\(\int \frac {(a+b x+c x^2)^2}{(b d+2 c d x)^7} \, dx\) [1132]

Optimal result

Integrand size = 24, antiderivative size = 37 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(b d+2 c d x)^7} \, dx=\frac {\left (a+b x+c x^2\right )^3}{3 \left (b^2-4 a c\right ) d^7 (b+2 c x)^6} \]

[Out]

1/3*(c*x^2+b*x+a)^3/(-4*a*c+b^2)/d^7/(2*c*x+b)^6

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {696} \[ \int \frac {\left (a+b x+c x^2\right )^2}{(b d+2 c d x)^7} \, dx=\frac {\left (a+b x+c x^2\right )^3}{3 d^7 \left (b^2-4 a c\right ) (b+2 c x)^6} \]

[In]

Int[(a + b*x + c*x^2)^2/(b*d + 2*c*d*x)^7,x]

[Out]

(a + b*x + c*x^2)^3/(3*(b^2 - 4*a*c)*d^7*(b + 2*c*x)^6)

Rule 696

Int[((d_) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[2*c*(d + e*x)^(m + 1
)*((a + b*x + c*x^2)^(p + 1)/(e*(p + 1)*(b^2 - 4*a*c))), x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b^2 - 4*
a*c, 0] && EqQ[2*c*d - b*e, 0] && EqQ[m + 2*p + 3, 0] && NeQ[p, -1]

Rubi steps \begin{align*} \text {integral}& = \frac {\left (a+b x+c x^2\right )^3}{3 \left (b^2-4 a c\right ) d^7 (b+2 c x)^6} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.76 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(b d+2 c d x)^7} \, dx=-\frac {b^4-8 a b^2 c+16 a^2 c^2-3 \left (b^2-4 a c\right ) (b+2 c x)^2+3 (b+2 c x)^4}{192 c^3 d^7 (b+2 c x)^6} \]

[In]

Integrate[(a + b*x + c*x^2)^2/(b*d + 2*c*d*x)^7,x]

[Out]

-1/192*(b^4 - 8*a*b^2*c + 16*a^2*c^2 - 3*(b^2 - 4*a*c)*(b + 2*c*x)^2 + 3*(b + 2*c*x)^4)/(c^3*d^7*(b + 2*c*x)^6
)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(73\) vs. \(2(35)=70\).

Time = 2.93 (sec) , antiderivative size = 74, normalized size of antiderivative = 2.00

method	result	size
default	\(\frac {-\frac {4 a c -b^{2}}{64 c^{3} \left (2 c x +b \right )^{4}}-\frac {16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}}{192 c^{3} \left (2 c x +b \right )^{6}}-\frac {1}{64 c^{3} \left (2 c x +b \right )^{2}}}{d^{7}}\)	\(74\)
risch	\(\frac {-\frac {x^{4} c}{4}-\frac {b \,x^{3}}{2}-\frac {\left (4 a c +5 b^{2}\right ) x^{2}}{16 c}-\frac {b \left (4 a c +b^{2}\right ) x}{16 c^{2}}-\frac {16 a^{2} c^{2}+4 a \,b^{2} c +b^{4}}{192 c^{3}}}{d^{7} \left (2 c x +b \right )^{6}}\)	\(83\)
gosper	\(-\frac {48 c^{4} x^{4}+96 b \,c^{3} x^{3}+48 x^{2} a \,c^{3}+60 b^{2} c^{2} x^{2}+48 a b \,c^{2} x +12 b^{3} c x +16 a^{2} c^{2}+4 a \,b^{2} c +b^{4}}{192 \left (2 c x +b \right )^{6} d^{7} c^{3}}\)	\(88\)
norman	\(\frac {\frac {a^{2} x}{b d}+\frac {\left (5 a^{2} c +a \,b^{2}\right ) x^{2}}{b^{2} d}+\frac {c^{2} \left (16 a^{2} c^{2}+4 a \,b^{2} c +b^{4}\right ) x^{5}}{b^{5} d}+\frac {c \left (20 a^{2} c^{2}+5 a \,b^{2} c +b^{4}\right ) x^{4}}{b^{4} d}+\frac {\left (40 a^{2} c^{2}+10 a \,b^{2} c +b^{4}\right ) x^{3}}{3 d \,b^{3}}+\frac {c^{3} \left (16 a^{2} c^{2}+4 a \,b^{2} c +b^{4}\right ) x^{6}}{3 b^{6} d}}{d^{6} \left (2 c x +b \right )^{6}}\)	\(172\)
parallelrisch	\(\frac {16 x^{6} a^{2} c^{5}+4 x^{6} a \,b^{2} c^{4}+x^{6} b^{4} c^{3}+48 a^{2} b \,c^{4} x^{5}+12 x^{5} a \,b^{3} c^{3}+3 x^{5} b^{5} c^{2}+60 a^{2} b^{2} c^{3} x^{4}+15 c^{2} x^{4} a \,b^{4}+3 x^{4} b^{6} c +40 x^{3} a^{2} b^{3} c^{2}+10 x^{3} a \,b^{5} c +x^{3} b^{7}+15 x^{2} a^{2} b^{4} c +3 a \,b^{6} x^{2}+3 a^{2} x \,b^{5}}{3 b^{6} d^{7} \left (2 c x +b \right )^{6}}\)	\(182\)

[In]

int((c*x^2+b*x+a)^2/(2*c*d*x+b*d)^7,x,method=_RETURNVERBOSE)

[Out]

1/d^7*(-1/64*(4*a*c-b^2)/c^3/(2*c*x+b)^4-1/192*(16*a^2*c^2-8*a*b^2*c+b^4)/c^3/(2*c*x+b)^6-1/64/c^3/(2*c*x+b)^2
)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 164 vs. \(2 (35) = 70\).

Time = 0.55 (sec) , antiderivative size = 164, normalized size of antiderivative = 4.43 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(b d+2 c d x)^7} \, dx=-\frac {48 \, c^{4} x^{4} + 96 \, b c^{3} x^{3} + b^{4} + 4 \, a b^{2} c + 16 \, a^{2} c^{2} + 12 \, {\left (5 \, b^{2} c^{2} + 4 \, a c^{3}\right )} x^{2} + 12 \, {\left (b^{3} c + 4 \, a b c^{2}\right )} x}{192 \, {\left (64 \, c^{9} d^{7} x^{6} + 192 \, b c^{8} d^{7} x^{5} + 240 \, b^{2} c^{7} d^{7} x^{4} + 160 \, b^{3} c^{6} d^{7} x^{3} + 60 \, b^{4} c^{5} d^{7} x^{2} + 12 \, b^{5} c^{4} d^{7} x + b^{6} c^{3} d^{7}\right )}} \]

[In]

integrate((c*x^2+b*x+a)^2/(2*c*d*x+b*d)^7,x, algorithm="fricas")

[Out]

-1/192*(48*c^4*x^4 + 96*b*c^3*x^3 + b^4 + 4*a*b^2*c + 16*a^2*c^2 + 12*(5*b^2*c^2 + 4*a*c^3)*x^2 + 12*(b^3*c +
4*a*b*c^2)*x)/(64*c^9*d^7*x^6 + 192*b*c^8*d^7*x^5 + 240*b^2*c^7*d^7*x^4 + 160*b^3*c^6*d^7*x^3 + 60*b^4*c^5*d^7
*x^2 + 12*b^5*c^4*d^7*x + b^6*c^3*d^7)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 173 vs. \(2 (32) = 64\).

Time = 1.28 (sec) , antiderivative size = 173, normalized size of antiderivative = 4.68 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(b d+2 c d x)^7} \, dx=\frac {- 16 a^{2} c^{2} - 4 a b^{2} c - b^{4} - 96 b c^{3} x^{3} - 48 c^{4} x^{4} + x^{2} \left (- 48 a c^{3} - 60 b^{2} c^{2}\right ) + x \left (- 48 a b c^{2} - 12 b^{3} c\right )}{192 b^{6} c^{3} d^{7} + 2304 b^{5} c^{4} d^{7} x + 11520 b^{4} c^{5} d^{7} x^{2} + 30720 b^{3} c^{6} d^{7} x^{3} + 46080 b^{2} c^{7} d^{7} x^{4} + 36864 b c^{8} d^{7} x^{5} + 12288 c^{9} d^{7} x^{6}} \]

[In]

integrate((c*x**2+b*x+a)**2/(2*c*d*x+b*d)**7,x)

[Out]

(-16*a**2*c**2 - 4*a*b**2*c - b**4 - 96*b*c**3*x**3 - 48*c**4*x**4 + x**2*(-48*a*c**3 - 60*b**2*c**2) + x*(-48
*a*b*c**2 - 12*b**3*c))/(192*b**6*c**3*d**7 + 2304*b**5*c**4*d**7*x + 11520*b**4*c**5*d**7*x**2 + 30720*b**3*c
**6*d**7*x**3 + 46080*b**2*c**7*d**7*x**4 + 36864*b*c**8*d**7*x**5 + 12288*c**9*d**7*x**6)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 164 vs. \(2 (35) = 70\).

Time = 0.21 (sec) , antiderivative size = 164, normalized size of antiderivative = 4.43 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(b d+2 c d x)^7} \, dx=-\frac {48 \, c^{4} x^{4} + 96 \, b c^{3} x^{3} + b^{4} + 4 \, a b^{2} c + 16 \, a^{2} c^{2} + 12 \, {\left (5 \, b^{2} c^{2} + 4 \, a c^{3}\right )} x^{2} + 12 \, {\left (b^{3} c + 4 \, a b c^{2}\right )} x}{192 \, {\left (64 \, c^{9} d^{7} x^{6} + 192 \, b c^{8} d^{7} x^{5} + 240 \, b^{2} c^{7} d^{7} x^{4} + 160 \, b^{3} c^{6} d^{7} x^{3} + 60 \, b^{4} c^{5} d^{7} x^{2} + 12 \, b^{5} c^{4} d^{7} x + b^{6} c^{3} d^{7}\right )}} \]

[In]

integrate((c*x^2+b*x+a)^2/(2*c*d*x+b*d)^7,x, algorithm="maxima")

[Out]

-1/192*(48*c^4*x^4 + 96*b*c^3*x^3 + b^4 + 4*a*b^2*c + 16*a^2*c^2 + 12*(5*b^2*c^2 + 4*a*c^3)*x^2 + 12*(b^3*c +
4*a*b*c^2)*x)/(64*c^9*d^7*x^6 + 192*b*c^8*d^7*x^5 + 240*b^2*c^7*d^7*x^4 + 160*b^3*c^6*d^7*x^3 + 60*b^4*c^5*d^7
*x^2 + 12*b^5*c^4*d^7*x + b^6*c^3*d^7)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 87 vs. \(2 (35) = 70\).

Time = 0.27 (sec) , antiderivative size = 87, normalized size of antiderivative = 2.35 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(b d+2 c d x)^7} \, dx=-\frac {48 \, c^{4} x^{4} + 96 \, b c^{3} x^{3} + 60 \, b^{2} c^{2} x^{2} + 48 \, a c^{3} x^{2} + 12 \, b^{3} c x + 48 \, a b c^{2} x + b^{4} + 4 \, a b^{2} c + 16 \, a^{2} c^{2}}{192 \, {\left (2 \, c x + b\right )}^{6} c^{3} d^{7}} \]

[In]

integrate((c*x^2+b*x+a)^2/(2*c*d*x+b*d)^7,x, algorithm="giac")

[Out]

-1/192*(48*c^4*x^4 + 96*b*c^3*x^3 + 60*b^2*c^2*x^2 + 48*a*c^3*x^2 + 12*b^3*c*x + 48*a*b*c^2*x + b^4 + 4*a*b^2*
c + 16*a^2*c^2)/((2*c*x + b)^6*c^3*d^7)

Mupad [B] (verification not implemented)

Time = 9.88 (sec) , antiderivative size = 157, normalized size of antiderivative = 4.24 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(b d+2 c d x)^7} \, dx=-\frac {\frac {16\,a^2\,c^2+4\,a\,b^2\,c+b^4}{192\,c^3}+\frac {b\,x^3}{2}+\frac {c\,x^4}{4}+\frac {x^2\,\left (5\,b^2+4\,a\,c\right )}{16\,c}+\frac {b\,x\,\left (b^2+4\,a\,c\right )}{16\,c^2}}{b^6\,d^7+12\,b^5\,c\,d^7\,x+60\,b^4\,c^2\,d^7\,x^2+160\,b^3\,c^3\,d^7\,x^3+240\,b^2\,c^4\,d^7\,x^4+192\,b\,c^5\,d^7\,x^5+64\,c^6\,d^7\,x^6} \]

[In]

int((a + b*x + c*x^2)^2/(b*d + 2*c*d*x)^7,x)

[Out]

-((b^4 + 16*a^2*c^2 + 4*a*b^2*c)/(192*c^3) + (b*x^3)/2 + (c*x^4)/4 + (x^2*(4*a*c + 5*b^2))/(16*c) + (b*x*(4*a*
c + b^2))/(16*c^2))/(b^6*d^7 + 64*c^6*d^7*x^6 + 192*b*c^5*d^7*x^5 + 60*b^4*c^2*d^7*x^2 + 160*b^3*c^3*d^7*x^3 +
240*b^2*c^4*d^7*x^4 + 12*b^5*c*d^7*x)