3.8.65 \(\int \frac {1}{x (a+b x)^{3/2} (c+d x)^{5/2}} \, dx\)

Optimal. Leaf size=180 \[ -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{a^{3/2} c^{5/2}}+\frac {2 d \sqrt {a+b x} (3 b c-a d) (3 a d+b c)}{3 a c^2 \sqrt {c+d x} (b c-a d)^3}+\frac {2 b}{a \sqrt {a+b x} (c+d x)^{3/2} (b c-a d)}+\frac {2 d \sqrt {a+b x} (a d+3 b c)}{3 a c (c+d x)^{3/2} (b c-a d)^2} \]

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Rubi [A]  time = 0.15, antiderivative size = 180, 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 = {104, 152, 12, 93, 208} \begin {gather*} -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{a^{3/2} c^{5/2}}+\frac {2 d \sqrt {a+b x} (3 b c-a d) (3 a d+b c)}{3 a c^2 \sqrt {c+d x} (b c-a d)^3}+\frac {2 b}{a \sqrt {a+b x} (c+d x)^{3/2} (b c-a d)}+\frac {2 d \sqrt {a+b x} (a d+3 b c)}{3 a c (c+d x)^{3/2} (b c-a d)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x*(a + b*x)^(3/2)*(c + d*x)^(5/2)),x]

[Out]

(2*b)/(a*(b*c - a*d)*Sqrt[a + b*x]*(c + d*x)^(3/2)) + (2*d*(3*b*c + a*d)*Sqrt[a + b*x])/(3*a*c*(b*c - a*d)^2*(
c + d*x)^(3/2)) + (2*d*(3*b*c - a*d)*(b*c + 3*a*d)*Sqrt[a + b*x])/(3*a*c^2*(b*c - a*d)^3*Sqrt[c + d*x]) - (2*A
rcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(a^(3/2)*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)^{3/2} (c+d x)^{5/2}} \, dx &=\frac {2 b}{a (b c-a d) \sqrt {a+b x} (c+d x)^{3/2}}+\frac {2 \int \frac {\frac {1}{2} (b c-a d)+2 b d x}{x \sqrt {a+b x} (c+d x)^{5/2}} \, dx}{a (b c-a d)}\\ &=\frac {2 b}{a (b c-a d) \sqrt {a+b x} (c+d x)^{3/2}}+\frac {2 d (3 b c+a d) \sqrt {a+b x}}{3 a c (b c-a d)^2 (c+d x)^{3/2}}-\frac {4 \int \frac {-\frac {3}{4} (b c-a d)^2-\frac {1}{2} b d (3 b c+a d) x}{x \sqrt {a+b x} (c+d x)^{3/2}} \, dx}{3 a c (b c-a d)^2}\\ &=\frac {2 b}{a (b c-a d) \sqrt {a+b x} (c+d x)^{3/2}}+\frac {2 d (3 b c+a d) \sqrt {a+b x}}{3 a c (b c-a d)^2 (c+d x)^{3/2}}+\frac {2 d (3 b c-a d) (b c+3 a d) \sqrt {a+b x}}{3 a c^2 (b c-a d)^3 \sqrt {c+d x}}+\frac {8 \int \frac {3 (b c-a d)^3}{8 x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{3 a c^2 (b c-a d)^3}\\ &=\frac {2 b}{a (b c-a d) \sqrt {a+b x} (c+d x)^{3/2}}+\frac {2 d (3 b c+a d) \sqrt {a+b x}}{3 a c (b c-a d)^2 (c+d x)^{3/2}}+\frac {2 d (3 b c-a d) (b c+3 a d) \sqrt {a+b x}}{3 a c^2 (b c-a d)^3 \sqrt {c+d x}}+\frac {\int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{a c^2}\\ &=\frac {2 b}{a (b c-a d) \sqrt {a+b x} (c+d x)^{3/2}}+\frac {2 d (3 b c+a d) \sqrt {a+b x}}{3 a c (b c-a d)^2 (c+d x)^{3/2}}+\frac {2 d (3 b c-a d) (b c+3 a d) \sqrt {a+b x}}{3 a c^2 (b c-a d)^3 \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 c^2}\\ &=\frac {2 b}{a (b c-a d) \sqrt {a+b x} (c+d x)^{3/2}}+\frac {2 d (3 b c+a d) \sqrt {a+b x}}{3 a c (b c-a d)^2 (c+d x)^{3/2}}+\frac {2 d (3 b c-a d) (b c+3 a d) \sqrt {a+b x}}{3 a c^2 (b c-a d)^3 \sqrt {c+d x}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{a^{3/2} c^{5/2}}\\ \end {align*}

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Mathematica [A]  time = 0.41, size = 158, normalized size = 0.88 \begin {gather*} \frac {2 \left (-a^3 d^3 (4 c+3 d x)+a^2 b d^2 \left (9 c^2+4 c d x-3 d^2 x^2\right )+a b^2 c d^2 x (9 c+8 d x)+3 b^3 c^2 (c+d x)^2\right )}{3 a c^2 \sqrt {a+b x} (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^{3/2} c^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x*(a + b*x)^(3/2)*(c + d*x)^(5/2)),x]

[Out]

(2*(3*b^3*c^2*(c + d*x)^2 - a^3*d^3*(4*c + 3*d*x) + a*b^2*c*d^2*x*(9*c + 8*d*x) + a^2*b*d^2*(9*c^2 + 4*c*d*x -
3*d^2*x^2)))/(3*a*c^2*(b*c - a*d)^3*Sqrt[a + b*x]*(c + d*x)^(3/2)) - (2*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt
[a]*Sqrt[c + d*x])])/(a^(3/2)*c^(5/2))

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IntegrateAlgebraic [A]  time = 0.24, size = 150, normalized size = 0.83 \begin {gather*} \frac {2 (a+b x)^{3/2} \left (\frac {3 a^2 d^3 (c+d x)}{a+b x}-\frac {3 b^3 c^2 (c+d x)^2}{(a+b x)^2}-\frac {9 a b c d^2 (c+d x)}{a+b x}+a c d^3\right )}{3 a c^2 (c+d x)^{3/2} (a d-b c)^3}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {c+d x}}{\sqrt {c} \sqrt {a+b x}}\right )}{a^{3/2} c^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[1/(x*(a + b*x)^(3/2)*(c + d*x)^(5/2)),x]

[Out]

(2*(a + b*x)^(3/2)*(a*c*d^3 - (9*a*b*c*d^2*(c + d*x))/(a + b*x) + (3*a^2*d^3*(c + d*x))/(a + b*x) - (3*b^3*c^2
*(c + d*x)^2)/(a + b*x)^2))/(3*a*c^2*(-(b*c) + a*d)^3*(c + d*x)^(3/2)) - (2*ArcTanh[(Sqrt[a]*Sqrt[c + d*x])/(S
qrt[c]*Sqrt[a + b*x])])/(a^(3/2)*c^(5/2))

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fricas [B]  time = 4.76, size = 1336, normalized size = 7.42

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(b*x+a)^(3/2)/(d*x+c)^(5/2),x, algorithm="fricas")

[Out]

[1/6*(3*(a*b^3*c^5 - 3*a^2*b^2*c^4*d + 3*a^3*b*c^3*d^2 - a^4*c^2*d^3 + (b^4*c^3*d^2 - 3*a*b^3*c^2*d^3 + 3*a^2*
b^2*c*d^4 - a^3*b*d^5)*x^3 + (2*b^4*c^4*d - 5*a*b^3*c^3*d^2 + 3*a^2*b^2*c^2*d^3 + a^3*b*c*d^4 - a^4*d^5)*x^2 +
(b^4*c^5 - a*b^3*c^4*d - 3*a^2*b^2*c^3*d^2 + 5*a^3*b*c^2*d^3 - 2*a^4*c*d^4)*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*b^3*c^5 + 9*a^3*b*c^3*d^2 - 4*a^4*c^2*d^3 + (3*a*b^3*c^3*d^2 + 8*a^2*b^2*c^2*d^3
- 3*a^3*b*c*d^4)*x^2 + (6*a*b^3*c^4*d + 9*a^2*b^2*c^3*d^2 + 4*a^3*b*c^2*d^3 - 3*a^4*c*d^4)*x)*sqrt(b*x + a)*s
qrt(d*x + c))/(a^3*b^3*c^8 - 3*a^4*b^2*c^7*d + 3*a^5*b*c^6*d^2 - a^6*c^5*d^3 + (a^2*b^4*c^6*d^2 - 3*a^3*b^3*c^
5*d^3 + 3*a^4*b^2*c^4*d^4 - a^5*b*c^3*d^5)*x^3 + (2*a^2*b^4*c^7*d - 5*a^3*b^3*c^6*d^2 + 3*a^4*b^2*c^5*d^3 + a^
5*b*c^4*d^4 - a^6*c^3*d^5)*x^2 + (a^2*b^4*c^8 - a^3*b^3*c^7*d - 3*a^4*b^2*c^6*d^2 + 5*a^5*b*c^5*d^3 - 2*a^6*c^
4*d^4)*x), 1/3*(3*(a*b^3*c^5 - 3*a^2*b^2*c^4*d + 3*a^3*b*c^3*d^2 - a^4*c^2*d^3 + (b^4*c^3*d^2 - 3*a*b^3*c^2*d^
3 + 3*a^2*b^2*c*d^4 - a^3*b*d^5)*x^3 + (2*b^4*c^4*d - 5*a*b^3*c^3*d^2 + 3*a^2*b^2*c^2*d^3 + a^3*b*c*d^4 - a^4*
d^5)*x^2 + (b^4*c^5 - a*b^3*c^4*d - 3*a^2*b^2*c^3*d^2 + 5*a^3*b*c^2*d^3 - 2*a^4*c*d^4)*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^3*c^5 + 9*a^3*b*c^3*d^2 - 4*a^4*c^2*d^3 + (3*a*b^3*c^3*d^2 + 8*a^2*b^2*c^2*d^3 - 3*a^3*b*c*d^4)
*x^2 + (6*a*b^3*c^4*d + 9*a^2*b^2*c^3*d^2 + 4*a^3*b*c^2*d^3 - 3*a^4*c*d^4)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(a^
3*b^3*c^8 - 3*a^4*b^2*c^7*d + 3*a^5*b*c^6*d^2 - a^6*c^5*d^3 + (a^2*b^4*c^6*d^2 - 3*a^3*b^3*c^5*d^3 + 3*a^4*b^2
*c^4*d^4 - a^5*b*c^3*d^5)*x^3 + (2*a^2*b^4*c^7*d - 5*a^3*b^3*c^6*d^2 + 3*a^4*b^2*c^5*d^3 + a^5*b*c^4*d^4 - a^6
*c^3*d^5)*x^2 + (a^2*b^4*c^8 - a^3*b^3*c^7*d - 3*a^4*b^2*c^6*d^2 + 5*a^5*b*c^5*d^3 - 2*a^6*c^4*d^4)*x)]

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giac [B]  time = 5.13, size = 520, normalized size = 2.89 \begin {gather*} \frac {4 \, \sqrt {b d} b^{4}}{{\left (a b^{2} c^{2} {\left | b \right |} - 2 \, a^{2} b c d {\left | b \right |} + a^{3} d^{2} {\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 )}} + \frac {2 \, \sqrt {b x + a} {\left (\frac {{\left (8 \, b^{6} c^{5} d^{4} {\left | b \right |} - 19 \, a b^{5} c^{4} d^{5} {\left | b \right |} + 14 \, a^{2} b^{4} c^{3} d^{6} {\left | b \right |} - 3 \, a^{3} b^{3} c^{2} d^{7} {\left | b \right |}\right )} {\left (b x + a\right )}}{b^{7} c^{9} d - 5 \, a b^{6} c^{8} d^{2} + 10 \, a^{2} b^{5} c^{7} d^{3} - 10 \, a^{3} b^{4} c^{6} d^{4} + 5 \, a^{4} b^{3} c^{5} d^{5} - a^{5} b^{2} c^{4} d^{6}} + \frac {3 \, {\left (3 \, b^{7} c^{6} d^{3} {\left | b \right |} - 10 \, a b^{6} c^{5} d^{4} {\left | b \right |} + 12 \, a^{2} b^{5} c^{4} d^{5} {\left | b \right |} - 6 \, a^{3} b^{4} c^{3} d^{6} {\left | b \right |} + a^{4} b^{3} c^{2} d^{7} {\left | b \right |}\right )}}{b^{7} c^{9} d - 5 \, a b^{6} c^{8} d^{2} + 10 \, a^{2} b^{5} c^{7} d^{3} - 10 \, a^{3} b^{4} c^{6} d^{4} + 5 \, a^{4} b^{3} c^{5} d^{5} - a^{5} b^{2} c^{4} d^{6}}\right )}}{3 \, {\left (b^{2} c + {\left (b x + a\right )} b d - a b d\right )}^{\frac {3}{2}}} - \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 c^{2} {\left | b \right |}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(b*x+a)^(3/2)/(d*x+c)^(5/2),x, algorithm="giac")

[Out]

4*sqrt(b*d)*b^4/((a*b^2*c^2*abs(b) - 2*a^2*b*c*d*abs(b) + a^3*d^2*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)) + 2/3*sqrt(b*x + a)*((8*b^6*c^5*d^4*abs(b) - 19*a*b^5*c^4*d^5
*abs(b) + 14*a^2*b^4*c^3*d^6*abs(b) - 3*a^3*b^3*c^2*d^7*abs(b))*(b*x + a)/(b^7*c^9*d - 5*a*b^6*c^8*d^2 + 10*a^
2*b^5*c^7*d^3 - 10*a^3*b^4*c^6*d^4 + 5*a^4*b^3*c^5*d^5 - a^5*b^2*c^4*d^6) + 3*(3*b^7*c^6*d^3*abs(b) - 10*a*b^6
*c^5*d^4*abs(b) + 12*a^2*b^5*c^4*d^5*abs(b) - 6*a^3*b^4*c^3*d^6*abs(b) + a^4*b^3*c^2*d^7*abs(b))/(b^7*c^9*d -
5*a*b^6*c^8*d^2 + 10*a^2*b^5*c^7*d^3 - 10*a^3*b^4*c^6*d^4 + 5*a^4*b^3*c^5*d^5 - a^5*b^2*c^4*d^6))/(b^2*c + (b*
x + a)*b*d - a*b*d)^(3/2) - 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*c^2*abs(b))

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maple [B]  time = 0.04, size = 1253, normalized size = 6.96

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x/(b*x+a)^(3/2)/(d*x+c)^(5/2),x)

[Out]

-1/3*(-9*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^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*a^4*c*d^4+6*b^3*c^4*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+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*a^4*d^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*b^4*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)*a^4*c
^2*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)*a*b^3*c^5-9*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*c^3*d^2+9*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^2*c^4*d-6*x*a^3*d^4*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-8*a^3*c*d^3*(a*c)^(1/2)*((b*x+a)*(d*x+c
))^(1/2)+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^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^3*b^4*c^3*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^2*b^4*c^4*d+9*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^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^2*a^3*b*c*d^4-9*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^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^2*a*b^3*c^3*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*a^3
*b*c^2*d^3+9*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^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*b^3*c^4*d-6*x^2*a^2*b*d^4*(a*c)^(1/2)*((b*x+a)*(d*x+c))
^(1/2)+6*x^2*b^3*c^2*d^2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+12*x*b^3*c^3*d*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2
)+18*a^2*b*c^2*d^2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+16*x^2*a*b^2*c*d^3*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+
8*x*a^2*b*c*d^3*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+18*x*a*b^2*c^2*d^2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2))/a/
c^2/(a*c)^(1/2)/(a*d-b*c)^3/((b*x+a)*(d*x+c))^(1/2)/(d*x+c)^(3/2)/(b*x+a)^(1/2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{{\left (b x + a\right )}^{\frac {3}{2}} {\left (d x + c\right )}^{\frac {5}{2}} x}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(b*x+a)^(3/2)/(d*x+c)^(5/2),x, algorithm="maxima")

[Out]

integrate(1/((b*x + a)^(3/2)*(d*x + c)^(5/2)*x), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{x\,{\left (a+b\,x\right )}^{3/2}\,{\left (c+d\,x\right )}^{5/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x*(a + b*x)^(3/2)*(c + d*x)^(5/2)),x)

[Out]

int(1/(x*(a + b*x)^(3/2)*(c + d*x)^(5/2)), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x \left (a + b x\right )^{\frac {3}{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/(b*x+a)**(3/2)/(d*x+c)**(5/2),x)

[Out]

Integral(1/(x*(a + b*x)**(3/2)*(c + d*x)**(5/2)), x)

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