Integrand size = 17, antiderivative size = 86 \[ \int \frac {(c+d x)^{3/2}}{a+b x} \, dx=\frac {2 (b c-a d) \sqrt {c+d x}}{b^2}+\frac {2 (c+d x)^{3/2}}{3 b}-\frac {2 (b c-a d)^{3/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{b^{5/2}} \] Output:
2*(-a*d+b*c)*(d*x+c)^(1/2)/b^2+2/3*(d*x+c)^(3/2)/b-2*(-a*d+b*c)^(3/2)*arct anh(b^(1/2)*(d*x+c)^(1/2)/(-a*d+b*c)^(1/2))/b^(5/2)
Time = 0.14 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.90 \[ \int \frac {(c+d x)^{3/2}}{a+b x} \, dx=\frac {2 \sqrt {c+d x} (4 b c-3 a d+b d x)}{3 b^2}+\frac {2 (-b c+a d)^{3/2} \arctan \left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {-b c+a d}}\right )}{b^{5/2}} \] Input:
Integrate[(c + d*x)^(3/2)/(a + b*x),x]
Output:
(2*Sqrt[c + d*x]*(4*b*c - 3*a*d + b*d*x))/(3*b^2) + (2*(-(b*c) + a*d)^(3/2 )*ArcTan[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[-(b*c) + a*d]])/b^(5/2)
Time = 0.20 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.06, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {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.
\(\displaystyle \int \frac {(c+d x)^{3/2}}{a+b x} \, dx\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {(b c-a d) \int \frac {\sqrt {c+d x}}{a+b x}dx}{b}+\frac {2 (c+d x)^{3/2}}{3 b}\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {(b c-a d) \left (\frac {(b c-a d) \int \frac {1}{(a+b x) \sqrt {c+d x}}dx}{b}+\frac {2 \sqrt {c+d x}}{b}\right )}{b}+\frac {2 (c+d x)^{3/2}}{3 b}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {(b c-a d) \left (\frac {2 (b c-a d) \int \frac {1}{a+\frac {b (c+d x)}{d}-\frac {b c}{d}}d\sqrt {c+d x}}{b d}+\frac {2 \sqrt {c+d x}}{b}\right )}{b}+\frac {2 (c+d x)^{3/2}}{3 b}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {(b c-a d) \left (\frac {2 \sqrt {c+d x}}{b}-\frac {2 \sqrt {b c-a d} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{b^{3/2}}\right )}{b}+\frac {2 (c+d x)^{3/2}}{3 b}\) |
Input:
Int[(c + d*x)^(3/2)/(a + b*x),x]
Output:
(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
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]
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]
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]
Time = 0.31 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.02
method | result | size |
risch | \(-\frac {2 \left (-b d x +3 a d -4 b c \right ) \sqrt {x d +c}}{3 b^{2}}+\frac {2 \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \arctan \left (\frac {b \sqrt {x d +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{b^{2} \sqrt {\left (a d -b c \right ) b}}\) | \(88\) |
pseudoelliptic | \(-\frac {2 \left (-\left (a d -b c \right )^{2} \arctan \left (\frac {b \sqrt {x d +c}}{\sqrt {\left (a d -b c \right ) b}}\right )+\left (\frac {\left (-x d -4 c \right ) b}{3}+a d \right ) \sqrt {x d +c}\, \sqrt {\left (a d -b c \right ) b}\right )}{\sqrt {\left (a d -b c \right ) b}\, b^{2}}\) | \(88\) |
derivativedivides | \(-\frac {2 \left (-\frac {b \left (x d +c \right )^{\frac {3}{2}}}{3}+\sqrt {x d +c}\, a d -\sqrt {x d +c}\, b c \right )}{b^{2}}+\frac {2 \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \arctan \left (\frac {b \sqrt {x d +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{b^{2} \sqrt {\left (a d -b c \right ) b}}\) | \(99\) |
default | \(-\frac {2 \left (-\frac {b \left (x d +c \right )^{\frac {3}{2}}}{3}+\sqrt {x d +c}\, a d -\sqrt {x d +c}\, b c \right )}{b^{2}}+\frac {2 \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \arctan \left (\frac {b \sqrt {x d +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{b^{2} \sqrt {\left (a d -b c \right ) b}}\) | \(99\) |
Input:
int((d*x+c)^(3/2)/(b*x+a),x,method=_RETURNVERBOSE)
Output:
-2/3*(-b*d*x+3*a*d-4*b*c)*(d*x+c)^(1/2)/b^2+2*(a^2*d^2-2*a*b*c*d+b^2*c^2)/ b^2/((a*d-b*c)*b)^(1/2)*arctan(b*(d*x+c)^(1/2)/((a*d-b*c)*b)^(1/2))
Time = 0.09 (sec) , antiderivative size = 188, normalized size of antiderivative = 2.19 \[ \int \frac {(c+d x)^{3/2}}{a+b x} \, dx=\left [-\frac {3 \, {\left (b c - a d\right )} \sqrt {\frac {b c - a d}{b}} \log \left (\frac {b d x + 2 \, b c - a d + 2 \, \sqrt {d x + c} b \sqrt {\frac {b c - a d}{b}}}{b x + a}\right ) - 2 \, {\left (b d x + 4 \, b c - 3 \, a d\right )} \sqrt {d x + c}}{3 \, b^{2}}, -\frac {2 \, {\left (3 \, {\left (b c - a d\right )} \sqrt {-\frac {b c - a d}{b}} \arctan \left (-\frac {\sqrt {d x + c} b \sqrt {-\frac {b c - a d}{b}}}{b c - a d}\right ) - {\left (b d x + 4 \, b c - 3 \, a d\right )} \sqrt {d x + c}\right )}}{3 \, b^{2}}\right ] \] Input:
integrate((d*x+c)^(3/2)/(b*x+a),x, algorithm="fricas")
Output:
[-1/3*(3*(b*c - a*d)*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*(b*d*x + 4*b*c - 3*a*d)*sq rt(d*x + c))/b^2, -2/3*(3*(b*c - a*d)*sqrt(-(b*c - a*d)/b)*arctan(-sqrt(d* x + c)*b*sqrt(-(b*c - a*d)/b)/(b*c - a*d)) - (b*d*x + 4*b*c - 3*a*d)*sqrt( d*x + c))/b^2]
Time = 1.15 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.22 \[ \int \frac {(c+d x)^{3/2}}{a+b x} \, dx=\begin {cases} \frac {2 \left (\frac {d \left (c + d x\right )^{\frac {3}{2}}}{3 b} + \frac {\sqrt {c + d x} \left (- a d^{2} + b c d\right )}{b^{2}} + \frac {d \left (a d - b c\right )^{2} \operatorname {atan}{\left (\frac {\sqrt {c + d x}}{\sqrt {\frac {a d - b c}{b}}} \right )}}{b^{3} \sqrt {\frac {a d - b c}{b}}}\right )}{d} & \text {for}\: d \neq 0 \\c^{\frac {3}{2}} \left (\begin {cases} \frac {x}{a} & \text {for}\: b = 0 \\\frac {\log {\left (a + b x \right )}}{b} & \text {otherwise} \end {cases}\right ) & \text {otherwise} \end {cases} \] Input:
integrate((d*x+c)**(3/2)/(b*x+a),x)
Output:
Piecewise((2*(d*(c + d*x)**(3/2)/(3*b) + sqrt(c + d*x)*(-a*d**2 + b*c*d)/b **2 + d*(a*d - b*c)**2*atan(sqrt(c + d*x)/sqrt((a*d - b*c)/b))/(b**3*sqrt( (a*d - b*c)/b)))/d, Ne(d, 0)), (c**(3/2)*Piecewise((x/a, Eq(b, 0)), (log(a + b*x)/b, True)), True))
Exception generated. \[ \int \frac {(c+d x)^{3/2}}{a+b x} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((d*x+c)^(3/2)/(b*x+a),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
Time = 0.13 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.22 \[ \int \frac {(c+d x)^{3/2}}{a+b x} \, dx=\frac {2 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \arctan \left (\frac {\sqrt {d x + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{\sqrt {-b^{2} c + a b d} b^{2}} + \frac {2 \, {\left ({\left (d x + c\right )}^{\frac {3}{2}} b^{2} + 3 \, \sqrt {d x + c} b^{2} c - 3 \, \sqrt {d x + c} a b d\right )}}{3 \, b^{3}} \] Input:
integrate((d*x+c)^(3/2)/(b*x+a),x, algorithm="giac")
Output:
2*(b^2*c^2 - 2*a*b*c*d + a^2*d^2)*arctan(sqrt(d*x + c)*b/sqrt(-b^2*c + a*b *d))/(sqrt(-b^2*c + a*b*d)*b^2) + 2/3*((d*x + c)^(3/2)*b^2 + 3*sqrt(d*x + c)*b^2*c - 3*sqrt(d*x + c)*a*b*d)/b^3
Time = 0.16 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.08 \[ \int \frac {(c+d x)^{3/2}}{a+b x} \, dx=\frac {2\,{\left (c+d\,x\right )}^{3/2}}{3\,b}-\frac {2\,\left (a\,d-b\,c\right )\,\sqrt {c+d\,x}}{b^2}+\frac {2\,\mathrm {atan}\left (\frac {\sqrt {b}\,{\left (a\,d-b\,c\right )}^{3/2}\,\sqrt {c+d\,x}}{a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2}\right )\,{\left (a\,d-b\,c\right )}^{3/2}}{b^{5/2}} \] Input:
int((c + d*x)^(3/2)/(a + b*x),x)
Output:
(2*(c + d*x)^(3/2))/(3*b) - (2*(a*d - b*c)*(c + d*x)^(1/2))/b^2 + (2*atan( (b^(1/2)*(a*d - b*c)^(3/2)*(c + d*x)^(1/2))/(a^2*d^2 + b^2*c^2 - 2*a*b*c*d ))*(a*d - b*c)^(3/2))/b^(5/2)
Time = 0.17 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.38 \[ \int \frac {(c+d x)^{3/2}}{a+b x} \, dx=\frac {2 \sqrt {b}\, \sqrt {a d -b c}\, \mathit {atan} \left (\frac {\sqrt {d x +c}\, b}{\sqrt {b}\, \sqrt {a d -b c}}\right ) a d -2 \sqrt {b}\, \sqrt {a d -b c}\, \mathit {atan} \left (\frac {\sqrt {d x +c}\, b}{\sqrt {b}\, \sqrt {a d -b c}}\right ) b c -2 \sqrt {d x +c}\, a b d +\frac {8 \sqrt {d x +c}\, b^{2} c}{3}+\frac {2 \sqrt {d x +c}\, b^{2} d x}{3}}{b^{3}} \] Input:
int((d*x+c)^(3/2)/(b*x+a),x)
Output:
(2*(3*sqrt(b)*sqrt(a*d - b*c)*atan((sqrt(c + d*x)*b)/(sqrt(b)*sqrt(a*d - b *c)))*a*d - 3*sqrt(b)*sqrt(a*d - b*c)*atan((sqrt(c + d*x)*b)/(sqrt(b)*sqrt (a*d - b*c)))*b*c - 3*sqrt(c + d*x)*a*b*d + 4*sqrt(c + d*x)*b**2*c + sqrt( c + d*x)*b**2*d*x))/(3*b**3)