Integrand size = 33, antiderivative size = 39 \[ \int \frac {(a+b x) (d+e x)^3}{\sqrt {a^2+2 a b x+b^2 x^2}} \, dx=\frac {(a+b x) (d+e x)^4}{4 e \sqrt {a^2+2 a b x+b^2 x^2}} \]
Time = 0.01 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.77 \[ \int \frac {(a+b x) (d+e x)^3}{\sqrt {a^2+2 a b x+b^2 x^2}} \, dx=\frac {(a+b x) (d+e x)^4}{4 e \sqrt {(a+b x)^2}} \]
Time = 0.18 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.061, Rules used = {1187, 17}
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.
\(\displaystyle \int \frac {(a+b x) (d+e x)^3}{\sqrt {a^2+2 a b x+b^2 x^2}} \, dx\) |
\(\Big \downarrow \) 1187 |
\(\displaystyle \frac {b (a+b x) \int \frac {(d+e x)^3}{b}dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
\(\Big \downarrow \) 17 |
\(\displaystyle \frac {(a+b x) (d+e x)^4}{4 e \sqrt {a^2+2 a b x+b^2 x^2}}\) |
3.21.16.3.1 Defintions of rubi rules used
Int[(c_.)*((a_.) + (b_.)*(x_))^(m_.), x_Symbol] :> Simp[c*((a + b*x)^(m + 1 )/(b*(m + 1))), x] /; FreeQ[{a, b, c, m}, x] && NeQ[m, -1]
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_) + (b_.)*(x_ ) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(a + b*x + c*x^2)^FracPart[p]/(c^ IntPart[p]*(b/2 + c*x)^(2*FracPart[p])) Int[(d + e*x)^m*(f + g*x)^n*(b/2 + c*x)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p}, x] && EqQ[b^2 - 4*a*c, 0] && !IntegerQ[p]
Time = 0.41 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.69
method | result | size |
default | \(\frac {\left (b x +a \right ) \left (e x +d \right )^{4}}{4 e \sqrt {\left (b x +a \right )^{2}}}\) | \(27\) |
risch | \(\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (e x +d \right )^{4}}{4 \left (b x +a \right ) e}\) | \(29\) |
gosper | \(\frac {x \left (e^{3} x^{3}+4 d \,e^{2} x^{2}+6 d^{2} e x +4 d^{3}\right ) \left (b x +a \right )}{4 \sqrt {\left (b x +a \right )^{2}}}\) | \(47\) |
Time = 0.42 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.79 \[ \int \frac {(a+b x) (d+e x)^3}{\sqrt {a^2+2 a b x+b^2 x^2}} \, dx=\frac {1}{4} \, e^{3} x^{4} + d e^{2} x^{3} + \frac {3}{2} \, d^{2} e x^{2} + d^{3} x \]
Leaf count of result is larger than twice the leaf count of optimal. 1015 vs. \(2 (26) = 52\).
Time = 2.40 (sec) , antiderivative size = 1015, normalized size of antiderivative = 26.03 \[ \int \frac {(a+b x) (d+e x)^3}{\sqrt {a^2+2 a b x+b^2 x^2}} \, dx=\begin {cases} \sqrt {a^{2} + 2 a b x + b^{2} x^{2}} \left (\frac {e^{3} x^{3}}{4 b} + \frac {x^{2} \left (- \frac {3 a e^{3}}{4} + 3 b d e^{2}\right )}{3 b^{2}} + \frac {x \left (- \frac {3 a^{2} e^{3}}{4 b} + 3 a d e^{2} - \frac {5 a \left (- \frac {3 a e^{3}}{4} + 3 b d e^{2}\right )}{3 b} + 3 b d^{2} e\right )}{2 b^{2}} + \frac {- \frac {2 a^{2} \left (- \frac {3 a e^{3}}{4} + 3 b d e^{2}\right )}{3 b^{2}} + 3 a d^{2} e - \frac {3 a \left (- \frac {3 a^{2} e^{3}}{4 b} + 3 a d e^{2} - \frac {5 a \left (- \frac {3 a e^{3}}{4} + 3 b d e^{2}\right )}{3 b} + 3 b d^{2} e\right )}{2 b} + b d^{3}}{b^{2}}\right ) + \frac {\left (\frac {a}{b} + x\right ) \left (- \frac {a^{2} \left (- \frac {3 a^{2} e^{3}}{4 b} + 3 a d e^{2} - \frac {5 a \left (- \frac {3 a e^{3}}{4} + 3 b d e^{2}\right )}{3 b} + 3 b d^{2} e\right )}{2 b^{2}} + a d^{3} - \frac {a \left (- \frac {2 a^{2} \left (- \frac {3 a e^{3}}{4} + 3 b d e^{2}\right )}{3 b^{2}} + 3 a d^{2} e - \frac {3 a \left (- \frac {3 a^{2} e^{3}}{4 b} + 3 a d e^{2} - \frac {5 a \left (- \frac {3 a e^{3}}{4} + 3 b d e^{2}\right )}{3 b} + 3 b d^{2} e\right )}{2 b} + b d^{3}\right )}{b}\right ) \log {\left (\frac {a}{b} + x \right )}}{\sqrt {b^{2} \left (\frac {a}{b} + x\right )^{2}}} & \text {for}\: b^{2} \neq 0 \\\frac {2 a d^{3} \sqrt {a^{2} + 2 a b x} + \frac {3 d^{2} e \left (- a^{2} \sqrt {a^{2} + 2 a b x} + \frac {\left (a^{2} + 2 a b x\right )^{\frac {3}{2}}}{3}\right )}{b} + \frac {d^{3} \left (- a^{2} \sqrt {a^{2} + 2 a b x} + \frac {\left (a^{2} + 2 a b x\right )^{\frac {3}{2}}}{3}\right )}{a} + \frac {3 d e^{2} \left (a^{4} \sqrt {a^{2} + 2 a b x} - \frac {2 a^{2} \left (a^{2} + 2 a b x\right )^{\frac {3}{2}}}{3} + \frac {\left (a^{2} + 2 a b x\right )^{\frac {5}{2}}}{5}\right )}{2 a b^{2}} + \frac {3 d^{2} e \left (a^{4} \sqrt {a^{2} + 2 a b x} - \frac {2 a^{2} \left (a^{2} + 2 a b x\right )^{\frac {3}{2}}}{3} + \frac {\left (a^{2} + 2 a b x\right )^{\frac {5}{2}}}{5}\right )}{2 a^{2} b} + \frac {e^{3} \left (- a^{6} \sqrt {a^{2} + 2 a b x} + a^{4} \left (a^{2} + 2 a b x\right )^{\frac {3}{2}} - \frac {3 a^{2} \left (a^{2} + 2 a b x\right )^{\frac {5}{2}}}{5} + \frac {\left (a^{2} + 2 a b x\right )^{\frac {7}{2}}}{7}\right )}{4 a^{2} b^{3}} + \frac {3 d e^{2} \left (- a^{6} \sqrt {a^{2} + 2 a b x} + a^{4} \left (a^{2} + 2 a b x\right )^{\frac {3}{2}} - \frac {3 a^{2} \left (a^{2} + 2 a b x\right )^{\frac {5}{2}}}{5} + \frac {\left (a^{2} + 2 a b x\right )^{\frac {7}{2}}}{7}\right )}{4 a^{3} b^{2}} + \frac {e^{3} \left (a^{8} \sqrt {a^{2} + 2 a b x} - \frac {4 a^{6} \left (a^{2} + 2 a b x\right )^{\frac {3}{2}}}{3} + \frac {6 a^{4} \left (a^{2} + 2 a b x\right )^{\frac {5}{2}}}{5} - \frac {4 a^{2} \left (a^{2} + 2 a b x\right )^{\frac {7}{2}}}{7} + \frac {\left (a^{2} + 2 a b x\right )^{\frac {9}{2}}}{9}\right )}{8 a^{4} b^{3}}}{2 a b} & \text {for}\: a b \neq 0 \\\frac {a d^{3} x + \frac {3 a d^{2} e x^{2}}{2} + a d e^{2} x^{3} + \frac {a e^{3} x^{4}}{4} + \frac {b d^{3} x^{2}}{2} + b d^{2} e x^{3} + \frac {3 b d e^{2} x^{4}}{4} + \frac {b e^{3} x^{5}}{5}}{\sqrt {a^{2}}} & \text {otherwise} \end {cases} \]
Piecewise((sqrt(a**2 + 2*a*b*x + b**2*x**2)*(e**3*x**3/(4*b) + x**2*(-3*a* e**3/4 + 3*b*d*e**2)/(3*b**2) + x*(-3*a**2*e**3/(4*b) + 3*a*d*e**2 - 5*a*( -3*a*e**3/4 + 3*b*d*e**2)/(3*b) + 3*b*d**2*e)/(2*b**2) + (-2*a**2*(-3*a*e* *3/4 + 3*b*d*e**2)/(3*b**2) + 3*a*d**2*e - 3*a*(-3*a**2*e**3/(4*b) + 3*a*d *e**2 - 5*a*(-3*a*e**3/4 + 3*b*d*e**2)/(3*b) + 3*b*d**2*e)/(2*b) + b*d**3) /b**2) + (a/b + x)*(-a**2*(-3*a**2*e**3/(4*b) + 3*a*d*e**2 - 5*a*(-3*a*e** 3/4 + 3*b*d*e**2)/(3*b) + 3*b*d**2*e)/(2*b**2) + a*d**3 - a*(-2*a**2*(-3*a *e**3/4 + 3*b*d*e**2)/(3*b**2) + 3*a*d**2*e - 3*a*(-3*a**2*e**3/(4*b) + 3* a*d*e**2 - 5*a*(-3*a*e**3/4 + 3*b*d*e**2)/(3*b) + 3*b*d**2*e)/(2*b) + b*d* *3)/b)*log(a/b + x)/sqrt(b**2*(a/b + x)**2), Ne(b**2, 0)), ((2*a*d**3*sqrt (a**2 + 2*a*b*x) + 3*d**2*e*(-a**2*sqrt(a**2 + 2*a*b*x) + (a**2 + 2*a*b*x) **(3/2)/3)/b + d**3*(-a**2*sqrt(a**2 + 2*a*b*x) + (a**2 + 2*a*b*x)**(3/2)/ 3)/a + 3*d*e**2*(a**4*sqrt(a**2 + 2*a*b*x) - 2*a**2*(a**2 + 2*a*b*x)**(3/2 )/3 + (a**2 + 2*a*b*x)**(5/2)/5)/(2*a*b**2) + 3*d**2*e*(a**4*sqrt(a**2 + 2 *a*b*x) - 2*a**2*(a**2 + 2*a*b*x)**(3/2)/3 + (a**2 + 2*a*b*x)**(5/2)/5)/(2 *a**2*b) + e**3*(-a**6*sqrt(a**2 + 2*a*b*x) + a**4*(a**2 + 2*a*b*x)**(3/2) - 3*a**2*(a**2 + 2*a*b*x)**(5/2)/5 + (a**2 + 2*a*b*x)**(7/2)/7)/(4*a**2*b **3) + 3*d*e**2*(-a**6*sqrt(a**2 + 2*a*b*x) + a**4*(a**2 + 2*a*b*x)**(3/2) - 3*a**2*(a**2 + 2*a*b*x)**(5/2)/5 + (a**2 + 2*a*b*x)**(7/2)/7)/(4*a**3*b **2) + e**3*(a**8*sqrt(a**2 + 2*a*b*x) - 4*a**6*(a**2 + 2*a*b*x)**(3/2)...
Leaf count of result is larger than twice the leaf count of optimal. 432 vs. \(2 (26) = 52\).
Time = 0.20 (sec) , antiderivative size = 432, normalized size of antiderivative = 11.08 \[ \int \frac {(a+b x) (d+e x)^3}{\sqrt {a^2+2 a b x+b^2 x^2}} \, dx=\frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} e^{3} x^{3}}{4 \, b} + \frac {13 \, a^{2} e^{3} x^{2}}{12 \, b^{2}} - \frac {7 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} a e^{3} x^{2}}{12 \, b^{2}} - \frac {13 \, a^{3} e^{3} x}{6 \, b^{3}} + \frac {a d^{3} \log \left (x + \frac {a}{b}\right )}{b} + \frac {a^{4} e^{3} \log \left (x + \frac {a}{b}\right )}{b^{4}} + \frac {7 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} a^{3} e^{3}}{6 \, b^{4}} - \frac {5 \, {\left (3 \, b d e^{2} + a e^{3}\right )} a x^{2}}{6 \, b^{2}} + \frac {3 \, {\left (b d^{2} e + a d e^{2}\right )} x^{2}}{2 \, b} + \frac {{\left (3 \, b d e^{2} + a e^{3}\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} x^{2}}{3 \, b^{2}} + \frac {5 \, {\left (3 \, b d e^{2} + a e^{3}\right )} a^{2} x}{3 \, b^{3}} - \frac {3 \, {\left (b d^{2} e + a d e^{2}\right )} a x}{b^{2}} - \frac {{\left (3 \, b d e^{2} + a e^{3}\right )} a^{3} \log \left (x + \frac {a}{b}\right )}{b^{4}} + \frac {3 \, {\left (b d^{2} e + a d e^{2}\right )} a^{2} \log \left (x + \frac {a}{b}\right )}{b^{3}} - \frac {{\left (b d^{3} + 3 \, a d^{2} e\right )} a \log \left (x + \frac {a}{b}\right )}{b^{2}} - \frac {2 \, {\left (3 \, b d e^{2} + a e^{3}\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} a^{2}}{3 \, b^{4}} + \frac {{\left (b d^{3} + 3 \, a d^{2} e\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}}}{b^{2}} \]
1/4*sqrt(b^2*x^2 + 2*a*b*x + a^2)*e^3*x^3/b + 13/12*a^2*e^3*x^2/b^2 - 7/12 *sqrt(b^2*x^2 + 2*a*b*x + a^2)*a*e^3*x^2/b^2 - 13/6*a^3*e^3*x/b^3 + a*d^3* log(x + a/b)/b + a^4*e^3*log(x + a/b)/b^4 + 7/6*sqrt(b^2*x^2 + 2*a*b*x + a ^2)*a^3*e^3/b^4 - 5/6*(3*b*d*e^2 + a*e^3)*a*x^2/b^2 + 3/2*(b*d^2*e + a*d*e ^2)*x^2/b + 1/3*(3*b*d*e^2 + a*e^3)*sqrt(b^2*x^2 + 2*a*b*x + a^2)*x^2/b^2 + 5/3*(3*b*d*e^2 + a*e^3)*a^2*x/b^3 - 3*(b*d^2*e + a*d*e^2)*a*x/b^2 - (3*b *d*e^2 + a*e^3)*a^3*log(x + a/b)/b^4 + 3*(b*d^2*e + a*d*e^2)*a^2*log(x + a /b)/b^3 - (b*d^3 + 3*a*d^2*e)*a*log(x + a/b)/b^2 - 2/3*(3*b*d*e^2 + a*e^3) *sqrt(b^2*x^2 + 2*a*b*x + a^2)*a^2/b^4 + (b*d^3 + 3*a*d^2*e)*sqrt(b^2*x^2 + 2*a*b*x + a^2)/b^2
Leaf count of result is larger than twice the leaf count of optimal. 82 vs. \(2 (26) = 52\).
Time = 0.27 (sec) , antiderivative size = 82, normalized size of antiderivative = 2.10 \[ \int \frac {(a+b x) (d+e x)^3}{\sqrt {a^2+2 a b x+b^2 x^2}} \, dx=\frac {{\left (e x + d\right )}^{4} \mathrm {sgn}\left (b x + a\right )}{4 \, e} - \frac {{\left (b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4}\right )} \mathrm {sgn}\left (b x + a\right )}{4 \, b^{4} e} \]
1/4*(e*x + d)^4*sgn(b*x + a)/e - 1/4*(b^4*d^4 - 4*a*b^3*d^3*e + 6*a^2*b^2* d^2*e^2 - 4*a^3*b*d*e^3 + a^4*e^4)*sgn(b*x + a)/(b^4*e)
Timed out. \[ \int \frac {(a+b x) (d+e x)^3}{\sqrt {a^2+2 a b x+b^2 x^2}} \, dx=\int \frac {\left (a+b\,x\right )\,{\left (d+e\,x\right )}^3}{\sqrt {{\left (a+b\,x\right )}^2}} \,d x \]