3.8.57 \(\int \frac {1}{x^2 (a+b x)^{3/2} (c+d x)^{3/2}} \, dx\)

Optimal. Leaf size=185 \[ \frac {3 (a d+b c) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{a^{5/2} c^{5/2}}-\frac {d \sqrt {a+b x} \left (3 a^2 d^2-2 a b c d+3 b^2 c^2\right )}{a^2 c^2 \sqrt {c+d x} (b c-a d)^2}-\frac {b (3 b c-a d)}{a^2 c \sqrt {a+b x} \sqrt {c+d x} (b c-a d)}-\frac {1}{a c x \sqrt {a+b x} \sqrt {c+d x}} \]

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Rubi [A]  time = 0.15, antiderivative size = 185, normalized size of antiderivative = 1.00, number of steps used = 6, 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} \begin {gather*} -\frac {d \sqrt {a+b x} \left (3 a^2 d^2-2 a b c d+3 b^2 c^2\right )}{a^2 c^2 \sqrt {c+d x} (b c-a d)^2}+\frac {3 (a d+b c) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{a^{5/2} c^{5/2}}-\frac {b (3 b c-a d)}{a^2 c \sqrt {a+b x} \sqrt {c+d x} (b c-a d)}-\frac {1}{a c x \sqrt {a+b x} \sqrt {c+d x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^2*(a + b*x)^(3/2)*(c + d*x)^(3/2)),x]

[Out]

-((b*(3*b*c - a*d))/(a^2*c*(b*c - a*d)*Sqrt[a + b*x]*Sqrt[c + d*x])) - 1/(a*c*x*Sqrt[a + b*x]*Sqrt[c + d*x]) -
(d*(3*b^2*c^2 - 2*a*b*c*d + 3*a^2*d^2)*Sqrt[a + b*x])/(a^2*c^2*(b*c - a*d)^2*Sqrt[c + d*x]) + (3*(b*c + a*d)*
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 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)^{3/2} (c+d x)^{3/2}} \, dx &=-\frac {1}{a c x \sqrt {a+b x} \sqrt {c+d x}}-\frac {\int \frac {\frac {3}{2} (b c+a d)+2 b d x}{x (a+b x)^{3/2} (c+d x)^{3/2}} \, dx}{a c}\\ &=-\frac {b (3 b c-a d)}{a^2 c (b c-a d) \sqrt {a+b x} \sqrt {c+d x}}-\frac {1}{a c x \sqrt {a+b x} \sqrt {c+d x}}-\frac {2 \int \frac {\frac {3}{4} (b c-a d) (b c+a d)+\frac {1}{2} b d (3 b c-a d) x}{x \sqrt {a+b x} (c+d x)^{3/2}} \, dx}{a^2 c (b c-a d)}\\ &=-\frac {b (3 b c-a d)}{a^2 c (b c-a d) \sqrt {a+b x} \sqrt {c+d x}}-\frac {1}{a c x \sqrt {a+b x} \sqrt {c+d x}}-\frac {d \left (3 b^2 c^2-2 a b c d+3 a^2 d^2\right ) \sqrt {a+b x}}{a^2 c^2 (b c-a d)^2 \sqrt {c+d x}}+\frac {4 \int -\frac {3 (b c-a d)^2 (b c+a d)}{8 x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{a^2 c^2 (b c-a d)^2}\\ &=-\frac {b (3 b c-a d)}{a^2 c (b c-a d) \sqrt {a+b x} \sqrt {c+d x}}-\frac {1}{a c x \sqrt {a+b x} \sqrt {c+d x}}-\frac {d \left (3 b^2 c^2-2 a b c d+3 a^2 d^2\right ) \sqrt {a+b x}}{a^2 c^2 (b c-a d)^2 \sqrt {c+d x}}-\frac {(3 (b c+a d)) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{2 a^2 c^2}\\ &=-\frac {b (3 b c-a d)}{a^2 c (b c-a d) \sqrt {a+b x} \sqrt {c+d x}}-\frac {1}{a c x \sqrt {a+b x} \sqrt {c+d x}}-\frac {d \left (3 b^2 c^2-2 a b c d+3 a^2 d^2\right ) \sqrt {a+b x}}{a^2 c^2 (b c-a d)^2 \sqrt {c+d x}}-\frac {(3 (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^2 c^2}\\ &=-\frac {b (3 b c-a d)}{a^2 c (b c-a d) \sqrt {a+b x} \sqrt {c+d x}}-\frac {1}{a c x \sqrt {a+b x} \sqrt {c+d x}}-\frac {d \left (3 b^2 c^2-2 a b c d+3 a^2 d^2\right ) \sqrt {a+b x}}{a^2 c^2 (b c-a d)^2 \sqrt {c+d x}}+\frac {3 (b c+a d) \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 = 0.21, size = 165, normalized size = 0.89 \begin {gather*} \frac {3 (a d+b c) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{a^{5/2} c^{5/2}}+\frac {a^3 \left (-d^2\right ) (c+3 d x)+a^2 b d \left (2 c^2+c d x-3 d^2 x^2\right )+a b^2 c \left (-c^2+c d x+2 d^2 x^2\right )-3 b^3 c^2 x (c+d x)}{a^2 c^2 x \sqrt {a+b x} \sqrt {c+d x} (b c-a d)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^2*(a + b*x)^(3/2)*(c + d*x)^(3/2)),x]

[Out]

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

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IntegrateAlgebraic [A]  time = 0.33, size = 222, normalized size = 1.20 \begin {gather*} \frac {3 (a d+b c) \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {c+d x}}{\sqrt {c} \sqrt {a+b x}}\right )}{a^{5/2} c^{5/2}}+\frac {\sqrt {a+b x} \left (-\frac {3 a^3 d^3 (c+d x)}{a+b x}+\frac {3 a^2 b c d^2 (c+d x)}{a+b x}+2 a^2 c d^3+\frac {3 b^3 c^3 (c+d x)}{a+b x}-\frac {2 a b^3 c^2 (c+d x)^2}{(a+b x)^2}-\frac {3 a b^2 c^2 d (c+d x)}{a+b x}\right )}{a^2 c^2 \sqrt {c+d x} (a d-b c)^2 \left (\frac {a (c+d x)}{a+b x}-c\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[1/(x^2*(a + b*x)^(3/2)*(c + d*x)^(3/2)),x]

[Out]

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

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fricas [B]  time = 3.40, size = 940, normalized size = 5.08 \begin {gather*} \left [\frac {3 \, {\left ({\left (b^{4} c^{3} d - a b^{3} c^{2} d^{2} - a^{2} b^{2} c d^{3} + a^{3} b d^{4}\right )} x^{3} + {\left (b^{4} c^{4} - 2 \, a^{2} b^{2} c^{2} d^{2} + a^{4} d^{4}\right )} x^{2} + {\left (a b^{3} c^{4} - a^{2} b^{2} c^{3} d - a^{3} b c^{2} d^{2} + a^{4} c 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 (a^{2} b^{2} c^{4} - 2 \, a^{3} b c^{3} d + a^{4} c^{2} d^{2} + {\left (3 \, a b^{3} c^{3} d - 2 \, a^{2} b^{2} c^{2} d^{2} + 3 \, a^{3} b c d^{3}\right )} x^{2} + {\left (3 \, a b^{3} c^{4} - a^{2} b^{2} c^{3} d - a^{3} b c^{2} d^{2} + 3 \, a^{4} c d^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{4 \, {\left ({\left (a^{3} b^{3} c^{5} d - 2 \, a^{4} b^{2} c^{4} d^{2} + a^{5} b c^{3} d^{3}\right )} x^{3} + {\left (a^{3} b^{3} c^{6} - a^{4} b^{2} c^{5} d - a^{5} b c^{4} d^{2} + a^{6} c^{3} d^{3}\right )} x^{2} + {\left (a^{4} b^{2} c^{6} - 2 \, a^{5} b c^{5} d + a^{6} c^{4} d^{2}\right )} x\right )}}, -\frac {3 \, {\left ({\left (b^{4} c^{3} d - a b^{3} c^{2} d^{2} - a^{2} b^{2} c d^{3} + a^{3} b d^{4}\right )} x^{3} + {\left (b^{4} c^{4} - 2 \, a^{2} b^{2} c^{2} d^{2} + a^{4} d^{4}\right )} x^{2} + {\left (a b^{3} c^{4} - a^{2} b^{2} c^{3} d - a^{3} b c^{2} d^{2} + a^{4} c 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 (a^{2} b^{2} c^{4} - 2 \, a^{3} b c^{3} d + a^{4} c^{2} d^{2} + {\left (3 \, a b^{3} c^{3} d - 2 \, a^{2} b^{2} c^{2} d^{2} + 3 \, a^{3} b c d^{3}\right )} x^{2} + {\left (3 \, a b^{3} c^{4} - a^{2} b^{2} c^{3} d - a^{3} b c^{2} d^{2} + 3 \, a^{4} c d^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left ({\left (a^{3} b^{3} c^{5} d - 2 \, a^{4} b^{2} c^{4} d^{2} + a^{5} b c^{3} d^{3}\right )} x^{3} + {\left (a^{3} b^{3} c^{6} - a^{4} b^{2} c^{5} d - a^{5} b c^{4} d^{2} + a^{6} c^{3} d^{3}\right )} x^{2} + {\left (a^{4} b^{2} c^{6} - 2 \, a^{5} b c^{5} d + a^{6} c^{4} d^{2}\right )} x\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/(b*x+a)^(3/2)/(d*x+c)^(3/2),x, algorithm="fricas")

[Out]

[1/4*(3*((b^4*c^3*d - a*b^3*c^2*d^2 - a^2*b^2*c*d^3 + a^3*b*d^4)*x^3 + (b^4*c^4 - 2*a^2*b^2*c^2*d^2 + a^4*d^4)
*x^2 + (a*b^3*c^4 - a^2*b^2*c^3*d - a^3*b*c^2*d^2 + a^4*c*d^3)*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*(a^2*b^2*c^4 - 2*a^3*b*c^3*d + a^4*c^2*d^2 + (3*a*b^3*c^3*d - 2*a^2*b^2*c^2*d^2 + 3*a^3*b*c*d^3)*x^
2 + (3*a*b^3*c^4 - a^2*b^2*c^3*d - a^3*b*c^2*d^2 + 3*a^4*c*d^3)*x)*sqrt(b*x + a)*sqrt(d*x + c))/((a^3*b^3*c^5*
d - 2*a^4*b^2*c^4*d^2 + a^5*b*c^3*d^3)*x^3 + (a^3*b^3*c^6 - a^4*b^2*c^5*d - a^5*b*c^4*d^2 + a^6*c^3*d^3)*x^2 +
(a^4*b^2*c^6 - 2*a^5*b*c^5*d + a^6*c^4*d^2)*x), -1/2*(3*((b^4*c^3*d - a*b^3*c^2*d^2 - a^2*b^2*c*d^3 + a^3*b*d
^4)*x^3 + (b^4*c^4 - 2*a^2*b^2*c^2*d^2 + a^4*d^4)*x^2 + (a*b^3*c^4 - a^2*b^2*c^3*d - a^3*b*c^2*d^2 + a^4*c*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*(a^2*b^2*c^4 - 2*a^3*b*c^3*d + a^4*c^2*d^2 + (3*a*b^3*c^3*d - 2*a^2*b^2*c^2*d
^2 + 3*a^3*b*c*d^3)*x^2 + (3*a*b^3*c^4 - a^2*b^2*c^3*d - a^3*b*c^2*d^2 + 3*a^4*c*d^3)*x)*sqrt(b*x + a)*sqrt(d*
x + c))/((a^3*b^3*c^5*d - 2*a^4*b^2*c^4*d^2 + a^5*b*c^3*d^3)*x^3 + (a^3*b^3*c^6 - a^4*b^2*c^5*d - a^5*b*c^4*d^
2 + a^6*c^3*d^3)*x^2 + (a^4*b^2*c^6 - 2*a^5*b*c^5*d + a^6*c^4*d^2)*x)]

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giac [B]  time = 14.86, size = 817, normalized size = 4.42 \begin {gather*} -\frac {2 \, \sqrt {b x + a} b^{2} d^{3}}{{\left (b^{2} c^{4} {\left | b \right |} - 2 \, a b c^{3} d {\left | b \right |} + a^{2} c^{2} d^{2} {\left | b \right |}\right )} \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}} - \frac {2 \, {\left (3 \, \sqrt {b d} b^{8} c^{4} - 8 \, \sqrt {b d} a b^{7} c^{3} d + 8 \, \sqrt {b d} a^{2} b^{6} c^{2} d^{2} - 4 \, \sqrt {b d} a^{3} b^{5} c d^{3} + \sqrt {b d} a^{4} b^{4} d^{4} - 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^{6} c^{3} - 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} a^{2} b^{4} c 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^{4} c^{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 )}^{4} a^{2} b^{2} d^{2}\right )}}{{\left (b^{6} c^{3} - 3 \, a b^{5} c^{2} d + 3 \, a^{2} b^{4} c d^{2} - a^{3} b^{3} d^{3} - 3 \, {\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^{4} c^{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} a b^{3} c 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^{2} d^{2} + 3 \, {\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^{2} c + {\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 d - {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{6}\right )} {\left (a^{2} b c^{3} {\left | b \right |} - a^{3} c^{2} d {\left | b \right |}\right )}} + \frac {3 \, {\left (\sqrt {b d} b^{3} c + \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^{2} b c^{2} {\left | b \right |}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/(b*x+a)^(3/2)/(d*x+c)^(3/2),x, algorithm="giac")

[Out]

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

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maple [B]  time = 0.04, size = 897, normalized size = 4.85 \begin {gather*} \frac {3 a^{3} b \,d^{4} x^{3} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}}{x}\right )-3 a^{2} b^{2} c \,d^{3} x^{3} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}}{x}\right )-3 a \,b^{3} c^{2} d^{2} x^{3} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}}{x}\right )+3 b^{4} c^{3} d \,x^{3} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}}{x}\right )+3 a^{4} d^{4} x^{2} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}}{x}\right )-6 a^{2} b^{2} c^{2} d^{2} x^{2} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}}{x}\right )+3 b^{4} c^{4} x^{2} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}}{x}\right )+3 a^{4} c \,d^{3} x \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}}{x}\right )-3 a^{3} b \,c^{2} d^{2} x \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}}{x}\right )-3 a^{2} b^{2} c^{3} d x \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}}{x}\right )+3 a \,b^{3} c^{4} x \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}}{x}\right )-6 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a^{2} b \,d^{3} x^{2}+4 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a \,b^{2} c \,d^{2} x^{2}-6 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, b^{3} c^{2} d \,x^{2}-6 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a^{3} d^{3} x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a^{2} b c \,d^{2} x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a \,b^{2} c^{2} d x -6 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, b^{3} c^{3} x -2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a^{3} c \,d^{2}+4 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a^{2} b \,c^{2} d -2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a \,b^{2} c^{3}}{2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \left (a d -b c \right )^{2} \sqrt {a c}\, \sqrt {b x +a}\, \sqrt {d x +c}\, a^{2} c^{2} x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^2/(b*x+a)^(3/2)/(d*x+c)^(3/2),x)

[Out]

1/2/a^2/c^2*(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^3*a^3*b*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^3*a^2*b^2*c*d^3-3*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^3*a*b^3*c^2*d^2+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^3
*b^4*c^3*d+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^4*d^4-6*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^2*a^2*b^2*c^2*d^2+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^4*c^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*a^4*c*d^3
-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*a^3*b*c^2*d^2-3*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*a^2*b^2*c^3*d+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*a*b^3*c^4-6*x^2*a^2*b*d^3*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+4*x^2*a*b^2*c*d^2*(a*c)^(1/2)*((b*x
+a)*(d*x+c))^(1/2)-6*x^2*b^3*c^2*d*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-6*x*a^3*d^3*(a*c)^(1/2)*((b*x+a)*(d*x+c
))^(1/2)+2*x*a^2*b*c*d^2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*x*a*b^2*c^2*d*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/
2)-6*x*b^3*c^3*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-2*a^3*c*d^2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+4*a^2*b*c^2
*d*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-2*a*b^2*c^3*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2))/((b*x+a)*(d*x+c))^(1/2
)/(a*d-b*c)^2/x/(a*c)^(1/2)/(b*x+a)^(1/2)/(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)^(3/2)/(d*x+c)^(3/2),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 = -0.01 \begin {gather*} \int \frac {1}{x^2\,{\left (a+b\,x\right )}^{3/2}\,{\left (c+d\,x\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^2*(a + b*x)^(3/2)*(c + d*x)^(3/2)),x)

[Out]

int(1/(x^2*(a + b*x)^(3/2)*(c + d*x)^(3/2)), x)

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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 )^{\frac {3}{2}} \left (c + d x\right )^{\frac {3}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**2/(b*x+a)**(3/2)/(d*x+c)**(3/2),x)

[Out]

Integral(1/(x**2*(a + b*x)**(3/2)*(c + d*x)**(3/2)), x)

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