a^6*b^3*c^5*d^4 - 3*a^7*b^2*c^4*d^5 + a^8*b
*c^3*d^6)*x^3 + (a^3*b^6*c^9 - 9*a^5*b^4*c^7*d^2 + 16*a^6*b^3*c^6*d^3 - 9*a^7*b^2*c^5*d^4 + a^9*c^3*d^6)*x^2 +
2*(a^4*b^5*c^9 - 3*a^5*b^4*c^8*d + 2*a^6*b^3*c^7*d^2 + 2*a^7*b^2*c^6*d^3 - 3*a^8*b*c^5*d^4 + a^9*c^4*d^5)*x),
1/3*(3*(a^2*b^4*c^6 - 4*a^3*b^3*c^5*d + 6*a^4*b^2*c^4*d^2 - 4*a^5*b*c^3*d^3 + a^6*c^2*d^4 + (b^6*c^4*d^2 - 4*
a*b^5*c^3*d^3 + 6*a^2*b^4*c^2*d^4 - 4*a^3*b^3*c*d^5 + a^4*b^2*d^6)*x^4 + 2*(b^6*c^5*d - 3*a*b^5*c^4*d^2 + 2*a^
2*b^4*c^3*d^3 + 2*a^3*b^3*c^2*d^4 - 3*a^4*b^2*c*d^5 + a^5*b*d^6)*x^3 + (b^6*c^6 - 9*a^2*b^4*c^4*d^2 + 16*a^3*b
^3*c^3*d^3 - 9*a^4*b^2*c^2*d^4 + a^6*d^6)*x^2 + 2*(a*b^5*c^6 - 3*a^2*b^4*c^5*d + 2*a^3*b^3*c^4*d^2 + 2*a^4*b^2
*c^3*d^3 - 3*a^5*b*c^2*d^4 + a^6*c*d^5)*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*(4*a^2*b^4*c^6 - 12*a^3*b^3*c^5*d - 12*
a^5*b*c^3*d^3 + 4*a^6*c^2*d^4 + (3*a*b^5*c^4*d^2 - 11*a^2*b^4*c^3*d^3 - 11*a^3*b^3*c^2*d^4 + 3*a^4*b^2*c*d^5)*
x^3 + 6*(a*b^5*c^5*d - 3*a^2*b^4*c^4*d^2 - 4*a^3*b^3*c^3*d^3 - 3*a^4*b^2*c^2*d^4 + a^5*b*c*d^5)*x^2 + 3*(a*b^5
*c^6 - a^2*b^4*c^5*d - 8*a^3*b^3*c^4*d^2 - 8*a^4*b^2*c^3*d^3 - a^5*b*c^2*d^4 + a^6*c*d^5)*x)*sqrt(b*x + a)*sqr
t(d*x + c))/(a^5*b^4*c^9 - 4*a^6*b^3*c^8*d + 6*a^7*b^2*c^7*d^2 - 4*a^8*b*c^6*d^3 + a^9*c^5*d^4 + (a^3*b^6*c^7*
d^2 - 4*a^4*b^5*c^6*d^3 + 6*a^5*b^4*c^5*d^4 - 4*a^6*b^3*c^4*d^5 + a^7*b^2*c^3*d^6)*x^4 + 2*(a^3*b^6*c^8*d - 3*
a^4*b^5*c^7*d^2 + 2*a^5*b^4*c^6*d^3 + 2*a^6*b^3*c^5*d^4 - 3*a^7*b^2*c^4*d^5 + a^8*b*c^3*d^6)*x^3 + (a^3*b^6*c^
9 - 9*a^5*b^4*c^7*d^2 + 16*a^6*b^3*c^6*d^3 - 9*a^7*b^2*c^5*d^4 + a^9*c^3*d^6)*x^2 + 2*(a^4*b^5*c^9 - 3*a^5*b^4
*c^8*d + 2*a^6*b^3*c^7*d^2 + 2*a^7*b^2*c^6*d^3 - 3*a^8*b*c^5*d^4 + a^9*c^4*d^5)*x)]

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giac [B]  time = 10.68, size = 942, normalized size = 3.74 \[ -\frac {2 \, \sqrt {b x + a} {\left (\frac {{\left (11 \, b^{7} c^{6} d^{5} {\left | b \right |} - 36 \, a b^{6} c^{5} d^{6} {\left | b \right |} + 42 \, a^{2} b^{5} c^{4} d^{7} {\left | b \right |} - 20 \, a^{3} b^{4} c^{3} d^{8} {\left | b \right |} + 3 \, a^{4} b^{3} c^{2} d^{9} {\left | b \right |}\right )} {\left (b x + a\right )}}{b^{9} c^{11} d - 7 \, a b^{8} c^{10} d^{2} + 21 \, a^{2} b^{7} c^{9} d^{3} - 35 \, a^{3} b^{6} c^{8} d^{4} + 35 \, a^{4} b^{5} c^{7} d^{5} - 21 \, a^{5} b^{4} c^{6} d^{6} + 7 \, a^{6} b^{3} c^{5} d^{7} - a^{7} b^{2} c^{4} d^{8}} + \frac {3 \, {\left (4 \, b^{8} c^{7} d^{4} {\left | b \right |} - 17 \, a b^{7} c^{6} d^{5} {\left | b \right |} + 28 \, a^{2} b^{6} c^{5} d^{6} {\left | b \right |} - 22 \, a^{3} b^{5} c^{4} d^{7} {\left | b \right |} + 8 \, a^{4} b^{4} c^{3} d^{8} {\left | b \right |} - a^{5} b^{3} c^{2} d^{9} {\left | b \right |}\right )}}{b^{9} c^{11} d - 7 \, a b^{8} c^{10} d^{2} + 21 \, a^{2} b^{7} c^{9} d^{3} - 35 \, a^{3} b^{6} c^{8} d^{4} + 35 \, a^{4} b^{5} c^{7} d^{5} - 21 \, a^{5} b^{4} c^{6} d^{6} + 7 \, a^{6} b^{3} c^{5} d^{7} - a^{7} b^{2} c^{4} d^{8}}\right )}}{3 \, {\left (b^{2} c + {\left (b x + a\right )} b d - a b d\right )}^{\frac {3}{2}}} + \frac {4 \, {\left (3 \, \sqrt {b d} b^{9} c^{3} - 17 \, \sqrt {b d} a b^{8} c^{2} d + 25 \, \sqrt {b d} a^{2} b^{7} c d^{2} - 11 \, \sqrt {b d} a^{3} b^{6} d^{3} - 6 \, \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^{7} c^{2} + 30 \, \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^{6} c d - 24 \, \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^{2} b^{5} d^{2} + 3 \, \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 )}^{4} b^{5} c - 9 \, \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 )}^{4} a b^{4} d\right )}}{3 \, {\left (a^{2} b^{3} c^{3} {\left | b \right |} - 3 \, a^{3} b^{2} c^{2} d {\left | b \right |} + 3 \, a^{4} b c d^{2} {\left | b \right |} - a^{5} d^{3} {\left | b \right |}\right )} {\left (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}\right )}^{3}} - \frac {2 \, \sqrt {b d} b \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^{2} c^{2} {\left | b \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*sqrt(b*x + a)*((11*b^7*c^6*d^5*abs(b) - 36*a*b^6*c^5*d^6*abs(b) + 42*a^2*b^5*c^4*d^7*abs(b) - 20*a^3*b^4*
c^3*d^8*abs(b) + 3*a^4*b^3*c^2*d^9*abs(b))*(b*x + a)/(b^9*c^11*d - 7*a*b^8*c^10*d^2 + 21*a^2*b^7*c^9*d^3 - 35*
a^3*b^6*c^8*d^4 + 35*a^4*b^5*c^7*d^5 - 21*a^5*b^4*c^6*d^6 + 7*a^6*b^3*c^5*d^7 - a^7*b^2*c^4*d^8) + 3*(4*b^8*c^
7*d^4*abs(b) - 17*a*b^7*c^6*d^5*abs(b) + 28*a^2*b^6*c^5*d^6*abs(b) - 22*a^3*b^5*c^4*d^7*abs(b) + 8*a^4*b^4*c^3
*d^8*abs(b) - a^5*b^3*c^2*d^9*abs(b))/(b^9*c^11*d - 7*a*b^8*c^10*d^2 + 21*a^2*b^7*c^9*d^3 - 35*a^3*b^6*c^8*d^4
+ 35*a^4*b^5*c^7*d^5 - 21*a^5*b^4*c^6*d^6 + 7*a^6*b^3*c^5*d^7 - a^7*b^2*c^4*d^8))/(b^2*c + (b*x + a)*b*d - a*
b*d)^(3/2) + 4/3*(3*sqrt(b*d)*b^9*c^3 - 17*sqrt(b*d)*a*b^8*c^2*d + 25*sqrt(b*d)*a^2*b^7*c*d^2 - 11*sqrt(b*d)*a
^3*b^6*d^3 - 6*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*b^7*c^2 + 30*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^6*c*d - 24*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^5*d^2 + 3*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b
^2*c + (b*x + a)*b*d - a*b*d))^4*b^5*c - 9*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^4*d)/((a^2*b^3*c^3*abs(b) - 3*a^3*b^2*c^2*d*abs(b) + 3*a^4*b*c*d^2*abs(b) - a^5*d^3*abs(b))*(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)^3) - 2*sqrt(b*d)*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^2*c^2*abs(b))

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maple [B]  time = 0.05, size = 2033, normalized size = 8.07 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3/c^2/a^2*(12*ln((a*d*x+b*c*x+2*a*c+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2))/x)*x*a^3*b^3*c^4*d^2-8*((b*x+a)*
(d*x+c))^(1/2)*(a*c)^(1/2)*a^5*c*d^4+3*ln((a*d*x+b*c*x+2*a*c+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2))/x)*x^2*a^6
*d^6+3*ln((a*d*x+b*c*x+2*a*c+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2))/x)*x^2*b^6*c^6+3*ln((a*d*x+b*c*x+2*a*c+2*(
a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2))/x)*a^6*c^2*d^4+3*ln((a*d*x+b*c*x+2*a*c+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/
2))/x)*a^2*b^4*c^6+6*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)*x*a^4*b*c*d^4+6*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)*x
*a*b^4*c^4*d+22*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)*x^3*a^2*b^3*c*d^4+22*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)*x
^3*a*b^4*c^2*d^3+36*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)*x^2*a^3*b^2*c*d^4+48*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/
2)*x^2*a^2*b^3*c^2*d^3+48*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)*x*a^3*b^2*c^2*d^3+48*((b*x+a)*(d*x+c))^(1/2)*(a*
c)^(1/2)*x*a^2*b^3*c^3*d^2+36*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)*x^2*a*b^4*c^3*d^2-18*ln((a*d*x+b*c*x+2*a*c+2
*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2))/x)*x*a^2*b^4*c^5*d-6*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)*x^3*a^3*b^2*d^5
-6*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)*x^3*b^5*c^3*d^2-12*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)*x^2*a^4*b*d^5-12
*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)*x^2*b^5*c^4*d+24*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)*a^4*b*c^2*d^3+24*((b
*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)*a^2*b^3*c^4*d-12*ln((a*d*x+b*c*x+2*a*c+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2))
/x)*x^4*a^3*b^3*c*d^5+18*ln((a*d*x+b*c*x+2*a*c+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2))/x)*x^4*a^2*b^4*c^2*d^4-1
2*ln((a*d*x+b*c*x+2*a*c+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2))/x)*x^4*a*b^5*c^3*d^3-18*ln((a*d*x+b*c*x+2*a*c+2
*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2))/x)*x^3*a^4*b^2*c*d^5+12*ln((a*d*x+b*c*x+2*a*c+2*(a*c)^(1/2)*((b*x+a)*(d*
x+c))^(1/2))/x)*x^3*a^3*b^3*c^2*d^4+12*ln((a*d*x+b*c*x+2*a*c+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2))/x)*x^3*a^2
*b^4*c^3*d^3-18*ln((a*d*x+b*c*x+2*a*c+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2))/x)*x^3*a*b^5*c^4*d^2-27*ln((a*d*x
+b*c*x+2*a*c+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2))/x)*x^2*a^4*b^2*c^2*d^4+48*ln((a*d*x+b*c*x+2*a*c+2*(a*c)^(1
/2)*((b*x+a)*(d*x+c))^(1/2))/x)*x^2*a^3*b^3*c^3*d^3-27*ln((a*d*x+b*c*x+2*a*c+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(
1/2))/x)*x^2*a^2*b^4*c^4*d^2-18*ln((a*d*x+b*c*x+2*a*c+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2))/x)*x*a^5*b*c^2*d^
4+12*ln((a*d*x+b*c*x+2*a*c+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2))/x)*x*a^4*b^2*c^3*d^3-8*((b*x+a)*(d*x+c))^(1/
2)*(a*c)^(1/2)*a*b^4*c^5-6*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)*x*a^5*d^5-6*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)
*x*b^5*c^5+3*ln((a*d*x+b*c*x+2*a*c+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2))/x)*x^4*a^4*b^2*d^6+3*ln((a*d*x+b*c*x
+2*a*c+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2))/x)*x^4*b^6*c^4*d^2+6*ln((a*d*x+b*c*x+2*a*c+2*(a*c)^(1/2)*((b*x+a
)*(d*x+c))^(1/2))/x)*x^3*a^5*b*d^6+6*ln((a*d*x+b*c*x+2*a*c+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2))/x)*x^3*b^6*c
^5*d+6*ln((a*d*x+b*c*x+2*a*c+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2))/x)*x*a^6*c*d^5+6*ln((a*d*x+b*c*x+2*a*c+2*(
a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2))/x)*x*a*b^5*c^6-12*ln((a*d*x+b*c*x+2*a*c+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1
/2))/x)*a^5*b*c^3*d^3+18*ln((a*d*x+b*c*x+2*a*c+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2))/x)*a^4*b^2*c^4*d^2-12*ln
((a*d*x+b*c*x+2*a*c+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2))/x)*a^3*b^3*c^5*d)/((b*x+a)*(d*x+c))^(1/2)/(a*d-b*c)
^4/(a*c)^(1/2)/(b*x+a)^(3/2)/(d*x+c)^(3/2)

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(b*x+a)^(5/2)/(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 details)Is a*d-b*c zero or nonzero?

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {1}{x\,{\left (a+b\,x\right )}^{5/2}\,{\left (c+d\,x\right )}^{5/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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