3.5.65 \(\int \frac {\sqrt {c+d x}}{x^2 (a+b x)^2} \, dx\) [465]

Optimal. Leaf size=140 \[ -\frac {2 b \sqrt {c+d x}}{a^2 (a+b x)}-\frac {\sqrt {c+d x}}{a x (a+b x)}+\frac {(4 b c-a d) \tanh ^{-1}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{a^3 \sqrt {c}}-\frac {\sqrt {b} (4 b c-3 a d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{a^3 \sqrt {b c-a d}} \]

[Out]

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

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Rubi [A]
time = 0.11, antiderivative size = 140, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {101, 156, 162, 65, 214} \begin {gather*} -\frac {\sqrt {b} (4 b c-3 a d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{a^3 \sqrt {b c-a d}}+\frac {(4 b c-a d) \tanh ^{-1}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{a^3 \sqrt {c}}-\frac {2 b \sqrt {c+d x}}{a^2 (a+b x)}-\frac {\sqrt {c+d x}}{a x (a+b x)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*b*Sqrt[c + d*x])/(a^2*(a + b*x)) - Sqrt[c + d*x]/(a*x*(a + b*x)) + ((4*b*c - a*d)*ArcTanh[Sqrt[c + d*x]/Sq
rt[c]])/(a^3*Sqrt[c]) - (Sqrt[b]*(4*b*c - 3*a*d)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[b*c - a*d]])/(a^3*Sqrt[b
*c - a*d])

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 101

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(a + b*
x)^(m + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/((m + 1)*(b*e - a*f))), x] - Dist[1/((m + 1)*(b*e - a*f)), Int[(a +
b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[d*e*n + c*f*(m + p + 2) + d*f*(m + n + p + 2)*x, x], x], x] /;
FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 0] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p
] || IntegersQ[p, m + n])

Rule 156

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] && ILtQ[m, -1]

Rule 162

Int[(((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :>
Dist[(b*g - a*h)/(b*c - a*d), Int[(e + f*x)^p/(a + b*x), x], x] - Dist[(d*g - c*h)/(b*c - a*d), Int[(e + f*x)
^p/(c + d*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x]

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

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Mathematica [A]
time = 0.67, size = 119, normalized size = 0.85 \begin {gather*} \frac {-\frac {a (a+2 b x) \sqrt {c+d x}}{x (a+b x)}+\frac {\sqrt {b} (4 b c-3 a d) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {-b c+a d}}\right )}{\sqrt {-b c+a d}}+\frac {(4 b c-a d) \tanh ^{-1}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{\sqrt {c}}}{a^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.09, size = 136, normalized size = 0.97

method	result	size
derivativedivides	\(2 d^{3} \left (-\frac {b \left (\frac {a d \sqrt {d x +c}}{2 b \left (d x +c \right )+2 a d -2 b c}+\frac {\left (3 a d -4 b c \right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{2 \sqrt {\left (a d -b c \right ) b}}\right )}{a^{3} d^{3}}+\frac {-\frac {a \sqrt {d x +c}}{2 x}-\frac {\left (a d -4 b c \right ) \arctanh \left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )}{2 \sqrt {c}}}{a^{3} d^{3}}\right )\)	\(136\)
default	\(2 d^{3} \left (-\frac {b \left (\frac {a d \sqrt {d x +c}}{2 b \left (d x +c \right )+2 a d -2 b c}+\frac {\left (3 a d -4 b c \right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{2 \sqrt {\left (a d -b c \right ) b}}\right )}{a^{3} d^{3}}+\frac {-\frac {a \sqrt {d x +c}}{2 x}-\frac {\left (a d -4 b c \right ) \arctanh \left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )}{2 \sqrt {c}}}{a^{3} d^{3}}\right )\)	\(136\)
risch	\(-\frac {\sqrt {d x +c}}{a^{2} x}-\frac {d b \sqrt {d x +c}}{a^{2} \left (b d x +a d \right )}-\frac {3 d b \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{a^{2} \sqrt {\left (a d -b c \right ) b}}+\frac {4 b^{2} \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right ) c}{a^{3} \sqrt {\left (a d -b c \right ) b}}-\frac {d \arctanh \left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )}{a^{2} \sqrt {c}}+\frac {4 \sqrt {c}\, \arctanh \left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right ) b}{a^{3}}\)	\(167\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*d^3*(-1/a^3/d^3*b*(1/2*a*d*(d*x+c)^(1/2)/(b*(d*x+c)+a*d-b*c)+1/2*(3*a*d-4*b*c)/((a*d-b*c)*b)^(1/2)*arctan(b*
(d*x+c)^(1/2)/((a*d-b*c)*b)^(1/2)))+1/a^3/d^3*(-1/2*a*(d*x+c)^(1/2)/x-1/2*(a*d-4*b*c)/c^(1/2)*arctanh((d*x+c)^
(1/2)/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((d*x+c)^(1/2)/x^2/(b*x+a)^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 detail

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Fricas [A]
time = 1.08, size = 798, normalized size = 5.70 \begin {gather*} \left [-\frac {{\left ({\left (4 \, b^{2} c^{2} - 3 \, a b c d\right )} x^{2} + {\left (4 \, a b c^{2} - 3 \, a^{2} c d\right )} x\right )} \sqrt {\frac {b}{b c - a d}} \log \left (\frac {b d x + 2 \, b c - a d + 2 \, {\left (b c - a d\right )} \sqrt {d x + c} \sqrt {\frac {b}{b c - a d}}}{b x + a}\right ) + {\left ({\left (4 \, b^{2} c - a b d\right )} x^{2} + {\left (4 \, a b c - a^{2} d\right )} x\right )} \sqrt {c} \log \left (\frac {d x - 2 \, \sqrt {d x + c} \sqrt {c} + 2 \, c}{x}\right ) + 2 \, {\left (2 \, a b c x + a^{2} c\right )} \sqrt {d x + c}}{2 \, {\left (a^{3} b c x^{2} + a^{4} c x\right )}}, -\frac {2 \, {\left ({\left (4 \, b^{2} c^{2} - 3 \, a b c d\right )} x^{2} + {\left (4 \, a b c^{2} - 3 \, a^{2} c d\right )} x\right )} \sqrt {-\frac {b}{b c - a d}} \arctan \left (-\frac {{\left (b c - a d\right )} \sqrt {d x + c} \sqrt {-\frac {b}{b c - a d}}}{b d x + b c}\right ) + {\left ({\left (4 \, b^{2} c - a b d\right )} x^{2} + {\left (4 \, a b c - a^{2} d\right )} x\right )} \sqrt {c} \log \left (\frac {d x - 2 \, \sqrt {d x + c} \sqrt {c} + 2 \, c}{x}\right ) + 2 \, {\left (2 \, a b c x + a^{2} c\right )} \sqrt {d x + c}}{2 \, {\left (a^{3} b c x^{2} + a^{4} c x\right )}}, -\frac {2 \, {\left ({\left (4 \, b^{2} c - a b d\right )} x^{2} + {\left (4 \, a b c - a^{2} d\right )} x\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {d x + c} \sqrt {-c}}{c}\right ) + {\left ({\left (4 \, b^{2} c^{2} - 3 \, a b c d\right )} x^{2} + {\left (4 \, a b c^{2} - 3 \, a^{2} c d\right )} x\right )} \sqrt {\frac {b}{b c - a d}} \log \left (\frac {b d x + 2 \, b c - a d + 2 \, {\left (b c - a d\right )} \sqrt {d x + c} \sqrt {\frac {b}{b c - a d}}}{b x + a}\right ) + 2 \, {\left (2 \, a b c x + a^{2} c\right )} \sqrt {d x + c}}{2 \, {\left (a^{3} b c x^{2} + a^{4} c x\right )}}, -\frac {{\left ({\left (4 \, b^{2} c^{2} - 3 \, a b c d\right )} x^{2} + {\left (4 \, a b c^{2} - 3 \, a^{2} c d\right )} x\right )} \sqrt {-\frac {b}{b c - a d}} \arctan \left (-\frac {{\left (b c - a d\right )} \sqrt {d x + c} \sqrt {-\frac {b}{b c - a d}}}{b d x + b c}\right ) + {\left ({\left (4 \, b^{2} c - a b d\right )} x^{2} + {\left (4 \, a b c - a^{2} d\right )} x\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {d x + c} \sqrt {-c}}{c}\right ) + {\left (2 \, a b c x + a^{2} c\right )} \sqrt {d x + c}}{a^{3} b c x^{2} + a^{4} c x}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 790 vs. \(2 (122) = 244\).
time = 27.29, size = 790, normalized size = 5.64 \begin {gather*} \frac {2 b^{2} c d \sqrt {c + d x}}{2 a^{4} d^{2} - 2 a^{3} b c d + 2 a^{3} b d^{2} x - 2 a^{2} b^{2} c d x} - \frac {2 b d^{2} \sqrt {c + d x}}{2 a^{3} d^{2} - 2 a^{2} b c d + 2 a^{2} b d^{2} x - 2 a b^{2} c d x} + \frac {b d^{2} \sqrt {- \frac {1}{b \left (a d - b c\right )^{3}}} \log {\left (- a^{2} d^{2} \sqrt {- \frac {1}{b \left (a d - b c\right )^{3}}} + 2 a b c d \sqrt {- \frac {1}{b \left (a d - b c\right )^{3}}} - b^{2} c^{2} \sqrt {- \frac {1}{b \left (a d - b c\right )^{3}}} + \sqrt {c + d x} \right )}}{2 a} - \frac {b d^{2} \sqrt {- \frac {1}{b \left (a d - b c\right )^{3}}} \log {\left (a^{2} d^{2} \sqrt {- \frac {1}{b \left (a d - b c\right )^{3}}} - 2 a b c d \sqrt {- \frac {1}{b \left (a d - b c\right )^{3}}} + b^{2} c^{2} \sqrt {- \frac {1}{b \left (a d - b c\right )^{3}}} + \sqrt {c + d x} \right )}}{2 a} - \frac {b^{2} c d \sqrt {- \frac {1}{b \left (a d - b c\right )^{3}}} \log {\left (- a^{2} d^{2} \sqrt {- \frac {1}{b \left (a d - b c\right )^{3}}} + 2 a b c d \sqrt {- \frac {1}{b \left (a d - b c\right )^{3}}} - b^{2} c^{2} \sqrt {- \frac {1}{b \left (a d - b c\right )^{3}}} + \sqrt {c + d x} \right )}}{2 a^{2}} + \frac {b^{2} c d \sqrt {- \frac {1}{b \left (a d - b c\right )^{3}}} \log {\left (a^{2} d^{2} \sqrt {- \frac {1}{b \left (a d - b c\right )^{3}}} - 2 a b c d \sqrt {- \frac {1}{b \left (a d - b c\right )^{3}}} + b^{2} c^{2} \sqrt {- \frac {1}{b \left (a d - b c\right )^{3}}} + \sqrt {c + d x} \right )}}{2 a^{2}} - \frac {c d \sqrt {\frac {1}{c^{3}}} \log {\left (- c^{2} \sqrt {\frac {1}{c^{3}}} + \sqrt {c + d x} \right )}}{2 a^{2}} + \frac {c d \sqrt {\frac {1}{c^{3}}} \log {\left (c^{2} \sqrt {\frac {1}{c^{3}}} + \sqrt {c + d x} \right )}}{2 a^{2}} - \frac {2 d \operatorname {atan}{\left (\frac {\sqrt {c + d x}}{\sqrt {\frac {a d}{b} - c}} \right )}}{a^{2} \sqrt {\frac {a d}{b} - c}} + \frac {2 d \operatorname {atan}{\left (\frac {\sqrt {c + d x}}{\sqrt {- c}} \right )}}{a^{2} \sqrt {- c}} - \frac {\sqrt {c + d x}}{a^{2} x} + \frac {4 b c \operatorname {atan}{\left (\frac {\sqrt {c + d x}}{\sqrt {\frac {a d}{b} - c}} \right )}}{a^{3} \sqrt {\frac {a d}{b} - c}} - \frac {4 b c \operatorname {atan}{\left (\frac {\sqrt {c + d x}}{\sqrt {- c}} \right )}}{a^{3} \sqrt {- c}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*b**2*c*d*sqrt(c + d*x)/(2*a**4*d**2 - 2*a**3*b*c*d + 2*a**3*b*d**2*x - 2*a**2*b**2*c*d*x) - 2*b*d**2*sqrt(c
+ d*x)/(2*a**3*d**2 - 2*a**2*b*c*d + 2*a**2*b*d**2*x - 2*a*b**2*c*d*x) + b*d**2*sqrt(-1/(b*(a*d - b*c)**3))*lo
g(-a**2*d**2*sqrt(-1/(b*(a*d - b*c)**3)) + 2*a*b*c*d*sqrt(-1/(b*(a*d - b*c)**3)) - b**2*c**2*sqrt(-1/(b*(a*d -
b*c)**3)) + sqrt(c + d*x))/(2*a) - b*d**2*sqrt(-1/(b*(a*d - b*c)**3))*log(a**2*d**2*sqrt(-1/(b*(a*d - b*c)**3
)) - 2*a*b*c*d*sqrt(-1/(b*(a*d - b*c)**3)) + b**2*c**2*sqrt(-1/(b*(a*d - b*c)**3)) + sqrt(c + d*x))/(2*a) - b*
*2*c*d*sqrt(-1/(b*(a*d - b*c)**3))*log(-a**2*d**2*sqrt(-1/(b*(a*d - b*c)**3)) + 2*a*b*c*d*sqrt(-1/(b*(a*d - b*
c)**3)) - b**2*c**2*sqrt(-1/(b*(a*d - b*c)**3)) + sqrt(c + d*x))/(2*a**2) + b**2*c*d*sqrt(-1/(b*(a*d - b*c)**3
))*log(a**2*d**2*sqrt(-1/(b*(a*d - b*c)**3)) - 2*a*b*c*d*sqrt(-1/(b*(a*d - b*c)**3)) + b**2*c**2*sqrt(-1/(b*(a
*d - b*c)**3)) + sqrt(c + d*x))/(2*a**2) - c*d*sqrt(c**(-3))*log(-c**2*sqrt(c**(-3)) + sqrt(c + d*x))/(2*a**2)
+ c*d*sqrt(c**(-3))*log(c**2*sqrt(c**(-3)) + sqrt(c + d*x))/(2*a**2) - 2*d*atan(sqrt(c + d*x)/sqrt(a*d/b - c)
)/(a**2*sqrt(a*d/b - c)) + 2*d*atan(sqrt(c + d*x)/sqrt(-c))/(a**2*sqrt(-c)) - sqrt(c + d*x)/(a**2*x) + 4*b*c*a
tan(sqrt(c + d*x)/sqrt(a*d/b - c))/(a**3*sqrt(a*d/b - c)) - 4*b*c*atan(sqrt(c + d*x)/sqrt(-c))/(a**3*sqrt(-c))

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Giac [A]
time = 0.56, size = 166, normalized size = 1.19 \begin {gather*} \frac {{\left (4 \, b^{2} c - 3 \, a b d\right )} \arctan \left (\frac {\sqrt {d x + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{\sqrt {-b^{2} c + a b d} a^{3}} - \frac {{\left (4 \, b c - a d\right )} \arctan \left (\frac {\sqrt {d x + c}}{\sqrt {-c}}\right )}{a^{3} \sqrt {-c}} - \frac {2 \, {\left (d x + c\right )}^{\frac {3}{2}} b d - 2 \, \sqrt {d x + c} b c d + \sqrt {d x + c} a d^{2}}{{\left ({\left (d x + c\right )}^{2} b - 2 \, {\left (d x + c\right )} b c + b c^{2} + {\left (d x + c\right )} a d - a c d\right )} a^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(4*b^2*c - 3*a*b*d)*arctan(sqrt(d*x + c)*b/sqrt(-b^2*c + a*b*d))/(sqrt(-b^2*c + a*b*d)*a^3) - (4*b*c - a*d)*ar
ctan(sqrt(d*x + c)/sqrt(-c))/(a^3*sqrt(-c)) - (2*(d*x + c)^(3/2)*b*d - 2*sqrt(d*x + c)*b*c*d + sqrt(d*x + c)*a
*d^2)/(((d*x + c)^2*b - 2*(d*x + c)*b*c + b*c^2 + (d*x + c)*a*d - a*c*d)*a^2)

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Mupad [B]
time = 0.81, size = 1175, normalized size = 8.39 \begin {gather*} -\frac {\frac {2\,b\,d\,{\left (c+d\,x\right )}^{3/2}}{a^2}+\frac {d\,\left (a\,d-2\,b\,c\right )\,\sqrt {c+d\,x}}{a^2}}{\left (a\,d-2\,b\,c\right )\,\left (c+d\,x\right )+b\,{\left (c+d\,x\right )}^2+b\,c^2-a\,c\,d}-\frac {\mathrm {atanh}\left (\frac {8\,b^3\,d^5\,\sqrt {c+d\,x}}{\sqrt {c}\,\left (8\,b^3\,d^5-\frac {2\,a\,b^2\,d^6}{c}\right )}-\frac {2\,b^2\,d^6\,\sqrt {c+d\,x}}{c^{3/2}\,\left (\frac {8\,b^3\,d^5}{a}-\frac {2\,b^2\,d^6}{c}\right )}\right )\,\left (a\,d-4\,b\,c\right )}{a^3\,\sqrt {c}}+\frac {\mathrm {atan}\left (\frac {\frac {\sqrt {-b\,\left (a\,d-b\,c\right )}\,\left (3\,a\,d-4\,b\,c\right )\,\left (\frac {4\,\sqrt {c+d\,x}\,\left (5\,a^2\,b^3\,d^4-16\,a\,b^4\,c\,d^3+16\,b^5\,c^2\,d^2\right )}{a^4}-\frac {\sqrt {-b\,\left (a\,d-b\,c\right )}\,\left (3\,a\,d-4\,b\,c\right )\,\left (\frac {2\,\left (2\,a^7\,b^2\,d^4-4\,a^6\,b^3\,c\,d^3\right )}{a^6}-\frac {2\,\left (2\,a^7\,b^2\,d^3-4\,a^6\,b^3\,c\,d^2\right )\,\sqrt {-b\,\left (a\,d-b\,c\right )}\,\left (3\,a\,d-4\,b\,c\right )\,\sqrt {c+d\,x}}{a^4\,\left (a^4\,d-a^3\,b\,c\right )}\right )}{2\,\left (a^4\,d-a^3\,b\,c\right )}\right )\,1{}\mathrm {i}}{2\,\left (a^4\,d-a^3\,b\,c\right )}+\frac {\sqrt {-b\,\left (a\,d-b\,c\right )}\,\left (3\,a\,d-4\,b\,c\right )\,\left (\frac {4\,\sqrt {c+d\,x}\,\left (5\,a^2\,b^3\,d^4-16\,a\,b^4\,c\,d^3+16\,b^5\,c^2\,d^2\right )}{a^4}+\frac {\sqrt {-b\,\left (a\,d-b\,c\right )}\,\left (3\,a\,d-4\,b\,c\right )\,\left (\frac {2\,\left (2\,a^7\,b^2\,d^4-4\,a^6\,b^3\,c\,d^3\right )}{a^6}+\frac {2\,\left (2\,a^7\,b^2\,d^3-4\,a^6\,b^3\,c\,d^2\right )\,\sqrt {-b\,\left (a\,d-b\,c\right )}\,\left (3\,a\,d-4\,b\,c\right )\,\sqrt {c+d\,x}}{a^4\,\left (a^4\,d-a^3\,b\,c\right )}\right )}{2\,\left (a^4\,d-a^3\,b\,c\right )}\right )\,1{}\mathrm {i}}{2\,\left (a^4\,d-a^3\,b\,c\right )}}{\frac {4\,\left (3\,a^2\,b^3\,d^5-16\,a\,b^4\,c\,d^4+16\,b^5\,c^2\,d^3\right )}{a^6}-\frac {\sqrt {-b\,\left (a\,d-b\,c\right )}\,\left (3\,a\,d-4\,b\,c\right )\,\left (\frac {4\,\sqrt {c+d\,x}\,\left (5\,a^2\,b^3\,d^4-16\,a\,b^4\,c\,d^3+16\,b^5\,c^2\,d^2\right )}{a^4}-\frac {\sqrt {-b\,\left (a\,d-b\,c\right )}\,\left (3\,a\,d-4\,b\,c\right )\,\left (\frac {2\,\left (2\,a^7\,b^2\,d^4-4\,a^6\,b^3\,c\,d^3\right )}{a^6}-\frac {2\,\left (2\,a^7\,b^2\,d^3-4\,a^6\,b^3\,c\,d^2\right )\,\sqrt {-b\,\left (a\,d-b\,c\right )}\,\left (3\,a\,d-4\,b\,c\right )\,\sqrt {c+d\,x}}{a^4\,\left (a^4\,d-a^3\,b\,c\right )}\right )}{2\,\left (a^4\,d-a^3\,b\,c\right )}\right )}{2\,\left (a^4\,d-a^3\,b\,c\right )}+\frac {\sqrt {-b\,\left (a\,d-b\,c\right )}\,\left (3\,a\,d-4\,b\,c\right )\,\left (\frac {4\,\sqrt {c+d\,x}\,\left (5\,a^2\,b^3\,d^4-16\,a\,b^4\,c\,d^3+16\,b^5\,c^2\,d^2\right )}{a^4}+\frac {\sqrt {-b\,\left (a\,d-b\,c\right )}\,\left (3\,a\,d-4\,b\,c\right )\,\left (\frac {2\,\left (2\,a^7\,b^2\,d^4-4\,a^6\,b^3\,c\,d^3\right )}{a^6}+\frac {2\,\left (2\,a^7\,b^2\,d^3-4\,a^6\,b^3\,c\,d^2\right )\,\sqrt {-b\,\left (a\,d-b\,c\right )}\,\left (3\,a\,d-4\,b\,c\right )\,\sqrt {c+d\,x}}{a^4\,\left (a^4\,d-a^3\,b\,c\right )}\right )}{2\,\left (a^4\,d-a^3\,b\,c\right )}\right )}{2\,\left (a^4\,d-a^3\,b\,c\right )}}\right )\,\sqrt {-b\,\left (a\,d-b\,c\right )}\,\left (3\,a\,d-4\,b\,c\right )\,1{}\mathrm {i}}{a^4\,d-a^3\,b\,c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(atan((((-b*(a*d - b*c))^(1/2)*(3*a*d - 4*b*c)*((4*(c + d*x)^(1/2)*(5*a^2*b^3*d^4 + 16*b^5*c^2*d^2 - 16*a*b^4*
c*d^3))/a^4 - ((-b*(a*d - b*c))^(1/2)*(3*a*d - 4*b*c)*((2*(2*a^7*b^2*d^4 - 4*a^6*b^3*c*d^3))/a^6 - (2*(2*a^7*b
^2*d^3 - 4*a^6*b^3*c*d^2)*(-b*(a*d - b*c))^(1/2)*(3*a*d - 4*b*c)*(c + d*x)^(1/2))/(a^4*(a^4*d - a^3*b*c))))/(2
*(a^4*d - a^3*b*c)))*1i)/(2*(a^4*d - a^3*b*c)) + ((-b*(a*d - b*c))^(1/2)*(3*a*d - 4*b*c)*((4*(c + d*x)^(1/2)*(
5*a^2*b^3*d^4 + 16*b^5*c^2*d^2 - 16*a*b^4*c*d^3))/a^4 + ((-b*(a*d - b*c))^(1/2)*(3*a*d - 4*b*c)*((2*(2*a^7*b^2
*d^4 - 4*a^6*b^3*c*d^3))/a^6 + (2*(2*a^7*b^2*d^3 - 4*a^6*b^3*c*d^2)*(-b*(a*d - b*c))^(1/2)*(3*a*d - 4*b*c)*(c
+ d*x)^(1/2))/(a^4*(a^4*d - a^3*b*c))))/(2*(a^4*d - a^3*b*c)))*1i)/(2*(a^4*d - a^3*b*c)))/((4*(3*a^2*b^3*d^5 +
16*b^5*c^2*d^3 - 16*a*b^4*c*d^4))/a^6 - ((-b*(a*d - b*c))^(1/2)*(3*a*d - 4*b*c)*((4*(c + d*x)^(1/2)*(5*a^2*b^
3*d^4 + 16*b^5*c^2*d^2 - 16*a*b^4*c*d^3))/a^4 - ((-b*(a*d - b*c))^(1/2)*(3*a*d - 4*b*c)*((2*(2*a^7*b^2*d^4 - 4
*a^6*b^3*c*d^3))/a^6 - (2*(2*a^7*b^2*d^3 - 4*a^6*b^3*c*d^2)*(-b*(a*d - b*c))^(1/2)*(3*a*d - 4*b*c)*(c + d*x)^(
1/2))/(a^4*(a^4*d - a^3*b*c))))/(2*(a^4*d - a^3*b*c))))/(2*(a^4*d - a^3*b*c)) + ((-b*(a*d - b*c))^(1/2)*(3*a*d
- 4*b*c)*((4*(c + d*x)^(1/2)*(5*a^2*b^3*d^4 + 16*b^5*c^2*d^2 - 16*a*b^4*c*d^3))/a^4 + ((-b*(a*d - b*c))^(1/2)
*(3*a*d - 4*b*c)*((2*(2*a^7*b^2*d^4 - 4*a^6*b^3*c*d^3))/a^6 + (2*(2*a^7*b^2*d^3 - 4*a^6*b^3*c*d^2)*(-b*(a*d -
b*c))^(1/2)*(3*a*d - 4*b*c)*(c + d*x)^(1/2))/(a^4*(a^4*d - a^3*b*c))))/(2*(a^4*d - a^3*b*c))))/(2*(a^4*d - a^3
*b*c))))*(-b*(a*d - b*c))^(1/2)*(3*a*d - 4*b*c)*1i)/(a^4*d - a^3*b*c) - (atanh((8*b^3*d^5*(c + d*x)^(1/2))/(c^
(1/2)*(8*b^3*d^5 - (2*a*b^2*d^6)/c)) - (2*b^2*d^6*(c + d*x)^(1/2))/(c^(3/2)*((8*b^3*d^5)/a - (2*b^2*d^6)/c)))*
(a*d - 4*b*c))/(a^3*c^(1/2)) - ((2*b*d*(c + d*x)^(3/2))/a^2 + (d*(a*d - 2*b*c)*(c + d*x)^(1/2))/a^2)/((a*d - 2
*b*c)*(c + d*x) + b*(c + d*x)^2 + b*c^2 - a*c*d)

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