3.1.20 \(\int \frac {\sqrt {a+b x} (e+f x)}{x (c+d x)^2} \, dx\)

Optimal. Leaf size=128 \[ -\frac {\left (2 a d^2 e-b c (c f+d e)\right ) \tan ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b c-a d}}\right )}{c^2 d^{3/2} \sqrt {b c-a d}}-\frac {2 \sqrt {a} e \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{c^2}+\frac {\sqrt {a+b x} (d e-c f)}{c d (c+d x)} \]

________________________________________________________________________________________

Rubi [A]  time = 0.12, antiderivative size = 128, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {149, 156, 63, 208, 205} \begin {gather*} -\frac {\left (2 a d^2 e-b c (c f+d e)\right ) \tan ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b c-a d}}\right )}{c^2 d^{3/2} \sqrt {b c-a d}}-\frac {2 \sqrt {a} e \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{c^2}+\frac {\sqrt {a+b x} (d e-c f)}{c d (c+d x)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

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

________________________________________________________________________________________

Mathematica [A]  time = 0.20, size = 122, normalized size = 0.95 \begin {gather*} \frac {\frac {\left (b c (c f+d e)-2 a d^2 e\right ) \tan ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b c-a d}}\right )}{d^{3/2} \sqrt {b c-a d}}+\frac {c \sqrt {a+b x} (d e-c f)}{d (c+d x)}-2 \sqrt {a} e \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{c^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.48, size = 139, normalized size = 1.09 \begin {gather*} \frac {\left (-2 a d^2 e+b c^2 f+b c d e\right ) \tan ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b c-a d}}\right )}{c^2 d^{3/2} \sqrt {b c-a d}}-\frac {2 \sqrt {a} e \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{c^2}-\frac {b \sqrt {a+b x} (c f-d e)}{c d (d (a+b x)-a d+b c)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [B]  time = 1.36, size = 1008, normalized size = 7.88 \begin {gather*} \left [-\frac {{\left (b c^{3} f + {\left (b c^{2} d - 2 \, a c d^{2}\right )} e + {\left (b c^{2} d f + {\left (b c d^{2} - 2 \, a d^{3}\right )} e\right )} x\right )} \sqrt {-b c d + a d^{2}} \log \left (\frac {b d x - b c + 2 \, a d - 2 \, \sqrt {-b c d + a d^{2}} \sqrt {b x + a}}{d x + c}\right ) - 2 \, {\left ({\left (b c d^{3} - a d^{4}\right )} e x + {\left (b c^{2} d^{2} - a c d^{3}\right )} e\right )} \sqrt {a} \log \left (\frac {b x - 2 \, \sqrt {b x + a} \sqrt {a} + 2 \, a}{x}\right ) - 2 \, {\left ({\left (b c^{2} d^{2} - a c d^{3}\right )} e - {\left (b c^{3} d - a c^{2} d^{2}\right )} f\right )} \sqrt {b x + a}}{2 \, {\left (b c^{4} d^{2} - a c^{3} d^{3} + {\left (b c^{3} d^{3} - a c^{2} d^{4}\right )} x\right )}}, \frac {4 \, {\left ({\left (b c d^{3} - a d^{4}\right )} e x + {\left (b c^{2} d^{2} - a c d^{3}\right )} e\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {b x + a} \sqrt {-a}}{a}\right ) - {\left (b c^{3} f + {\left (b c^{2} d - 2 \, a c d^{2}\right )} e + {\left (b c^{2} d f + {\left (b c d^{2} - 2 \, a d^{3}\right )} e\right )} x\right )} \sqrt {-b c d + a d^{2}} \log \left (\frac {b d x - b c + 2 \, a d - 2 \, \sqrt {-b c d + a d^{2}} \sqrt {b x + a}}{d x + c}\right ) + 2 \, {\left ({\left (b c^{2} d^{2} - a c d^{3}\right )} e - {\left (b c^{3} d - a c^{2} d^{2}\right )} f\right )} \sqrt {b x + a}}{2 \, {\left (b c^{4} d^{2} - a c^{3} d^{3} + {\left (b c^{3} d^{3} - a c^{2} d^{4}\right )} x\right )}}, -\frac {{\left (b c^{3} f + {\left (b c^{2} d - 2 \, a c d^{2}\right )} e + {\left (b c^{2} d f + {\left (b c d^{2} - 2 \, a d^{3}\right )} e\right )} x\right )} \sqrt {b c d - a d^{2}} \arctan \left (\frac {\sqrt {b c d - a d^{2}} \sqrt {b x + a}}{b d x + a d}\right ) - {\left ({\left (b c d^{3} - a d^{4}\right )} e x + {\left (b c^{2} d^{2} - a c d^{3}\right )} e\right )} \sqrt {a} \log \left (\frac {b x - 2 \, \sqrt {b x + a} \sqrt {a} + 2 \, a}{x}\right ) - {\left ({\left (b c^{2} d^{2} - a c d^{3}\right )} e - {\left (b c^{3} d - a c^{2} d^{2}\right )} f\right )} \sqrt {b x + a}}{b c^{4} d^{2} - a c^{3} d^{3} + {\left (b c^{3} d^{3} - a c^{2} d^{4}\right )} x}, -\frac {{\left (b c^{3} f + {\left (b c^{2} d - 2 \, a c d^{2}\right )} e + {\left (b c^{2} d f + {\left (b c d^{2} - 2 \, a d^{3}\right )} e\right )} x\right )} \sqrt {b c d - a d^{2}} \arctan \left (\frac {\sqrt {b c d - a d^{2}} \sqrt {b x + a}}{b d x + a d}\right ) - 2 \, {\left ({\left (b c d^{3} - a d^{4}\right )} e x + {\left (b c^{2} d^{2} - a c d^{3}\right )} e\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {b x + a} \sqrt {-a}}{a}\right ) - {\left ({\left (b c^{2} d^{2} - a c d^{3}\right )} e - {\left (b c^{3} d - a c^{2} d^{2}\right )} f\right )} \sqrt {b x + a}}{b c^{4} d^{2} - a c^{3} d^{3} + {\left (b c^{3} d^{3} - a c^{2} d^{4}\right )} x}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/2*((b*c^3*f + (b*c^2*d - 2*a*c*d^2)*e + (b*c^2*d*f + (b*c*d^2 - 2*a*d^3)*e)*x)*sqrt(-b*c*d + a*d^2)*log((b
*d*x - b*c + 2*a*d - 2*sqrt(-b*c*d + a*d^2)*sqrt(b*x + a))/(d*x + c)) - 2*((b*c*d^3 - a*d^4)*e*x + (b*c^2*d^2
- a*c*d^3)*e)*sqrt(a)*log((b*x - 2*sqrt(b*x + a)*sqrt(a) + 2*a)/x) - 2*((b*c^2*d^2 - a*c*d^3)*e - (b*c^3*d - a
*c^2*d^2)*f)*sqrt(b*x + a))/(b*c^4*d^2 - a*c^3*d^3 + (b*c^3*d^3 - a*c^2*d^4)*x), 1/2*(4*((b*c*d^3 - a*d^4)*e*x
+ (b*c^2*d^2 - a*c*d^3)*e)*sqrt(-a)*arctan(sqrt(b*x + a)*sqrt(-a)/a) - (b*c^3*f + (b*c^2*d - 2*a*c*d^2)*e + (
b*c^2*d*f + (b*c*d^2 - 2*a*d^3)*e)*x)*sqrt(-b*c*d + a*d^2)*log((b*d*x - b*c + 2*a*d - 2*sqrt(-b*c*d + a*d^2)*s
qrt(b*x + a))/(d*x + c)) + 2*((b*c^2*d^2 - a*c*d^3)*e - (b*c^3*d - a*c^2*d^2)*f)*sqrt(b*x + a))/(b*c^4*d^2 - a
*c^3*d^3 + (b*c^3*d^3 - a*c^2*d^4)*x), -((b*c^3*f + (b*c^2*d - 2*a*c*d^2)*e + (b*c^2*d*f + (b*c*d^2 - 2*a*d^3)
*e)*x)*sqrt(b*c*d - a*d^2)*arctan(sqrt(b*c*d - a*d^2)*sqrt(b*x + a)/(b*d*x + a*d)) - ((b*c*d^3 - a*d^4)*e*x +
(b*c^2*d^2 - a*c*d^3)*e)*sqrt(a)*log((b*x - 2*sqrt(b*x + a)*sqrt(a) + 2*a)/x) - ((b*c^2*d^2 - a*c*d^3)*e - (b*
c^3*d - a*c^2*d^2)*f)*sqrt(b*x + a))/(b*c^4*d^2 - a*c^3*d^3 + (b*c^3*d^3 - a*c^2*d^4)*x), -((b*c^3*f + (b*c^2*
d - 2*a*c*d^2)*e + (b*c^2*d*f + (b*c*d^2 - 2*a*d^3)*e)*x)*sqrt(b*c*d - a*d^2)*arctan(sqrt(b*c*d - a*d^2)*sqrt(
b*x + a)/(b*d*x + a*d)) - 2*((b*c*d^3 - a*d^4)*e*x + (b*c^2*d^2 - a*c*d^3)*e)*sqrt(-a)*arctan(sqrt(b*x + a)*sq
rt(-a)/a) - ((b*c^2*d^2 - a*c*d^3)*e - (b*c^3*d - a*c^2*d^2)*f)*sqrt(b*x + a))/(b*c^4*d^2 - a*c^3*d^3 + (b*c^3
*d^3 - a*c^2*d^4)*x)]

________________________________________________________________________________________

giac [A]  time = 1.32, size = 142, normalized size = 1.11 \begin {gather*} \frac {2 \, a \arctan \left (\frac {\sqrt {b x + a}}{\sqrt {-a}}\right ) e}{\sqrt {-a} c^{2}} + \frac {{\left (b c^{2} f + b c d e - 2 \, a d^{2} e\right )} \arctan \left (\frac {\sqrt {b x + a} d}{\sqrt {b c d - a d^{2}}}\right )}{\sqrt {b c d - a d^{2}} c^{2} d} - \frac {\sqrt {b x + a} b c f - \sqrt {b x + a} b d e}{{\left (b c + {\left (b x + a\right )} d - a d\right )} c d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*a*arctan(sqrt(b*x + a)/sqrt(-a))*e/(sqrt(-a)*c^2) + (b*c^2*f + b*c*d*e - 2*a*d^2*e)*arctan(sqrt(b*x + a)*d/s
qrt(b*c*d - a*d^2))/(sqrt(b*c*d - a*d^2)*c^2*d) - (sqrt(b*x + a)*b*c*f - sqrt(b*x + a)*b*d*e)/((b*c + (b*x + a
)*d - a*d)*c*d)

________________________________________________________________________________________

maple [A]  time = 0.02, size = 137, normalized size = 1.07 \begin {gather*} 2 \left (-\frac {\sqrt {a}\, e \arctanh \left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{b \,c^{2}}-\frac {\frac {\left (c f -d e \right ) \sqrt {b x +a}\, b c}{2 \left (-a d +b c +\left (b x +a \right ) d \right ) d}-\frac {\left (2 a \,d^{2} e -b \,c^{2} f -b c d e \right ) \arctanh \left (\frac {\sqrt {b x +a}\, d}{\sqrt {\left (a d -b c \right ) d}}\right )}{2 \sqrt {\left (a d -b c \right ) d}\, d}}{b \,c^{2}}\right ) b \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*b*(-a^(1/2)/b*e/c^2*arctanh((b*x+a)^(1/2)/a^(1/2))-1/c^2/b*(1/2*b*c*(c*f-d*e)/d*(b*x+a)^(1/2)/((b*x+a)*d-a*d
+b*c)-1/2*(2*a*d^2*e-b*c^2*f-b*c*d*e)/d/((a*d-b*c)*d)^(1/2)*arctanh((b*x+a)^(1/2)/((a*d-b*c)*d)^(1/2)*d)))

________________________________________________________________________________________

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

________________________________________________________________________________________

mupad [B]  time = 2.95, size = 1814, normalized size = 14.17

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [B]  time = 142.24, size = 1149, normalized size = 8.98

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*a*b*d*e*sqrt(a + b*x)/(2*a*b*c**2*d + 2*a*b*c*d**2*x - 2*b**2*c**3 - 2*b**2*c**2*d*x) - a*b*f*sqrt(1/(d*(a*d
- b*c)**3))*log(-a**2*d**2*sqrt(1/(d*(a*d - b*c)**3)) + 2*a*b*c*d*sqrt(1/(d*(a*d - b*c)**3)) - b**2*c**2*sqrt
(1/(d*(a*d - b*c)**3)) + sqrt(a + b*x))/2 + a*b*f*sqrt(1/(d*(a*d - b*c)**3))*log(a**2*d**2*sqrt(1/(d*(a*d - b*
c)**3)) - 2*a*b*c*d*sqrt(1/(d*(a*d - b*c)**3)) + b**2*c**2*sqrt(1/(d*(a*d - b*c)**3)) + sqrt(a + b*x))/2 - 2*a
*b*f*sqrt(a + b*x)/(2*a*b*c*d + 2*a*b*d**2*x - 2*b**2*c**2 - 2*b**2*c*d*x) + a*b*d*e*sqrt(1/(d*(a*d - b*c)**3)
)*log(-a**2*d**2*sqrt(1/(d*(a*d - b*c)**3)) + 2*a*b*c*d*sqrt(1/(d*(a*d - b*c)**3)) - b**2*c**2*sqrt(1/(d*(a*d
- b*c)**3)) + sqrt(a + b*x))/(2*c) - a*b*d*e*sqrt(1/(d*(a*d - b*c)**3))*log(a**2*d**2*sqrt(1/(d*(a*d - b*c)**3
)) - 2*a*b*c*d*sqrt(1/(d*(a*d - b*c)**3)) + b**2*c**2*sqrt(1/(d*(a*d - b*c)**3)) + sqrt(a + b*x))/(2*c) - 2*a*
e*atan(sqrt(a + b*x)/sqrt(-a + b*c/d))/(c**2*sqrt(-a + b*c/d)) + 2*a*e*atan(sqrt(a + b*x)/sqrt(-a))/(c**2*sqrt
(-a)) + 2*b**2*c*f*sqrt(a + b*x)/(2*a*b*c*d**2 + 2*a*b*d**3*x - 2*b**2*c**2*d - 2*b**2*c*d**2*x) + b**2*c*f*sq
rt(1/(d*(a*d - b*c)**3))*log(-a**2*d**2*sqrt(1/(d*(a*d - b*c)**3)) + 2*a*b*c*d*sqrt(1/(d*(a*d - b*c)**3)) - b*
*2*c**2*sqrt(1/(d*(a*d - b*c)**3)) + sqrt(a + b*x))/(2*d) - b**2*c*f*sqrt(1/(d*(a*d - b*c)**3))*log(a**2*d**2*
sqrt(1/(d*(a*d - b*c)**3)) - 2*a*b*c*d*sqrt(1/(d*(a*d - b*c)**3)) + b**2*c**2*sqrt(1/(d*(a*d - b*c)**3)) + sqr
t(a + b*x))/(2*d) - b**2*e*sqrt(1/(d*(a*d - b*c)**3))*log(-a**2*d**2*sqrt(1/(d*(a*d - b*c)**3)) + 2*a*b*c*d*sq
rt(1/(d*(a*d - b*c)**3)) - b**2*c**2*sqrt(1/(d*(a*d - b*c)**3)) + sqrt(a + b*x))/2 + b**2*e*sqrt(1/(d*(a*d - b
*c)**3))*log(a**2*d**2*sqrt(1/(d*(a*d - b*c)**3)) - 2*a*b*c*d*sqrt(1/(d*(a*d - b*c)**3)) + b**2*c**2*sqrt(1/(d
*(a*d - b*c)**3)) + sqrt(a + b*x))/2 - 2*b**2*e*sqrt(a + b*x)/(2*a*b*c*d + 2*a*b*d**2*x - 2*b**2*c**2 - 2*b**2
*c*d*x) + 2*b*f*atan(sqrt(a + b*x)/sqrt(-a + b*c/d))/(d**2*sqrt(-a + b*c/d))

________________________________________________________________________________________