\(\int \frac {(c+d x)^{5/2}}{(a+b x)^2} \, dx\) [236]

Optimal result

Integrand size = 17, antiderivative size = 110 \[ \int \frac {(c+d x)^{5/2}}{(a+b x)^2} \, dx=\frac {5 d (b c-a d) \sqrt {c+d x}}{b^3}+\frac {5 d (c+d x)^{3/2}}{3 b^2}-\frac {(c+d x)^{5/2}}{b (a+b x)}-\frac {5 d (b c-a d)^{3/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{b^{7/2}} \] Output:

5*d*(-a*d+b*c)*(d*x+c)^(1/2)/b^3+5/3*d*(d*x+c)^(3/2)/b^2-(d*x+c)^(5/2)/b/( 
b*x+a)-5*d*(-a*d+b*c)^(3/2)*arctanh(b^(1/2)*(d*x+c)^(1/2)/(-a*d+b*c)^(1/2) 
)/b^(7/2)
 

Mathematica [A] (verified)

Time = 0.27 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.05 \[ \int \frac {(c+d x)^{5/2}}{(a+b x)^2} \, dx=-\frac {\sqrt {c+d x} \left (15 a^2 d^2+10 a b d (-2 c+d x)+b^2 \left (3 c^2-14 c d x-2 d^2 x^2\right )\right )}{3 b^3 (a+b x)}+\frac {5 d (-b c+a d)^{3/2} \arctan \left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {-b c+a d}}\right )}{b^{7/2}} \] Input:

Integrate[(c + d*x)^(5/2)/(a + b*x)^2,x]
 

Output:

-1/3*(Sqrt[c + d*x]*(15*a^2*d^2 + 10*a*b*d*(-2*c + d*x) + b^2*(3*c^2 - 14* 
c*d*x - 2*d^2*x^2)))/(b^3*(a + b*x)) + (5*d*(-(b*c) + a*d)^(3/2)*ArcTan[(S 
qrt[b]*Sqrt[c + d*x])/Sqrt[-(b*c) + a*d]])/b^(7/2)
 

Rubi [A] (verified)

Time = 0.21 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.10, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {51, 60, 60, 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.

\(\displaystyle \int \frac {(c+d x)^{5/2}}{(a+b x)^2} \, dx\)

\(\Big \downarrow \) 51

\(\displaystyle \frac {5 d \int \frac {(c+d x)^{3/2}}{a+b x}dx}{2 b}-\frac {(c+d x)^{5/2}}{b (a+b x)}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {5 d \left (\frac {(b c-a d) \int \frac {\sqrt {c+d x}}{a+b x}dx}{b}+\frac {2 (c+d x)^{3/2}}{3 b}\right )}{2 b}-\frac {(c+d x)^{5/2}}{b (a+b x)}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {5 d \left (\frac {(b c-a d) \left (\frac {(b c-a d) \int \frac {1}{(a+b x) \sqrt {c+d x}}dx}{b}+\frac {2 \sqrt {c+d x}}{b}\right )}{b}+\frac {2 (c+d x)^{3/2}}{3 b}\right )}{2 b}-\frac {(c+d x)^{5/2}}{b (a+b x)}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {5 d \left (\frac {(b c-a d) \left (\frac {2 (b c-a d) \int \frac {1}{a+\frac {b (c+d x)}{d}-\frac {b c}{d}}d\sqrt {c+d x}}{b d}+\frac {2 \sqrt {c+d x}}{b}\right )}{b}+\frac {2 (c+d x)^{3/2}}{3 b}\right )}{2 b}-\frac {(c+d x)^{5/2}}{b (a+b x)}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {5 d \left (\frac {(b c-a d) \left (\frac {2 \sqrt {c+d x}}{b}-\frac {2 \sqrt {b c-a d} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{b^{3/2}}\right )}{b}+\frac {2 (c+d x)^{3/2}}{3 b}\right )}{2 b}-\frac {(c+d x)^{5/2}}{b (a+b x)}\)

Input:

Int[(c + d*x)^(5/2)/(a + b*x)^2,x]
 

Output:

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

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 60

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] + Simp[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] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !Integer 
Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinear 
Q[a, b, c, d, m, n, x]
 

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 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 = 0.33 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.09

method	result	size
risch	\(-\frac {2 d \left (-b d x +6 a d -7 b c \right ) \sqrt {x d +c}}{3 b^{3}}+\frac {\left (2 a^{2} d^{2}-4 a b c d +2 b^{2} c^{2}\right ) d \left (-\frac {\sqrt {x d +c}}{2 \left (\left (x d +c \right ) b +a d -b c \right )}+\frac {5 \arctan \left (\frac {b \sqrt {x d +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{2 \sqrt {\left (a d -b c \right ) b}}\right )}{b^{3}}\)	\(120\)
pseudoelliptic	\(-\frac {5 \left (-d \left (a d -b c \right )^{2} \left (b x +a \right ) \arctan \left (\frac {b \sqrt {x d +c}}{\sqrt {\left (a d -b c \right ) b}}\right )+\sqrt {\left (a d -b c \right ) b}\, \left (\frac {\left (-\frac {2}{3} d^{2} x^{2}-\frac {14}{3} c d x +c^{2}\right ) b^{2}}{5}-\frac {4 \left (-\frac {x d}{2}+c \right ) a d b}{3}+a^{2} d^{2}\right ) \sqrt {x d +c}\right )}{\sqrt {\left (a d -b c \right ) b}\, b^{3} \left (b x +a \right )}\)	\(127\)
derivativedivides	\(2 d \left (-\frac {-\frac {b \left (x d +c \right )^{\frac {3}{2}}}{3}+2 \sqrt {x d +c}\, a d -2 \sqrt {x d +c}\, b c}{b^{3}}+\frac {\frac {\left (-\frac {1}{2} a^{2} d^{2}+a b c d -\frac {1}{2} b^{2} c^{2}\right ) \sqrt {x d +c}}{\left (x d +c \right ) b +a d -b c}+\frac {5 \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \arctan \left (\frac {b \sqrt {x d +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{2 \sqrt {\left (a d -b c \right ) b}}}{b^{3}}\right )\)	\(152\)
default	\(2 d \left (-\frac {-\frac {b \left (x d +c \right )^{\frac {3}{2}}}{3}+2 \sqrt {x d +c}\, a d -2 \sqrt {x d +c}\, b c}{b^{3}}+\frac {\frac {\left (-\frac {1}{2} a^{2} d^{2}+a b c d -\frac {1}{2} b^{2} c^{2}\right ) \sqrt {x d +c}}{\left (x d +c \right ) b +a d -b c}+\frac {5 \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \arctan \left (\frac {b \sqrt {x d +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{2 \sqrt {\left (a d -b c \right ) b}}}{b^{3}}\right )\)	\(152\)

Input:

int((d*x+c)^(5/2)/(b*x+a)^2,x,method=_RETURNVERBOSE)
 

Output:

-2/3*d*(-b*d*x+6*a*d-7*b*c)*(d*x+c)^(1/2)/b^3+1/b^3*(2*a^2*d^2-4*a*b*c*d+2 
*b^2*c^2)*d*(-1/2*(d*x+c)^(1/2)/((d*x+c)*b+a*d-b*c)+5/2/((a*d-b*c)*b)^(1/2 
)*arctan(b*(d*x+c)^(1/2)/((a*d-b*c)*b)^(1/2)))
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 330, normalized size of antiderivative = 3.00 \[ \int \frac {(c+d x)^{5/2}}{(a+b x)^2} \, dx=\left [-\frac {15 \, {\left (a b c d - a^{2} d^{2} + {\left (b^{2} c d - a b d^{2}\right )} x\right )} \sqrt {\frac {b c - a d}{b}} \log \left (\frac {b d x + 2 \, b c - a d + 2 \, \sqrt {d x + c} b \sqrt {\frac {b c - a d}{b}}}{b x + a}\right ) - 2 \, {\left (2 \, b^{2} d^{2} x^{2} - 3 \, b^{2} c^{2} + 20 \, a b c d - 15 \, a^{2} d^{2} + 2 \, {\left (7 \, b^{2} c d - 5 \, a b d^{2}\right )} x\right )} \sqrt {d x + c}}{6 \, {\left (b^{4} x + a b^{3}\right )}}, -\frac {15 \, {\left (a b c d - a^{2} d^{2} + {\left (b^{2} c d - a b d^{2}\right )} x\right )} \sqrt {-\frac {b c - a d}{b}} \arctan \left (-\frac {\sqrt {d x + c} b \sqrt {-\frac {b c - a d}{b}}}{b c - a d}\right ) - {\left (2 \, b^{2} d^{2} x^{2} - 3 \, b^{2} c^{2} + 20 \, a b c d - 15 \, a^{2} d^{2} + 2 \, {\left (7 \, b^{2} c d - 5 \, a b d^{2}\right )} x\right )} \sqrt {d x + c}}{3 \, {\left (b^{4} x + a b^{3}\right )}}\right ] \] Input:

integrate((d*x+c)^(5/2)/(b*x+a)^2,x, algorithm="fricas")
 

Output:

[-1/6*(15*(a*b*c*d - a^2*d^2 + (b^2*c*d - a*b*d^2)*x)*sqrt((b*c - a*d)/b)* 
log((b*d*x + 2*b*c - a*d + 2*sqrt(d*x + c)*b*sqrt((b*c - a*d)/b))/(b*x + a 
)) - 2*(2*b^2*d^2*x^2 - 3*b^2*c^2 + 20*a*b*c*d - 15*a^2*d^2 + 2*(7*b^2*c*d 
 - 5*a*b*d^2)*x)*sqrt(d*x + c))/(b^4*x + a*b^3), -1/3*(15*(a*b*c*d - a^2*d 
^2 + (b^2*c*d - a*b*d^2)*x)*sqrt(-(b*c - a*d)/b)*arctan(-sqrt(d*x + c)*b*s 
qrt(-(b*c - a*d)/b)/(b*c - a*d)) - (2*b^2*d^2*x^2 - 3*b^2*c^2 + 20*a*b*c*d 
 - 15*a^2*d^2 + 2*(7*b^2*c*d - 5*a*b*d^2)*x)*sqrt(d*x + c))/(b^4*x + a*b^3 
)]
 

Sympy [F]

\[ \int \frac {(c+d x)^{5/2}}{(a+b x)^2} \, dx=\int \frac {\left (c + d x\right )^{\frac {5}{2}}}{\left (a + b x\right )^{2}}\, dx \] Input:

integrate((d*x+c)**(5/2)/(b*x+a)**2,x)
 

Output:

Integral((c + d*x)**(5/2)/(a + b*x)**2, x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {(c+d x)^{5/2}}{(a+b x)^2} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((d*x+c)^(5/2)/(b*x+a)^2,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*d-b*c>0)', see `assume?` for m 
ore detail
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.65 \[ \int \frac {(c+d x)^{5/2}}{(a+b x)^2} \, dx=\frac {5 \, {\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} \arctan \left (\frac {\sqrt {d x + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{\sqrt {-b^{2} c + a b d} b^{3}} - \frac {\sqrt {d x + c} b^{2} c^{2} d - 2 \, \sqrt {d x + c} a b c d^{2} + \sqrt {d x + c} a^{2} d^{3}}{{\left ({\left (d x + c\right )} b - b c + a d\right )} b^{3}} + \frac {2 \, {\left ({\left (d x + c\right )}^{\frac {3}{2}} b^{4} d + 6 \, \sqrt {d x + c} b^{4} c d - 6 \, \sqrt {d x + c} a b^{3} d^{2}\right )}}{3 \, b^{6}} \] Input:

integrate((d*x+c)^(5/2)/(b*x+a)^2,x, algorithm="giac")
 

Output:

5*(b^2*c^2*d - 2*a*b*c*d^2 + a^2*d^3)*arctan(sqrt(d*x + c)*b/sqrt(-b^2*c + 
 a*b*d))/(sqrt(-b^2*c + a*b*d)*b^3) - (sqrt(d*x + c)*b^2*c^2*d - 2*sqrt(d* 
x + c)*a*b*c*d^2 + sqrt(d*x + c)*a^2*d^3)/(((d*x + c)*b - b*c + a*d)*b^3) 
+ 2/3*((d*x + c)^(3/2)*b^4*d + 6*sqrt(d*x + c)*b^4*c*d - 6*sqrt(d*x + c)*a 
*b^3*d^2)/b^6
 

Mupad [B] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.46 \[ \int \frac {(c+d x)^{5/2}}{(a+b x)^2} \, dx=\frac {2\,d\,{\left (c+d\,x\right )}^{3/2}}{3\,b^2}-\frac {\sqrt {c+d\,x}\,\left (a^2\,d^3-2\,a\,b\,c\,d^2+b^2\,c^2\,d\right )}{b^4\,\left (c+d\,x\right )-b^4\,c+a\,b^3\,d}+\frac {5\,d\,\mathrm {atan}\left (\frac {\sqrt {b}\,d\,{\left (a\,d-b\,c\right )}^{3/2}\,\sqrt {c+d\,x}}{a^2\,d^3-2\,a\,b\,c\,d^2+b^2\,c^2\,d}\right )\,{\left (a\,d-b\,c\right )}^{3/2}}{b^{7/2}}+\frac {2\,d\,\left (2\,b^2\,c-2\,a\,b\,d\right )\,\sqrt {c+d\,x}}{b^4} \] Input:

int((c + d*x)^(5/2)/(a + b*x)^2,x)
 

Output:

(2*d*(c + d*x)^(3/2))/(3*b^2) - ((c + d*x)^(1/2)*(a^2*d^3 + b^2*c^2*d - 2* 
a*b*c*d^2))/(b^4*(c + d*x) - b^4*c + a*b^3*d) + (5*d*atan((b^(1/2)*d*(a*d 
- b*c)^(3/2)*(c + d*x)^(1/2))/(a^2*d^3 + b^2*c^2*d - 2*a*b*c*d^2))*(a*d - 
b*c)^(3/2))/b^(7/2) + (2*d*(2*b^2*c - 2*a*b*d)*(c + d*x)^(1/2))/b^4
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 273, normalized size of antiderivative = 2.48 \[ \int \frac {(c+d x)^{5/2}}{(a+b x)^2} \, dx=\frac {15 \sqrt {b}\, \sqrt {a d -b c}\, \mathit {atan} \left (\frac {\sqrt {d x +c}\, b}{\sqrt {b}\, \sqrt {a d -b c}}\right ) a^{2} d^{2}-15 \sqrt {b}\, \sqrt {a d -b c}\, \mathit {atan} \left (\frac {\sqrt {d x +c}\, b}{\sqrt {b}\, \sqrt {a d -b c}}\right ) a b c d +15 \sqrt {b}\, \sqrt {a d -b c}\, \mathit {atan} \left (\frac {\sqrt {d x +c}\, b}{\sqrt {b}\, \sqrt {a d -b c}}\right ) a b \,d^{2} x -15 \sqrt {b}\, \sqrt {a d -b c}\, \mathit {atan} \left (\frac {\sqrt {d x +c}\, b}{\sqrt {b}\, \sqrt {a d -b c}}\right ) b^{2} c d x -15 \sqrt {d x +c}\, a^{2} b \,d^{2}+20 \sqrt {d x +c}\, a \,b^{2} c d -10 \sqrt {d x +c}\, a \,b^{2} d^{2} x -3 \sqrt {d x +c}\, b^{3} c^{2}+14 \sqrt {d x +c}\, b^{3} c d x +2 \sqrt {d x +c}\, b^{3} d^{2} x^{2}}{3 b^{4} \left (b x +a \right )} \] Input:

int((d*x+c)^(5/2)/(b*x+a)^2,x)
 

Output:

(15*sqrt(b)*sqrt(a*d - b*c)*atan((sqrt(c + d*x)*b)/(sqrt(b)*sqrt(a*d - b*c 
)))*a**2*d**2 - 15*sqrt(b)*sqrt(a*d - b*c)*atan((sqrt(c + d*x)*b)/(sqrt(b) 
*sqrt(a*d - b*c)))*a*b*c*d + 15*sqrt(b)*sqrt(a*d - b*c)*atan((sqrt(c + d*x 
)*b)/(sqrt(b)*sqrt(a*d - b*c)))*a*b*d**2*x - 15*sqrt(b)*sqrt(a*d - b*c)*at 
an((sqrt(c + d*x)*b)/(sqrt(b)*sqrt(a*d - b*c)))*b**2*c*d*x - 15*sqrt(c + d 
*x)*a**2*b*d**2 + 20*sqrt(c + d*x)*a*b**2*c*d - 10*sqrt(c + d*x)*a*b**2*d* 
*2*x - 3*sqrt(c + d*x)*b**3*c**2 + 14*sqrt(c + d*x)*b**3*c*d*x + 2*sqrt(c 
+ d*x)*b**3*d**2*x**2)/(3*b**4*(a + b*x))