3.18.53 \(\int \frac {(A+B x) \sqrt {d+e x}}{(a+b x)^2} \, dx\) [1753]

Optimal. Leaf size=140 \[ \frac {(2 b B d+A b e-3 a B e) \sqrt {d+e x}}{b^2 (b d-a e)}-\frac {(A b-a B) (d+e x)^{3/2}}{b (b d-a e) (a+b x)}-\frac {(2 b B d+A b e-3 a B e) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{b^{5/2} \sqrt {b d-a e}} \]

[Out]

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

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Rubi [A]
time = 0.07, antiderivative size = 140, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {79, 52, 65, 214} \begin {gather*} -\frac {(-3 a B e+A b e+2 b B d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{b^{5/2} \sqrt {b d-a e}}+\frac {\sqrt {d+e x} (-3 a B e+A b e+2 b B d)}{b^2 (b d-a e)}-\frac {(d+e x)^{3/2} (A b-a B)}{b (a+b x) (b d-a e)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((2*b*B*d + A*b*e - 3*a*B*e)*Sqrt[d + e*x])/(b^2*(b*d - a*e)) - ((A*b - a*B)*(d + e*x)^(3/2))/(b*(b*d - a*e)*(
a + b*x)) - ((2*b*B*d + A*b*e - 3*a*B*e)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(b^(5/2)*Sqrt[b*d -
a*e])

Rule 52

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^n/(b*(
m + n + 1))), x] + Dist[n*((b*c - a*d)/(b*(m + n + 1))), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

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 79

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

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

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Mathematica [A]
time = 0.36, size = 96, normalized size = 0.69 \begin {gather*} \frac {(-A b+3 a B+2 b B x) \sqrt {d+e x}}{b^2 (a+b x)}+\frac {(2 b B d+A b e-3 a B e) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {-b d+a e}}\right )}{b^{5/2} \sqrt {-b d+a e}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-(A*b) + 3*a*B + 2*b*B*x)*Sqrt[d + e*x])/(b^2*(a + b*x)) + ((2*b*B*d + A*b*e - 3*a*B*e)*ArcTan[(Sqrt[b]*Sqrt
[d + e*x])/Sqrt[-(b*d) + a*e]])/(b^(5/2)*Sqrt[-(b*d) + a*e])

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Maple [A]
time = 0.11, size = 108, normalized size = 0.77

method	result	size
derivativedivides	\(\frac {2 B \sqrt {e x +d}}{b^{2}}+\frac {\frac {2 \left (-\frac {1}{2} A b e +\frac {1}{2} B a e \right ) \sqrt {e x +d}}{b \left (e x +d \right )+a e -b d}+\frac {\left (A b e -3 B a e +2 B b d \right ) \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{\sqrt {\left (a e -b d \right ) b}}}{b^{2}}\)	\(108\)
default	\(\frac {2 B \sqrt {e x +d}}{b^{2}}+\frac {\frac {2 \left (-\frac {1}{2} A b e +\frac {1}{2} B a e \right ) \sqrt {e x +d}}{b \left (e x +d \right )+a e -b d}+\frac {\left (A b e -3 B a e +2 B b d \right ) \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{\sqrt {\left (a e -b d \right ) b}}}{b^{2}}\)	\(108\)
risch	\(\frac {2 B \sqrt {e x +d}}{b^{2}}-\frac {\sqrt {e x +d}\, A e}{b \left (b e x +a e \right )}+\frac {\sqrt {e x +d}\, B a e}{b^{2} \left (b e x +a e \right )}+\frac {\arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right ) A e}{b \sqrt {\left (a e -b d \right ) b}}-\frac {3 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right ) B a e}{b^{2} \sqrt {\left (a e -b d \right ) b}}+\frac {2 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right ) B d}{b \sqrt {\left (a e -b d \right ) b}}\)	\(186\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*B/b^2*(e*x+d)^(1/2)+2/b^2*((-1/2*A*b*e+1/2*B*a*e)*(e*x+d)^(1/2)/(b*(e*x+d)+a*e-b*d)+1/2*(A*b*e-3*B*a*e+2*B*b
*d)/((a*e-b*d)*b)^(1/2)*arctan(b*(e*x+d)^(1/2)/((a*e-b*d)*b)^(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((B*x+A)*(e*x+d)^(1/2)/(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(b*d-%e*a>0)', see `assume?` fo
r more detai

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Fricas [A]
time = 1.31, size = 398, normalized size = 2.84 \begin {gather*} \left [\frac {{\left (2 \, B b^{2} d x + 2 \, B a b d - {\left (3 \, B a^{2} - A a b + {\left (3 \, B a b - A b^{2}\right )} x\right )} e\right )} \sqrt {b^{2} d - a b e} \log \left (\frac {2 \, b d + {\left (b x - a\right )} e - 2 \, \sqrt {b^{2} d - a b e} \sqrt {x e + d}}{b x + a}\right ) + 2 \, {\left (2 \, B b^{3} d x + {\left (3 \, B a b^{2} - A b^{3}\right )} d - {\left (2 \, B a b^{2} x + 3 \, B a^{2} b - A a b^{2}\right )} e\right )} \sqrt {x e + d}}{2 \, {\left (b^{5} d x + a b^{4} d - {\left (a b^{4} x + a^{2} b^{3}\right )} e\right )}}, \frac {{\left (2 \, B b^{2} d x + 2 \, B a b d - {\left (3 \, B a^{2} - A a b + {\left (3 \, B a b - A b^{2}\right )} x\right )} e\right )} \sqrt {-b^{2} d + a b e} \arctan \left (\frac {\sqrt {-b^{2} d + a b e} \sqrt {x e + d}}{b x e + b d}\right ) + {\left (2 \, B b^{3} d x + {\left (3 \, B a b^{2} - A b^{3}\right )} d - {\left (2 \, B a b^{2} x + 3 \, B a^{2} b - A a b^{2}\right )} e\right )} \sqrt {x e + d}}{b^{5} d x + a b^{4} d - {\left (a b^{4} x + a^{2} b^{3}\right )} e}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/2*((2*B*b^2*d*x + 2*B*a*b*d - (3*B*a^2 - A*a*b + (3*B*a*b - A*b^2)*x)*e)*sqrt(b^2*d - a*b*e)*log((2*b*d + (
b*x - a)*e - 2*sqrt(b^2*d - a*b*e)*sqrt(x*e + d))/(b*x + a)) + 2*(2*B*b^3*d*x + (3*B*a*b^2 - A*b^3)*d - (2*B*a
*b^2*x + 3*B*a^2*b - A*a*b^2)*e)*sqrt(x*e + d))/(b^5*d*x + a*b^4*d - (a*b^4*x + a^2*b^3)*e), ((2*B*b^2*d*x + 2
*B*a*b*d - (3*B*a^2 - A*a*b + (3*B*a*b - A*b^2)*x)*e)*sqrt(-b^2*d + a*b*e)*arctan(sqrt(-b^2*d + a*b*e)*sqrt(x*
e + d)/(b*x*e + b*d)) + (2*B*b^3*d*x + (3*B*a*b^2 - A*b^3)*d - (2*B*a*b^2*x + 3*B*a^2*b - A*a*b^2)*e)*sqrt(x*e
+ d))/(b^5*d*x + a*b^4*d - (a*b^4*x + a^2*b^3)*e)]

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 1251 vs. \(2 (124) = 248\).
time = 33.16, size = 1251, normalized size = 8.94 \begin {gather*} - \frac {2 A a e^{2} \sqrt {d + e x}}{2 a^{2} b e^{2} - 2 a b^{2} d e + 2 a b^{2} e^{2} x - 2 b^{3} d e x} + \frac {A a e^{2} \sqrt {- \frac {1}{b \left (a e - b d\right )^{3}}} \log {\left (- a^{2} e^{2} \sqrt {- \frac {1}{b \left (a e - b d\right )^{3}}} + 2 a b d e \sqrt {- \frac {1}{b \left (a e - b d\right )^{3}}} - b^{2} d^{2} \sqrt {- \frac {1}{b \left (a e - b d\right )^{3}}} + \sqrt {d + e x} \right )}}{2 b} - \frac {A a e^{2} \sqrt {- \frac {1}{b \left (a e - b d\right )^{3}}} \log {\left (a^{2} e^{2} \sqrt {- \frac {1}{b \left (a e - b d\right )^{3}}} - 2 a b d e \sqrt {- \frac {1}{b \left (a e - b d\right )^{3}}} + b^{2} d^{2} \sqrt {- \frac {1}{b \left (a e - b d\right )^{3}}} + \sqrt {d + e x} \right )}}{2 b} - \frac {A d e \sqrt {- \frac {1}{b \left (a e - b d\right )^{3}}} \log {\left (- a^{2} e^{2} \sqrt {- \frac {1}{b \left (a e - b d\right )^{3}}} + 2 a b d e \sqrt {- \frac {1}{b \left (a e - b d\right )^{3}}} - b^{2} d^{2} \sqrt {- \frac {1}{b \left (a e - b d\right )^{3}}} + \sqrt {d + e x} \right )}}{2} + \frac {A d e \sqrt {- \frac {1}{b \left (a e - b d\right )^{3}}} \log {\left (a^{2} e^{2} \sqrt {- \frac {1}{b \left (a e - b d\right )^{3}}} - 2 a b d e \sqrt {- \frac {1}{b \left (a e - b d\right )^{3}}} + b^{2} d^{2} \sqrt {- \frac {1}{b \left (a e - b d\right )^{3}}} + \sqrt {d + e x} \right )}}{2} + \frac {2 A d e \sqrt {d + e x}}{2 a^{2} e^{2} - 2 a b d e + 2 a b e^{2} x - 2 b^{2} d e x} + \frac {2 A e \operatorname {atan}{\left (\frac {\sqrt {d + e x}}{\sqrt {\frac {a e}{b} - d}} \right )}}{b^{2} \sqrt {\frac {a e}{b} - d}} + \frac {2 B a^{2} e^{2} \sqrt {d + e x}}{2 a^{2} b^{2} e^{2} - 2 a b^{3} d e + 2 a b^{3} e^{2} x - 2 b^{4} d e x} - \frac {B a^{2} e^{2} \sqrt {- \frac {1}{b \left (a e - b d\right )^{3}}} \log {\left (- a^{2} e^{2} \sqrt {- \frac {1}{b \left (a e - b d\right )^{3}}} + 2 a b d e \sqrt {- \frac {1}{b \left (a e - b d\right )^{3}}} - b^{2} d^{2} \sqrt {- \frac {1}{b \left (a e - b d\right )^{3}}} + \sqrt {d + e x} \right )}}{2 b^{2}} + \frac {B a^{2} e^{2} \sqrt {- \frac {1}{b \left (a e - b d\right )^{3}}} \log {\left (a^{2} e^{2} \sqrt {- \frac {1}{b \left (a e - b d\right )^{3}}} - 2 a b d e \sqrt {- \frac {1}{b \left (a e - b d\right )^{3}}} + b^{2} d^{2} \sqrt {- \frac {1}{b \left (a e - b d\right )^{3}}} + \sqrt {d + e x} \right )}}{2 b^{2}} - \frac {2 B a d e \sqrt {d + e x}}{2 a^{2} b e^{2} - 2 a b^{2} d e + 2 a b^{2} e^{2} x - 2 b^{3} d e x} + \frac {B a d e \sqrt {- \frac {1}{b \left (a e - b d\right )^{3}}} \log {\left (- a^{2} e^{2} \sqrt {- \frac {1}{b \left (a e - b d\right )^{3}}} + 2 a b d e \sqrt {- \frac {1}{b \left (a e - b d\right )^{3}}} - b^{2} d^{2} \sqrt {- \frac {1}{b \left (a e - b d\right )^{3}}} + \sqrt {d + e x} \right )}}{2 b} - \frac {B a d e \sqrt {- \frac {1}{b \left (a e - b d\right )^{3}}} \log {\left (a^{2} e^{2} \sqrt {- \frac {1}{b \left (a e - b d\right )^{3}}} - 2 a b d e \sqrt {- \frac {1}{b \left (a e - b d\right )^{3}}} + b^{2} d^{2} \sqrt {- \frac {1}{b \left (a e - b d\right )^{3}}} + \sqrt {d + e x} \right )}}{2 b} - \frac {4 B a e \operatorname {atan}{\left (\frac {\sqrt {d + e x}}{\sqrt {\frac {a e}{b} - d}} \right )}}{b^{3} \sqrt {\frac {a e}{b} - d}} + \frac {2 B d \operatorname {atan}{\left (\frac {\sqrt {d + e x}}{\sqrt {\frac {a e}{b} - d}} \right )}}{b^{2} \sqrt {\frac {a e}{b} - d}} + \frac {2 B \sqrt {d + e x}}{b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)**(1/2)/(b*x+a)**2,x)

[Out]

-2*A*a*e**2*sqrt(d + e*x)/(2*a**2*b*e**2 - 2*a*b**2*d*e + 2*a*b**2*e**2*x - 2*b**3*d*e*x) + A*a*e**2*sqrt(-1/(
b*(a*e - b*d)**3))*log(-a**2*e**2*sqrt(-1/(b*(a*e - b*d)**3)) + 2*a*b*d*e*sqrt(-1/(b*(a*e - b*d)**3)) - b**2*d
**2*sqrt(-1/(b*(a*e - b*d)**3)) + sqrt(d + e*x))/(2*b) - A*a*e**2*sqrt(-1/(b*(a*e - b*d)**3))*log(a**2*e**2*sq
rt(-1/(b*(a*e - b*d)**3)) - 2*a*b*d*e*sqrt(-1/(b*(a*e - b*d)**3)) + b**2*d**2*sqrt(-1/(b*(a*e - b*d)**3)) + sq
rt(d + e*x))/(2*b) - A*d*e*sqrt(-1/(b*(a*e - b*d)**3))*log(-a**2*e**2*sqrt(-1/(b*(a*e - b*d)**3)) + 2*a*b*d*e*
sqrt(-1/(b*(a*e - b*d)**3)) - b**2*d**2*sqrt(-1/(b*(a*e - b*d)**3)) + sqrt(d + e*x))/2 + A*d*e*sqrt(-1/(b*(a*e
- b*d)**3))*log(a**2*e**2*sqrt(-1/(b*(a*e - b*d)**3)) - 2*a*b*d*e*sqrt(-1/(b*(a*e - b*d)**3)) + b**2*d**2*sqr
t(-1/(b*(a*e - b*d)**3)) + sqrt(d + e*x))/2 + 2*A*d*e*sqrt(d + e*x)/(2*a**2*e**2 - 2*a*b*d*e + 2*a*b*e**2*x -
2*b**2*d*e*x) + 2*A*e*atan(sqrt(d + e*x)/sqrt(a*e/b - d))/(b**2*sqrt(a*e/b - d)) + 2*B*a**2*e**2*sqrt(d + e*x)
/(2*a**2*b**2*e**2 - 2*a*b**3*d*e + 2*a*b**3*e**2*x - 2*b**4*d*e*x) - B*a**2*e**2*sqrt(-1/(b*(a*e - b*d)**3))*
log(-a**2*e**2*sqrt(-1/(b*(a*e - b*d)**3)) + 2*a*b*d*e*sqrt(-1/(b*(a*e - b*d)**3)) - b**2*d**2*sqrt(-1/(b*(a*e
- b*d)**3)) + sqrt(d + e*x))/(2*b**2) + B*a**2*e**2*sqrt(-1/(b*(a*e - b*d)**3))*log(a**2*e**2*sqrt(-1/(b*(a*e
- b*d)**3)) - 2*a*b*d*e*sqrt(-1/(b*(a*e - b*d)**3)) + b**2*d**2*sqrt(-1/(b*(a*e - b*d)**3)) + sqrt(d + e*x))/
(2*b**2) - 2*B*a*d*e*sqrt(d + e*x)/(2*a**2*b*e**2 - 2*a*b**2*d*e + 2*a*b**2*e**2*x - 2*b**3*d*e*x) + B*a*d*e*s
qrt(-1/(b*(a*e - b*d)**3))*log(-a**2*e**2*sqrt(-1/(b*(a*e - b*d)**3)) + 2*a*b*d*e*sqrt(-1/(b*(a*e - b*d)**3))
- b**2*d**2*sqrt(-1/(b*(a*e - b*d)**3)) + sqrt(d + e*x))/(2*b) - B*a*d*e*sqrt(-1/(b*(a*e - b*d)**3))*log(a**2*
e**2*sqrt(-1/(b*(a*e - b*d)**3)) - 2*a*b*d*e*sqrt(-1/(b*(a*e - b*d)**3)) + b**2*d**2*sqrt(-1/(b*(a*e - b*d)**3
)) + sqrt(d + e*x))/(2*b) - 4*B*a*e*atan(sqrt(d + e*x)/sqrt(a*e/b - d))/(b**3*sqrt(a*e/b - d)) + 2*B*d*atan(sq
rt(d + e*x)/sqrt(a*e/b - d))/(b**2*sqrt(a*e/b - d)) + 2*B*sqrt(d + e*x)/b**2

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Giac [A]
time = 0.92, size = 126, normalized size = 0.90 \begin {gather*} \frac {2 \, \sqrt {x e + d} B}{b^{2}} + \frac {{\left (2 \, B b d - 3 \, B a e + A b e\right )} \arctan \left (\frac {\sqrt {x e + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{\sqrt {-b^{2} d + a b e} b^{2}} + \frac {\sqrt {x e + d} B a e - \sqrt {x e + d} A b e}{{\left ({\left (x e + d\right )} b - b d + a e\right )} b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(x*e + d)*B/b^2 + (2*B*b*d - 3*B*a*e + A*b*e)*arctan(sqrt(x*e + d)*b/sqrt(-b^2*d + a*b*e))/(sqrt(-b^2*d
+ a*b*e)*b^2) + (sqrt(x*e + d)*B*a*e - sqrt(x*e + d)*A*b*e)/(((x*e + d)*b - b*d + a*e)*b^2)

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Mupad [B]
time = 0.14, size = 108, normalized size = 0.77 \begin {gather*} \frac {2\,B\,\sqrt {d+e\,x}}{b^2}-\frac {\left (A\,b\,e-B\,a\,e\right )\,\sqrt {d+e\,x}}{b^3\,\left (d+e\,x\right )-b^3\,d+a\,b^2\,e}+\frac {\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {d+e\,x}}{\sqrt {a\,e-b\,d}}\right )\,\left (A\,b\,e-3\,B\,a\,e+2\,B\,b\,d\right )}{b^{5/2}\,\sqrt {a\,e-b\,d}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x)*(d + e*x)^(1/2))/(a + b*x)^2,x)

[Out]

(2*B*(d + e*x)^(1/2))/b^2 - ((A*b*e - B*a*e)*(d + e*x)^(1/2))/(b^3*(d + e*x) - b^3*d + a*b^2*e) + (atan((b^(1/
2)*(d + e*x)^(1/2))/(a*e - b*d)^(1/2))*(A*b*e - 3*B*a*e + 2*B*b*d))/(b^(5/2)*(a*e - b*d)^(1/2))

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