Integrand size = 22, antiderivative size = 157 \[ \int \frac {(a+b x)^{5/2}}{x^4 \sqrt {c+d x}} \, dx=-\frac {5 (b c-a d)^2 \sqrt {a+b x} \sqrt {c+d x}}{8 c^3 x}-\frac {5 (b c-a d) (a+b x)^{3/2} \sqrt {c+d x}}{12 c^2 x^2}-\frac {(a+b x)^{5/2} \sqrt {c+d x}}{3 c x^3}-\frac {5 (b c-a d)^3 \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{8 \sqrt {a} c^{7/2}} \]
-5/8*(-a*d+b*c)^3*arctanh(c^(1/2)*(b*x+a)^(1/2)/a^(1/2)/(d*x+c)^(1/2))/c^( 7/2)/a^(1/2)-5/12*(-a*d+b*c)*(b*x+a)^(3/2)*(d*x+c)^(1/2)/c^2/x^2-1/3*(b*x+ a)^(5/2)*(d*x+c)^(1/2)/c/x^3-5/8*(-a*d+b*c)^2*(b*x+a)^(1/2)*(d*x+c)^(1/2)/ c^3/x
Time = 0.23 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.84 \[ \int \frac {(a+b x)^{5/2}}{x^4 \sqrt {c+d x}} \, dx=-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (33 b^2 c^2 x^2+2 a b c x (13 c-20 d x)+a^2 \left (8 c^2-10 c d x+15 d^2 x^2\right )\right )}{24 c^3 x^3}+\frac {5 (-b c+a d)^3 \text {arctanh}\left (\frac {\sqrt {a} \sqrt {c+d x}}{\sqrt {c} \sqrt {a+b x}}\right )}{8 \sqrt {a} c^{7/2}} \]
-1/24*(Sqrt[a + b*x]*Sqrt[c + d*x]*(33*b^2*c^2*x^2 + 2*a*b*c*x*(13*c - 20* d*x) + a^2*(8*c^2 - 10*c*d*x + 15*d^2*x^2)))/(c^3*x^3) + (5*(-(b*c) + a*d) ^3*ArcTanh[(Sqrt[a]*Sqrt[c + d*x])/(Sqrt[c]*Sqrt[a + b*x])])/(8*Sqrt[a]*c^ (7/2))
Time = 0.25 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.05, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {105, 105, 105, 104, 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.
\(\displaystyle \int \frac {(a+b x)^{5/2}}{x^4 \sqrt {c+d x}} \, dx\) |
\(\Big \downarrow \) 105 |
\(\displaystyle \frac {5 (b c-a d) \int \frac {(a+b x)^{3/2}}{x^3 \sqrt {c+d x}}dx}{6 c}-\frac {(a+b x)^{5/2} \sqrt {c+d x}}{3 c x^3}\) |
\(\Big \downarrow \) 105 |
\(\displaystyle \frac {5 (b c-a d) \left (\frac {3 (b c-a d) \int \frac {\sqrt {a+b x}}{x^2 \sqrt {c+d x}}dx}{4 c}-\frac {(a+b x)^{3/2} \sqrt {c+d x}}{2 c x^2}\right )}{6 c}-\frac {(a+b x)^{5/2} \sqrt {c+d x}}{3 c x^3}\) |
\(\Big \downarrow \) 105 |
\(\displaystyle \frac {5 (b c-a d) \left (\frac {3 (b c-a d) \left (\frac {(b c-a d) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}}dx}{2 c}-\frac {\sqrt {a+b x} \sqrt {c+d x}}{c x}\right )}{4 c}-\frac {(a+b x)^{3/2} \sqrt {c+d x}}{2 c x^2}\right )}{6 c}-\frac {(a+b x)^{5/2} \sqrt {c+d x}}{3 c x^3}\) |
\(\Big \downarrow \) 104 |
\(\displaystyle \frac {5 (b c-a d) \left (\frac {3 (b c-a d) \left (\frac {(b c-a d) \int \frac {1}{\frac {c (a+b x)}{c+d x}-a}d\frac {\sqrt {a+b x}}{\sqrt {c+d x}}}{c}-\frac {\sqrt {a+b x} \sqrt {c+d x}}{c x}\right )}{4 c}-\frac {(a+b x)^{3/2} \sqrt {c+d x}}{2 c x^2}\right )}{6 c}-\frac {(a+b x)^{5/2} \sqrt {c+d x}}{3 c x^3}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {5 (b c-a d) \left (\frac {3 (b c-a d) \left (-\frac {(b c-a d) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{\sqrt {a} c^{3/2}}-\frac {\sqrt {a+b x} \sqrt {c+d x}}{c x}\right )}{4 c}-\frac {(a+b x)^{3/2} \sqrt {c+d x}}{2 c x^2}\right )}{6 c}-\frac {(a+b x)^{5/2} \sqrt {c+d x}}{3 c x^3}\) |
-1/3*((a + b*x)^(5/2)*Sqrt[c + d*x])/(c*x^3) + (5*(b*c - a*d)*(-1/2*((a + b*x)^(3/2)*Sqrt[c + d*x])/(c*x^2) + (3*(b*c - a*d)*(-((Sqrt[a + b*x]*Sqrt[ c + d*x])/(c*x)) - ((b*c - a*d)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*S qrt[c + d*x])])/(Sqrt[a]*c^(3/2))))/(4*c)))/(6*c)
3.7.80.3.1 Defintions of rubi rules used
Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x _)), x_] :> With[{q = Denominator[m]}, Simp[q Subst[Int[x^(q*(m + 1) - 1) /(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^(1/q)], x] ] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && L tQ[-1, m, 0] && SimplerQ[a + b*x, c + d*x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/((m + 1)*(b*e - a*f))), x] - Simp[n*((d*e - c*f)/((m + 1)*(b*e - a*f))) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[m + n + p + 2, 0] && GtQ[n, 0] && (SumSimplerQ[m, 1] || !SumSimplerQ[p, 1]) && NeQ[m, -1]
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]
Leaf count of result is larger than twice the leaf count of optimal. \(404\) vs. \(2(125)=250\).
Time = 1.60 (sec) , antiderivative size = 405, normalized size of antiderivative = 2.58
method | result | size |
default | \(\frac {\sqrt {b x +a}\, \sqrt {d x +c}\, \left (15 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a^{3} d^{3} x^{3}-45 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a^{2} b c \,d^{2} x^{3}+45 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a \,b^{2} c^{2} d \,x^{3}-15 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) b^{3} c^{3} x^{3}-30 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a^{2} d^{2} x^{2}+80 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a b c d \,x^{2}-66 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, b^{2} c^{2} x^{2}+20 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a^{2} c d x -52 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a b \,c^{2} x -16 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a^{2} c^{2} \sqrt {a c}\right )}{48 c^{3} \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, x^{3} \sqrt {a c}}\) | \(405\) |
1/48*(b*x+a)^(1/2)*(d*x+c)^(1/2)/c^3*(15*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b *x+a)*(d*x+c))^(1/2)+2*a*c)/x)*a^3*d^3*x^3-45*ln((a*d*x+b*c*x+2*(a*c)^(1/2 )*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*a^2*b*c*d^2*x^3+45*ln((a*d*x+b*c*x+2*( a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*a*b^2*c^2*d*x^3-15*ln((a*d*x+ b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*b^3*c^3*x^3-30*(a*c) ^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a^2*d^2*x^2+80*(a*c)^(1/2)*((b*x+a)*(d*x+c) )^(1/2)*a*b*c*d*x^2-66*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*b^2*c^2*x^2+20* (a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a^2*c*d*x-52*(a*c)^(1/2)*((b*x+a)*(d*x +c))^(1/2)*a*b*c^2*x-16*((b*x+a)*(d*x+c))^(1/2)*a^2*c^2*(a*c)^(1/2))/((b*x +a)*(d*x+c))^(1/2)/x^3/(a*c)^(1/2)
Time = 0.57 (sec) , antiderivative size = 438, normalized size of antiderivative = 2.79 \[ \int \frac {(a+b x)^{5/2}}{x^4 \sqrt {c+d x}} \, dx=\left [-\frac {15 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \sqrt {a c} x^{3} \log \left (\frac {8 \, a^{2} c^{2} + {\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} + 4 \, {\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {a c} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (a b c^{2} + a^{2} c d\right )} x}{x^{2}}\right ) + 4 \, {\left (8 \, a^{3} c^{3} + {\left (33 \, a b^{2} c^{3} - 40 \, a^{2} b c^{2} d + 15 \, a^{3} c d^{2}\right )} x^{2} + 2 \, {\left (13 \, a^{2} b c^{3} - 5 \, a^{3} c^{2} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{96 \, a c^{4} x^{3}}, \frac {15 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \sqrt {-a c} x^{3} \arctan \left (\frac {{\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {-a c} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (a b c d x^{2} + a^{2} c^{2} + {\left (a b c^{2} + a^{2} c d\right )} x\right )}}\right ) - 2 \, {\left (8 \, a^{3} c^{3} + {\left (33 \, a b^{2} c^{3} - 40 \, a^{2} b c^{2} d + 15 \, a^{3} c d^{2}\right )} x^{2} + 2 \, {\left (13 \, a^{2} b c^{3} - 5 \, a^{3} c^{2} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{48 \, a c^{4} x^{3}}\right ] \]
[-1/96*(15*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*sqrt(a*c)*x ^3*log((8*a^2*c^2 + (b^2*c^2 + 6*a*b*c*d + a^2*d^2)*x^2 + 4*(2*a*c + (b*c + a*d)*x)*sqrt(a*c)*sqrt(b*x + a)*sqrt(d*x + c) + 8*(a*b*c^2 + a^2*c*d)*x) /x^2) + 4*(8*a^3*c^3 + (33*a*b^2*c^3 - 40*a^2*b*c^2*d + 15*a^3*c*d^2)*x^2 + 2*(13*a^2*b*c^3 - 5*a^3*c^2*d)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(a*c^4*x^ 3), 1/48*(15*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*sqrt(-a*c )*x^3*arctan(1/2*(2*a*c + (b*c + a*d)*x)*sqrt(-a*c)*sqrt(b*x + a)*sqrt(d*x + c)/(a*b*c*d*x^2 + a^2*c^2 + (a*b*c^2 + a^2*c*d)*x)) - 2*(8*a^3*c^3 + (3 3*a*b^2*c^3 - 40*a^2*b*c^2*d + 15*a^3*c*d^2)*x^2 + 2*(13*a^2*b*c^3 - 5*a^3 *c^2*d)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(a*c^4*x^3)]
\[ \int \frac {(a+b x)^{5/2}}{x^4 \sqrt {c+d x}} \, dx=\int \frac {\left (a + b x\right )^{\frac {5}{2}}}{x^{4} \sqrt {c + d x}}\, dx \]
Exception generated. \[ \int \frac {(a+b x)^{5/2}}{x^4 \sqrt {c+d x}} \, dx=\text {Exception raised: ValueError} \]
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. 2071 vs. \(2 (125) = 250\).
Time = 2.28 (sec) , antiderivative size = 2071, normalized size of antiderivative = 13.19 \[ \int \frac {(a+b x)^{5/2}}{x^4 \sqrt {c+d x}} \, dx=\text {Too large to display} \]
-1/24*b*(15*(sqrt(b*d)*b^4*c^3 - 3*sqrt(b*d)*a*b^3*c^2*d + 3*sqrt(b*d)*a^2 *b^2*c*d^2 - sqrt(b*d)*a^3*b*d^3)*arctan(-1/2*(b^2*c + a*b*d - (sqrt(b*d)* sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2)/(sqrt(-a*b*c*d)*b) )/(sqrt(-a*b*c*d)*b*c^3) + 2*(33*sqrt(b*d)*b^14*c^8 - 238*sqrt(b*d)*a*b^13 *c^7*d + 750*sqrt(b*d)*a^2*b^12*c^6*d^2 - 1350*sqrt(b*d)*a^3*b^11*c^5*d^3 + 1520*sqrt(b*d)*a^4*b^10*c^4*d^4 - 1098*sqrt(b*d)*a^5*b^9*c^3*d^5 + 498*s qrt(b*d)*a^6*b^8*c^2*d^6 - 130*sqrt(b*d)*a^7*b^7*c*d^7 + 15*sqrt(b*d)*a^8* b^6*d^8 - 165*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)* b*d - a*b*d))^2*b^12*c^7 + 747*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b ^2*c + (b*x + a)*b*d - a*b*d))^2*a*b^11*c^6*d - 1233*sqrt(b*d)*(sqrt(b*d)* sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^2*b^10*c^5*d^2 + 687*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b* d))^2*a^3*b^9*c^4*d^3 + 417*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2* c + (b*x + a)*b*d - a*b*d))^2*a^4*b^8*c^3*d^4 - 783*sqrt(b*d)*(sqrt(b*d)*s qrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^5*b^7*c^2*d^5 + 40 5*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d) )^2*a^6*b^6*c*d^6 - 75*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + ( b*x + a)*b*d - a*b*d))^2*a^7*b^5*d^7 + 330*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*b^10*c^6 - 768*sqrt(b*d)*(sqr t(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*a*b^9*c^5...
Timed out. \[ \int \frac {(a+b x)^{5/2}}{x^4 \sqrt {c+d x}} \, dx=\int \frac {{\left (a+b\,x\right )}^{5/2}}{x^4\,\sqrt {c+d\,x}} \,d x \]