Integrand size = 23, antiderivative size = 293 \[ \int \frac {x \left (a+b x+c x^2\right )^{3/2}}{(d+e x)^6} \, dx=\frac {3 \left (b^2-4 a c\right ) (b d-2 a e) (b d-2 a e+(2 c d-b e) x) \sqrt {a+b x+c x^2}}{128 \left (c d^2-b d e+a e^2\right )^3 (d+e x)^2}-\frac {(b d-2 a e) (b d-2 a e+(2 c d-b e) x) \left (a+b x+c x^2\right )^{3/2}}{16 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^4}+\frac {d \left (a+b x+c x^2\right )^{5/2}}{5 \left (c d^2-b d e+a e^2\right ) (d+e x)^5}-\frac {3 \left (b^2-4 a c\right )^2 (b d-2 a e) \text {arctanh}\left (\frac {b d-2 a e+(2 c d-b e) x}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x+c x^2}}\right )}{256 \left (c d^2-b d e+a e^2\right )^{7/2}} \] Output:
3/128*(-4*a*c+b^2)*(-2*a*e+b*d)*(b*d-2*a*e+(-b*e+2*c*d)*x)*(c*x^2+b*x+a)^( 1/2)/(a*e^2-b*d*e+c*d^2)^3/(e*x+d)^2-1/16*(-2*a*e+b*d)*(b*d-2*a*e+(-b*e+2* c*d)*x)*(c*x^2+b*x+a)^(3/2)/(a*e^2-b*d*e+c*d^2)^2/(e*x+d)^4+1/5*d*(c*x^2+b *x+a)^(5/2)/(a*e^2-b*d*e+c*d^2)/(e*x+d)^5-3/256*(-4*a*c+b^2)^2*(-2*a*e+b*d )*arctanh(1/2*(b*d-2*a*e+(-b*e+2*c*d)*x)/(a*e^2-b*d*e+c*d^2)^(1/2)/(c*x^2+ b*x+a)^(1/2))/(a*e^2-b*d*e+c*d^2)^(7/2)
Time = 11.71 (sec) , antiderivative size = 274, normalized size of antiderivative = 0.94 \[ \int \frac {x \left (a+b x+c x^2\right )^{3/2}}{(d+e x)^6} \, dx=\frac {d (a+x (b+c x))^{5/2}}{5 \left (c d^2+e (-b d+a e)\right ) (d+e x)^5}+\frac {(b d-2 a e) \left (\frac {2 (-b d+2 a e-2 c d x+b e x) (a+x (b+c x))^{3/2}}{(d+e x)^4}+3 \left (b^2-4 a c\right ) \left (\frac {\sqrt {a+x (b+c x)} (-2 a e+2 c d x+b (d-e x))}{4 \left (c d^2+e (-b d+a e)\right ) (d+e x)^2}+\frac {\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {-b d+2 a e-2 c d x+b e x}{2 \sqrt {c d^2+e (-b d+a e)} \sqrt {a+x (b+c x)}}\right )}{8 \left (c d^2+e (-b d+a e)\right )^{3/2}}\right )\right )}{32 \left (c d^2+e (-b d+a e)\right )^2} \] Input:
Integrate[(x*(a + b*x + c*x^2)^(3/2))/(d + e*x)^6,x]
Output:
(d*(a + x*(b + c*x))^(5/2))/(5*(c*d^2 + e*(-(b*d) + a*e))*(d + e*x)^5) + ( (b*d - 2*a*e)*((2*(-(b*d) + 2*a*e - 2*c*d*x + b*e*x)*(a + x*(b + c*x))^(3/ 2))/(d + e*x)^4 + 3*(b^2 - 4*a*c)*((Sqrt[a + x*(b + c*x)]*(-2*a*e + 2*c*d* x + b*(d - e*x)))/(4*(c*d^2 + e*(-(b*d) + a*e))*(d + e*x)^2) + ((b^2 - 4*a *c)*ArcTanh[(-(b*d) + 2*a*e - 2*c*d*x + b*e*x)/(2*Sqrt[c*d^2 + e*(-(b*d) + a*e)]*Sqrt[a + x*(b + c*x)])])/(8*(c*d^2 + e*(-(b*d) + a*e))^(3/2)))))/(3 2*(c*d^2 + e*(-(b*d) + a*e))^2)
Time = 0.47 (sec) , antiderivative size = 321, normalized size of antiderivative = 1.10, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {1228, 1152, 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.
\(\displaystyle \int \frac {x \left (a+b x+c x^2\right )^{3/2}}{(d+e x)^6} \, dx\) |
\(\Big \downarrow \) 1228 |
\(\displaystyle \frac {d \left (a+b x+c x^2\right )^{5/2}}{5 (d+e x)^5 \left (a e^2-b d e+c d^2\right )}-\frac {(b d-2 a e) \int \frac {\left (c x^2+b x+a\right )^{3/2}}{(d+e x)^5}dx}{2 \left (a e^2-b d e+c d^2\right )}\) |
\(\Big \downarrow \) 1152 |
\(\displaystyle \frac {d \left (a+b x+c x^2\right )^{5/2}}{5 (d+e x)^5 \left (a e^2-b d e+c d^2\right )}-\frac {(b d-2 a e) \left (\frac {\left (a+b x+c x^2\right )^{3/2} (-2 a e+x (2 c d-b e)+b d)}{8 (d+e x)^4 \left (a e^2-b d e+c d^2\right )}-\frac {3 \left (b^2-4 a c\right ) \int \frac {\sqrt {c x^2+b x+a}}{(d+e x)^3}dx}{16 \left (a e^2-b d e+c d^2\right )}\right )}{2 \left (a e^2-b d e+c d^2\right )}\) |
\(\Big \downarrow \) 1152 |
\(\displaystyle \frac {d \left (a+b x+c x^2\right )^{5/2}}{5 (d+e x)^5 \left (a e^2-b d e+c d^2\right )}-\frac {(b d-2 a e) \left (\frac {\left (a+b x+c x^2\right )^{3/2} (-2 a e+x (2 c d-b e)+b d)}{8 (d+e x)^4 \left (a e^2-b d e+c d^2\right )}-\frac {3 \left (b^2-4 a c\right ) \left (\frac {\sqrt {a+b x+c x^2} (-2 a e+x (2 c d-b e)+b d)}{4 (d+e x)^2 \left (a e^2-b d e+c d^2\right )}-\frac {\left (b^2-4 a c\right ) \int \frac {1}{(d+e x) \sqrt {c x^2+b x+a}}dx}{8 \left (a e^2-b d e+c d^2\right )}\right )}{16 \left (a e^2-b d e+c d^2\right )}\right )}{2 \left (a e^2-b d e+c d^2\right )}\) |
\(\Big \downarrow \) 1154 |
\(\displaystyle \frac {d \left (a+b x+c x^2\right )^{5/2}}{5 (d+e x)^5 \left (a e^2-b d e+c d^2\right )}-\frac {(b d-2 a e) \left (\frac {\left (a+b x+c x^2\right )^{3/2} (-2 a e+x (2 c d-b e)+b d)}{8 (d+e x)^4 \left (a e^2-b d e+c d^2\right )}-\frac {3 \left (b^2-4 a c\right ) \left (\frac {\left (b^2-4 a c\right ) \int \frac {1}{4 \left (c d^2-b e d+a e^2\right )-\frac {(b d-2 a e+(2 c d-b e) x)^2}{c x^2+b x+a}}d\left (-\frac {b d-2 a e+(2 c d-b e) x}{\sqrt {c x^2+b x+a}}\right )}{4 \left (a e^2-b d e+c d^2\right )}+\frac {\sqrt {a+b x+c x^2} (-2 a e+x (2 c d-b e)+b d)}{4 (d+e x)^2 \left (a e^2-b d e+c d^2\right )}\right )}{16 \left (a e^2-b d e+c d^2\right )}\right )}{2 \left (a e^2-b d e+c d^2\right )}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {d \left (a+b x+c x^2\right )^{5/2}}{5 (d+e x)^5 \left (a e^2-b d e+c d^2\right )}-\frac {(b d-2 a e) \left (\frac {\left (a+b x+c x^2\right )^{3/2} (-2 a e+x (2 c d-b e)+b d)}{8 (d+e x)^4 \left (a e^2-b d e+c d^2\right )}-\frac {3 \left (b^2-4 a c\right ) \left (\frac {\sqrt {a+b x+c x^2} (-2 a e+x (2 c d-b e)+b d)}{4 (d+e x)^2 \left (a e^2-b d e+c d^2\right )}-\frac {\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {-2 a e+x (2 c d-b e)+b d}{2 \sqrt {a+b x+c x^2} \sqrt {a e^2-b d e+c d^2}}\right )}{8 \left (a e^2-b d e+c d^2\right )^{3/2}}\right )}{16 \left (a e^2-b d e+c d^2\right )}\right )}{2 \left (a e^2-b d e+c d^2\right )}\) |
Input:
Int[(x*(a + b*x + c*x^2)^(3/2))/(d + e*x)^6,x]
Output:
(d*(a + b*x + c*x^2)^(5/2))/(5*(c*d^2 - b*d*e + a*e^2)*(d + e*x)^5) - ((b* d - 2*a*e)*(((b*d - 2*a*e + (2*c*d - b*e)*x)*(a + b*x + c*x^2)^(3/2))/(8*( c*d^2 - b*d*e + a*e^2)*(d + e*x)^4) - (3*(b^2 - 4*a*c)*(((b*d - 2*a*e + (2 *c*d - b*e)*x)*Sqrt[a + b*x + c*x^2])/(4*(c*d^2 - b*d*e + a*e^2)*(d + e*x) ^2) - ((b^2 - 4*a*c)*ArcTanh[(b*d - 2*a*e + (2*c*d - b*e)*x)/(2*Sqrt[c*d^2 - b*d*e + a*e^2]*Sqrt[a + b*x + c*x^2])])/(8*(c*d^2 - b*d*e + a*e^2)^(3/2 ))))/(16*(c*d^2 - b*d*e + a*e^2))))/(2*(c*d^2 - b*d*e + a*e^2))
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]
Leaf count of result is larger than twice the leaf count of optimal. \(10264\) vs. \(2(271)=542\).
Time = 1.86 (sec) , antiderivative size = 10265, normalized size of antiderivative = 35.03
Input:
int(x*(c*x^2+b*x+a)^(3/2)/(e*x+d)^6,x,method=_RETURNVERBOSE)
Output:
result too large to display
Leaf count of result is larger than twice the leaf count of optimal. 2391 vs. \(2 (271) = 542\).
Time = 43.12 (sec) , antiderivative size = 4824, normalized size of antiderivative = 16.46 \[ \int \frac {x \left (a+b x+c x^2\right )^{3/2}}{(d+e x)^6} \, dx=\text {Too large to display} \] Input:
integrate(x*(c*x^2+b*x+a)^(3/2)/(e*x+d)^6,x, algorithm="fricas")
Output:
Too large to include
\[ \int \frac {x \left (a+b x+c x^2\right )^{3/2}}{(d+e x)^6} \, dx=\int \frac {x \left (a + b x + c x^{2}\right )^{\frac {3}{2}}}{\left (d + e x\right )^{6}}\, dx \] Input:
integrate(x*(c*x**2+b*x+a)**(3/2)/(e*x+d)**6,x)
Output:
Integral(x*(a + b*x + c*x**2)**(3/2)/(d + e*x)**6, x)
Exception generated. \[ \int \frac {x \left (a+b x+c x^2\right )^{3/2}}{(d+e x)^6} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(x*(c*x^2+b*x+a)^(3/2)/(e*x+d)^6,x, algorithm="maxima")
Output:
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^2-b*d*e>0)', see `assume?` f or more de
Leaf count of result is larger than twice the leaf count of optimal. 9517 vs. \(2 (271) = 542\).
Time = 1.41 (sec) , antiderivative size = 9517, normalized size of antiderivative = 32.48 \[ \int \frac {x \left (a+b x+c x^2\right )^{3/2}}{(d+e x)^6} \, dx=\text {Too large to display} \] Input:
integrate(x*(c*x^2+b*x+a)^(3/2)/(e*x+d)^6,x, algorithm="giac")
Output:
-3/128*(b^5*d - 8*a*b^3*c*d + 16*a^2*b*c^2*d - 2*a*b^4*e + 16*a^2*b^2*c*e - 32*a^3*c^2*e)*arctan(-((sqrt(c)*x - sqrt(c*x^2 + b*x + a))*e + sqrt(c)*d )/sqrt(-c*d^2 + b*d*e - a*e^2))/((c^3*d^6 - 3*b*c^2*d^5*e + 3*b^2*c*d^4*e^ 2 + 3*a*c^2*d^4*e^2 - b^3*d^3*e^3 - 6*a*b*c*d^3*e^3 + 3*a*b^2*d^2*e^4 + 3* a^2*c*d^2*e^4 - 3*a^2*b*d*e^5 + a^3*e^6)*sqrt(-c*d^2 + b*d*e - a*e^2)) + 1 /640*(1280*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*c^5*d^6*e^4 - 3840*(sqrt( c)*x - sqrt(c*x^2 + b*x + a))^9*b*c^4*d^5*e^5 + 3840*(sqrt(c)*x - sqrt(c*x ^2 + b*x + a))^9*b^2*c^3*d^4*e^6 + 3840*(sqrt(c)*x - sqrt(c*x^2 + b*x + a) )^9*a*c^4*d^4*e^6 - 1280*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*b^3*c^2*d^3 *e^7 - 7680*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*a*b*c^3*d^3*e^7 + 3840*( sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*a*b^2*c^2*d^2*e^8 + 3840*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*a^2*c^3*d^2*e^8 + 15*(sqrt(c)*x - sqrt(c*x^2 + b *x + a))^9*b^5*d*e^9 - 120*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*a*b^3*c*d *e^9 - 3600*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*a^2*b*c^2*d*e^9 - 30*(sq rt(c)*x - sqrt(c*x^2 + b*x + a))^9*a*b^4*e^10 + 240*(sqrt(c)*x - sqrt(c*x^ 2 + b*x + a))^9*a^2*b^2*c*e^10 + 800*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9 *a^3*c^2*e^10 + 5120*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^8*c^(11/2)*d^7*e^ 3 - 12800*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^8*b*c^(9/2)*d^6*e^4 + 7680*( sqrt(c)*x - sqrt(c*x^2 + b*x + a))^8*b^2*c^(7/2)*d^5*e^5 + 15360*(sqrt(c)* x - sqrt(c*x^2 + b*x + a))^8*a*c^(9/2)*d^5*e^5 + 2560*(sqrt(c)*x - sqrt...
Timed out. \[ \int \frac {x \left (a+b x+c x^2\right )^{3/2}}{(d+e x)^6} \, dx=\int \frac {x\,{\left (c\,x^2+b\,x+a\right )}^{3/2}}{{\left (d+e\,x\right )}^6} \,d x \] Input:
int((x*(a + b*x + c*x^2)^(3/2))/(d + e*x)^6,x)
Output:
int((x*(a + b*x + c*x^2)^(3/2))/(d + e*x)^6, x)
Time = 96.50 (sec) , antiderivative size = 7687, normalized size of antiderivative = 26.24 \[ \int \frac {x \left (a+b x+c x^2\right )^{3/2}}{(d+e x)^6} \, dx =\text {Too large to display} \] Input:
int(x*(c*x^2+b*x+a)^(3/2)/(e*x+d)^6,x)
Output:
(480*sqrt(a*e**2 - b*d*e + c*d**2)*log(2*sqrt(a + b*x + c*x**2)*sqrt(a*e** 2 - b*d*e + c*d**2) - 2*a*e + b*d - b*e*x + 2*c*d*x)*a**3*c**2*d**5*e + 24 00*sqrt(a*e**2 - b*d*e + c*d**2)*log(2*sqrt(a + b*x + c*x**2)*sqrt(a*e**2 - b*d*e + c*d**2) - 2*a*e + b*d - b*e*x + 2*c*d*x)*a**3*c**2*d**4*e**2*x + 4800*sqrt(a*e**2 - b*d*e + c*d**2)*log(2*sqrt(a + b*x + c*x**2)*sqrt(a*e* *2 - b*d*e + c*d**2) - 2*a*e + b*d - b*e*x + 2*c*d*x)*a**3*c**2*d**3*e**3* x**2 + 4800*sqrt(a*e**2 - b*d*e + c*d**2)*log(2*sqrt(a + b*x + c*x**2)*sqr t(a*e**2 - b*d*e + c*d**2) - 2*a*e + b*d - b*e*x + 2*c*d*x)*a**3*c**2*d**2 *e**4*x**3 + 2400*sqrt(a*e**2 - b*d*e + c*d**2)*log(2*sqrt(a + b*x + c*x** 2)*sqrt(a*e**2 - b*d*e + c*d**2) - 2*a*e + b*d - b*e*x + 2*c*d*x)*a**3*c** 2*d*e**5*x**4 + 480*sqrt(a*e**2 - b*d*e + c*d**2)*log(2*sqrt(a + b*x + c*x **2)*sqrt(a*e**2 - b*d*e + c*d**2) - 2*a*e + b*d - b*e*x + 2*c*d*x)*a**3*c **2*e**6*x**5 - 240*sqrt(a*e**2 - b*d*e + c*d**2)*log(2*sqrt(a + b*x + c*x **2)*sqrt(a*e**2 - b*d*e + c*d**2) - 2*a*e + b*d - b*e*x + 2*c*d*x)*a**2*b **2*c*d**5*e - 1200*sqrt(a*e**2 - b*d*e + c*d**2)*log(2*sqrt(a + b*x + c*x **2)*sqrt(a*e**2 - b*d*e + c*d**2) - 2*a*e + b*d - b*e*x + 2*c*d*x)*a**2*b **2*c*d**4*e**2*x - 2400*sqrt(a*e**2 - b*d*e + c*d**2)*log(2*sqrt(a + b*x + c*x**2)*sqrt(a*e**2 - b*d*e + c*d**2) - 2*a*e + b*d - b*e*x + 2*c*d*x)*a **2*b**2*c*d**3*e**3*x**2 - 2400*sqrt(a*e**2 - b*d*e + c*d**2)*log(2*sqrt( a + b*x + c*x**2)*sqrt(a*e**2 - b*d*e + c*d**2) - 2*a*e + b*d - b*e*x +...