Integrand size = 22, antiderivative size = 119 \[ \int \frac {A+B x}{(a+b x) (d+e x)^{5/2}} \, dx=-\frac {2 (B d-A e)}{3 e (b d-a e) (d+e x)^{3/2}}+\frac {2 (A b-a B)}{(b d-a e)^2 \sqrt {d+e x}}-\frac {2 \sqrt {b} (A b-a B) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{(b d-a e)^{5/2}} \]
-2/3*(-A*e+B*d)/e/(-a*e+b*d)/(e*x+d)^(3/2)-2*(A*b-B*a)*arctanh(b^(1/2)*(e* x+d)^(1/2)/(-a*e+b*d)^(1/2))*b^(1/2)/(-a*e+b*d)^(5/2)+2*(A*b-B*a)/(-a*e+b* d)^2/(e*x+d)^(1/2)
Time = 0.25 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.99 \[ \int \frac {A+B x}{(a+b x) (d+e x)^{5/2}} \, dx=-\frac {2 \left (b B d^2-A b e (4 d+3 e x)+a e (2 B d+A e+3 B e x)\right )}{3 e (b d-a e)^2 (d+e x)^{3/2}}+\frac {2 \sqrt {b} (A b-a B) \arctan \left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {-b d+a e}}\right )}{(-b d+a e)^{5/2}} \]
(-2*(b*B*d^2 - A*b*e*(4*d + 3*e*x) + a*e*(2*B*d + A*e + 3*B*e*x)))/(3*e*(b *d - a*e)^2*(d + e*x)^(3/2)) + (2*Sqrt[b]*(A*b - a*B)*ArcTan[(Sqrt[b]*Sqrt [d + e*x])/Sqrt[-(b*d) + a*e]])/(-(b*d) + a*e)^(5/2)
Time = 0.23 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.03, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {87, 61, 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.
\(\displaystyle \int \frac {A+B x}{(a+b x) (d+e x)^{5/2}} \, dx\) |
\(\Big \downarrow \) 87 |
\(\displaystyle \frac {(A b-a B) \int \frac {1}{(a+b x) (d+e x)^{3/2}}dx}{b d-a e}-\frac {2 (B d-A e)}{3 e (d+e x)^{3/2} (b d-a e)}\) |
\(\Big \downarrow \) 61 |
\(\displaystyle \frac {(A b-a B) \left (\frac {b \int \frac {1}{(a+b x) \sqrt {d+e x}}dx}{b d-a e}+\frac {2}{\sqrt {d+e x} (b d-a e)}\right )}{b d-a e}-\frac {2 (B d-A e)}{3 e (d+e x)^{3/2} (b d-a e)}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {(A b-a B) \left (\frac {2 b \int \frac {1}{a+\frac {b (d+e x)}{e}-\frac {b d}{e}}d\sqrt {d+e x}}{e (b d-a e)}+\frac {2}{\sqrt {d+e x} (b d-a e)}\right )}{b d-a e}-\frac {2 (B d-A e)}{3 e (d+e x)^{3/2} (b d-a e)}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {(A b-a B) \left (\frac {2}{\sqrt {d+e x} (b d-a e)}-\frac {2 \sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{(b d-a e)^{3/2}}\right )}{b d-a e}-\frac {2 (B d-A e)}{3 e (d+e x)^{3/2} (b d-a e)}\) |
(-2*(B*d - A*e))/(3*e*(b*d - a*e)*(d + e*x)^(3/2)) + ((A*b - a*B)*(2/((b*d - a*e)*Sqrt[d + e*x]) - (2*Sqrt[b]*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b *d - a*e]])/(b*d - a*e)^(3/2)))/(b*d - a*e)
3.18.48.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( m + n + 2)/((b*c - a*d)*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && LtQ[m, -1] && !(LtQ[n, -1] && (EqQ[a, 0 ] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[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_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)) Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || Intege rQ[p] || !(IntegerQ[n] || !(EqQ[e, 0] || !(EqQ[c, 0] || LtQ[p, n]))))
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 = 1.02 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.97
method | result | size |
derivativedivides | \(\frac {\frac {2 b e \left (A b -B a \right ) \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{\left (a e -b d \right )^{2} \sqrt {\left (a e -b d \right ) b}}-\frac {2 \left (A e -B d \right )}{3 \left (a e -b d \right ) \left (e x +d \right )^{\frac {3}{2}}}+\frac {2 e \left (A b -B a \right )}{\left (a e -b d \right )^{2} \sqrt {e x +d}}}{e}\) | \(116\) |
default | \(\frac {\frac {2 b e \left (A b -B a \right ) \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{\left (a e -b d \right )^{2} \sqrt {\left (a e -b d \right ) b}}-\frac {2 \left (A e -B d \right )}{3 \left (a e -b d \right ) \left (e x +d \right )^{\frac {3}{2}}}+\frac {2 e \left (A b -B a \right )}{\left (a e -b d \right )^{2} \sqrt {e x +d}}}{e}\) | \(116\) |
pseudoelliptic | \(-\frac {2 \left (-3 \left (e x +d \right )^{\frac {3}{2}} \left (A b -B a \right ) e b \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )+\sqrt {\left (a e -b d \right ) b}\, \left (\left (-3 A b x +a \left (3 B x +A \right )\right ) e^{2}+2 d \left (-2 A b +B a \right ) e +b B \,d^{2}\right )\right )}{3 \sqrt {\left (a e -b d \right ) b}\, \left (e x +d \right )^{\frac {3}{2}} e \left (a e -b d \right )^{2}}\) | \(127\) |
2/e*(-1/3*(A*e-B*d)/(a*e-b*d)/(e*x+d)^(3/2)+e*(A*b-B*a)/(a*e-b*d)^2/(e*x+d )^(1/2)+b*e*(A*b-B*a)/(a*e-b*d)^2/((a*e-b*d)*b)^(1/2)*arctan(b*(e*x+d)^(1/ 2)/((a*e-b*d)*b)^(1/2)))
Leaf count of result is larger than twice the leaf count of optimal. 248 vs. \(2 (103) = 206\).
Time = 0.26 (sec) , antiderivative size = 506, normalized size of antiderivative = 4.25 \[ \int \frac {A+B x}{(a+b x) (d+e x)^{5/2}} \, dx=\left [-\frac {3 \, {\left ({\left (B a - A b\right )} e^{3} x^{2} + 2 \, {\left (B a - A b\right )} d e^{2} x + {\left (B a - A b\right )} d^{2} e\right )} \sqrt {\frac {b}{b d - a e}} \log \left (\frac {b e x + 2 \, b d - a e - 2 \, {\left (b d - a e\right )} \sqrt {e x + d} \sqrt {\frac {b}{b d - a e}}}{b x + a}\right ) + 2 \, {\left (B b d^{2} + A a e^{2} + 3 \, {\left (B a - A b\right )} e^{2} x + 2 \, {\left (B a - 2 \, A b\right )} d e\right )} \sqrt {e x + d}}{3 \, {\left (b^{2} d^{4} e - 2 \, a b d^{3} e^{2} + a^{2} d^{2} e^{3} + {\left (b^{2} d^{2} e^{3} - 2 \, a b d e^{4} + a^{2} e^{5}\right )} x^{2} + 2 \, {\left (b^{2} d^{3} e^{2} - 2 \, a b d^{2} e^{3} + a^{2} d e^{4}\right )} x\right )}}, \frac {2 \, {\left (3 \, {\left ({\left (B a - A b\right )} e^{3} x^{2} + 2 \, {\left (B a - A b\right )} d e^{2} x + {\left (B a - A b\right )} d^{2} e\right )} \sqrt {-\frac {b}{b d - a e}} \arctan \left (-\frac {{\left (b d - a e\right )} \sqrt {e x + d} \sqrt {-\frac {b}{b d - a e}}}{b e x + b d}\right ) - {\left (B b d^{2} + A a e^{2} + 3 \, {\left (B a - A b\right )} e^{2} x + 2 \, {\left (B a - 2 \, A b\right )} d e\right )} \sqrt {e x + d}\right )}}{3 \, {\left (b^{2} d^{4} e - 2 \, a b d^{3} e^{2} + a^{2} d^{2} e^{3} + {\left (b^{2} d^{2} e^{3} - 2 \, a b d e^{4} + a^{2} e^{5}\right )} x^{2} + 2 \, {\left (b^{2} d^{3} e^{2} - 2 \, a b d^{2} e^{3} + a^{2} d e^{4}\right )} x\right )}}\right ] \]
[-1/3*(3*((B*a - A*b)*e^3*x^2 + 2*(B*a - A*b)*d*e^2*x + (B*a - A*b)*d^2*e) *sqrt(b/(b*d - a*e))*log((b*e*x + 2*b*d - a*e - 2*(b*d - a*e)*sqrt(e*x + d )*sqrt(b/(b*d - a*e)))/(b*x + a)) + 2*(B*b*d^2 + A*a*e^2 + 3*(B*a - A*b)*e ^2*x + 2*(B*a - 2*A*b)*d*e)*sqrt(e*x + d))/(b^2*d^4*e - 2*a*b*d^3*e^2 + a^ 2*d^2*e^3 + (b^2*d^2*e^3 - 2*a*b*d*e^4 + a^2*e^5)*x^2 + 2*(b^2*d^3*e^2 - 2 *a*b*d^2*e^3 + a^2*d*e^4)*x), 2/3*(3*((B*a - A*b)*e^3*x^2 + 2*(B*a - A*b)* d*e^2*x + (B*a - A*b)*d^2*e)*sqrt(-b/(b*d - a*e))*arctan(-(b*d - a*e)*sqrt (e*x + d)*sqrt(-b/(b*d - a*e))/(b*e*x + b*d)) - (B*b*d^2 + A*a*e^2 + 3*(B* a - A*b)*e^2*x + 2*(B*a - 2*A*b)*d*e)*sqrt(e*x + d))/(b^2*d^4*e - 2*a*b*d^ 3*e^2 + a^2*d^2*e^3 + (b^2*d^2*e^3 - 2*a*b*d*e^4 + a^2*e^5)*x^2 + 2*(b^2*d ^3*e^2 - 2*a*b*d^2*e^3 + a^2*d*e^4)*x)]
Time = 3.71 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.16 \[ \int \frac {A+B x}{(a+b x) (d+e x)^{5/2}} \, dx=\begin {cases} \frac {2 \left (- \frac {e \left (- A b + B a\right )}{\sqrt {d + e x} \left (a e - b d\right )^{2}} - \frac {e \left (- A b + B a\right ) \operatorname {atan}{\left (\frac {\sqrt {d + e x}}{\sqrt {\frac {a e - b d}{b}}} \right )}}{\sqrt {\frac {a e - b d}{b}} \left (a e - b d\right )^{2}} + \frac {- A e + B d}{3 \left (d + e x\right )^{\frac {3}{2}} \left (a e - b d\right )}\right )}{e} & \text {for}\: e \neq 0 \\\frac {\frac {B x}{b} - \frac {\left (- A b + B a\right ) \left (\begin {cases} \frac {x}{a} & \text {for}\: b = 0 \\\frac {\log {\left (a + b x \right )}}{b} & \text {otherwise} \end {cases}\right )}{b}}{d^{\frac {5}{2}}} & \text {otherwise} \end {cases} \]
Piecewise((2*(-e*(-A*b + B*a)/(sqrt(d + e*x)*(a*e - b*d)**2) - e*(-A*b + B *a)*atan(sqrt(d + e*x)/sqrt((a*e - b*d)/b))/(sqrt((a*e - b*d)/b)*(a*e - b* d)**2) + (-A*e + B*d)/(3*(d + e*x)**(3/2)*(a*e - b*d)))/e, Ne(e, 0)), ((B* x/b - (-A*b + B*a)*Piecewise((x/a, Eq(b, 0)), (log(a + b*x)/b, True))/b)/d **(5/2), True))
Exception generated. \[ \int \frac {A+B x}{(a+b x) (d+e x)^{5/2}} \, dx=\text {Exception raised: ValueError} \]
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*e-b*d>0)', see `assume?` for m ore detail
Time = 0.27 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.29 \[ \int \frac {A+B x}{(a+b x) (d+e x)^{5/2}} \, dx=-\frac {2 \, {\left (B a b - A b^{2}\right )} \arctan \left (\frac {\sqrt {e x + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{{\left (b^{2} d^{2} - 2 \, a b d e + a^{2} e^{2}\right )} \sqrt {-b^{2} d + a b e}} - \frac {2 \, {\left (B b d^{2} + 3 \, {\left (e x + d\right )} B a e - 3 \, {\left (e x + d\right )} A b e - B a d e - A b d e + A a e^{2}\right )}}{3 \, {\left (b^{2} d^{2} e - 2 \, a b d e^{2} + a^{2} e^{3}\right )} {\left (e x + d\right )}^{\frac {3}{2}}} \]
-2*(B*a*b - A*b^2)*arctan(sqrt(e*x + d)*b/sqrt(-b^2*d + a*b*e))/((b^2*d^2 - 2*a*b*d*e + a^2*e^2)*sqrt(-b^2*d + a*b*e)) - 2/3*(B*b*d^2 + 3*(e*x + d)* B*a*e - 3*(e*x + d)*A*b*e - B*a*d*e - A*b*d*e + A*a*e^2)/((b^2*d^2*e - 2*a *b*d*e^2 + a^2*e^3)*(e*x + d)^(3/2))
Time = 1.66 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.08 \[ \int \frac {A+B x}{(a+b x) (d+e x)^{5/2}} \, dx=\frac {2\,\sqrt {b}\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {d+e\,x}\,\left (a^2\,e^2-2\,a\,b\,d\,e+b^2\,d^2\right )}{{\left (a\,e-b\,d\right )}^{5/2}}\right )\,\left (A\,b-B\,a\right )}{{\left (a\,e-b\,d\right )}^{5/2}}-\frac {\frac {2\,\left (A\,e-B\,d\right )}{3\,\left (a\,e-b\,d\right )}-\frac {2\,\left (A\,b\,e-B\,a\,e\right )\,\left (d+e\,x\right )}{{\left (a\,e-b\,d\right )}^2}}{e\,{\left (d+e\,x\right )}^{3/2}} \]