Integrand size = 33, antiderivative size = 210 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{d+e x} \, dx=-\frac {b (b d-a e)^3 x \sqrt {a^2+2 a b x+b^2 x^2}}{e^4 (a+b x)}+\frac {(b d-a e)^2 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}{2 e^3}-\frac {(b d-a e) (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^2}+\frac {(a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}{4 e}+\frac {(b d-a e)^4 \sqrt {a^2+2 a b x+b^2 x^2} \log (d+e x)}{e^5 (a+b x)} \]
-b*(-a*e+b*d)^3*x*((b*x+a)^2)^(1/2)/e^4/(b*x+a)+1/2*(-a*e+b*d)^2*(b*x+a)*( (b*x+a)^2)^(1/2)/e^3-1/3*(-a*e+b*d)*(b*x+a)^2*((b*x+a)^2)^(1/2)/e^2+1/4*(b *x+a)^3*((b*x+a)^2)^(1/2)/e+(-a*e+b*d)^4*ln(e*x+d)*((b*x+a)^2)^(1/2)/e^5/( b*x+a)
Time = 1.05 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.63 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{d+e x} \, dx=\frac {\sqrt {(a+b x)^2} \left (b e x \left (48 a^3 e^3+36 a^2 b e^2 (-2 d+e x)+8 a b^2 e \left (6 d^2-3 d e x+2 e^2 x^2\right )+b^3 \left (-12 d^3+6 d^2 e x-4 d e^2 x^2+3 e^3 x^3\right )\right )+12 (b d-a e)^4 \log (d+e x)\right )}{12 e^5 (a+b x)} \]
(Sqrt[(a + b*x)^2]*(b*e*x*(48*a^3*e^3 + 36*a^2*b*e^2*(-2*d + e*x) + 8*a*b^ 2*e*(6*d^2 - 3*d*e*x + 2*e^2*x^2) + b^3*(-12*d^3 + 6*d^2*e*x - 4*d*e^2*x^2 + 3*e^3*x^3)) + 12*(b*d - a*e)^4*Log[d + e*x]))/(12*e^5*(a + b*x))
Time = 0.29 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.60, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.121, Rules used = {1187, 27, 49, 2009}
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.
\(\displaystyle \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{d+e x} \, dx\) |
\(\Big \downarrow \) 1187 |
\(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {b^3 (a+b x)^4}{d+e x}dx}{b^3 (a+b x)}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {(a+b x)^4}{d+e x}dx}{a+b x}\) |
\(\Big \downarrow \) 49 |
\(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (\frac {(a e-b d)^4}{e^4 (d+e x)}-\frac {b (b d-a e)^3}{e^4}+\frac {b (a+b x)^3}{e}-\frac {b (b d-a e) (a+b x)^2}{e^2}+\frac {b (b d-a e)^2 (a+b x)}{e^3}\right )dx}{a+b x}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \left (\frac {(b d-a e)^4 \log (d+e x)}{e^5}-\frac {b x (b d-a e)^3}{e^4}+\frac {(a+b x)^2 (b d-a e)^2}{2 e^3}-\frac {(a+b x)^3 (b d-a e)}{3 e^2}+\frac {(a+b x)^4}{4 e}\right )}{a+b x}\) |
(Sqrt[a^2 + 2*a*b*x + b^2*x^2]*(-((b*(b*d - a*e)^3*x)/e^4) + ((b*d - a*e)^ 2*(a + b*x)^2)/(2*e^3) - ((b*d - a*e)*(a + b*x)^3)/(3*e^2) + (a + b*x)^4/( 4*e) + ((b*d - a*e)^4*Log[d + e*x])/e^5))/(a + b*x)
3.20.76.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 0] && IGtQ[m + n + 2, 0]
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 = 0.33 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.05
method | result | size |
risch | \(\frac {\sqrt {\left (b x +a \right )^{2}}\, b \left (\frac {b^{3} x^{4} e^{3}}{4}+\frac {\left (\left (2 a e -b d \right ) b^{2} e^{2}+2 a \,b^{2} e^{3}\right ) x^{3}}{3}+\frac {\left (2 \left (2 a e -b d \right ) a b \,e^{2}+b e \left (2 e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right )\right ) x^{2}}{2}+\left (2 a e -b d \right ) \left (2 e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) x \right )}{\left (b x +a \right ) e^{4}}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (e^{4} a^{4}-4 b d \,e^{3} a^{3}+6 b^{2} d^{2} e^{2} a^{2}-4 b^{3} d^{3} e a +b^{4} d^{4}\right ) \ln \left (e x +d \right )}{\left (b x +a \right ) e^{5}}\) | \(221\) |
default | \(\frac {\left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}} \left (3 e^{4} x^{4} b^{4}+16 x^{3} a \,b^{3} e^{4}-4 x^{3} b^{4} d \,e^{3}+36 x^{2} a^{2} b^{2} e^{4}-24 x^{2} a \,b^{3} d \,e^{3}+6 x^{2} b^{4} d^{2} e^{2}+12 \ln \left (e x +d \right ) a^{4} e^{4}-48 \ln \left (e x +d \right ) a^{3} b d \,e^{3}+72 \ln \left (e x +d \right ) a^{2} b^{2} d^{2} e^{2}-48 \ln \left (e x +d \right ) a \,b^{3} d^{3} e +12 \ln \left (e x +d \right ) b^{4} d^{4}+48 x \,a^{3} b \,e^{4}-72 x \,a^{2} b^{2} d \,e^{3}+48 x a \,b^{3} d^{2} e^{2}-12 x \,b^{4} d^{3} e \right )}{12 \left (b x +a \right )^{3} e^{5}}\) | \(225\) |
((b*x+a)^2)^(1/2)/(b*x+a)*b/e^4*(1/4*b^3*x^4*e^3+1/3*((2*a*e-b*d)*b^2*e^2+ 2*a*b^2*e^3)*x^3+1/2*(2*(2*a*e-b*d)*a*b*e^2+b*e*(2*a^2*e^2-2*a*b*d*e+b^2*d ^2))*x^2+(2*a*e-b*d)*(2*a^2*e^2-2*a*b*d*e+b^2*d^2)*x)+((b*x+a)^2)^(1/2)/(b *x+a)*(a^4*e^4-4*a^3*b*d*e^3+6*a^2*b^2*d^2*e^2-4*a*b^3*d^3*e+b^4*d^4)/e^5* ln(e*x+d)
Time = 0.34 (sec) , antiderivative size = 179, normalized size of antiderivative = 0.85 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{d+e x} \, dx=\frac {3 \, b^{4} e^{4} x^{4} - 4 \, {\left (b^{4} d e^{3} - 4 \, a b^{3} e^{4}\right )} x^{3} + 6 \, {\left (b^{4} d^{2} e^{2} - 4 \, a b^{3} d e^{3} + 6 \, a^{2} b^{2} e^{4}\right )} x^{2} - 12 \, {\left (b^{4} d^{3} e - 4 \, a b^{3} d^{2} e^{2} + 6 \, a^{2} b^{2} d e^{3} - 4 \, a^{3} b e^{4}\right )} x + 12 \, {\left (b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4}\right )} \log \left (e x + d\right )}{12 \, e^{5}} \]
1/12*(3*b^4*e^4*x^4 - 4*(b^4*d*e^3 - 4*a*b^3*e^4)*x^3 + 6*(b^4*d^2*e^2 - 4 *a*b^3*d*e^3 + 6*a^2*b^2*e^4)*x^2 - 12*(b^4*d^3*e - 4*a*b^3*d^2*e^2 + 6*a^ 2*b^2*d*e^3 - 4*a^3*b*e^4)*x + 12*(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)*log(e*x + d))/e^5
\[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{d+e x} \, dx=\int \frac {\left (a + b x\right ) \left (\left (a + b x\right )^{2}\right )^{\frac {3}{2}}}{d + e x}\, dx \]
Exception generated. \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{d+e x} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more de tails)Is e
Time = 0.27 (sec) , antiderivative size = 274, normalized size of antiderivative = 1.30 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{d+e x} \, dx=\frac {3 \, b^{4} e^{3} x^{4} \mathrm {sgn}\left (b x + a\right ) - 4 \, b^{4} d e^{2} x^{3} \mathrm {sgn}\left (b x + a\right ) + 16 \, a b^{3} e^{3} x^{3} \mathrm {sgn}\left (b x + a\right ) + 6 \, b^{4} d^{2} e x^{2} \mathrm {sgn}\left (b x + a\right ) - 24 \, a b^{3} d e^{2} x^{2} \mathrm {sgn}\left (b x + a\right ) + 36 \, a^{2} b^{2} e^{3} x^{2} \mathrm {sgn}\left (b x + a\right ) - 12 \, b^{4} d^{3} x \mathrm {sgn}\left (b x + a\right ) + 48 \, a b^{3} d^{2} e x \mathrm {sgn}\left (b x + a\right ) - 72 \, a^{2} b^{2} d e^{2} x \mathrm {sgn}\left (b x + a\right ) + 48 \, a^{3} b e^{3} x \mathrm {sgn}\left (b x + a\right )}{12 \, e^{4}} + \frac {{\left (b^{4} d^{4} \mathrm {sgn}\left (b x + a\right ) - 4 \, a b^{3} d^{3} e \mathrm {sgn}\left (b x + a\right ) + 6 \, a^{2} b^{2} d^{2} e^{2} \mathrm {sgn}\left (b x + a\right ) - 4 \, a^{3} b d e^{3} \mathrm {sgn}\left (b x + a\right ) + a^{4} e^{4} \mathrm {sgn}\left (b x + a\right )\right )} \log \left ({\left | e x + d \right |}\right )}{e^{5}} \]
1/12*(3*b^4*e^3*x^4*sgn(b*x + a) - 4*b^4*d*e^2*x^3*sgn(b*x + a) + 16*a*b^3 *e^3*x^3*sgn(b*x + a) + 6*b^4*d^2*e*x^2*sgn(b*x + a) - 24*a*b^3*d*e^2*x^2* sgn(b*x + a) + 36*a^2*b^2*e^3*x^2*sgn(b*x + a) - 12*b^4*d^3*x*sgn(b*x + a) + 48*a*b^3*d^2*e*x*sgn(b*x + a) - 72*a^2*b^2*d*e^2*x*sgn(b*x + a) + 48*a^ 3*b*e^3*x*sgn(b*x + a))/e^4 + (b^4*d^4*sgn(b*x + a) - 4*a*b^3*d^3*e*sgn(b* x + a) + 6*a^2*b^2*d^2*e^2*sgn(b*x + a) - 4*a^3*b*d*e^3*sgn(b*x + a) + a^4 *e^4*sgn(b*x + a))*log(abs(e*x + d))/e^5
Timed out. \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{d+e x} \, dx=\int \frac {\left (a+b\,x\right )\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2}}{d+e\,x} \,d x \]