Integrand size = 26, antiderivative size = 200 \[ \int \frac {(A+B x) \sqrt {b x+c x^2}}{(d+e x)^4} \, dx=-\frac {(b B d-2 A c d+A b e) (b d+(2 c d-b e) x) \sqrt {b x+c x^2}}{8 d^2 (c d-b e)^2 (d+e x)^2}+\frac {(B d-A e) \left (b x+c x^2\right )^{3/2}}{3 d (c d-b e) (d+e x)^3}+\frac {b^2 (b B d-2 A c d+A b e) \text {arctanh}\left (\frac {b d+(2 c d-b e) x}{2 \sqrt {d} \sqrt {c d-b e} \sqrt {b x+c x^2}}\right )}{16 d^{5/2} (c d-b e)^{5/2}} \]
1/3*(-A*e+B*d)*(c*x^2+b*x)^(3/2)/d/(-b*e+c*d)/(e*x+d)^3+1/16*b^2*(A*b*e-2* A*c*d+B*b*d)*arctanh(1/2*(b*d+(-b*e+2*c*d)*x)/d^(1/2)/(-b*e+c*d)^(1/2)/(c* x^2+b*x)^(1/2))/d^(5/2)/(-b*e+c*d)^(5/2)-1/8*(A*b*e-2*A*c*d+B*b*d)*(b*d+(- b*e+2*c*d)*x)*(c*x^2+b*x)^(1/2)/d^2/(-b*e+c*d)^2/(e*x+d)^2
Time = 10.26 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.00 \[ \int \frac {(A+B x) \sqrt {b x+c x^2}}{(d+e x)^4} \, dx=\frac {\sqrt {x (b+c x)} \left (8 (-B d+A e) x^{3/2} (b+c x)-\frac {3 (b B d-2 A c d+A b e) (d+e x) \left (\sqrt {d} \sqrt {c d-b e} \sqrt {x} \sqrt {b+c x} (-b d-2 c d x+b e x)+b^2 (d+e x)^2 \text {arctanh}\left (\frac {\sqrt {c d-b e} \sqrt {x}}{\sqrt {d} \sqrt {b+c x}}\right )\right )}{d^{3/2} (c d-b e)^{3/2} \sqrt {b+c x}}\right )}{24 d (-c d+b e) \sqrt {x} (d+e x)^3} \]
(Sqrt[x*(b + c*x)]*(8*(-(B*d) + A*e)*x^(3/2)*(b + c*x) - (3*(b*B*d - 2*A*c *d + A*b*e)*(d + e*x)*(Sqrt[d]*Sqrt[c*d - b*e]*Sqrt[x]*Sqrt[b + c*x]*(-(b* d) - 2*c*d*x + b*e*x) + b^2*(d + e*x)^2*ArcTanh[(Sqrt[c*d - b*e]*Sqrt[x])/ (Sqrt[d]*Sqrt[b + c*x])]))/(d^(3/2)*(c*d - b*e)^(3/2)*Sqrt[b + c*x])))/(24 *d*(-(c*d) + b*e)*Sqrt[x]*(d + e*x)^3)
Time = 0.34 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.02, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {1228, 1152, 1154, 219}
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.
\(\displaystyle \int \frac {(A+B x) \sqrt {b x+c x^2}}{(d+e x)^4} \, dx\) |
\(\Big \downarrow \) 1228 |
\(\displaystyle \frac {\left (b x+c x^2\right )^{3/2} (B d-A e)}{3 d (d+e x)^3 (c d-b e)}-\frac {(A b e-2 A c d+b B d) \int \frac {\sqrt {c x^2+b x}}{(d+e x)^3}dx}{2 d (c d-b e)}\) |
\(\Big \downarrow \) 1152 |
\(\displaystyle \frac {\left (b x+c x^2\right )^{3/2} (B d-A e)}{3 d (d+e x)^3 (c d-b e)}-\frac {(A b e-2 A c d+b B d) \left (\frac {\sqrt {b x+c x^2} (x (2 c d-b e)+b d)}{4 d (d+e x)^2 (c d-b e)}-\frac {b^2 \int \frac {1}{(d+e x) \sqrt {c x^2+b x}}dx}{8 d (c d-b e)}\right )}{2 d (c d-b e)}\) |
\(\Big \downarrow \) 1154 |
\(\displaystyle \frac {\left (b x+c x^2\right )^{3/2} (B d-A e)}{3 d (d+e x)^3 (c d-b e)}-\frac {(A b e-2 A c d+b B d) \left (\frac {b^2 \int \frac {1}{4 d (c d-b e)-\frac {(b d+(2 c d-b e) x)^2}{c x^2+b x}}d\left (-\frac {b d+(2 c d-b e) x}{\sqrt {c x^2+b x}}\right )}{4 d (c d-b e)}+\frac {\sqrt {b x+c x^2} (x (2 c d-b e)+b d)}{4 d (d+e x)^2 (c d-b e)}\right )}{2 d (c d-b e)}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {\left (b x+c x^2\right )^{3/2} (B d-A e)}{3 d (d+e x)^3 (c d-b e)}-\frac {(A b e-2 A c d+b B d) \left (\frac {\sqrt {b x+c x^2} (x (2 c d-b e)+b d)}{4 d (d+e x)^2 (c d-b e)}-\frac {b^2 \text {arctanh}\left (\frac {x (2 c d-b e)+b d}{2 \sqrt {d} \sqrt {b x+c x^2} \sqrt {c d-b e}}\right )}{8 d^{3/2} (c d-b e)^{3/2}}\right )}{2 d (c d-b e)}\) |
((B*d - A*e)*(b*x + c*x^2)^(3/2))/(3*d*(c*d - b*e)*(d + e*x)^3) - ((b*B*d - 2*A*c*d + A*b*e)*(((b*d + (2*c*d - b*e)*x)*Sqrt[b*x + c*x^2])/(4*d*(c*d - b*e)*(d + e*x)^2) - (b^2*ArcTanh[(b*d + (2*c*d - b*e)*x)/(2*Sqrt[d]*Sqrt [c*d - b*e]*Sqrt[b*x + c*x^2])])/(8*d^(3/2)*(c*d - b*e)^(3/2))))/(2*d*(c*d - b*e))
3.12.69.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> Simp[(-(d + e*x)^(m + 1))*(d*b - 2*a*e + (2*c*d - b*e)*x)*((a + b *x + c*x^2)^p/(2*(m + 1)*(c*d^2 - b*d*e + a*e^2))), x] + Simp[p*((b^2 - 4*a *c)/(2*(m + 1)*(c*d^2 - b*d*e + a*e^2))) Int[(d + e*x)^(m + 2)*(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[m + 2*p + 2, 0] && GtQ[p, 0]
Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Sym bol] :> Simp[-2 Subst[Int[1/(4*c*d^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, ( 2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c , d, e}, x]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(-(e*f - d*g))*(d + e*x)^(m + 1)*((a + b*x + c*x^2)^(p + 1)/(2*(p + 1)*(c*d^2 - b*d*e + a*e^2))), x] - Simp[(b*(e *f + d*g) - 2*(c*d*f + a*e*g))/(2*(c*d^2 - b*d*e + a*e^2)) Int[(d + e*x)^ (m + 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x ] && EqQ[Simplify[m + 2*p + 3], 0]
Time = 0.59 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.06
method | result | size |
pseudoelliptic | \(-\frac {\left (e x +d \right )^{3} \left (\left (-2 A c +B b \right ) d +A b e \right ) b^{2} \arctan \left (\frac {\sqrt {x \left (c x +b \right )}\, d}{x \sqrt {d \left (b e -c d \right )}}\right )+\sqrt {d \left (b e -c d \right )}\, \sqrt {x \left (c x +b \right )}\, \left (\left (B \,b^{2}-2 c \left (\frac {B x}{3}+A \right ) b -4 c^{2} x \left (\frac {2 B x}{3}+A \right )\right ) d^{3}+\left (\left (A +\frac {8 B x}{3}\right ) b^{2}+\frac {14 c x \left (B x +A \right ) b}{3}-\frac {4 A \,c^{2} x^{2}}{3}\right ) e \,d^{2}-\frac {8 \left (\left (\frac {3 B x}{8}+A \right ) b -\frac {A c x}{2}\right ) x \,e^{2} b d}{3}-A \,b^{2} e^{3} x^{2}\right )}{8 \sqrt {d \left (b e -c d \right )}\, \left (e x +d \right )^{3} \left (b e -c d \right )^{2} d^{2}}\) | \(213\) |
default | \(\text {Expression too large to display}\) | \(2080\) |
-1/8/(d*(b*e-c*d))^(1/2)*((e*x+d)^3*((-2*A*c+B*b)*d+A*b*e)*b^2*arctan((x*( c*x+b))^(1/2)/x*d/(d*(b*e-c*d))^(1/2))+(d*(b*e-c*d))^(1/2)*(x*(c*x+b))^(1/ 2)*((B*b^2-2*c*(1/3*B*x+A)*b-4*c^2*x*(2/3*B*x+A))*d^3+((A+8/3*B*x)*b^2+14/ 3*c*x*(B*x+A)*b-4/3*A*c^2*x^2)*e*d^2-8/3*((3/8*B*x+A)*b-1/2*A*c*x)*x*e^2*b *d-A*b^2*e^3*x^2))/(e*x+d)^3/(b*e-c*d)^2/d^2
Leaf count of result is larger than twice the leaf count of optimal. 602 vs. \(2 (178) = 356\).
Time = 0.33 (sec) , antiderivative size = 1216, normalized size of antiderivative = 6.08 \[ \int \frac {(A+B x) \sqrt {b x+c x^2}}{(d+e x)^4} \, dx=\text {Too large to display} \]
[1/48*(3*(A*b^3*d^3*e + (B*b^3 - 2*A*b^2*c)*d^4 + (A*b^3*e^4 + (B*b^3 - 2* A*b^2*c)*d*e^3)*x^3 + 3*(A*b^3*d*e^3 + (B*b^3 - 2*A*b^2*c)*d^2*e^2)*x^2 + 3*(A*b^3*d^2*e^2 + (B*b^3 - 2*A*b^2*c)*d^3*e)*x)*sqrt(c*d^2 - b*d*e)*log(( b*d + (2*c*d - b*e)*x + 2*sqrt(c*d^2 - b*d*e)*sqrt(c*x^2 + b*x))/(e*x + d) ) + 2*(3*A*b^3*d^3*e^2 - 3*(B*b^2*c - 2*A*b*c^2)*d^5 + 3*(B*b^3 - 3*A*b^2* c)*d^4*e + (8*B*c^3*d^5 - 3*A*b^3*d*e^4 - 2*(11*B*b*c^2 - 2*A*c^3)*d^4*e + (17*B*b^2*c - 8*A*b*c^2)*d^3*e^2 - (3*B*b^3 - 7*A*b^2*c)*d^2*e^3)*x^2 - 2 *(4*A*b^3*d^2*e^3 - (B*b*c^2 + 6*A*c^3)*d^5 + (5*B*b^2*c + 13*A*b*c^2)*d^4 *e - (4*B*b^3 + 11*A*b^2*c)*d^3*e^2)*x)*sqrt(c*x^2 + b*x))/(c^3*d^9 - 3*b* c^2*d^8*e + 3*b^2*c*d^7*e^2 - b^3*d^6*e^3 + (c^3*d^6*e^3 - 3*b*c^2*d^5*e^4 + 3*b^2*c*d^4*e^5 - b^3*d^3*e^6)*x^3 + 3*(c^3*d^7*e^2 - 3*b*c^2*d^6*e^3 + 3*b^2*c*d^5*e^4 - b^3*d^4*e^5)*x^2 + 3*(c^3*d^8*e - 3*b*c^2*d^7*e^2 + 3*b ^2*c*d^6*e^3 - b^3*d^5*e^4)*x), 1/24*(3*(A*b^3*d^3*e + (B*b^3 - 2*A*b^2*c) *d^4 + (A*b^3*e^4 + (B*b^3 - 2*A*b^2*c)*d*e^3)*x^3 + 3*(A*b^3*d*e^3 + (B*b ^3 - 2*A*b^2*c)*d^2*e^2)*x^2 + 3*(A*b^3*d^2*e^2 + (B*b^3 - 2*A*b^2*c)*d^3* e)*x)*sqrt(-c*d^2 + b*d*e)*arctan(-sqrt(-c*d^2 + b*d*e)*sqrt(c*x^2 + b*x)/ ((c*d - b*e)*x)) + (3*A*b^3*d^3*e^2 - 3*(B*b^2*c - 2*A*b*c^2)*d^5 + 3*(B*b ^3 - 3*A*b^2*c)*d^4*e + (8*B*c^3*d^5 - 3*A*b^3*d*e^4 - 2*(11*B*b*c^2 - 2*A *c^3)*d^4*e + (17*B*b^2*c - 8*A*b*c^2)*d^3*e^2 - (3*B*b^3 - 7*A*b^2*c)*d^2 *e^3)*x^2 - 2*(4*A*b^3*d^2*e^3 - (B*b*c^2 + 6*A*c^3)*d^5 + (5*B*b^2*c +...
\[ \int \frac {(A+B x) \sqrt {b x+c x^2}}{(d+e x)^4} \, dx=\int \frac {\sqrt {x \left (b + c x\right )} \left (A + B x\right )}{\left (d + e x\right )^{4}}\, dx \]
Exception generated. \[ \int \frac {(A+B x) \sqrt {b x+c x^2}}{(d+e x)^4} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(b*e-c*d>0)', see `assume?` for m ore detail
Leaf count of result is larger than twice the leaf count of optimal. 1576 vs. \(2 (178) = 356\).
Time = 0.32 (sec) , antiderivative size = 1576, normalized size of antiderivative = 7.88 \[ \int \frac {(A+B x) \sqrt {b x+c x^2}}{(d+e x)^4} \, dx=\text {Too large to display} \]
1/8*(B*b^3*d - 2*A*b^2*c*d + A*b^3*e)*arctan(-((sqrt(c)*x - sqrt(c*x^2 + b *x))*e + sqrt(c)*d)/sqrt(-c*d^2 + b*d*e))/((c^2*d^4 - 2*b*c*d^3*e + b^2*d^ 2*e^2)*sqrt(-c*d^2 + b*d*e)) + 1/24*(48*(sqrt(c)*x - sqrt(c*x^2 + b*x))^5* B*c^3*d^4*e^2 - 96*(sqrt(c)*x - sqrt(c*x^2 + b*x))^5*B*b*c^2*d^3*e^3 + 48* (sqrt(c)*x - sqrt(c*x^2 + b*x))^5*B*b^2*c*d^2*e^4 - 3*(sqrt(c)*x - sqrt(c* x^2 + b*x))^5*B*b^3*d*e^5 + 6*(sqrt(c)*x - sqrt(c*x^2 + b*x))^5*A*b^2*c*d* e^5 - 3*(sqrt(c)*x - sqrt(c*x^2 + b*x))^5*A*b^3*e^6 + 96*(sqrt(c)*x - sqrt (c*x^2 + b*x))^4*B*c^(7/2)*d^5*e - 144*(sqrt(c)*x - sqrt(c*x^2 + b*x))^4*B *b*c^(5/2)*d^4*e^2 + 48*(sqrt(c)*x - sqrt(c*x^2 + b*x))^4*A*c^(7/2)*d^4*e^ 2 - 96*(sqrt(c)*x - sqrt(c*x^2 + b*x))^4*A*b*c^(5/2)*d^3*e^3 + 33*(sqrt(c) *x - sqrt(c*x^2 + b*x))^4*B*b^3*sqrt(c)*d^2*e^4 + 78*(sqrt(c)*x - sqrt(c*x ^2 + b*x))^4*A*b^2*c^(3/2)*d^2*e^4 - 15*(sqrt(c)*x - sqrt(c*x^2 + b*x))^4* A*b^3*sqrt(c)*d*e^5 + 64*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*B*c^4*d^6 - 16* (sqrt(c)*x - sqrt(c*x^2 + b*x))^3*B*b*c^3*d^5*e + 32*(sqrt(c)*x - sqrt(c*x ^2 + b*x))^3*A*c^4*d^5*e - 144*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*B*b^2*c^2 *d^4*e^2 + 16*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*A*b*c^3*d^4*e^2 + 58*(sqrt (c)*x - sqrt(c*x^2 + b*x))^3*B*b^3*c*d^3*e^3 - 84*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*A*b^2*c^2*d^3*e^3 + 8*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*B*b^4*d^ 2*e^4 + 74*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*A*b^3*c*d^2*e^4 - 8*(sqrt(c)* x - sqrt(c*x^2 + b*x))^3*A*b^4*d*e^5 + 96*(sqrt(c)*x - sqrt(c*x^2 + b*x...
Timed out. \[ \int \frac {(A+B x) \sqrt {b x+c x^2}}{(d+e x)^4} \, dx=\int \frac {\sqrt {c\,x^2+b\,x}\,\left (A+B\,x\right )}{{\left (d+e\,x\right )}^4} \,d x \]