Integrand size = 19, antiderivative size = 113 \[ \int \frac {(c+d x)^{3/2}}{\sqrt {a+b x}} \, dx=\frac {3 (b c-a d) \sqrt {a+b x} \sqrt {c+d x}}{4 b^2}+\frac {\sqrt {a+b x} (c+d x)^{3/2}}{2 b}+\frac {3 (b c-a d)^2 \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{4 b^{5/2} \sqrt {d}} \] Output:
3/4*(-a*d+b*c)*(b*x+a)^(1/2)*(d*x+c)^(1/2)/b^2+1/2*(b*x+a)^(1/2)*(d*x+c)^( 3/2)/b+3/4*(-a*d+b*c)^2*arctanh(d^(1/2)*(b*x+a)^(1/2)/b^(1/2)/(d*x+c)^(1/2 ))/b^(5/2)/d^(1/2)
Time = 0.14 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.83 \[ \int \frac {(c+d x)^{3/2}}{\sqrt {a+b x}} \, dx=\frac {\sqrt {a+b x} \sqrt {c+d x} (5 b c-3 a d+2 b d x)}{4 b^2}+\frac {3 (b c-a d)^2 \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {a+b x}}\right )}{4 b^{5/2} \sqrt {d}} \] Input:
Integrate[(c + d*x)^(3/2)/Sqrt[a + b*x],x]
Output:
(Sqrt[a + b*x]*Sqrt[c + d*x]*(5*b*c - 3*a*d + 2*b*d*x))/(4*b^2) + (3*(b*c - a*d)^2*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[a + b*x])])/(4*b^(5 /2)*Sqrt[d])
Time = 0.18 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {60, 60, 66, 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.
\(\displaystyle \int \frac {(c+d x)^{3/2}}{\sqrt {a+b x}} \, dx\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {3 (b c-a d) \int \frac {\sqrt {c+d x}}{\sqrt {a+b x}}dx}{4 b}+\frac {\sqrt {a+b x} (c+d x)^{3/2}}{2 b}\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {3 (b c-a d) \left (\frac {(b c-a d) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}}dx}{2 b}+\frac {\sqrt {a+b x} \sqrt {c+d x}}{b}\right )}{4 b}+\frac {\sqrt {a+b x} (c+d x)^{3/2}}{2 b}\) |
\(\Big \downarrow \) 66 |
\(\displaystyle \frac {3 (b c-a d) \left (\frac {(b c-a d) \int \frac {1}{b-\frac {d (a+b x)}{c+d x}}d\frac {\sqrt {a+b x}}{\sqrt {c+d x}}}{b}+\frac {\sqrt {a+b x} \sqrt {c+d x}}{b}\right )}{4 b}+\frac {\sqrt {a+b x} (c+d x)^{3/2}}{2 b}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {3 (b c-a d) \left (\frac {(b c-a d) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{b^{3/2} \sqrt {d}}+\frac {\sqrt {a+b x} \sqrt {c+d x}}{b}\right )}{4 b}+\frac {\sqrt {a+b x} (c+d x)^{3/2}}{2 b}\) |
Input:
Int[(c + d*x)^(3/2)/Sqrt[a + b*x],x]
Output:
(Sqrt[a + b*x]*(c + d*x)^(3/2))/(2*b) + (3*(b*c - a*d)*((Sqrt[a + b*x]*Sqr t[c + d*x])/b + ((b*c - a*d)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt [c + d*x])])/(b^(3/2)*Sqrt[d])))/(4*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[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[ 2 Subst[Int[1/(b - d*x^2), x], x, Sqrt[a + b*x]/Sqrt[c + d*x]], x] /; Fre eQ[{a, b, c, d}, x] && !GtQ[c - a*(d/b), 0]
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 = 0.25 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.24
method | result | size |
default | \(\frac {\sqrt {b x +a}\, \left (x d +c \right )^{\frac {3}{2}}}{2 b}-\frac {3 \left (a d -b c \right ) \left (\frac {\sqrt {b x +a}\, \sqrt {x d +c}}{b}-\frac {\left (a d -b c \right ) \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \ln \left (\frac {\frac {1}{2} a d +\frac {1}{2} b c +b d x}{\sqrt {d b}}+\sqrt {b d \,x^{2}+\left (a d +b c \right ) x +a c}\right )}{2 b \sqrt {x d +c}\, \sqrt {b x +a}\, \sqrt {d b}}\right )}{4 b}\) | \(140\) |
Input:
int((d*x+c)^(3/2)/(b*x+a)^(1/2),x,method=_RETURNVERBOSE)
Output:
1/2*(b*x+a)^(1/2)*(d*x+c)^(3/2)/b-3/4*(a*d-b*c)/b*((b*x+a)^(1/2)*(d*x+c)^( 1/2)/b-1/2*(a*d-b*c)/b*((b*x+a)*(d*x+c))^(1/2)/(d*x+c)^(1/2)/(b*x+a)^(1/2) *ln((1/2*a*d+1/2*b*c+b*d*x)/(d*b)^(1/2)+(b*d*x^2+(a*d+b*c)*x+a*c)^(1/2))/( d*b)^(1/2))
Time = 0.09 (sec) , antiderivative size = 306, normalized size of antiderivative = 2.71 \[ \int \frac {(c+d x)^{3/2}}{\sqrt {a+b x}} \, dx=\left [\frac {3 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt {b d} \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 4 \, {\left (2 \, b d x + b c + a d\right )} \sqrt {b d} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x\right ) + 4 \, {\left (2 \, b^{2} d^{2} x + 5 \, b^{2} c d - 3 \, a b d^{2}\right )} \sqrt {b x + a} \sqrt {d x + c}}{16 \, b^{3} d}, -\frac {3 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt {-b d} \arctan \left (\frac {{\left (2 \, b d x + b c + a d\right )} \sqrt {-b d} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (b^{2} d^{2} x^{2} + a b c d + {\left (b^{2} c d + a b d^{2}\right )} x\right )}}\right ) - 2 \, {\left (2 \, b^{2} d^{2} x + 5 \, b^{2} c d - 3 \, a b d^{2}\right )} \sqrt {b x + a} \sqrt {d x + c}}{8 \, b^{3} d}\right ] \] Input:
integrate((d*x+c)^(3/2)/(b*x+a)^(1/2),x, algorithm="fricas")
Output:
[1/16*(3*(b^2*c^2 - 2*a*b*c*d + a^2*d^2)*sqrt(b*d)*log(8*b^2*d^2*x^2 + b^2 *c^2 + 6*a*b*c*d + a^2*d^2 + 4*(2*b*d*x + b*c + a*d)*sqrt(b*d)*sqrt(b*x + a)*sqrt(d*x + c) + 8*(b^2*c*d + a*b*d^2)*x) + 4*(2*b^2*d^2*x + 5*b^2*c*d - 3*a*b*d^2)*sqrt(b*x + a)*sqrt(d*x + c))/(b^3*d), -1/8*(3*(b^2*c^2 - 2*a*b *c*d + a^2*d^2)*sqrt(-b*d)*arctan(1/2*(2*b*d*x + b*c + a*d)*sqrt(-b*d)*sqr t(b*x + a)*sqrt(d*x + c)/(b^2*d^2*x^2 + a*b*c*d + (b^2*c*d + a*b*d^2)*x)) - 2*(2*b^2*d^2*x + 5*b^2*c*d - 3*a*b*d^2)*sqrt(b*x + a)*sqrt(d*x + c))/(b^ 3*d)]
\[ \int \frac {(c+d x)^{3/2}}{\sqrt {a+b x}} \, dx=\int \frac {\left (c + d x\right )^{\frac {3}{2}}}{\sqrt {a + b x}}\, dx \] Input:
integrate((d*x+c)**(3/2)/(b*x+a)**(1/2),x)
Output:
Integral((c + d*x)**(3/2)/sqrt(a + b*x), x)
Exception generated. \[ \int \frac {(c+d x)^{3/2}}{\sqrt {a+b x}} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((d*x+c)^(3/2)/(b*x+a)^(1/2),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
Leaf count of result is larger than twice the leaf count of optimal. 233 vs. \(2 (87) = 174\).
Time = 0.16 (sec) , antiderivative size = 233, normalized size of antiderivative = 2.06 \[ \int \frac {(c+d x)^{3/2}}{\sqrt {a+b x}} \, dx=-\frac {\frac {4 \, {\left (\frac {{\left (b^{2} c - a b d\right )} \log \left ({\left | -\sqrt {b d} \sqrt {b x + a} + \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} \right |}\right )}{\sqrt {b d}} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} \sqrt {b x + a}\right )} c {\left | b \right |}}{b^{2}} - \frac {{\left (\sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} {\left (2 \, b x + 2 \, a + \frac {b c d - 5 \, a d^{2}}{d^{2}}\right )} \sqrt {b x + a} + \frac {{\left (b^{3} c^{2} + 2 \, a b^{2} c d - 3 \, a^{2} b d^{2}\right )} \log \left ({\left | -\sqrt {b d} \sqrt {b x + a} + \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} \right |}\right )}{\sqrt {b d} d}\right )} d {\left | b \right |}}{b^{3}}}{4 \, b} \] Input:
integrate((d*x+c)^(3/2)/(b*x+a)^(1/2),x, algorithm="giac")
Output:
-1/4*(4*((b^2*c - a*b*d)*log(abs(-sqrt(b*d)*sqrt(b*x + a) + sqrt(b^2*c + ( b*x + a)*b*d - a*b*d)))/sqrt(b*d) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d)*sq rt(b*x + a))*c*abs(b)/b^2 - (sqrt(b^2*c + (b*x + a)*b*d - a*b*d)*(2*b*x + 2*a + (b*c*d - 5*a*d^2)/d^2)*sqrt(b*x + a) + (b^3*c^2 + 2*a*b^2*c*d - 3*a^ 2*b*d^2)*log(abs(-sqrt(b*d)*sqrt(b*x + a) + sqrt(b^2*c + (b*x + a)*b*d - a *b*d)))/(sqrt(b*d)*d))*d*abs(b)/b^3)/b
Timed out. \[ \int \frac {(c+d x)^{3/2}}{\sqrt {a+b x}} \, dx=\int \frac {{\left (c+d\,x\right )}^{3/2}}{\sqrt {a+b\,x}} \,d x \] Input:
int((c + d*x)^(3/2)/(a + b*x)^(1/2),x)
Output:
int((c + d*x)^(3/2)/(a + b*x)^(1/2), x)
Time = 0.17 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.75 \[ \int \frac {(c+d x)^{3/2}}{\sqrt {a+b x}} \, dx=\frac {-3 \sqrt {d x +c}\, \sqrt {b x +a}\, a b \,d^{2}+5 \sqrt {d x +c}\, \sqrt {b x +a}\, b^{2} c d +2 \sqrt {d x +c}\, \sqrt {b x +a}\, b^{2} d^{2} x +3 \sqrt {d}\, \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {d}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {d x +c}}{\sqrt {a d -b c}}\right ) a^{2} d^{2}-6 \sqrt {d}\, \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {d}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {d x +c}}{\sqrt {a d -b c}}\right ) a b c d +3 \sqrt {d}\, \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {d}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {d x +c}}{\sqrt {a d -b c}}\right ) b^{2} c^{2}}{4 b^{3} d} \] Input:
int((d*x+c)^(3/2)/(b*x+a)^(1/2),x)
Output:
( - 3*sqrt(c + d*x)*sqrt(a + b*x)*a*b*d**2 + 5*sqrt(c + d*x)*sqrt(a + b*x) *b**2*c*d + 2*sqrt(c + d*x)*sqrt(a + b*x)*b**2*d**2*x + 3*sqrt(d)*sqrt(b)* log((sqrt(d)*sqrt(a + b*x) + sqrt(b)*sqrt(c + d*x))/sqrt(a*d - b*c))*a**2* d**2 - 6*sqrt(d)*sqrt(b)*log((sqrt(d)*sqrt(a + b*x) + sqrt(b)*sqrt(c + d*x ))/sqrt(a*d - b*c))*a*b*c*d + 3*sqrt(d)*sqrt(b)*log((sqrt(d)*sqrt(a + b*x) + sqrt(b)*sqrt(c + d*x))/sqrt(a*d - b*c))*b**2*c**2)/(4*b**3*d)