Integrand size = 41, antiderivative size = 25 \[ \int \frac {(b d-2 a e+(2 c d-b e) x) \left (a+b x+c x^2\right )^{3/2}}{(d+e x)^6} \, dx=\frac {2 \left (a+b x+c x^2\right )^{5/2}}{5 (d+e x)^5} \] Output:
2/5*(c*x^2+b*x+a)^(5/2)/(e*x+d)^5
Time = 10.40 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.96 \[ \int \frac {(b d-2 a e+(2 c d-b e) x) \left (a+b x+c x^2\right )^{3/2}}{(d+e x)^6} \, dx=\frac {2 (a+x (b+c x))^{5/2}}{5 (d+e x)^5} \] Input:
Integrate[((b*d - 2*a*e + (2*c*d - b*e)*x)*(a + b*x + c*x^2)^(3/2))/(d + e *x)^6,x]
Output:
(2*(a + x*(b + c*x))^(5/2))/(5*(d + e*x)^5)
Time = 0.27 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.024, Rules used = {1226}
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.
\(\displaystyle \int \frac {\left (a+b x+c x^2\right )^{3/2} (-2 a e+x (2 c d-b e)+b d)}{(d+e x)^6} \, dx\) |
\(\Big \downarrow \) 1226 |
\(\displaystyle \frac {2 \left (a+b x+c x^2\right )^{5/2}}{5 (d+e x)^5}\) |
Input:
Int[((b*d - 2*a*e + (2*c*d - b*e)*x)*(a + b*x + c*x^2)^(3/2))/(d + e*x)^6, x]
Output:
(2*(a + b*x + c*x^2)^(5/2))/(5*(d + e*x)^5)
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] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && EqQ[Simplify[m + 2*p + 3], 0] && EqQ[b*(e*f + d*g) - 2*(c*d*f + a*e*g), 0]
Time = 2.54 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.88
method | result | size |
gosper | \(\frac {2 \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}}}{5 \left (e x +d \right )^{5}}\) | \(22\) |
trager | \(\frac {2 \left (c^{2} x^{4}+2 b c \,x^{3}+2 a c \,x^{2}+b^{2} x^{2}+2 a b x +a^{2}\right ) \sqrt {c \,x^{2}+b x +a}}{5 \left (e x +d \right )^{5}}\) | \(59\) |
orering | \(-\frac {2 \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} \left (b d -2 a e +\left (-b e +2 c d \right ) x \right )}{5 \left (e x +d \right )^{5} \left (b e x -2 c d x +2 a e -b d \right )}\) | \(61\) |
default | \(\text {Expression too large to display}\) | \(10289\) |
Input:
int((b*d-2*a*e+(-b*e+2*c*d)*x)*(c*x^2+b*x+a)^(3/2)/(e*x+d)^6,x,method=_RET URNVERBOSE)
Output:
2/5*(c*x^2+b*x+a)^(5/2)/(e*x+d)^5
Leaf count of result is larger than twice the leaf count of optimal. 100 vs. \(2 (21) = 42\).
Time = 33.59 (sec) , antiderivative size = 100, normalized size of antiderivative = 4.00 \[ \int \frac {(b d-2 a e+(2 c d-b e) x) \left (a+b x+c x^2\right )^{3/2}}{(d+e x)^6} \, dx=\frac {2 \, {\left (c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x + {\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}\right )} \sqrt {c x^{2} + b x + a}}{5 \, {\left (e^{5} x^{5} + 5 \, d e^{4} x^{4} + 10 \, d^{2} e^{3} x^{3} + 10 \, d^{3} e^{2} x^{2} + 5 \, d^{4} e x + d^{5}\right )}} \] Input:
integrate((b*d-2*a*e+(-b*e+2*c*d)*x)*(c*x^2+b*x+a)^(3/2)/(e*x+d)^6,x, algo rithm="fricas")
Output:
2/5*(c^2*x^4 + 2*b*c*x^3 + 2*a*b*x + (b^2 + 2*a*c)*x^2 + a^2)*sqrt(c*x^2 + b*x + a)/(e^5*x^5 + 5*d*e^4*x^4 + 10*d^2*e^3*x^3 + 10*d^3*e^2*x^2 + 5*d^4 *e*x + d^5)
\[ \int \frac {(b d-2 a e+(2 c d-b e) x) \left (a+b x+c x^2\right )^{3/2}}{(d+e x)^6} \, dx=- \int \frac {2 a^{2} e \sqrt {a + b x + c x^{2}}}{d^{6} + 6 d^{5} e x + 15 d^{4} e^{2} x^{2} + 20 d^{3} e^{3} x^{3} + 15 d^{2} e^{4} x^{4} + 6 d e^{5} x^{5} + e^{6} x^{6}}\, dx - \int \left (- \frac {a b d \sqrt {a + b x + c x^{2}}}{d^{6} + 6 d^{5} e x + 15 d^{4} e^{2} x^{2} + 20 d^{3} e^{3} x^{3} + 15 d^{2} e^{4} x^{4} + 6 d e^{5} x^{5} + e^{6} x^{6}}\right )\, dx - \int \left (- \frac {b^{2} d x \sqrt {a + b x + c x^{2}}}{d^{6} + 6 d^{5} e x + 15 d^{4} e^{2} x^{2} + 20 d^{3} e^{3} x^{3} + 15 d^{2} e^{4} x^{4} + 6 d e^{5} x^{5} + e^{6} x^{6}}\right )\, dx - \int \frac {b^{2} e x^{2} \sqrt {a + b x + c x^{2}}}{d^{6} + 6 d^{5} e x + 15 d^{4} e^{2} x^{2} + 20 d^{3} e^{3} x^{3} + 15 d^{2} e^{4} x^{4} + 6 d e^{5} x^{5} + e^{6} x^{6}}\, dx - \int \left (- \frac {2 c^{2} d x^{3} \sqrt {a + b x + c x^{2}}}{d^{6} + 6 d^{5} e x + 15 d^{4} e^{2} x^{2} + 20 d^{3} e^{3} x^{3} + 15 d^{2} e^{4} x^{4} + 6 d e^{5} x^{5} + e^{6} x^{6}}\right )\, dx - \int \frac {3 a b e x \sqrt {a + b x + c x^{2}}}{d^{6} + 6 d^{5} e x + 15 d^{4} e^{2} x^{2} + 20 d^{3} e^{3} x^{3} + 15 d^{2} e^{4} x^{4} + 6 d e^{5} x^{5} + e^{6} x^{6}}\, dx - \int \left (- \frac {2 a c d x \sqrt {a + b x + c x^{2}}}{d^{6} + 6 d^{5} e x + 15 d^{4} e^{2} x^{2} + 20 d^{3} e^{3} x^{3} + 15 d^{2} e^{4} x^{4} + 6 d e^{5} x^{5} + e^{6} x^{6}}\right )\, dx - \int \frac {2 a c e x^{2} \sqrt {a + b x + c x^{2}}}{d^{6} + 6 d^{5} e x + 15 d^{4} e^{2} x^{2} + 20 d^{3} e^{3} x^{3} + 15 d^{2} e^{4} x^{4} + 6 d e^{5} x^{5} + e^{6} x^{6}}\, dx - \int \left (- \frac {3 b c d x^{2} \sqrt {a + b x + c x^{2}}}{d^{6} + 6 d^{5} e x + 15 d^{4} e^{2} x^{2} + 20 d^{3} e^{3} x^{3} + 15 d^{2} e^{4} x^{4} + 6 d e^{5} x^{5} + e^{6} x^{6}}\right )\, dx - \int \frac {b c e x^{3} \sqrt {a + b x + c x^{2}}}{d^{6} + 6 d^{5} e x + 15 d^{4} e^{2} x^{2} + 20 d^{3} e^{3} x^{3} + 15 d^{2} e^{4} x^{4} + 6 d e^{5} x^{5} + e^{6} x^{6}}\, dx \] Input:
integrate((b*d-2*a*e+(-b*e+2*c*d)*x)*(c*x**2+b*x+a)**(3/2)/(e*x+d)**6,x)
Output:
-Integral(2*a**2*e*sqrt(a + b*x + c*x**2)/(d**6 + 6*d**5*e*x + 15*d**4*e** 2*x**2 + 20*d**3*e**3*x**3 + 15*d**2*e**4*x**4 + 6*d*e**5*x**5 + e**6*x**6 ), x) - Integral(-a*b*d*sqrt(a + b*x + c*x**2)/(d**6 + 6*d**5*e*x + 15*d** 4*e**2*x**2 + 20*d**3*e**3*x**3 + 15*d**2*e**4*x**4 + 6*d*e**5*x**5 + e**6 *x**6), x) - Integral(-b**2*d*x*sqrt(a + b*x + c*x**2)/(d**6 + 6*d**5*e*x + 15*d**4*e**2*x**2 + 20*d**3*e**3*x**3 + 15*d**2*e**4*x**4 + 6*d*e**5*x** 5 + e**6*x**6), x) - Integral(b**2*e*x**2*sqrt(a + b*x + c*x**2)/(d**6 + 6 *d**5*e*x + 15*d**4*e**2*x**2 + 20*d**3*e**3*x**3 + 15*d**2*e**4*x**4 + 6* d*e**5*x**5 + e**6*x**6), x) - Integral(-2*c**2*d*x**3*sqrt(a + b*x + c*x* *2)/(d**6 + 6*d**5*e*x + 15*d**4*e**2*x**2 + 20*d**3*e**3*x**3 + 15*d**2*e **4*x**4 + 6*d*e**5*x**5 + e**6*x**6), x) - Integral(3*a*b*e*x*sqrt(a + b* x + c*x**2)/(d**6 + 6*d**5*e*x + 15*d**4*e**2*x**2 + 20*d**3*e**3*x**3 + 1 5*d**2*e**4*x**4 + 6*d*e**5*x**5 + e**6*x**6), x) - Integral(-2*a*c*d*x*sq rt(a + b*x + c*x**2)/(d**6 + 6*d**5*e*x + 15*d**4*e**2*x**2 + 20*d**3*e**3 *x**3 + 15*d**2*e**4*x**4 + 6*d*e**5*x**5 + e**6*x**6), x) - Integral(2*a* c*e*x**2*sqrt(a + b*x + c*x**2)/(d**6 + 6*d**5*e*x + 15*d**4*e**2*x**2 + 2 0*d**3*e**3*x**3 + 15*d**2*e**4*x**4 + 6*d*e**5*x**5 + e**6*x**6), x) - In tegral(-3*b*c*d*x**2*sqrt(a + b*x + c*x**2)/(d**6 + 6*d**5*e*x + 15*d**4*e **2*x**2 + 20*d**3*e**3*x**3 + 15*d**2*e**4*x**4 + 6*d*e**5*x**5 + e**6*x* *6), x) - Integral(b*c*e*x**3*sqrt(a + b*x + c*x**2)/(d**6 + 6*d**5*e*x...
Exception generated. \[ \int \frac {(b d-2 a e+(2 c d-b e) x) \left (a+b x+c x^2\right )^{3/2}}{(d+e x)^6} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((b*d-2*a*e+(-b*e+2*c*d)*x)*(c*x^2+b*x+a)^(3/2)/(e*x+d)^6,x, algo rithm="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. 2245 vs. \(2 (21) = 42\).
Time = 1.20 (sec) , antiderivative size = 2245, normalized size of antiderivative = 89.80 \[ \int \frac {(b d-2 a e+(2 c d-b e) x) \left (a+b x+c x^2\right )^{3/2}}{(d+e x)^6} \, dx=\text {Too large to display} \] Input:
integrate((b*d-2*a*e+(-b*e+2*c*d)*x)*(c*x^2+b*x+a)^(3/2)/(e*x+d)^6,x, algo rithm="giac")
Output:
2/5*(10*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*c^3*d*e^4 - 5*(sqrt(c)*x - s qrt(c*x^2 + b*x + a))^9*b*c^2*e^5 + 40*(sqrt(c)*x - sqrt(c*x^2 + b*x + a)) ^8*c^(7/2)*d^2*e^3 + 5*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^8*b*c^(5/2)*d*e ^4 - 10*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^8*b^2*c^(3/2)*e^5 - 10*(sqrt(c )*x - sqrt(c*x^2 + b*x + a))^8*a*c^(5/2)*e^5 + 80*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*c^4*d^3*e^2 + 40*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*b*c^3 *d^2*e^3 - 40*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*a*c^3*d*e^4 - 10*(sqrt (c)*x - sqrt(c*x^2 + b*x + a))^7*b^3*c*e^5 - 20*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*a*b*c^2*e^5 + 80*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*c^(9/2) *d^4*e + 120*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*b*c^(7/2)*d^3*e^2 + 10* (sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*b^2*c^(5/2)*d^2*e^3 - 120*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*a*c^(7/2)*d^2*e^3 - 20*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*a*b*c^(5/2)*d*e^4 - 5*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6 *b^4*sqrt(c)*e^5 - 30*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*a*b^2*c^(3/2)* e^5 + 32*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*c^5*d^5 + 160*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*b*c^4*d^4*e + 60*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*b^2*c^3*d^3*e^2 - 160*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a*c^4*d^ 3*e^2 - 120*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a*b*c^3*d^2*e^3 + 60*(sq rt(c)*x - sqrt(c*x^2 + b*x + a))^5*a^2*c^3*d*e^4 - (sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*b^5*e^5 - 20*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a*b^3...
Time = 32.70 (sec) , antiderivative size = 14640, normalized size of antiderivative = 585.60 \[ \int \frac {(b d-2 a e+(2 c d-b e) x) \left (a+b x+c x^2\right )^{3/2}}{(d+e x)^6} \, dx=\text {Too large to display} \] Input:
int(-((2*a*e - b*d + x*(b*e - 2*c*d))*(a + b*x + c*x^2)^(3/2))/(d + e*x)^6 ,x)
Output:
(((d*((e^2*(4*a*b^2*d - 10*a^2*b*e + 4*a^2*c*d))/(10*a*e^3 - 10*b*d*e^2 + 10*c*d^2*e) + (d*((d*((d*((e^2*(4*a*c^2*e - 10*b*c^2*d + 4*b^2*c*e))/(10*a *e^3 - 10*b*d*e^2 + 10*c*d^2*e) + (d*e*(4*c^3*d - 2*b*c^2*e))/(10*a*e^3 - 10*b*d*e^2 + 10*c*d^2*e)))/e - (e^2*(2*b^3*e - 8*a*c^2*d - 8*b^2*c*d + 12* a*b*c*e))/(10*a*e^3 - 10*b*d*e^2 + 10*c*d^2*e)))/e - (e^2*(2*b^3*d - 8*a*b ^2*e - 8*a^2*c*e + 12*a*b*c*d))/(10*a*e^3 - 10*b*d*e^2 + 10*c*d^2*e)))/e)) /e + (e^2*(4*a^3*e - 2*a^2*b*d))/(10*a*e^3 - 10*b*d*e^2 + 10*c*d^2*e))*(a + b*x + c*x^2)^(1/2))/(d + e*x)^5 - (((d*((d*((d*((d*((d*((d*((d*((c^4*e^5 *(7*b^2*e + 4*a*c*e - 16*b*c*d))/(60*(4*a*e^3 - 4*b*d*e^2 + 4*c*d^2*e)*(a* e^2 + c*d^2 - b*d*e)^3) - (c^5*d*e^4*(b*e - 2*c*d))/(30*(4*a*e^3 - 4*b*d*e ^2 + 4*c*d^2*e)*(a*e^2 + c*d^2 - b*d*e)^3)))/e - (38*b^3*c^3*e^7 - 192*c^6 *d^3*e^4 + 288*b*c^5*d^2*e^5 - 200*b^2*c^4*d*e^6 + 168*a*b*c^4*e^7 - 224*a *c^5*d*e^6)/(240*e*(4*a*e^3 - 4*b*d*e^2 + 4*c*d^2*e)*(a*e^2 + c*d^2 - b*d* e)^3)))/e + (224*a^2*c^4*e^7 + 25*b^4*c^2*e^7 + 192*c^6*d^4*e^3 + 404*a*b^ 2*c^3*e^7 + 384*a*c^5*d^2*e^5 - 960*b*c^5*d^3*e^4 - 376*b^3*c^3*d*e^6 + 10 56*b^2*c^4*d^2*e^5 - 1056*a*b*c^4*d*e^6)/(240*e*(4*a*e^3 - 4*b*d*e^2 + 4*c *d^2*e)*(a*e^2 + c*d^2 - b*d*e)^3)))/e - (8*b^5*c*e^7 - 944*c^6*d^5*e^2 + 398*a*b^3*c^2*e^7 + 1232*a^2*b*c^3*e^7 - 2272*a*c^5*d^3*e^4 - 1344*a^2*c^4 *d*e^6 + 2840*b*c^5*d^4*e^3 - 353*b^4*c^2*d*e^6 - 3472*b^2*c^4*d^3*e^4 + 1 888*b^3*c^3*d^2*e^5 + 4368*a*b*c^4*d^2*e^5 - 2832*a*b^2*c^3*d*e^6)/(240...
Time = 56.32 (sec) , antiderivative size = 104, normalized size of antiderivative = 4.16 \[ \int \frac {(b d-2 a e+(2 c d-b e) x) \left (a+b x+c x^2\right )^{3/2}}{(d+e x)^6} \, dx=\frac {2 \sqrt {c \,x^{2}+b x +a}\, \left (c^{2} x^{4}+2 b c \,x^{3}+2 a c \,x^{2}+b^{2} x^{2}+2 a b x +a^{2}\right )}{5 e^{5} x^{5}+25 d \,e^{4} x^{4}+50 d^{2} e^{3} x^{3}+50 d^{3} e^{2} x^{2}+25 d^{4} e x +5 d^{5}} \] Input:
int((b*d-2*a*e+(-b*e+2*c*d)*x)*(c*x^2+b*x+a)^(3/2)/(e*x+d)^6,x)
Output:
(2*sqrt(a + b*x + c*x**2)*(a**2 + 2*a*b*x + 2*a*c*x**2 + b**2*x**2 + 2*b*c *x**3 + c**2*x**4))/(5*(d**5 + 5*d**4*e*x + 10*d**3*e**2*x**2 + 10*d**2*e* *3*x**3 + 5*d*e**4*x**4 + e**5*x**5))