3.8.53 \(\int \frac {1}{x \sqrt {a+b x} (c+d x)^{5/2}} \, dx\) [753]

Optimal. Leaf size=124 \[ -\frac {2 d \sqrt {a+b x}}{3 c (b c-a d) (c+d x)^{3/2}}-\frac {2 d (5 b c-3 a d) \sqrt {a+b x}}{3 c^2 (b c-a d)^2 \sqrt {c+d x}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{\sqrt {a} c^{5/2}} \]

[Out]

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

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Rubi [A]
time = 0.05, antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {106, 157, 12, 95, 214} \begin {gather*} -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{\sqrt {a} c^{5/2}}-\frac {2 d \sqrt {a+b x} (5 b c-3 a d)}{3 c^2 \sqrt {c+d x} (b c-a d)^2}-\frac {2 d \sqrt {a+b x}}{3 c (c+d x)^{3/2} (b c-a d)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*d*Sqrt[a + b*x])/(3*c*(b*c - a*d)*(c + d*x)^(3/2)) - (2*d*(5*b*c - 3*a*d)*Sqrt[a + b*x])/(3*c^2*(b*c - a*d
)^2*Sqrt[c + d*x]) - (2*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(Sqrt[a]*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 95

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 106

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

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Mathematica [A]
time = 0.17, size = 103, normalized size = 0.83 \begin {gather*} \frac {2 d \sqrt {a+b x} \left (-6 b c+3 a d+\frac {c d (a+b x)}{c+d x}\right )}{3 c^2 (b c-a d)^2 \sqrt {c+d x}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{\sqrt {a} c^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*d*Sqrt[a + b*x]*(-6*b*c + 3*a*d + (c*d*(a + b*x))/(c + d*x)))/(3*c^2*(b*c - a*d)^2*Sqrt[c + d*x]) - (2*ArcT
anh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(Sqrt[a]*c^(5/2))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(585\) vs. \(2(100)=200\).
time = 0.07, size = 586, normalized size = 4.73

method	result	size
default	\(-\frac {\sqrt {b x +a}\, \left (3 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) a^{2} d^{4} x^{2}-6 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) a b c \,d^{3} x^{2}+3 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) b^{2} c^{2} d^{2} x^{2}+6 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) a^{2} c \,d^{3} x -12 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) a b \,c^{2} d^{2} x +6 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) b^{2} c^{3} d x +3 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) a^{2} c^{2} d^{2}-6 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) a b \,c^{3} d +3 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) b^{2} c^{4}-6 a \,d^{3} x \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+10 \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \sqrt {a c}\, b c \,d^{2} x -8 a c \,d^{2} \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+12 \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \sqrt {a c}\, b \,c^{2} d \right )}{3 \sqrt {a c}\, \left (a d -b c \right )^{2} \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \left (d x +c \right )^{\frac {3}{2}} c^{2}}\)	\(586\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(b*x+a)^(1/2)*(3*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)+2*a*c)/x)*a^2*d^4*x^2-6*ln((a*d*x+
b*c*x+2*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)+2*a*c)/x)*a*b*c*d^3*x^2+3*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((d*x+c)*(
b*x+a))^(1/2)+2*a*c)/x)*b^2*c^2*d^2*x^2+6*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)+2*a*c)/x)*a^2*
c*d^3*x-12*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)+2*a*c)/x)*a*b*c^2*d^2*x+6*ln((a*d*x+b*c*x+2*(
a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)+2*a*c)/x)*b^2*c^3*d*x+3*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/
2)+2*a*c)/x)*a^2*c^2*d^2-6*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)+2*a*c)/x)*a*b*c^3*d+3*ln((a*d
*x+b*c*x+2*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)+2*a*c)/x)*b^2*c^4-6*a*d^3*x*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)
+10*((d*x+c)*(b*x+a))^(1/2)*(a*c)^(1/2)*b*c*d^2*x-8*a*c*d^2*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)+12*((d*x+c)*(b
*x+a))^(1/2)*(a*c)^(1/2)*b*c^2*d)/(a*c)^(1/2)/(a*d-b*c)^2/((d*x+c)*(b*x+a))^(1/2)/(d*x+c)^(3/2)/c^2

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 329 vs. \(2 (100) = 200\).
time = 1.63, size = 678, normalized size = 5.47 \begin {gather*} \left [\frac {3 \, {\left (b^{2} c^{4} - 2 \, a b c^{3} d + a^{2} c^{2} d^{2} + {\left (b^{2} c^{2} d^{2} - 2 \, a b c d^{3} + a^{2} d^{4}\right )} x^{2} + 2 \, {\left (b^{2} c^{3} d - 2 \, a b c^{2} d^{2} + a^{2} 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 (6 \, a b c^{3} d - 4 \, a^{2} c^{2} d^{2} + {\left (5 \, a b c^{2} d^{2} - 3 \, a^{2} c d^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{6 \, {\left (a b^{2} c^{7} - 2 \, a^{2} b c^{6} d + a^{3} c^{5} d^{2} + {\left (a b^{2} c^{5} d^{2} - 2 \, a^{2} b c^{4} d^{3} + a^{3} c^{3} d^{4}\right )} x^{2} + 2 \, {\left (a b^{2} c^{6} d - 2 \, a^{2} b c^{5} d^{2} + a^{3} c^{4} d^{3}\right )} x\right )}}, \frac {3 \, {\left (b^{2} c^{4} - 2 \, a b c^{3} d + a^{2} c^{2} d^{2} + {\left (b^{2} c^{2} d^{2} - 2 \, a b c d^{3} + a^{2} d^{4}\right )} x^{2} + 2 \, {\left (b^{2} c^{3} d - 2 \, a b c^{2} d^{2} + a^{2} 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 (6 \, a b c^{3} d - 4 \, a^{2} c^{2} d^{2} + {\left (5 \, a b c^{2} d^{2} - 3 \, a^{2} c d^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{3 \, {\left (a b^{2} c^{7} - 2 \, a^{2} b c^{6} d + a^{3} c^{5} d^{2} + {\left (a b^{2} c^{5} d^{2} - 2 \, a^{2} b c^{4} d^{3} + a^{3} c^{3} d^{4}\right )} x^{2} + 2 \, {\left (a b^{2} c^{6} d - 2 \, a^{2} b c^{5} d^{2} + a^{3} c^{4} d^{3}\right )} x\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 271 vs. \(2 (100) = 200\).
time = 0.62, size = 271, normalized size = 2.19 \begin {gather*} -\frac {2 \, \sqrt {b x + a} {\left (\frac {{\left (5 \, b^{4} c^{3} d^{3} {\left | b \right |} - 3 \, a b^{3} c^{2} d^{4} {\left | b \right |}\right )} {\left (b x + a\right )}}{b^{4} c^{6} d - 2 \, a b^{3} c^{5} d^{2} + a^{2} b^{2} c^{4} d^{3}} + \frac {3 \, {\left (2 \, b^{5} c^{4} d^{2} {\left | b \right |} - 3 \, a b^{4} c^{3} d^{3} {\left | b \right |} + a^{2} b^{3} c^{2} d^{4} {\left | b \right |}\right )}}{b^{4} c^{6} d - 2 \, a b^{3} c^{5} d^{2} + a^{2} b^{2} c^{4} d^{3}}\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} c^{2} {\left | b \right |}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

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