\(\int \frac {(A+B x) \sqrt {d+e x}}{(a+b x)^3} \, dx\) [167]

Optimal result

Integrand size = 22, antiderivative size = 146 \[ \int \frac {(A+B x) \sqrt {d+e x}}{(a+b x)^3} \, dx=-\frac {(A b-a B) \sqrt {d+e x}}{2 b^2 (a+b x)^2}-\frac {(4 b B d+A b e-5 a B e) \sqrt {d+e x}}{4 b^2 (b d-a e) (a+b x)}-\frac {e (4 b B d-A b e-3 a B e) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{4 b^{5/2} (b d-a e)^{3/2}} \] Output:

-1/2*(A*b-B*a)*(e*x+d)^(1/2)/b^2/(b*x+a)^2-1/4*(A*b*e-5*B*a*e+4*B*b*d)*(e* 
x+d)^(1/2)/b^2/(-a*e+b*d)/(b*x+a)-1/4*e*(-A*b*e-3*B*a*e+4*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)^(3/2)
 

Mathematica [A] (verified)

Time = 0.61 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.98 \[ \int \frac {(A+B x) \sqrt {d+e x}}{(a+b x)^3} \, dx=\frac {\frac {\sqrt {b} \sqrt {d+e x} \left (A b (2 b d-a e+b e x)+B \left (-3 a^2 e+4 b^2 d x+a b (2 d-5 e x)\right )\right )}{(-b d+a e) (a+b x)^2}+\frac {e (-4 b B d+A b e+3 a B e) \arctan \left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {-b d+a e}}\right )}{(-b d+a e)^{3/2}}}{4 b^{5/2}} \] Input:

Integrate[((A + B*x)*Sqrt[d + e*x])/(a + b*x)^3,x]
 

Output:

((Sqrt[b]*Sqrt[d + e*x]*(A*b*(2*b*d - a*e + b*e*x) + B*(-3*a^2*e + 4*b^2*d 
*x + a*b*(2*d - 5*e*x))))/((-(b*d) + a*e)*(a + b*x)^2) + (e*(-4*b*B*d + A* 
b*e + 3*a*B*e)*ArcTan[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[-(b*d) + a*e]])/(-(b*d) 
 + a*e)^(3/2))/(4*b^(5/2))
 

Rubi [A] (verified)

Time = 0.27 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.99, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {87, 51, 73, 221}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {(A+B x) \sqrt {d+e x}}{(a+b x)^3} \, dx\)

\(\Big \downarrow \) 87

\(\displaystyle \frac {(-3 a B e-A b e+4 b B d) \int \frac {\sqrt {d+e x}}{(a+b x)^2}dx}{4 b (b d-a e)}-\frac {(d+e x)^{3/2} (A b-a B)}{2 b (a+b x)^2 (b d-a e)}\)

\(\Big \downarrow \) 51

\(\displaystyle \frac {(-3 a B e-A b e+4 b B d) \left (\frac {e \int \frac {1}{(a+b x) \sqrt {d+e x}}dx}{2 b}-\frac {\sqrt {d+e x}}{b (a+b x)}\right )}{4 b (b d-a e)}-\frac {(d+e x)^{3/2} (A b-a B)}{2 b (a+b x)^2 (b d-a e)}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {(-3 a B e-A b e+4 b B d) \left (\frac {\int \frac {1}{a+\frac {b (d+e x)}{e}-\frac {b d}{e}}d\sqrt {d+e x}}{b}-\frac {\sqrt {d+e x}}{b (a+b x)}\right )}{4 b (b d-a e)}-\frac {(d+e x)^{3/2} (A b-a B)}{2 b (a+b x)^2 (b d-a e)}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {(-3 a B e-A b e+4 b B d) \left (-\frac {e \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{b^{3/2} \sqrt {b d-a e}}-\frac {\sqrt {d+e x}}{b (a+b x)}\right )}{4 b (b d-a e)}-\frac {(d+e x)^{3/2} (A b-a B)}{2 b (a+b x)^2 (b d-a e)}\)

Input:

Int[((A + B*x)*Sqrt[d + e*x])/(a + b*x)^3,x]
 

Output:

-1/2*((A*b - a*B)*(d + e*x)^(3/2))/(b*(b*d - a*e)*(a + b*x)^2) + ((4*b*B*d 
 - A*b*e - 3*a*B*e)*(-(Sqrt[d + e*x]/(b*(a + b*x))) - (e*ArcTanh[(Sqrt[b]* 
Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(b^(3/2)*Sqrt[b*d - a*e])))/(4*b*(b*d - a 
*e))
 

Defintions of rubi rules used

rule 51

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + 1))), x] - Simp[d*(n/(b*(m + 1))) 
Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d, n}, x 
] && ILtQ[m, -1] && FractionQ[n] && GtQ[n, 0]
 

rule 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[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] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, x]
 

rule 87

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p 
_.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p 
+ 1)*(c*f - d*e))), x] - Simp[(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] || Intege 
rQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || LtQ[p, n]))))
 

rule 221

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]
 

Maple [A] (verified)

Time = 0.42 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.01

method	result	size
pseudoelliptic	\(-\frac {-\left (b x +a \right )^{2} e \left (\left (A e -4 B d \right ) b +3 B a e \right ) \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -d b \right ) b}}\right )+\sqrt {e x +d}\, \sqrt {\left (a e -d b \right ) b}\, \left (\left (-A e x -2 d \left (2 B x +A \right )\right ) b^{2}+a \left (\left (5 B x +A \right ) e -2 B d \right ) b +3 B \,a^{2} e \right )}{4 \sqrt {\left (a e -d b \right ) b}\, b^{2} \left (a e -d b \right ) \left (b x +a \right )^{2}}\)	\(147\)
derivativedivides	\(2 e \left (\frac {\frac {\left (A b e -5 B a e +4 B b d \right ) \left (e x +d \right )^{\frac {3}{2}}}{8 \left (a e -d b \right ) b}-\frac {\left (A b e +3 B a e -4 B b d \right ) \sqrt {e x +d}}{8 b^{2}}}{\left (\left (e x +d \right ) b +a e -d b \right )^{2}}+\frac {\left (A b e +3 B a e -4 B b d \right ) \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -d b \right ) b}}\right )}{8 \left (a e -d b \right ) b^{2} \sqrt {\left (a e -d b \right ) b}}\right )\)	\(152\)
default	\(2 e \left (\frac {\frac {\left (A b e -5 B a e +4 B b d \right ) \left (e x +d \right )^{\frac {3}{2}}}{8 \left (a e -d b \right ) b}-\frac {\left (A b e +3 B a e -4 B b d \right ) \sqrt {e x +d}}{8 b^{2}}}{\left (\left (e x +d \right ) b +a e -d b \right )^{2}}+\frac {\left (A b e +3 B a e -4 B b d \right ) \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -d b \right ) b}}\right )}{8 \left (a e -d b \right ) b^{2} \sqrt {\left (a e -d b \right ) b}}\right )\)	\(152\)

Input:

int((B*x+A)*(e*x+d)^(1/2)/(b*x+a)^3,x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 354 vs. \(2 (126) = 252\).

Time = 0.10 (sec) , antiderivative size = 721, normalized size of antiderivative = 4.94 \[ \int \frac {(A+B x) \sqrt {d+e x}}{(a+b x)^3} \, dx=\left [\frac {{\left (4 \, B a^{2} b d e - {\left (3 \, B a^{3} + A a^{2} b\right )} e^{2} + {\left (4 \, B b^{3} d e - {\left (3 \, B a b^{2} + A b^{3}\right )} e^{2}\right )} x^{2} + 2 \, {\left (4 \, B a b^{2} d e - {\left (3 \, B a^{2} b + A a b^{2}\right )} e^{2}\right )} x\right )} \sqrt {b^{2} d - a b e} \log \left (\frac {b e x + 2 \, b d - a e - 2 \, \sqrt {b^{2} d - a b e} \sqrt {e x + d}}{b x + a}\right ) - 2 \, {\left (2 \, {\left (B a b^{3} + A b^{4}\right )} d^{2} - {\left (5 \, B a^{2} b^{2} + 3 \, A a b^{3}\right )} d e + {\left (3 \, B a^{3} b + A a^{2} b^{2}\right )} e^{2} + {\left (4 \, B b^{4} d^{2} - {\left (9 \, B a b^{3} - A b^{4}\right )} d e + {\left (5 \, B a^{2} b^{2} - A a b^{3}\right )} e^{2}\right )} x\right )} \sqrt {e x + d}}{8 \, {\left (a^{2} b^{5} d^{2} - 2 \, a^{3} b^{4} d e + a^{4} b^{3} e^{2} + {\left (b^{7} d^{2} - 2 \, a b^{6} d e + a^{2} b^{5} e^{2}\right )} x^{2} + 2 \, {\left (a b^{6} d^{2} - 2 \, a^{2} b^{5} d e + a^{3} b^{4} e^{2}\right )} x\right )}}, \frac {{\left (4 \, B a^{2} b d e - {\left (3 \, B a^{3} + A a^{2} b\right )} e^{2} + {\left (4 \, B b^{3} d e - {\left (3 \, B a b^{2} + A b^{3}\right )} e^{2}\right )} x^{2} + 2 \, {\left (4 \, B a b^{2} d e - {\left (3 \, B a^{2} b + A a b^{2}\right )} e^{2}\right )} x\right )} \sqrt {-b^{2} d + a b e} \arctan \left (\frac {\sqrt {-b^{2} d + a b e} \sqrt {e x + d}}{b e x + b d}\right ) - {\left (2 \, {\left (B a b^{3} + A b^{4}\right )} d^{2} - {\left (5 \, B a^{2} b^{2} + 3 \, A a b^{3}\right )} d e + {\left (3 \, B a^{3} b + A a^{2} b^{2}\right )} e^{2} + {\left (4 \, B b^{4} d^{2} - {\left (9 \, B a b^{3} - A b^{4}\right )} d e + {\left (5 \, B a^{2} b^{2} - A a b^{3}\right )} e^{2}\right )} x\right )} \sqrt {e x + d}}{4 \, {\left (a^{2} b^{5} d^{2} - 2 \, a^{3} b^{4} d e + a^{4} b^{3} e^{2} + {\left (b^{7} d^{2} - 2 \, a b^{6} d e + a^{2} b^{5} e^{2}\right )} x^{2} + 2 \, {\left (a b^{6} d^{2} - 2 \, a^{2} b^{5} d e + a^{3} b^{4} e^{2}\right )} x\right )}}\right ] \] Input:

integrate((B*x+A)*(e*x+d)^(1/2)/(b*x+a)^3,x, algorithm="fricas")
 

Output:

[1/8*((4*B*a^2*b*d*e - (3*B*a^3 + A*a^2*b)*e^2 + (4*B*b^3*d*e - (3*B*a*b^2 
 + A*b^3)*e^2)*x^2 + 2*(4*B*a*b^2*d*e - (3*B*a^2*b + A*a*b^2)*e^2)*x)*sqrt 
(b^2*d - a*b*e)*log((b*e*x + 2*b*d - a*e - 2*sqrt(b^2*d - a*b*e)*sqrt(e*x 
+ d))/(b*x + a)) - 2*(2*(B*a*b^3 + A*b^4)*d^2 - (5*B*a^2*b^2 + 3*A*a*b^3)* 
d*e + (3*B*a^3*b + A*a^2*b^2)*e^2 + (4*B*b^4*d^2 - (9*B*a*b^3 - A*b^4)*d*e 
 + (5*B*a^2*b^2 - A*a*b^3)*e^2)*x)*sqrt(e*x + d))/(a^2*b^5*d^2 - 2*a^3*b^4 
*d*e + a^4*b^3*e^2 + (b^7*d^2 - 2*a*b^6*d*e + a^2*b^5*e^2)*x^2 + 2*(a*b^6* 
d^2 - 2*a^2*b^5*d*e + a^3*b^4*e^2)*x), 1/4*((4*B*a^2*b*d*e - (3*B*a^3 + A* 
a^2*b)*e^2 + (4*B*b^3*d*e - (3*B*a*b^2 + A*b^3)*e^2)*x^2 + 2*(4*B*a*b^2*d* 
e - (3*B*a^2*b + A*a*b^2)*e^2)*x)*sqrt(-b^2*d + a*b*e)*arctan(sqrt(-b^2*d 
+ a*b*e)*sqrt(e*x + d)/(b*e*x + b*d)) - (2*(B*a*b^3 + A*b^4)*d^2 - (5*B*a^ 
2*b^2 + 3*A*a*b^3)*d*e + (3*B*a^3*b + A*a^2*b^2)*e^2 + (4*B*b^4*d^2 - (9*B 
*a*b^3 - A*b^4)*d*e + (5*B*a^2*b^2 - A*a*b^3)*e^2)*x)*sqrt(e*x + d))/(a^2* 
b^5*d^2 - 2*a^3*b^4*d*e + a^4*b^3*e^2 + (b^7*d^2 - 2*a*b^6*d*e + a^2*b^5*e 
^2)*x^2 + 2*(a*b^6*d^2 - 2*a^2*b^5*d*e + a^3*b^4*e^2)*x)]
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(A+B x) \sqrt {d+e x}}{(a+b x)^3} \, dx=\text {Timed out} \] Input:

integrate((B*x+A)*(e*x+d)**(1/2)/(b*x+a)**3,x)
 

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {(A+B x) \sqrt {d+e x}}{(a+b x)^3} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((B*x+A)*(e*x+d)^(1/2)/(b*x+a)^3,x, algorithm="maxima")
 

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(a*e-b*d>0)', see `assume?` for m 
ore detail
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 235, normalized size of antiderivative = 1.61 \[ \int \frac {(A+B x) \sqrt {d+e x}}{(a+b x)^3} \, dx=\frac {{\left (4 \, B b d e - 3 \, B a e^{2} - A b e^{2}\right )} \arctan \left (\frac {\sqrt {e x + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{4 \, {\left (b^{3} d - a b^{2} e\right )} \sqrt {-b^{2} d + a b e}} - \frac {4 \, {\left (e x + d\right )}^{\frac {3}{2}} B b^{2} d e - 4 \, \sqrt {e x + d} B b^{2} d^{2} e - 5 \, {\left (e x + d\right )}^{\frac {3}{2}} B a b e^{2} + {\left (e x + d\right )}^{\frac {3}{2}} A b^{2} e^{2} + 7 \, \sqrt {e x + d} B a b d e^{2} + \sqrt {e x + d} A b^{2} d e^{2} - 3 \, \sqrt {e x + d} B a^{2} e^{3} - \sqrt {e x + d} A a b e^{3}}{4 \, {\left (b^{3} d - a b^{2} e\right )} {\left ({\left (e x + d\right )} b - b d + a e\right )}^{2}} \] Input:

integrate((B*x+A)*(e*x+d)^(1/2)/(b*x+a)^3,x, algorithm="giac")
 

Output:

1/4*(4*B*b*d*e - 3*B*a*e^2 - A*b*e^2)*arctan(sqrt(e*x + d)*b/sqrt(-b^2*d + 
 a*b*e))/((b^3*d - a*b^2*e)*sqrt(-b^2*d + a*b*e)) - 1/4*(4*(e*x + d)^(3/2) 
*B*b^2*d*e - 4*sqrt(e*x + d)*B*b^2*d^2*e - 5*(e*x + d)^(3/2)*B*a*b*e^2 + ( 
e*x + d)^(3/2)*A*b^2*e^2 + 7*sqrt(e*x + d)*B*a*b*d*e^2 + sqrt(e*x + d)*A*b 
^2*d*e^2 - 3*sqrt(e*x + d)*B*a^2*e^3 - sqrt(e*x + d)*A*a*b*e^3)/((b^3*d - 
a*b^2*e)*((e*x + d)*b - b*d + a*e)^2)
 

Mupad [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 222, normalized size of antiderivative = 1.52 \[ \int \frac {(A+B x) \sqrt {d+e x}}{(a+b x)^3} \, dx=\frac {e\,\mathrm {atan}\left (\frac {\sqrt {b}\,e\,\sqrt {d+e\,x}\,\left (A\,b\,e+3\,B\,a\,e-4\,B\,b\,d\right )}{\sqrt {a\,e-b\,d}\,\left (A\,b\,e^2+3\,B\,a\,e^2-4\,B\,b\,d\,e\right )}\right )\,\left (A\,b\,e+3\,B\,a\,e-4\,B\,b\,d\right )}{4\,b^{5/2}\,{\left (a\,e-b\,d\right )}^{3/2}}-\frac {\frac {\sqrt {d+e\,x}\,\left (A\,b\,e^2+3\,B\,a\,e^2-4\,B\,b\,d\,e\right )}{4\,b^2}-\frac {{\left (d+e\,x\right )}^{3/2}\,\left (A\,b\,e^2-5\,B\,a\,e^2+4\,B\,b\,d\,e\right )}{4\,b\,\left (a\,e-b\,d\right )}}{b^2\,{\left (d+e\,x\right )}^2-\left (2\,b^2\,d-2\,a\,b\,e\right )\,\left (d+e\,x\right )+a^2\,e^2+b^2\,d^2-2\,a\,b\,d\,e} \] Input:

int(((A + B*x)*(d + e*x)^(1/2))/(a + b*x)^3,x)
 

Output:

(e*atan((b^(1/2)*e*(d + e*x)^(1/2)*(A*b*e + 3*B*a*e - 4*B*b*d))/((a*e - b* 
d)^(1/2)*(A*b*e^2 + 3*B*a*e^2 - 4*B*b*d*e)))*(A*b*e + 3*B*a*e - 4*B*b*d))/ 
(4*b^(5/2)*(a*e - b*d)^(3/2)) - (((d + e*x)^(1/2)*(A*b*e^2 + 3*B*a*e^2 - 4 
*B*b*d*e))/(4*b^2) - ((d + e*x)^(3/2)*(A*b*e^2 - 5*B*a*e^2 + 4*B*b*d*e))/( 
4*b*(a*e - b*d)))/(b^2*(d + e*x)^2 - (2*b^2*d - 2*a*b*e)*(d + e*x) + a^2*e 
^2 + b^2*d^2 - 2*a*b*d*e)
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.88 \[ \int \frac {(A+B x) \sqrt {d+e x}}{(a+b x)^3} \, dx=\frac {\sqrt {b}\, \sqrt {a e -b d}\, \mathit {atan} \left (\frac {\sqrt {e x +d}\, b}{\sqrt {b}\, \sqrt {a e -b d}}\right ) a e +\sqrt {b}\, \sqrt {a e -b d}\, \mathit {atan} \left (\frac {\sqrt {e x +d}\, b}{\sqrt {b}\, \sqrt {a e -b d}}\right ) b e x -\sqrt {e x +d}\, a b e +\sqrt {e x +d}\, b^{2} d}{b^{2} \left (a b e x -b^{2} d x +a^{2} e -a b d \right )} \] Input:

int((B*x+A)*(e*x+d)^(1/2)/(b*x+a)^3,x)
 

Output:

(sqrt(b)*sqrt(a*e - b*d)*atan((sqrt(d + e*x)*b)/(sqrt(b)*sqrt(a*e - b*d))) 
*a*e + sqrt(b)*sqrt(a*e - b*d)*atan((sqrt(d + e*x)*b)/(sqrt(b)*sqrt(a*e - 
b*d)))*b*e*x - sqrt(d + e*x)*a*b*e + sqrt(d + e*x)*b**2*d)/(b**2*(a**2*e - 
 a*b*d + a*b*e*x - b**2*d*x))