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

Optimal. Leaf size=352 \[ \frac {5 (a d+b c) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{a^{7/2} c^{7/2}}-\frac {b (5 b c-3 a d)}{3 a^2 c (a+b x)^{3/2} (c+d x)^{3/2} (b c-a d)}-\frac {b \left (a^2 d^2-10 a b c d+5 b^2 c^2\right )}{a^3 c \sqrt {a+b x} (c+d x)^{3/2} (b c-a d)^2}-\frac {d \sqrt {a+b x} \left (-5 a^3 d^3+9 a^2 b c d^2-35 a b^2 c^2 d+15 b^3 c^3\right )}{3 a^3 c^2 (c+d x)^{3/2} (b c-a d)^3}-\frac {d \sqrt {a+b x} \left (15 a^4 d^4-40 a^3 b c d^3+18 a^2 b^2 c^2 d^2-40 a b^3 c^3 d+15 b^4 c^4\right )}{3 a^3 c^3 \sqrt {c+d x} (b c-a d)^4}-\frac {1}{a c x (a+b x)^{3/2} (c+d x)^{3/2}} \]

[Out]

-1/3*b*(-3*a*d+5*b*c)/a^2/c/(-a*d+b*c)/(b*x+a)^(3/2)/(d*x+c)^(3/2)-1/a/c/x/(b*x+a)^(3/2)/(d*x+c)^(3/2)+5*(a*d+
b*c)*arctanh(c^(1/2)*(b*x+a)^(1/2)/a^(1/2)/(d*x+c)^(1/2))/a^(7/2)/c^(7/2)-b*(a^2*d^2-10*a*b*c*d+5*b^2*c^2)/a^3
/c/(-a*d+b*c)^2/(d*x+c)^(3/2)/(b*x+a)^(1/2)-1/3*d*(-5*a^3*d^3+9*a^2*b*c*d^2-35*a*b^2*c^2*d+15*b^3*c^3)*(b*x+a)
^(1/2)/a^3/c^2/(-a*d+b*c)^3/(d*x+c)^(3/2)-1/3*d*(15*a^4*d^4-40*a^3*b*c*d^3+18*a^2*b^2*c^2*d^2-40*a*b^3*c^3*d+1
5*b^4*c^4)*(b*x+a)^(1/2)/a^3/c^3/(-a*d+b*c)^4/(d*x+c)^(1/2)

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Rubi [A]  time = 0.40, antiderivative size = 352, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {103, 152, 12, 93, 208} \[ -\frac {d \sqrt {a+b x} \left (18 a^2 b^2 c^2 d^2-40 a^3 b c d^3+15 a^4 d^4-40 a b^3 c^3 d+15 b^4 c^4\right )}{3 a^3 c^3 \sqrt {c+d x} (b c-a d)^4}-\frac {d \sqrt {a+b x} \left (9 a^2 b c d^2-5 a^3 d^3-35 a b^2 c^2 d+15 b^3 c^3\right )}{3 a^3 c^2 (c+d x)^{3/2} (b c-a d)^3}-\frac {b \left (a^2 d^2-10 a b c d+5 b^2 c^2\right )}{a^3 c \sqrt {a+b x} (c+d x)^{3/2} (b c-a d)^2}+\frac {5 (a d+b c) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{a^{7/2} c^{7/2}}-\frac {b (5 b c-3 a d)}{3 a^2 c (a+b x)^{3/2} (c+d x)^{3/2} (b c-a d)}-\frac {1}{a c x (a+b x)^{3/2} (c+d x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(b*(5*b*c - 3*a*d))/(3*a^2*c*(b*c - a*d)*(a + b*x)^(3/2)*(c + d*x)^(3/2)) - 1/(a*c*x*(a + b*x)^(3/2)*(c + d*x
)^(3/2)) - (b*(5*b^2*c^2 - 10*a*b*c*d + a^2*d^2))/(a^3*c*(b*c - a*d)^2*Sqrt[a + b*x]*(c + d*x)^(3/2)) - (d*(15
*b^3*c^3 - 35*a*b^2*c^2*d + 9*a^2*b*c*d^2 - 5*a^3*d^3)*Sqrt[a + b*x])/(3*a^3*c^2*(b*c - a*d)^3*(c + d*x)^(3/2)
) - (d*(15*b^4*c^4 - 40*a*b^3*c^3*d + 18*a^2*b^2*c^2*d^2 - 40*a^3*b*c*d^3 + 15*a^4*d^4)*Sqrt[a + b*x])/(3*a^3*
c^3*(b*c - a*d)^4*Sqrt[c + d*x]) + (5*(b*c + a*d)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(a
^(7/2)*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 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 103

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] &&
IntegerQ[m] && (IntegerQ[n] || IntegersQ[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^2 (a+b x)^{5/2} (c+d x)^{5/2}} \, dx &=-\frac {1}{a c x (a+b x)^{3/2} (c+d x)^{3/2}}-\frac {\int \frac {\frac {5}{2} (b c+a d)+4 b d x}{x (a+b x)^{5/2} (c+d x)^{5/2}} \, dx}{a c}\\ &=-\frac {b (5 b c-3 a d)}{3 a^2 c (b c-a d) (a+b x)^{3/2} (c+d x)^{3/2}}-\frac {1}{a c x (a+b x)^{3/2} (c+d x)^{3/2}}-\frac {2 \int \frac {\frac {15}{4} (b c-a d) (b c+a d)+\frac {3}{2} b d (5 b c-3 a d) x}{x (a+b x)^{3/2} (c+d x)^{5/2}} \, dx}{3 a^2 c (b c-a d)}\\ &=-\frac {b (5 b c-3 a d)}{3 a^2 c (b c-a d) (a+b x)^{3/2} (c+d x)^{3/2}}-\frac {1}{a c x (a+b x)^{3/2} (c+d x)^{3/2}}-\frac {b \left (5 b^2 c^2-10 a b c d+a^2 d^2\right )}{a^3 c (b c-a d)^2 \sqrt {a+b x} (c+d x)^{3/2}}-\frac {4 \int \frac {\frac {15}{8} (b c-a d)^2 (b c+a d)+\frac {3}{2} b d \left (5 b^2 c^2-10 a b c d+a^2 d^2\right ) x}{x \sqrt {a+b x} (c+d x)^{5/2}} \, dx}{3 a^3 c (b c-a d)^2}\\ &=-\frac {b (5 b c-3 a d)}{3 a^2 c (b c-a d) (a+b x)^{3/2} (c+d x)^{3/2}}-\frac {1}{a c x (a+b x)^{3/2} (c+d x)^{3/2}}-\frac {b \left (5 b^2 c^2-10 a b c d+a^2 d^2\right )}{a^3 c (b c-a d)^2 \sqrt {a+b x} (c+d x)^{3/2}}-\frac {d \left (15 b^3 c^3-35 a b^2 c^2 d+9 a^2 b c d^2-5 a^3 d^3\right ) \sqrt {a+b x}}{3 a^3 c^2 (b c-a d)^3 (c+d x)^{3/2}}+\frac {8 \int \frac {-\frac {45}{16} (b c-a d)^3 (b c+a d)-\frac {3}{8} b d \left (15 b^3 c^3-35 a b^2 c^2 d+9 a^2 b c d^2-5 a^3 d^3\right ) x}{x \sqrt {a+b x} (c+d x)^{3/2}} \, dx}{9 a^3 c^2 (b c-a d)^3}\\ &=-\frac {b (5 b c-3 a d)}{3 a^2 c (b c-a d) (a+b x)^{3/2} (c+d x)^{3/2}}-\frac {1}{a c x (a+b x)^{3/2} (c+d x)^{3/2}}-\frac {b \left (5 b^2 c^2-10 a b c d+a^2 d^2\right )}{a^3 c (b c-a d)^2 \sqrt {a+b x} (c+d x)^{3/2}}-\frac {d \left (15 b^3 c^3-35 a b^2 c^2 d+9 a^2 b c d^2-5 a^3 d^3\right ) \sqrt {a+b x}}{3 a^3 c^2 (b c-a d)^3 (c+d x)^{3/2}}-\frac {d \left (15 b^4 c^4-40 a b^3 c^3 d+18 a^2 b^2 c^2 d^2-40 a^3 b c d^3+15 a^4 d^4\right ) \sqrt {a+b x}}{3 a^3 c^3 (b c-a d)^4 \sqrt {c+d x}}-\frac {16 \int \frac {45 (b c-a d)^4 (b c+a d)}{32 x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{9 a^3 c^3 (b c-a d)^4}\\ &=-\frac {b (5 b c-3 a d)}{3 a^2 c (b c-a d) (a+b x)^{3/2} (c+d x)^{3/2}}-\frac {1}{a c x (a+b x)^{3/2} (c+d x)^{3/2}}-\frac {b \left (5 b^2 c^2-10 a b c d+a^2 d^2\right )}{a^3 c (b c-a d)^2 \sqrt {a+b x} (c+d x)^{3/2}}-\frac {d \left (15 b^3 c^3-35 a b^2 c^2 d+9 a^2 b c d^2-5 a^3 d^3\right ) \sqrt {a+b x}}{3 a^3 c^2 (b c-a d)^3 (c+d x)^{3/2}}-\frac {d \left (15 b^4 c^4-40 a b^3 c^3 d+18 a^2 b^2 c^2 d^2-40 a^3 b c d^3+15 a^4 d^4\right ) \sqrt {a+b x}}{3 a^3 c^3 (b c-a d)^4 \sqrt {c+d x}}-\frac {(5 (b c+a d)) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{2 a^3 c^3}\\ &=-\frac {b (5 b c-3 a d)}{3 a^2 c (b c-a d) (a+b x)^{3/2} (c+d x)^{3/2}}-\frac {1}{a c x (a+b x)^{3/2} (c+d x)^{3/2}}-\frac {b \left (5 b^2 c^2-10 a b c d+a^2 d^2\right )}{a^3 c (b c-a d)^2 \sqrt {a+b x} (c+d x)^{3/2}}-\frac {d \left (15 b^3 c^3-35 a b^2 c^2 d+9 a^2 b c d^2-5 a^3 d^3\right ) \sqrt {a+b x}}{3 a^3 c^2 (b c-a d)^3 (c+d x)^{3/2}}-\frac {d \left (15 b^4 c^4-40 a b^3 c^3 d+18 a^2 b^2 c^2 d^2-40 a^3 b c d^3+15 a^4 d^4\right ) \sqrt {a+b x}}{3 a^3 c^3 (b c-a d)^4 \sqrt {c+d x}}-\frac {(5 (b c+a d)) \operatorname {Subst}\left (\int \frac {1}{-a+c x^2} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )}{a^3 c^3}\\ &=-\frac {b (5 b c-3 a d)}{3 a^2 c (b c-a d) (a+b x)^{3/2} (c+d x)^{3/2}}-\frac {1}{a c x (a+b x)^{3/2} (c+d x)^{3/2}}-\frac {b \left (5 b^2 c^2-10 a b c d+a^2 d^2\right )}{a^3 c (b c-a d)^2 \sqrt {a+b x} (c+d x)^{3/2}}-\frac {d \left (15 b^3 c^3-35 a b^2 c^2 d+9 a^2 b c d^2-5 a^3 d^3\right ) \sqrt {a+b x}}{3 a^3 c^2 (b c-a d)^3 (c+d x)^{3/2}}-\frac {d \left (15 b^4 c^4-40 a b^3 c^3 d+18 a^2 b^2 c^2 d^2-40 a^3 b c d^3+15 a^4 d^4\right ) \sqrt {a+b x}}{3 a^3 c^3 (b c-a d)^4 \sqrt {c+d x}}+\frac {5 (b c+a d) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{a^{7/2} c^{7/2}}\\ \end {align*}

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Mathematica [A]  time = 0.91, size = 382, normalized size = 1.09 \[ \frac {-3 a^{3/2} b c^{5/2} x (a d-b c)^3 (3 a d-5 b c)-9 a^{5/2} c^{5/2} (b c-a d)^4+x (a+b x) \left (18 a^{3/2} b c^{5/2} d (5 b c-3 a d) (b c-a d)^2-3 \sqrt {a} c^{3/2} d (a+b x) \left (5 a^3 d^3-9 a^2 b c d^2+35 a b^2 c^2 d-15 b^3 c^3\right ) (a d-b c)+\sqrt {a+b x} (c+d x) \left (45 \sqrt {c+d x} (b c-a d)^4 (a d+b c) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )-3 \sqrt {a} \sqrt {c} d \sqrt {a+b x} \left (15 a^4 d^4-40 a^3 b c d^3+18 a^2 b^2 c^2 d^2-40 a b^3 c^3 d+15 b^4 c^4\right )\right )+45 \sqrt {a} b c^{5/2} (a d+b c) (a d-b c)^3\right )}{9 a^{7/2} c^{7/2} x (a+b x)^{3/2} (c+d x)^{3/2} (b c-a d)^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-9*a^(5/2)*c^(5/2)*(b*c - a*d)^4 - 3*a^(3/2)*b*c^(5/2)*(-(b*c) + a*d)^3*(-5*b*c + 3*a*d)*x + x*(a + b*x)*(18*
a^(3/2)*b*c^(5/2)*d*(5*b*c - 3*a*d)*(b*c - a*d)^2 + 45*Sqrt[a]*b*c^(5/2)*(-(b*c) + a*d)^3*(b*c + a*d) - 3*Sqrt
[a]*c^(3/2)*d*(-(b*c) + a*d)*(-15*b^3*c^3 + 35*a*b^2*c^2*d - 9*a^2*b*c*d^2 + 5*a^3*d^3)*(a + b*x) + Sqrt[a + b
*x]*(c + d*x)*(-3*Sqrt[a]*Sqrt[c]*d*(15*b^4*c^4 - 40*a*b^3*c^3*d + 18*a^2*b^2*c^2*d^2 - 40*a^3*b*c*d^3 + 15*a^
4*d^4)*Sqrt[a + b*x] + 45*(b*c - a*d)^4*(b*c + a*d)*Sqrt[c + d*x]*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqr
t[c + d*x])])))/(9*a^(7/2)*c^(7/2)*(b*c - a*d)^4*x*(a + b*x)^(3/2)*(c + d*x)^(3/2))

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fricas [B]  time = 25.19, size = 2506, normalized size = 7.12 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/12*(15*((b^7*c^5*d^2 - 3*a*b^6*c^4*d^3 + 2*a^2*b^5*c^3*d^4 + 2*a^3*b^4*c^2*d^5 - 3*a^4*b^3*c*d^6 + a^5*b^2*
d^7)*x^5 + 2*(b^7*c^6*d - 2*a*b^6*c^5*d^2 - a^2*b^5*c^4*d^3 + 4*a^3*b^4*c^3*d^4 - a^4*b^3*c^2*d^5 - 2*a^5*b^2*
c*d^6 + a^6*b*d^7)*x^4 + (b^7*c^7 + a*b^6*c^6*d - 9*a^2*b^5*c^5*d^2 + 7*a^3*b^4*c^4*d^3 + 7*a^4*b^3*c^3*d^4 -
9*a^5*b^2*c^2*d^5 + a^6*b*c*d^6 + a^7*d^7)*x^3 + 2*(a*b^6*c^7 - 2*a^2*b^5*c^6*d - a^3*b^4*c^5*d^2 + 4*a^4*b^3*
c^4*d^3 - a^5*b^2*c^3*d^4 - 2*a^6*b*c^2*d^5 + a^7*c*d^6)*x^2 + (a^2*b^5*c^7 - 3*a^3*b^4*c^6*d + 2*a^4*b^3*c^5*
d^2 + 2*a^5*b^2*c^4*d^3 - 3*a^6*b*c^3*d^4 + a^7*c^2*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*(3*a^3*b^4*c^7 - 12*a^4*b^3*c^6*d + 18*a^5*b^2*c^5*d^2 - 12*a^6*b*c^4*d^3 + 3*a^7*c^3*d^4 + (15*a*b^6*c^5
*d^2 - 40*a^2*b^5*c^4*d^3 + 18*a^3*b^4*c^3*d^4 - 40*a^4*b^3*c^2*d^5 + 15*a^5*b^2*c*d^6)*x^4 + 6*(5*a*b^6*c^6*d
- 10*a^2*b^5*c^5*d^2 - 3*a^3*b^4*c^4*d^3 - 3*a^4*b^3*c^3*d^4 - 10*a^5*b^2*c^2*d^5 + 5*a^6*b*c*d^6)*x^3 + 3*(5
*a*b^6*c^7 - 29*a^3*b^4*c^5*d^2 + 16*a^4*b^3*c^4*d^3 - 29*a^5*b^2*c^3*d^4 + 5*a^7*c*d^6)*x^2 + 4*(5*a^2*b^5*c^
7 - 12*a^3*b^4*c^6*d + 3*a^4*b^3*c^5*d^2 + 3*a^5*b^2*c^4*d^3 - 12*a^6*b*c^3*d^4 + 5*a^7*c^2*d^5)*x)*sqrt(b*x +
a)*sqrt(d*x + c))/((a^4*b^6*c^8*d^2 - 4*a^5*b^5*c^7*d^3 + 6*a^6*b^4*c^6*d^4 - 4*a^7*b^3*c^5*d^5 + a^8*b^2*c^4
*d^6)*x^5 + 2*(a^4*b^6*c^9*d - 3*a^5*b^5*c^8*d^2 + 2*a^6*b^4*c^7*d^3 + 2*a^7*b^3*c^6*d^4 - 3*a^8*b^2*c^5*d^5 +
a^9*b*c^4*d^6)*x^4 + (a^4*b^6*c^10 - 9*a^6*b^4*c^8*d^2 + 16*a^7*b^3*c^7*d^3 - 9*a^8*b^2*c^6*d^4 + a^10*c^4*d^
6)*x^3 + 2*(a^5*b^5*c^10 - 3*a^6*b^4*c^9*d + 2*a^7*b^3*c^8*d^2 + 2*a^8*b^2*c^7*d^3 - 3*a^9*b*c^6*d^4 + a^10*c^
5*d^5)*x^2 + (a^6*b^4*c^10 - 4*a^7*b^3*c^9*d + 6*a^8*b^2*c^8*d^2 - 4*a^9*b*c^7*d^3 + a^10*c^6*d^4)*x), -1/6*(1
5*((b^7*c^5*d^2 - 3*a*b^6*c^4*d^3 + 2*a^2*b^5*c^3*d^4 + 2*a^3*b^4*c^2*d^5 - 3*a^4*b^3*c*d^6 + a^5*b^2*d^7)*x^5
+ 2*(b^7*c^6*d - 2*a*b^6*c^5*d^2 - a^2*b^5*c^4*d^3 + 4*a^3*b^4*c^3*d^4 - a^4*b^3*c^2*d^5 - 2*a^5*b^2*c*d^6 +
a^6*b*d^7)*x^4 + (b^7*c^7 + a*b^6*c^6*d - 9*a^2*b^5*c^5*d^2 + 7*a^3*b^4*c^4*d^3 + 7*a^4*b^3*c^3*d^4 - 9*a^5*b^
2*c^2*d^5 + a^6*b*c*d^6 + a^7*d^7)*x^3 + 2*(a*b^6*c^7 - 2*a^2*b^5*c^6*d - a^3*b^4*c^5*d^2 + 4*a^4*b^3*c^4*d^3
- a^5*b^2*c^3*d^4 - 2*a^6*b*c^2*d^5 + a^7*c*d^6)*x^2 + (a^2*b^5*c^7 - 3*a^3*b^4*c^6*d + 2*a^4*b^3*c^5*d^2 + 2*
a^5*b^2*c^4*d^3 - 3*a^6*b*c^3*d^4 + a^7*c^2*d^5)*x)*sqrt(-a*c)*arctan(1/2*(2*a*c + (b*c + a*d)*x)*sqrt(-a*c)*s
qrt(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^3*b^4*c^7 - 12*a^4*b^3*c^
6*d + 18*a^5*b^2*c^5*d^2 - 12*a^6*b*c^4*d^3 + 3*a^7*c^3*d^4 + (15*a*b^6*c^5*d^2 - 40*a^2*b^5*c^4*d^3 + 18*a^3*
b^4*c^3*d^4 - 40*a^4*b^3*c^2*d^5 + 15*a^5*b^2*c*d^6)*x^4 + 6*(5*a*b^6*c^6*d - 10*a^2*b^5*c^5*d^2 - 3*a^3*b^4*c
^4*d^3 - 3*a^4*b^3*c^3*d^4 - 10*a^5*b^2*c^2*d^5 + 5*a^6*b*c*d^6)*x^3 + 3*(5*a*b^6*c^7 - 29*a^3*b^4*c^5*d^2 + 1
6*a^4*b^3*c^4*d^3 - 29*a^5*b^2*c^3*d^4 + 5*a^7*c*d^6)*x^2 + 4*(5*a^2*b^5*c^7 - 12*a^3*b^4*c^6*d + 3*a^4*b^3*c^
5*d^2 + 3*a^5*b^2*c^4*d^3 - 12*a^6*b*c^3*d^4 + 5*a^7*c^2*d^5)*x)*sqrt(b*x + a)*sqrt(d*x + c))/((a^4*b^6*c^8*d^
2 - 4*a^5*b^5*c^7*d^3 + 6*a^6*b^4*c^6*d^4 - 4*a^7*b^3*c^5*d^5 + a^8*b^2*c^4*d^6)*x^5 + 2*(a^4*b^6*c^9*d - 3*a^
5*b^5*c^8*d^2 + 2*a^6*b^4*c^7*d^3 + 2*a^7*b^3*c^6*d^4 - 3*a^8*b^2*c^5*d^5 + a^9*b*c^4*d^6)*x^4 + (a^4*b^6*c^10
- 9*a^6*b^4*c^8*d^2 + 16*a^7*b^3*c^7*d^3 - 9*a^8*b^2*c^6*d^4 + a^10*c^4*d^6)*x^3 + 2*(a^5*b^5*c^10 - 3*a^6*b^
4*c^9*d + 2*a^7*b^3*c^8*d^2 + 2*a^8*b^2*c^7*d^3 - 3*a^9*b*c^6*d^4 + a^10*c^5*d^5)*x^2 + (a^6*b^4*c^10 - 4*a^7*
b^3*c^9*d + 6*a^8*b^2*c^8*d^2 - 4*a^9*b*c^7*d^3 + a^10*c^6*d^4)*x)]

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giac [B]  time = 30.09, size = 1272, normalized size = 3.61 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*sqrt(b*x + a)*(2*(7*b^7*c^7*d^6*abs(b) - 24*a*b^6*c^6*d^7*abs(b) + 30*a^2*b^5*c^5*d^8*abs(b) - 16*a^3*b^4*
c^4*d^9*abs(b) + 3*a^4*b^3*c^3*d^10*abs(b))*(b*x + a)/(b^9*c^13*d - 7*a*b^8*c^12*d^2 + 21*a^2*b^7*c^11*d^3 - 3
5*a^3*b^6*c^10*d^4 + 35*a^4*b^5*c^9*d^5 - 21*a^5*b^4*c^8*d^6 + 7*a^6*b^3*c^7*d^7 - a^7*b^2*c^6*d^8) + 3*(5*b^8
*c^8*d^5*abs(b) - 22*a*b^7*c^7*d^6*abs(b) + 38*a^2*b^6*c^6*d^7*abs(b) - 32*a^3*b^5*c^5*d^8*abs(b) + 13*a^4*b^4
*c^4*d^9*abs(b) - 2*a^5*b^3*c^3*d^10*abs(b))/(b^9*c^13*d - 7*a*b^8*c^12*d^2 + 21*a^2*b^7*c^11*d^3 - 35*a^3*b^6
*c^10*d^4 + 35*a^4*b^5*c^9*d^5 - 21*a^5*b^4*c^8*d^6 + 7*a^6*b^3*c^7*d^7 - a^7*b^2*c^6*d^8))/(b^2*c + (b*x + a)
*b*d - a*b*d)^(3/2) - 8/3*(3*sqrt(b*d)*b^10*c^3 - 13*sqrt(b*d)*a*b^9*c^2*d + 17*sqrt(b*d)*a^2*b^8*c*d^2 - 7*sq
rt(b*d)*a^3*b^7*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^8*c^2 +
21*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^7*c*d - 15*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^6*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^6*c - 6*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^5*d)/((a^3*b^3*c^3*abs(b) - 3*a^4*b^2*c^2*d*abs(b) + 3*a^5*b*c*d^2*abs(b) - a^6*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) + 5*(sqrt(b*d)*b^3*
c + 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^3*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*(sqrt(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^3*c^3*abs(b))

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maple [B]  time = 0.05, size = 2703, normalized size = 7.68 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6/a^3/c^3*(15*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^5*c^7-60*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*b*c^2*d^5+15*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^7*d^7+15*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
^7*c^7+120*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^3*c^4*d^3-30*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^4*c^5*d^2-60*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^5*c^6*d-45*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*b*c^3*d^4+30*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^2*c^4*d^3+30*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^3*c^5*d^2-45*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^4*c^6*d-30*x^4*a^4*b^2*d^6*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)-30
*x^4*b^6*c^4*d^2*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)-60*x^3*a^5*b*d^6*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)-60*x
^3*b^6*c^5*d*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)-40*x*a^6*c*d^5*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)-40*x*a*b^5
*c^6*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)+24*a^5*b*c^3*d^3*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)-36*a^4*b^2*c^4*d
^2*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)+24*a^3*b^3*c^5*d*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)-45*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^5*a^4*b^3*c*d^6+30*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^5*a^3*b^4*c^2*d^5+30*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^5*a^2*b^5*c^3*d^4-45*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^5*a*b^6*c^4*d^3-60*l
n((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^5*b^2*c*d^6-30*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^3*c^2*d^5+120*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^4*c^3*d^4-30*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^5*c^4*d^3-60*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^6*c^5*d^2+15*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^6*b*c*d^6-135*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^2*c^2*d^5+105*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^3*c^3*d^4+105*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^4*c^4
*d^3-135*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^5*c^5*d^2+15*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^6*c^6*d+80*x^4*a^3*b^3*c*d^5*((b*x+a)*(d*x+c))^(1/2)
*(a*c)^(1/2)-36*x^4*a^2*b^4*c^2*d^4*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)+80*x^4*a*b^5*c^3*d^3*((b*x+a)*(d*x+c))
^(1/2)*(a*c)^(1/2)+120*x^3*a^4*b^2*c*d^5*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)+36*x^3*a^3*b^3*c^2*d^4*((b*x+a)*(
d*x+c))^(1/2)*(a*c)^(1/2)+36*x^3*a^2*b^4*c^3*d^3*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)+120*x^3*a*b^5*c^4*d^2*((b
*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)+174*x^2*a^4*b^2*c^2*d^4*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)-96*x^2*a^3*b^3*c^
3*d^3*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)+174*x^2*a^2*b^4*c^4*d^2*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)+96*x*a^5
*b*c^2*d^4*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)-24*x*a^4*b^2*c^3*d^3*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)-24*x*a
^3*b^3*c^4*d^2*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)+96*x*a^2*b^4*c^5*d*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)-30*l
n((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^5*b^2*c^3*d^4-6*a^2*b^4*c^6*((b*x+a)*(d*x
+c))^(1/2)*(a*c)^(1/2)+15*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^5*a^5*b^2*d^7+15*l
n((a*d*x+b*c*x+2*a*c+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2))/x)*x^5*b^7*c^5*d^2+30*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^6*b*d^7+30*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^7*c^6*d+30*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^7*c*d^6+30*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*b^6*c^7+15*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^7*c^2*d^5-30*x^2*a^6*d^6*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)-30*x^2*b^6*c^6*
((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)-6*a^6*c^2*d^4*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2))/((b*x+a)*(d*x+c))^(1/2)
/(a*d-b*c)^4/x/(a*c)^(1/2)/(b*x+a)^(3/2)/(d*x+c)^(3/2)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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