3.1.13 \(\int \frac {\sqrt {c+d x} (e+f x)}{x (a+b x)^2} \, dx\)
Optimal. Leaf size=127 \[ \frac {\left (2 b^2 c e-a d (a f+b e)\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{a^2 b^{3/2} \sqrt {b c-a d}}-\frac {2 \sqrt {c} e \tanh ^{-1}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{a^2}+\frac {\sqrt {c+d x} (b e-a f)}{a b (a+b x)} \]
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Rubi [A] time = 0.11, antiderivative size = 127, normalized size of antiderivative = 1.00,
number of steps used = 6, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used =
{149, 156, 63, 208} \begin {gather*} \frac {\left (2 b^2 c e-a d (a f+b e)\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{a^2 b^{3/2} \sqrt {b c-a d}}-\frac {2 \sqrt {c} e \tanh ^{-1}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{a^2}+\frac {\sqrt {c+d x} (b e-a f)}{a b (a+b x)} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[(Sqrt[c + d*x]*(e + f*x))/(x*(a + b*x)^2),x]
[Out]
((b*e - a*f)*Sqrt[c + d*x])/(a*b*(a + b*x)) - (2*Sqrt[c]*e*ArcTanh[Sqrt[c + d*x]/Sqrt[c]])/a^2 + ((2*b^2*c*e -
a*d*(b*e + a*f))*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[b*c - a*d]])/(a^2*b^(3/2)*Sqrt[b*c - a*d])
Rule 63
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 149
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*(e + f*x)^(p + 1))/(b*(b*e - a*f)*(m + 1)), x] - Dist[1
/(b*(b*e - a*f)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[b*c*(f*g - e*h)*(m + 1) + (
b*g - a*h)*(d*e*n + c*f*(p + 1)) + d*(b*(f*g - e*h)*(m + 1) + f*(b*g - a*h)*(n + p + 1))*x, x], x], x] /; Free
Q[{a, b, c, d, e, f, g, h, p}, x] && LtQ[m, -1] && GtQ[n, 0] && IntegerQ[m]
Rule 156
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 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 {\sqrt {c+d x} (e+f x)}{x (a+b x)^2} \, dx &=\frac {(b e-a f) \sqrt {c+d x}}{a b (a+b x)}-\frac {\int \frac {-b c e-\frac {1}{2} d (b e+a f) x}{x (a+b x) \sqrt {c+d x}} \, dx}{a b}\\ &=\frac {(b e-a f) \sqrt {c+d x}}{a b (a+b x)}+\frac {(c e) \int \frac {1}{x \sqrt {c+d x}} \, dx}{a^2}+\frac {\left (-b^2 c e+\frac {1}{2} a d (b e+a f)\right ) \int \frac {1}{(a+b x) \sqrt {c+d x}} \, dx}{a^2 b}\\ &=\frac {(b e-a f) \sqrt {c+d x}}{a b (a+b x)}+\frac {(2 c e) \operatorname {Subst}\left (\int \frac {1}{-\frac {c}{d}+\frac {x^2}{d}} \, dx,x,\sqrt {c+d x}\right )}{a^2 d}+\frac {\left (2 \left (-b^2 c e+\frac {1}{2} a d (b e+a f)\right )\right ) \operatorname {Subst}\left (\int \frac {1}{a-\frac {b c}{d}+\frac {b x^2}{d}} \, dx,x,\sqrt {c+d x}\right )}{a^2 b d}\\ &=\frac {(b e-a f) \sqrt {c+d x}}{a b (a+b x)}-\frac {2 \sqrt {c} e \tanh ^{-1}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{a^2}+\frac {\left (2 b^2 c e-a d (b e+a f)\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{a^2 b^{3/2} \sqrt {b c-a d}}\\ \end {align*}
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Mathematica [A] time = 0.44, size = 124, normalized size = 0.98 \begin {gather*} \frac {-\frac {\left (a^2 d f+a b d e-2 b^2 c e\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{b^{3/2} \sqrt {b c-a d}}+\frac {a \sqrt {c+d x} (b e-a f)}{b (a+b x)}-2 \sqrt {c} e \tanh ^{-1}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{a^2} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(Sqrt[c + d*x]*(e + f*x))/(x*(a + b*x)^2),x]
[Out]
((a*(b*e - a*f)*Sqrt[c + d*x])/(b*(a + b*x)) - 2*Sqrt[c]*e*ArcTanh[Sqrt[c + d*x]/Sqrt[c]] - ((-2*b^2*c*e + a*b
*d*e + a^2*d*f)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[b*c - a*d]])/(b^(3/2)*Sqrt[b*c - a*d]))/a^2
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IntegrateAlgebraic [A] time = 0.50, size = 151, normalized size = 1.19 \begin {gather*} \frac {\left (a^2 (-d) f-a b d e+2 b^2 c e\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x} \sqrt {a d-b c}}{b c-a d}\right )}{a^2 b^{3/2} \sqrt {a d-b c}}-\frac {2 \sqrt {c} e \tanh ^{-1}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{a^2}-\frac {d \sqrt {c+d x} (a f-b e)}{a b (a d+b (c+d x)-b c)} \end {gather*}
Antiderivative was successfully verified.
[In]
IntegrateAlgebraic[(Sqrt[c + d*x]*(e + f*x))/(x*(a + b*x)^2),x]
[Out]
-((d*(-(b*e) + a*f)*Sqrt[c + d*x])/(a*b*(-(b*c) + a*d + b*(c + d*x)))) + ((2*b^2*c*e - a*b*d*e - a^2*d*f)*ArcT
an[(Sqrt[b]*Sqrt[-(b*c) + a*d]*Sqrt[c + d*x])/(b*c - a*d)])/(a^2*b^(3/2)*Sqrt[-(b*c) + a*d]) - (2*Sqrt[c]*e*Ar
cTanh[Sqrt[c + d*x]/Sqrt[c]])/a^2
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fricas [B] time = 1.35, size = 1018, normalized size = 8.02 \begin {gather*} \left [\frac {{\left (a^{3} d f - {\left (2 \, a b^{2} c - a^{2} b d\right )} e + {\left (a^{2} b d f - {\left (2 \, b^{3} c - a b^{2} d\right )} e\right )} x\right )} \sqrt {b^{2} c - a b d} \log \left (\frac {b d x + 2 \, b c - a d - 2 \, \sqrt {b^{2} c - a b d} \sqrt {d x + c}}{b x + a}\right ) + 2 \, {\left ({\left (b^{4} c - a b^{3} d\right )} e x + {\left (a b^{3} c - a^{2} b^{2} d\right )} e\right )} \sqrt {c} \log \left (\frac {d x - 2 \, \sqrt {d x + c} \sqrt {c} + 2 \, c}{x}\right ) + 2 \, {\left ({\left (a b^{3} c - a^{2} b^{2} d\right )} e - {\left (a^{2} b^{2} c - a^{3} b d\right )} f\right )} \sqrt {d x + c}}{2 \, {\left (a^{3} b^{3} c - a^{4} b^{2} d + {\left (a^{2} b^{4} c - a^{3} b^{3} d\right )} x\right )}}, \frac {{\left (a^{3} d f - {\left (2 \, a b^{2} c - a^{2} b d\right )} e + {\left (a^{2} b d f - {\left (2 \, b^{3} c - a b^{2} d\right )} e\right )} x\right )} \sqrt {-b^{2} c + a b d} \arctan \left (\frac {\sqrt {-b^{2} c + a b d} \sqrt {d x + c}}{b d x + b c}\right ) + {\left ({\left (b^{4} c - a b^{3} d\right )} e x + {\left (a b^{3} c - a^{2} b^{2} d\right )} e\right )} \sqrt {c} \log \left (\frac {d x - 2 \, \sqrt {d x + c} \sqrt {c} + 2 \, c}{x}\right ) + {\left ({\left (a b^{3} c - a^{2} b^{2} d\right )} e - {\left (a^{2} b^{2} c - a^{3} b d\right )} f\right )} \sqrt {d x + c}}{a^{3} b^{3} c - a^{4} b^{2} d + {\left (a^{2} b^{4} c - a^{3} b^{3} d\right )} x}, \frac {4 \, {\left ({\left (b^{4} c - a b^{3} d\right )} e x + {\left (a b^{3} c - a^{2} b^{2} d\right )} e\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {d x + c} \sqrt {-c}}{c}\right ) + {\left (a^{3} d f - {\left (2 \, a b^{2} c - a^{2} b d\right )} e + {\left (a^{2} b d f - {\left (2 \, b^{3} c - a b^{2} d\right )} e\right )} x\right )} \sqrt {b^{2} c - a b d} \log \left (\frac {b d x + 2 \, b c - a d - 2 \, \sqrt {b^{2} c - a b d} \sqrt {d x + c}}{b x + a}\right ) + 2 \, {\left ({\left (a b^{3} c - a^{2} b^{2} d\right )} e - {\left (a^{2} b^{2} c - a^{3} b d\right )} f\right )} \sqrt {d x + c}}{2 \, {\left (a^{3} b^{3} c - a^{4} b^{2} d + {\left (a^{2} b^{4} c - a^{3} b^{3} d\right )} x\right )}}, \frac {{\left (a^{3} d f - {\left (2 \, a b^{2} c - a^{2} b d\right )} e + {\left (a^{2} b d f - {\left (2 \, b^{3} c - a b^{2} d\right )} e\right )} x\right )} \sqrt {-b^{2} c + a b d} \arctan \left (\frac {\sqrt {-b^{2} c + a b d} \sqrt {d x + c}}{b d x + b c}\right ) + 2 \, {\left ({\left (b^{4} c - a b^{3} d\right )} e x + {\left (a b^{3} c - a^{2} b^{2} d\right )} e\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {d x + c} \sqrt {-c}}{c}\right ) + {\left ({\left (a b^{3} c - a^{2} b^{2} d\right )} e - {\left (a^{2} b^{2} c - a^{3} b d\right )} f\right )} \sqrt {d x + c}}{a^{3} b^{3} c - a^{4} b^{2} d + {\left (a^{2} b^{4} c - a^{3} b^{3} d\right )} x}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((f*x+e)*(d*x+c)^(1/2)/x/(b*x+a)^2,x, algorithm="fricas")
[Out]
[1/2*((a^3*d*f - (2*a*b^2*c - a^2*b*d)*e + (a^2*b*d*f - (2*b^3*c - a*b^2*d)*e)*x)*sqrt(b^2*c - a*b*d)*log((b*d
*x + 2*b*c - a*d - 2*sqrt(b^2*c - a*b*d)*sqrt(d*x + c))/(b*x + a)) + 2*((b^4*c - a*b^3*d)*e*x + (a*b^3*c - a^2
*b^2*d)*e)*sqrt(c)*log((d*x - 2*sqrt(d*x + c)*sqrt(c) + 2*c)/x) + 2*((a*b^3*c - a^2*b^2*d)*e - (a^2*b^2*c - a^
3*b*d)*f)*sqrt(d*x + c))/(a^3*b^3*c - a^4*b^2*d + (a^2*b^4*c - a^3*b^3*d)*x), ((a^3*d*f - (2*a*b^2*c - a^2*b*d
)*e + (a^2*b*d*f - (2*b^3*c - a*b^2*d)*e)*x)*sqrt(-b^2*c + a*b*d)*arctan(sqrt(-b^2*c + a*b*d)*sqrt(d*x + c)/(b
*d*x + b*c)) + ((b^4*c - a*b^3*d)*e*x + (a*b^3*c - a^2*b^2*d)*e)*sqrt(c)*log((d*x - 2*sqrt(d*x + c)*sqrt(c) +
2*c)/x) + ((a*b^3*c - a^2*b^2*d)*e - (a^2*b^2*c - a^3*b*d)*f)*sqrt(d*x + c))/(a^3*b^3*c - a^4*b^2*d + (a^2*b^4
*c - a^3*b^3*d)*x), 1/2*(4*((b^4*c - a*b^3*d)*e*x + (a*b^3*c - a^2*b^2*d)*e)*sqrt(-c)*arctan(sqrt(d*x + c)*sqr
t(-c)/c) + (a^3*d*f - (2*a*b^2*c - a^2*b*d)*e + (a^2*b*d*f - (2*b^3*c - a*b^2*d)*e)*x)*sqrt(b^2*c - a*b*d)*log
((b*d*x + 2*b*c - a*d - 2*sqrt(b^2*c - a*b*d)*sqrt(d*x + c))/(b*x + a)) + 2*((a*b^3*c - a^2*b^2*d)*e - (a^2*b^
2*c - a^3*b*d)*f)*sqrt(d*x + c))/(a^3*b^3*c - a^4*b^2*d + (a^2*b^4*c - a^3*b^3*d)*x), ((a^3*d*f - (2*a*b^2*c -
a^2*b*d)*e + (a^2*b*d*f - (2*b^3*c - a*b^2*d)*e)*x)*sqrt(-b^2*c + a*b*d)*arctan(sqrt(-b^2*c + a*b*d)*sqrt(d*x
+ c)/(b*d*x + b*c)) + 2*((b^4*c - a*b^3*d)*e*x + (a*b^3*c - a^2*b^2*d)*e)*sqrt(-c)*arctan(sqrt(d*x + c)*sqrt(
-c)/c) + ((a*b^3*c - a^2*b^2*d)*e - (a^2*b^2*c - a^3*b*d)*f)*sqrt(d*x + c))/(a^3*b^3*c - a^4*b^2*d + (a^2*b^4*
c - a^3*b^3*d)*x)]
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giac [A] time = 1.42, size = 142, normalized size = 1.12 \begin {gather*} \frac {2 \, c \arctan \left (\frac {\sqrt {d x + c}}{\sqrt {-c}}\right ) e}{a^{2} \sqrt {-c}} + \frac {{\left (a^{2} d f - 2 \, b^{2} c e + a b d e\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^{2} b} - \frac {\sqrt {d x + c} a d f - \sqrt {d x + c} b d e}{{\left ({\left (d x + c\right )} b - b c + a d\right )} a b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((f*x+e)*(d*x+c)^(1/2)/x/(b*x+a)^2,x, algorithm="giac")
[Out]
2*c*arctan(sqrt(d*x + c)/sqrt(-c))*e/(a^2*sqrt(-c)) + (a^2*d*f - 2*b^2*c*e + a*b*d*e)*arctan(sqrt(d*x + c)*b/s
qrt(-b^2*c + a*b*d))/(sqrt(-b^2*c + a*b*d)*a^2*b) - (sqrt(d*x + c)*a*d*f - sqrt(d*x + c)*b*d*e)/(((d*x + c)*b
- b*c + a*d)*a*b)
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maple [A] time = 0.02, size = 192, normalized size = 1.51 \begin {gather*} \frac {d e \arctan \left (\frac {\sqrt {d x +c}\, b}{\sqrt {\left (a d -b c \right ) b}}\right )}{\sqrt {\left (a d -b c \right ) b}\, a}-\frac {2 b c e \arctan \left (\frac {\sqrt {d x +c}\, b}{\sqrt {\left (a d -b c \right ) b}}\right )}{\sqrt {\left (a d -b c \right ) b}\, a^{2}}+\frac {d f \arctan \left (\frac {\sqrt {d x +c}\, b}{\sqrt {\left (a d -b c \right ) b}}\right )}{\sqrt {\left (a d -b c \right ) b}\, b}+\frac {\sqrt {d x +c}\, d e}{\left (b d x +a d \right ) a}-\frac {\sqrt {d x +c}\, d f}{\left (b d x +a d \right ) b}-\frac {2 \sqrt {c}\, e \arctanh \left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )}{a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((f*x+e)*(d*x+c)^(1/2)/x/(b*x+a)^2,x)
[Out]
-d/b*(d*x+c)^(1/2)/(b*d*x+a*d)*f+d/a*(d*x+c)^(1/2)/(b*d*x+a*d)*e+d/b/((a*d-b*c)*b)^(1/2)*arctan((d*x+c)^(1/2)/
((a*d-b*c)*b)^(1/2)*b)*f+d/a/((a*d-b*c)*b)^(1/2)*arctan((d*x+c)^(1/2)/((a*d-b*c)*b)^(1/2)*b)*e-2/a^2*b/((a*d-b
*c)*b)^(1/2)*arctan((d*x+c)^(1/2)/((a*d-b*c)*b)^(1/2)*b)*c*e-2*e*arctanh((d*x+c)^(1/2)/c^(1/2))*c^(1/2)/a^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((f*x+e)*(d*x+c)^(1/2)/x/(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 details)Is a*d-b*c positive or negative?
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mupad [B] time = 0.60, size = 1827, normalized size = 14.39
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(((e + f*x)*(c + d*x)^(1/2))/(x*(a + b*x)^2),x)
[Out]
(atan(((((((2*(2*a^4*b^3*c*d^3*e - 2*a^5*b^2*c*d^3*f))/(a^3*b) + ((4*a^5*b^3*d^3 - 8*a^4*b^4*c*d^2)*(-b^3*(a*d
- b*c))^(1/2)*(c + d*x)^(1/2)*(a^2*d*f - 2*b^2*c*e + a*b*d*e))/(a^2*b*(a^2*b^4*c - a^3*b^3*d)))*(-b^3*(a*d -
b*c))^(1/2)*(a^2*d*f - 2*b^2*c*e + a*b*d*e))/(2*(a^2*b^4*c - a^3*b^3*d)) + (2*(c + d*x)^(1/2)*(a^4*d^4*f^2 + a
^2*b^2*d^4*e^2 + 8*b^4*c^2*d^2*e^2 + 2*a^3*b*d^4*e*f - 4*a*b^3*c*d^3*e^2 - 4*a^2*b^2*c*d^3*e*f))/(a^2*b))*(-b^
3*(a*d - b*c))^(1/2)*(a^2*d*f - 2*b^2*c*e + a*b*d*e)*1i)/(2*(a^2*b^4*c - a^3*b^3*d)) - (((((2*(2*a^4*b^3*c*d^3
*e - 2*a^5*b^2*c*d^3*f))/(a^3*b) - ((4*a^5*b^3*d^3 - 8*a^4*b^4*c*d^2)*(-b^3*(a*d - b*c))^(1/2)*(c + d*x)^(1/2)
*(a^2*d*f - 2*b^2*c*e + a*b*d*e))/(a^2*b*(a^2*b^4*c - a^3*b^3*d)))*(-b^3*(a*d - b*c))^(1/2)*(a^2*d*f - 2*b^2*c
*e + a*b*d*e))/(2*(a^2*b^4*c - a^3*b^3*d)) - (2*(c + d*x)^(1/2)*(a^4*d^4*f^2 + a^2*b^2*d^4*e^2 + 8*b^4*c^2*d^2
*e^2 + 2*a^3*b*d^4*e*f - 4*a*b^3*c*d^3*e^2 - 4*a^2*b^2*c*d^3*e*f))/(a^2*b))*(-b^3*(a*d - b*c))^(1/2)*(a^2*d*f
- 2*b^2*c*e + a*b*d*e)*1i)/(2*(a^2*b^4*c - a^3*b^3*d)))/((4*(a*b^2*c*d^4*e^3 - 2*b^3*c^2*d^3*e^3 + a^3*c*d^4*e
*f^2 - 2*a*b^2*c^2*d^3*e^2*f + 2*a^2*b*c*d^4*e^2*f))/(a^3*b) + (((((2*(2*a^4*b^3*c*d^3*e - 2*a^5*b^2*c*d^3*f))
/(a^3*b) + ((4*a^5*b^3*d^3 - 8*a^4*b^4*c*d^2)*(-b^3*(a*d - b*c))^(1/2)*(c + d*x)^(1/2)*(a^2*d*f - 2*b^2*c*e +
a*b*d*e))/(a^2*b*(a^2*b^4*c - a^3*b^3*d)))*(-b^3*(a*d - b*c))^(1/2)*(a^2*d*f - 2*b^2*c*e + a*b*d*e))/(2*(a^2*b
^4*c - a^3*b^3*d)) + (2*(c + d*x)^(1/2)*(a^4*d^4*f^2 + a^2*b^2*d^4*e^2 + 8*b^4*c^2*d^2*e^2 + 2*a^3*b*d^4*e*f -
4*a*b^3*c*d^3*e^2 - 4*a^2*b^2*c*d^3*e*f))/(a^2*b))*(-b^3*(a*d - b*c))^(1/2)*(a^2*d*f - 2*b^2*c*e + a*b*d*e))/
(2*(a^2*b^4*c - a^3*b^3*d)) + (((((2*(2*a^4*b^3*c*d^3*e - 2*a^5*b^2*c*d^3*f))/(a^3*b) - ((4*a^5*b^3*d^3 - 8*a^
4*b^4*c*d^2)*(-b^3*(a*d - b*c))^(1/2)*(c + d*x)^(1/2)*(a^2*d*f - 2*b^2*c*e + a*b*d*e))/(a^2*b*(a^2*b^4*c - a^3
*b^3*d)))*(-b^3*(a*d - b*c))^(1/2)*(a^2*d*f - 2*b^2*c*e + a*b*d*e))/(2*(a^2*b^4*c - a^3*b^3*d)) - (2*(c + d*x)
^(1/2)*(a^4*d^4*f^2 + a^2*b^2*d^4*e^2 + 8*b^4*c^2*d^2*e^2 + 2*a^3*b*d^4*e*f - 4*a*b^3*c*d^3*e^2 - 4*a^2*b^2*c*
d^3*e*f))/(a^2*b))*(-b^3*(a*d - b*c))^(1/2)*(a^2*d*f - 2*b^2*c*e + a*b*d*e))/(2*(a^2*b^4*c - a^3*b^3*d))))*(-b
^3*(a*d - b*c))^(1/2)*(a^2*d*f - 2*b^2*c*e + a*b*d*e)*1i)/(a^2*b^4*c - a^3*b^3*d) - (2*c^(1/2)*e*atanh((4*c^(1
/2)*d^4*e*f^2*(c + d*x)^(1/2))/(4*c*d^4*e*f^2 + (4*b^2*c*d^4*e^3)/a^2 - (16*b^2*c^2*d^3*e^2*f)/a^2 + (8*b*c*d^
4*e^2*f)/a) + (8*c^(1/2)*d^4*e^2*f*(c + d*x)^(1/2))/(8*c*d^4*e^2*f + (4*b*c*d^4*e^3)/a - (16*b*c^2*d^3*e^2*f)/
a + (4*a*c*d^4*e*f^2)/b) + (4*b*c^(1/2)*d^4*e^3*(c + d*x)^(1/2))/(4*b*c*d^4*e^3 + 8*a*c*d^4*e^2*f - 16*b*c^2*d
^3*e^2*f + (4*a^2*c*d^4*e*f^2)/b) - (16*b*c^(3/2)*d^3*e^2*f*(c + d*x)^(1/2))/(4*b*c*d^4*e^3 + 8*a*c*d^4*e^2*f
- 16*b*c^2*d^3*e^2*f + (4*a^2*c*d^4*e*f^2)/b)))/a^2 - ((a*d*f - b*d*e)*(c + d*x)^(1/2))/(a*b*(a*d - b*c + b*(c
+ d*x)))
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sympy [B] time = 133.87, size = 1204, normalized size = 9.48
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((f*x+e)*(d*x+c)**(1/2)/x/(b*x+a)**2,x)
[Out]
-2*a*d**2*f*sqrt(c + d*x)/(2*a**2*b*d**2 - 2*a*b**2*c*d + 2*a*b**2*d**2*x - 2*b**3*c*d*x) + a*d**2*f*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*b) - a*d**2*f*sqrt(-1/(b*(a*d - b*c)**3))*log(a**2*d**2*sq
rt(-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)) + sq
rt(c + d*x))/(2*b) - 2*b*c*d*e*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) -
c*d*f*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 + c*d*f*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 + 2*c*d*f*sqrt(c + d*x)/(2*a**2*d**2 - 2*a*b*c*d + 2*a*b*d**2*x - 2*b**2*c*d*x) - d**2
*e*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 + d**2*e*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 + 2*d**2*e*sqrt(c + d*x)/(2*a**2*d**2 - 2*a*b*c*d + 2*a*b*d**2*x - 2*b**2*c*d*x) + 2*d*f*
atan(sqrt(c + d*x)/sqrt(a*d/b - c))/(b**2*sqrt(a*d/b - c)) + b*c*d*e*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*c*d*e*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*e*atan(s
qrt(c + d*x)/sqrt(a*d/b - c))/(a**2*sqrt(a*d/b - c)) + 2*c*e*atan(sqrt(c + d*x)/sqrt(-c))/(a**2*sqrt(-c))
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