Integrand size = 22, antiderivative size = 198 \[ \int \frac {(a+b x)^{5/2} \sqrt {c+d x}}{x^5} \, dx=-\frac {5 (b c-a d)^3 \sqrt {a+b x} \sqrt {c+d x}}{64 a c^3 x}-\frac {5 (b c-a d)^2 \sqrt {a+b x} (c+d x)^{3/2}}{32 c^3 x^2}-\frac {5 (b c-a d) (a+b x)^{3/2} (c+d x)^{3/2}}{24 c^2 x^3}-\frac {(a+b x)^{5/2} (c+d x)^{3/2}}{4 c x^4}+\frac {5 (b c-a d)^4 \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{64 a^{3/2} c^{7/2}} \]
-5/24*(-a*d+b*c)*(b*x+a)^(3/2)*(d*x+c)^(3/2)/c^2/x^3-1/4*(b*x+a)^(5/2)*(d* x+c)^(3/2)/c/x^4+5/64*(-a*d+b*c)^4*arctanh(c^(1/2)*(b*x+a)^(1/2)/a^(1/2)/( d*x+c)^(1/2))/a^(3/2)/c^(7/2)-5/32*(-a*d+b*c)^2*(d*x+c)^(3/2)*(b*x+a)^(1/2 )/c^3/x^2-5/64*(-a*d+b*c)^3*(b*x+a)^(1/2)*(d*x+c)^(1/2)/a/c^3/x
Time = 0.35 (sec) , antiderivative size = 162, normalized size of antiderivative = 0.82 \[ \int \frac {(a+b x)^{5/2} \sqrt {c+d x}}{x^5} \, dx=\frac {(b c-a d)^4 \left (-\frac {\sqrt {a} \sqrt {c} (a+b x)^{7/2} \sqrt {c+d x} \left (15 c^3+\frac {73 a c^2 (c+d x)}{a+b x}-\frac {55 a^2 c (c+d x)^2}{(a+b x)^2}+\frac {15 a^3 (c+d x)^3}{(a+b x)^3}\right )}{(b c x-a d x)^4}+15 \text {arctanh}\left (\frac {\sqrt {a} \sqrt {c+d x}}{\sqrt {c} \sqrt {a+b x}}\right )\right )}{192 a^{3/2} c^{7/2}} \]
((b*c - a*d)^4*(-((Sqrt[a]*Sqrt[c]*(a + b*x)^(7/2)*Sqrt[c + d*x]*(15*c^3 + (73*a*c^2*(c + d*x))/(a + b*x) - (55*a^2*c*(c + d*x)^2)/(a + b*x)^2 + (15 *a^3*(c + d*x)^3)/(a + b*x)^3))/(b*c*x - a*d*x)^4) + 15*ArcTanh[(Sqrt[a]*S qrt[c + d*x])/(Sqrt[c]*Sqrt[a + b*x])]))/(192*a^(3/2)*c^(7/2))
Time = 0.26 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.05, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {105, 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.
| \(\displaystyle \int \frac {(a+b x)^{5/2} \sqrt {c+d x}}{x^5} \, dx\) |
| \(\Big \downarrow \) 105 |
| \(\displaystyle \frac {5 (b c-a d) \int \frac {(a+b x)^{3/2} \sqrt {c+d x}}{x^4}dx}{8 c}-\frac {(a+b x)^{5/2} (c+d x)^{3/2}}{4 c x^4}\) |
| \(\Big \downarrow \) 105 |
| \(\displaystyle \frac {5 (b c-a d) \left (\frac {(b c-a d) \int \frac {\sqrt {a+b x} \sqrt {c+d x}}{x^3}dx}{2 c}-\frac {(a+b x)^{3/2} (c+d x)^{3/2}}{3 c x^3}\right )}{8 c}-\frac {(a+b x)^{5/2} (c+d x)^{3/2}}{4 c x^4}\) |
| \(\Big \downarrow \) 105 |
| \(\displaystyle \frac {5 (b c-a d) \left (\frac {(b c-a d) \left (\frac {(b c-a d) \int \frac {\sqrt {c+d x}}{x^2 \sqrt {a+b x}}dx}{4 c}-\frac {\sqrt {a+b x} (c+d x)^{3/2}}{2 c x^2}\right )}{2 c}-\frac {(a+b x)^{3/2} (c+d x)^{3/2}}{3 c x^3}\right )}{8 c}-\frac {(a+b x)^{5/2} (c+d x)^{3/2}}{4 c x^4}\) |
| \(\Big \downarrow \) 105 |
| \(\displaystyle \frac {5 (b c-a d) \left (\frac {(b c-a d) \left (\frac {(b c-a d) \left (-\frac {(b c-a d) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}}dx}{2 a}-\frac {\sqrt {a+b x} \sqrt {c+d x}}{a x}\right )}{4 c}-\frac {\sqrt {a+b x} (c+d x)^{3/2}}{2 c x^2}\right )}{2 c}-\frac {(a+b x)^{3/2} (c+d x)^{3/2}}{3 c x^3}\right )}{8 c}-\frac {(a+b x)^{5/2} (c+d x)^{3/2}}{4 c x^4}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle \frac {5 (b c-a d) \left (\frac {(b c-a d) \left (\frac {(b c-a d) \left (-\frac {(b c-a d) \int \frac {1}{\frac {c (a+b x)}{c+d x}-a}d\frac {\sqrt {a+b x}}{\sqrt {c+d x}}}{a}-\frac {\sqrt {a+b x} \sqrt {c+d x}}{a x}\right )}{4 c}-\frac {\sqrt {a+b x} (c+d x)^{3/2}}{2 c x^2}\right )}{2 c}-\frac {(a+b x)^{3/2} (c+d x)^{3/2}}{3 c x^3}\right )}{8 c}-\frac {(a+b x)^{5/2} (c+d x)^{3/2}}{4 c x^4}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {5 (b c-a d) \left (\frac {(b c-a d) \left (\frac {(b c-a d) \left (\frac {(b c-a d) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{a^{3/2} \sqrt {c}}-\frac {\sqrt {a+b x} \sqrt {c+d x}}{a x}\right )}{4 c}-\frac {\sqrt {a+b x} (c+d x)^{3/2}}{2 c x^2}\right )}{2 c}-\frac {(a+b x)^{3/2} (c+d x)^{3/2}}{3 c x^3}\right )}{8 c}-\frac {(a+b x)^{5/2} (c+d x)^{3/2}}{4 c x^4}\) |
-1/4*((a + b*x)^(5/2)*(c + d*x)^(3/2))/(c*x^4) + (5*(b*c - a*d)*(-1/3*((a + b*x)^(3/2)*(c + d*x)^(3/2))/(c*x^3) + ((b*c - a*d)*(-1/2*(Sqrt[a + b*x]* (c + d*x)^(3/2))/(c*x^2) + ((b*c - a*d)*(-((Sqrt[a + b*x]*Sqrt[c + d*x])/( a*x)) + ((b*c - a*d)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x ])])/(a^(3/2)*Sqrt[c])))/(4*c)))/(2*c)))/(8*c)
3.7.54.3.1 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. \(592\) vs. \(2(160)=320\).
Time = 1.58 (sec) , antiderivative size = 593, normalized size of antiderivative = 2.99
| method | result | size |
| default | \(\frac {\sqrt {b x +a}\, \sqrt {d x +c}\, \left (15 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a^{4} d^{4} x^{4}-60 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a^{3} b c \,d^{3} x^{4}+90 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a^{2} b^{2} c^{2} d^{2} x^{4}-60 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a \,b^{3} c^{3} d \,x^{4}+15 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) b^{4} c^{4} x^{4}-30 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a^{3} d^{3} x^{3}+110 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a^{2} b c \,d^{2} x^{3}-146 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a \,b^{2} c^{2} d \,x^{3}-30 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, b^{3} c^{3} x^{3}+20 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, a^{3} c \,d^{2} x^{2}-72 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, a^{2} b \,c^{2} d \,x^{2}-236 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, a \,b^{2} c^{3} x^{2}-16 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, a^{3} c^{2} d x -272 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, a^{2} b \,c^{3} x -96 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a^{3} c^{3} \sqrt {a c}\right )}{384 a \,c^{3} \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, x^{4} \sqrt {a c}}\) | \(593\) |
1/384*(b*x+a)^(1/2)*(d*x+c)^(1/2)/a/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^4*d^4*x^4-60*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*b*c*d^3*x^4+90*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^2*c^2*d^2*x^4-60*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^3*c^3*d*x ^4+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^4* c^4*x^4-30*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a^3*d^3*x^3+110*(a*c)^(1/2) *((b*x+a)*(d*x+c))^(1/2)*a^2*b*c*d^2*x^3-146*(a*c)^(1/2)*((b*x+a)*(d*x+c)) ^(1/2)*a*b^2*c^2*d*x^3-30*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*b^3*c^3*x^3+ 20*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)*a^3*c*d^2*x^2-72*((b*x+a)*(d*x+c))^ (1/2)*(a*c)^(1/2)*a^2*b*c^2*d*x^2-236*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)* a*b^2*c^3*x^2-16*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)*a^3*c^2*d*x-272*((b*x +a)*(d*x+c))^(1/2)*(a*c)^(1/2)*a^2*b*c^3*x-96*((b*x+a)*(d*x+c))^(1/2)*a^3* c^3*(a*c)^(1/2))/((b*x+a)*(d*x+c))^(1/2)/x^4/(a*c)^(1/2)
Time = 1.34 (sec) , antiderivative size = 566, normalized size of antiderivative = 2.86 \[ \int \frac {(a+b x)^{5/2} \sqrt {c+d x}}{x^5} \, dx=\left [\frac {15 \, {\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} \sqrt {a c} x^{4} \log \left (\frac {8 \, a^{2} c^{2} + {\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} + 4 \, {\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {a c} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (a b c^{2} + a^{2} c d\right )} x}{x^{2}}\right ) - 4 \, {\left (48 \, a^{4} c^{4} + {\left (15 \, a b^{3} c^{4} + 73 \, a^{2} b^{2} c^{3} d - 55 \, a^{3} b c^{2} d^{2} + 15 \, a^{4} c d^{3}\right )} x^{3} + 2 \, {\left (59 \, a^{2} b^{2} c^{4} + 18 \, a^{3} b c^{3} d - 5 \, a^{4} c^{2} d^{2}\right )} x^{2} + 8 \, {\left (17 \, a^{3} b c^{4} + a^{4} c^{3} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{768 \, a^{2} c^{4} x^{4}}, -\frac {15 \, {\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} \sqrt {-a c} x^{4} \arctan \left (\frac {{\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {-a c} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (a b c d x^{2} + a^{2} c^{2} + {\left (a b c^{2} + a^{2} c d\right )} x\right )}}\right ) + 2 \, {\left (48 \, a^{4} c^{4} + {\left (15 \, a b^{3} c^{4} + 73 \, a^{2} b^{2} c^{3} d - 55 \, a^{3} b c^{2} d^{2} + 15 \, a^{4} c d^{3}\right )} x^{3} + 2 \, {\left (59 \, a^{2} b^{2} c^{4} + 18 \, a^{3} b c^{3} d - 5 \, a^{4} c^{2} d^{2}\right )} x^{2} + 8 \, {\left (17 \, a^{3} b c^{4} + a^{4} c^{3} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{384 \, a^{2} c^{4} x^{4}}\right ] \]
[1/768*(15*(b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)*sqrt(a*c)*x^4*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*(48*a^4*c^4 + (15*a*b^3*c^4 + 73*a^2*b^2*c^3 *d - 55*a^3*b*c^2*d^2 + 15*a^4*c*d^3)*x^3 + 2*(59*a^2*b^2*c^4 + 18*a^3*b*c ^3*d - 5*a^4*c^2*d^2)*x^2 + 8*(17*a^3*b*c^4 + a^4*c^3*d)*x)*sqrt(b*x + a)* sqrt(d*x + c))/(a^2*c^4*x^4), -1/384*(15*(b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2* b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)*sqrt(-a*c)*x^4*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*(48*a^4*c^4 + (15*a*b^3*c^4 + 73*a^2*b^2* c^3*d - 55*a^3*b*c^2*d^2 + 15*a^4*c*d^3)*x^3 + 2*(59*a^2*b^2*c^4 + 18*a^3* b*c^3*d - 5*a^4*c^2*d^2)*x^2 + 8*(17*a^3*b*c^4 + a^4*c^3*d)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(a^2*c^4*x^4)]
\[ \int \frac {(a+b x)^{5/2} \sqrt {c+d x}}{x^5} \, dx=\int \frac {\left (a + b x\right )^{\frac {5}{2}} \sqrt {c + d x}}{x^{5}}\, dx \]
Exception generated. \[ \int \frac {(a+b x)^{5/2} \sqrt {c+d x}}{x^5} \, dx=\text {Exception raised: ValueError} \]
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. 3832 vs. \(2 (160) = 320\).
Time = 1.60 (sec) , antiderivative size = 3832, normalized size of antiderivative = 19.35 \[ \int \frac {(a+b x)^{5/2} \sqrt {c+d x}}{x^5} \, dx=\text {Too large to display} \]
1/192*(15*(sqrt(b*d)*b^5*c^4*abs(b) - 4*sqrt(b*d)*a*b^4*c^3*d*abs(b) + 6*s qrt(b*d)*a^2*b^3*c^2*d^2*abs(b) - 4*sqrt(b*d)*a^3*b^2*c*d^3*abs(b) + sqrt( b*d)*a^4*b*d^4*abs(b))*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)*a*b*c^3) - 2*(15*sqrt(b*d)*b^19*c^11*abs(b) - 47*sqrt(b*d)*a*b^18*c ^10*d*abs(b) - 219*sqrt(b*d)*a^2*b^17*c^9*d^2*abs(b) + 1659*sqrt(b*d)*a^3* b^16*c^8*d^3*abs(b) - 4698*sqrt(b*d)*a^4*b^15*c^7*d^4*abs(b) + 7770*sqrt(b *d)*a^5*b^14*c^6*d^5*abs(b) - 8358*sqrt(b*d)*a^6*b^13*c^5*d^6*abs(b) + 605 4*sqrt(b*d)*a^7*b^12*c^4*d^7*abs(b) - 2949*sqrt(b*d)*a^8*b^11*c^3*d^8*abs( b) + 933*sqrt(b*d)*a^9*b^10*c^2*d^9*abs(b) - 175*sqrt(b*d)*a^10*b^9*c*d^10 *abs(b) + 15*sqrt(b*d)*a^11*b^8*d^11*abs(b) - 105*sqrt(b*d)*(sqrt(b*d)*sqr t(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*b^17*c^10*abs(b) + 250 *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^16*c^9*d*abs(b) + 651*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^15*c^8*d^2*abs(b) - 3016*sqrt(b*d)*(s qrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^3*b^14*c ^7*d^3*abs(b) + 3550*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^13*c^6*d^4*abs(b) + 540*sqrt(b*d)*(sqrt(b*d)* sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^5*b^12*c^5*d^5*ab s(b) - 5410*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)...
Timed out. \[ \int \frac {(a+b x)^{5/2} \sqrt {c+d x}}{x^5} \, dx=\int \frac {{\left (a+b\,x\right )}^{5/2}\,\sqrt {c+d\,x}}{x^5} \,d x \]