\(\int \frac {(a+b x+c x^2)^3}{b d+2 c d x} \, dx\) [1142]

Optimal result

Integrand size = 24, antiderivative size = 100 \[ \int \frac {\left (a+b x+c x^2\right )^3}{b d+2 c d x} \, dx=\frac {3 \left (b^2-4 a c\right )^2 (b+2 c x)^2}{256 c^4 d}-\frac {3 \left (b^2-4 a c\right ) (b+2 c x)^4}{512 c^4 d}+\frac {(b+2 c x)^6}{768 c^4 d}-\frac {\left (b^2-4 a c\right )^3 \log (b+2 c x)}{128 c^4 d} \]

[Out]

3/256*(-4*a*c+b^2)^2*(2*c*x+b)^2/c^4/d-3/512*(-4*a*c+b^2)*(2*c*x+b)^4/c^4/d+1/768*(2*c*x+b)^6/c^4/d-1/128*(-4*
a*c+b^2)^3*ln(2*c*x+b)/c^4/d

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {697} \[ \int \frac {\left (a+b x+c x^2\right )^3}{b d+2 c d x} \, dx=-\frac {3 \left (b^2-4 a c\right ) (b+2 c x)^4}{512 c^4 d}+\frac {3 \left (b^2-4 a c\right )^2 (b+2 c x)^2}{256 c^4 d}-\frac {\left (b^2-4 a c\right )^3 \log (b+2 c x)}{128 c^4 d}+\frac {(b+2 c x)^6}{768 c^4 d} \]

[In]

Int[(a + b*x + c*x^2)^3/(b*d + 2*c*d*x),x]

[Out]

(3*(b^2 - 4*a*c)^2*(b + 2*c*x)^2)/(256*c^4*d) - (3*(b^2 - 4*a*c)*(b + 2*c*x)^4)/(512*c^4*d) + (b + 2*c*x)^6/(7
68*c^4*d) - ((b^2 - 4*a*c)^3*Log[b + 2*c*x])/(128*c^4*d)

Rule 697

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

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {\left (-b^2+4 a c\right )^3}{64 c^3 (b d+2 c d x)}+\frac {3 \left (-b^2+4 a c\right )^2 (b d+2 c d x)}{64 c^3 d^2}+\frac {3 \left (-b^2+4 a c\right ) (b d+2 c d x)^3}{64 c^3 d^4}+\frac {(b d+2 c d x)^5}{64 c^3 d^6}\right ) \, dx \\ & = \frac {3 \left (b^2-4 a c\right )^2 (b+2 c x)^2}{256 c^4 d}-\frac {3 \left (b^2-4 a c\right ) (b+2 c x)^4}{512 c^4 d}+\frac {(b+2 c x)^6}{768 c^4 d}-\frac {\left (b^2-4 a c\right )^3 \log (b+2 c x)}{128 c^4 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.11 \[ \int \frac {\left (a+b x+c x^2\right )^3}{b d+2 c d x} \, dx=\frac {2 c x (b+c x) \left (3 b^4-6 b^3 c x+8 b c^2 x \left (9 a+4 c x^2\right )+2 b^2 c \left (-18 a+5 c x^2\right )+8 c^2 \left (18 a^2+9 a c x^2+2 c^2 x^4\right )\right )-3 \left (b^2-4 a c\right )^3 \log (b+2 c x)}{384 c^4 d} \]

[In]

Integrate[(a + b*x + c*x^2)^3/(b*d + 2*c*d*x),x]

[Out]

(2*c*x*(b + c*x)*(3*b^4 - 6*b^3*c*x + 8*b*c^2*x*(9*a + 4*c*x^2) + 2*b^2*c*(-18*a + 5*c*x^2) + 8*c^2*(18*a^2 +
9*a*c*x^2 + 2*c^2*x^4)) - 3*(b^2 - 4*a*c)^3*Log[b + 2*c*x])/(384*c^4*d)

Maple [A] (verified)

Time = 2.52 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.69

method	result	size
norman	\(\frac {c^{2} x^{6}}{12 d}+\frac {\left (12 a c +7 b^{2}\right ) x^{4}}{32 d}+\frac {b c \,x^{5}}{4 d}+\frac {\left (48 a^{2} c^{2}+12 a \,b^{2} c -b^{4}\right ) x^{2}}{64 c^{2} d}+\frac {b \left (36 a c +b^{2}\right ) x^{3}}{48 c d}+\frac {b \left (48 a^{2} c^{2}-12 a \,b^{2} c +b^{4}\right ) x}{64 c^{3} d}+\frac {\left (64 c^{3} a^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right ) \ln \left (2 c x +b \right )}{128 c^{4} d}\)	\(169\)
parallelrisch	\(\frac {32 c^{6} x^{6}+96 b \,c^{5} x^{5}+144 a \,c^{5} x^{4}+84 b^{2} c^{4} x^{4}+288 a b \,c^{4} x^{3}+8 x^{3} b^{3} c^{3}+288 a^{2} c^{4} x^{2}+72 a \,b^{2} c^{3} x^{2}-6 x^{2} b^{4} c^{2}+192 \ln \left (\frac {b}{2}+c x \right ) a^{3} c^{3}-144 \ln \left (\frac {b}{2}+c x \right ) a^{2} b^{2} c^{2}+36 \ln \left (\frac {b}{2}+c x \right ) a \,b^{4} c -3 \ln \left (\frac {b}{2}+c x \right ) b^{6}+288 a^{2} b \,c^{3} x -72 x a \,b^{3} c^{2}+6 x \,b^{5} c}{384 c^{4} d}\)	\(192\)
default	\(\frac {\frac {\frac {16 c^{5} x^{6}}{3}+16 b \,c^{4} x^{5}+\frac {\left (32 b^{2} c^{3}+2 c \left (48 c^{3} a +12 b^{2} c^{2}\right )\right ) x^{4}}{4}+\frac {\left (b \left (48 c^{3} a +12 b^{2} c^{2}\right )+2 c \left (48 b \,c^{2} a -4 b^{3} c \right )\right ) x^{3}}{3}+\frac {\left (b \left (48 b \,c^{2} a -4 b^{3} c \right )+2 c \left (48 a^{2} c^{2}-12 a \,b^{2} c +b^{4}\right )\right ) x^{2}}{2}+b \left (48 a^{2} c^{2}-12 a \,b^{2} c +b^{4}\right ) x}{64 c^{3}}+\frac {\left (64 c^{3} a^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right ) \ln \left (2 c x +b \right )}{128 c^{4}}}{d}\)	\(211\)
risch	\(\frac {c^{2} x^{6}}{12 d}+\frac {b c \,x^{5}}{4 d}+\frac {3 c a \,x^{4}}{8 d}+\frac {7 b^{2} x^{4}}{32 d}+\frac {3 a b \,x^{3}}{4 d}+\frac {x^{3} b^{3}}{48 c d}+\frac {3 a^{2} x^{2}}{4 d}+\frac {3 a \,b^{2} x^{2}}{16 c d}-\frac {b^{4} x^{2}}{64 c^{2} d}+\frac {3 a^{2} b x}{4 c d}-\frac {3 a \,b^{3} x}{16 c^{2} d}+\frac {b^{5} x}{64 c^{3} d}+\frac {\ln \left (2 c x +b \right ) a^{3}}{2 c d}-\frac {3 \ln \left (2 c x +b \right ) a^{2} b^{2}}{8 c^{2} d}+\frac {3 \ln \left (2 c x +b \right ) a \,b^{4}}{32 c^{3} d}-\frac {\ln \left (2 c x +b \right ) b^{6}}{128 c^{4} d}\)	\(222\)

[In]

int((c*x^2+b*x+a)^3/(2*c*d*x+b*d),x,method=_RETURNVERBOSE)

[Out]

1/12*c^2/d*x^6+1/32*(12*a*c+7*b^2)/d*x^4+1/4*b*c/d*x^5+1/64*(48*a^2*c^2+12*a*b^2*c-b^4)/c^2/d*x^2+1/48*b*(36*a
*c+b^2)/c/d*x^3+1/64*b*(48*a^2*c^2-12*a*b^2*c+b^4)/c^3/d*x+1/128*(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)/c^
4/d*ln(2*c*x+b)

Fricas [A] (verification not implemented)

none

Time = 0.38 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.62 \[ \int \frac {\left (a+b x+c x^2\right )^3}{b d+2 c d x} \, dx=\frac {32 \, c^{6} x^{6} + 96 \, b c^{5} x^{5} + 12 \, {\left (7 \, b^{2} c^{4} + 12 \, a c^{5}\right )} x^{4} + 8 \, {\left (b^{3} c^{3} + 36 \, a b c^{4}\right )} x^{3} - 6 \, {\left (b^{4} c^{2} - 12 \, a b^{2} c^{3} - 48 \, a^{2} c^{4}\right )} x^{2} + 6 \, {\left (b^{5} c - 12 \, a b^{3} c^{2} + 48 \, a^{2} b c^{3}\right )} x - 3 \, {\left (b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} \log \left (2 \, c x + b\right )}{384 \, c^{4} d} \]

[In]

integrate((c*x^2+b*x+a)^3/(2*c*d*x+b*d),x, algorithm="fricas")

[Out]

1/384*(32*c^6*x^6 + 96*b*c^5*x^5 + 12*(7*b^2*c^4 + 12*a*c^5)*x^4 + 8*(b^3*c^3 + 36*a*b*c^4)*x^3 - 6*(b^4*c^2 -
12*a*b^2*c^3 - 48*a^2*c^4)*x^2 + 6*(b^5*c - 12*a*b^3*c^2 + 48*a^2*b*c^3)*x - 3*(b^6 - 12*a*b^4*c + 48*a^2*b^2
*c^2 - 64*a^3*c^3)*log(2*c*x + b))/(c^4*d)

Sympy [A] (verification not implemented)

Time = 0.30 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.56 \[ \int \frac {\left (a+b x+c x^2\right )^3}{b d+2 c d x} \, dx=\frac {b c x^{5}}{4 d} + \frac {c^{2} x^{6}}{12 d} + x^{4} \cdot \left (\frac {3 a c}{8 d} + \frac {7 b^{2}}{32 d}\right ) + x^{3} \cdot \left (\frac {3 a b}{4 d} + \frac {b^{3}}{48 c d}\right ) + x^{2} \cdot \left (\frac {3 a^{2}}{4 d} + \frac {3 a b^{2}}{16 c d} - \frac {b^{4}}{64 c^{2} d}\right ) + x \left (\frac {3 a^{2} b}{4 c d} - \frac {3 a b^{3}}{16 c^{2} d} + \frac {b^{5}}{64 c^{3} d}\right ) + \frac {\left (4 a c - b^{2}\right )^{3} \log {\left (b + 2 c x \right )}}{128 c^{4} d} \]

[In]

integrate((c*x**2+b*x+a)**3/(2*c*d*x+b*d),x)

[Out]

b*c*x**5/(4*d) + c**2*x**6/(12*d) + x**4*(3*a*c/(8*d) + 7*b**2/(32*d)) + x**3*(3*a*b/(4*d) + b**3/(48*c*d)) +
x**2*(3*a**2/(4*d) + 3*a*b**2/(16*c*d) - b**4/(64*c**2*d)) + x*(3*a**2*b/(4*c*d) - 3*a*b**3/(16*c**2*d) + b**5
/(64*c**3*d)) + (4*a*c - b**2)**3*log(b + 2*c*x)/(128*c**4*d)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.63 \[ \int \frac {\left (a+b x+c x^2\right )^3}{b d+2 c d x} \, dx=\frac {16 \, c^{5} x^{6} + 48 \, b c^{4} x^{5} + 6 \, {\left (7 \, b^{2} c^{3} + 12 \, a c^{4}\right )} x^{4} + 4 \, {\left (b^{3} c^{2} + 36 \, a b c^{3}\right )} x^{3} - 3 \, {\left (b^{4} c - 12 \, a b^{2} c^{2} - 48 \, a^{2} c^{3}\right )} x^{2} + 3 \, {\left (b^{5} - 12 \, a b^{3} c + 48 \, a^{2} b c^{2}\right )} x}{192 \, c^{3} d} - \frac {{\left (b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} \log \left (2 \, c x + b\right )}{128 \, c^{4} d} \]

[In]

integrate((c*x^2+b*x+a)^3/(2*c*d*x+b*d),x, algorithm="maxima")

[Out]

1/192*(16*c^5*x^6 + 48*b*c^4*x^5 + 6*(7*b^2*c^3 + 12*a*c^4)*x^4 + 4*(b^3*c^2 + 36*a*b*c^3)*x^3 - 3*(b^4*c - 12
*a*b^2*c^2 - 48*a^2*c^3)*x^2 + 3*(b^5 - 12*a*b^3*c + 48*a^2*b*c^2)*x)/(c^3*d) - 1/128*(b^6 - 12*a*b^4*c + 48*a
^2*b^2*c^2 - 64*a^3*c^3)*log(2*c*x + b)/(c^4*d)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 213 vs. \(2 (92) = 184\).

Time = 0.27 (sec) , antiderivative size = 213, normalized size of antiderivative = 2.13 \[ \int \frac {\left (a+b x+c x^2\right )^3}{b d+2 c d x} \, dx=-\frac {{\left (b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} \log \left ({\left | 2 \, c x + b \right |}\right )}{128 \, c^{4} d} + \frac {16 \, c^{8} d^{5} x^{6} + 48 \, b c^{7} d^{5} x^{5} + 42 \, b^{2} c^{6} d^{5} x^{4} + 72 \, a c^{7} d^{5} x^{4} + 4 \, b^{3} c^{5} d^{5} x^{3} + 144 \, a b c^{6} d^{5} x^{3} - 3 \, b^{4} c^{4} d^{5} x^{2} + 36 \, a b^{2} c^{5} d^{5} x^{2} + 144 \, a^{2} c^{6} d^{5} x^{2} + 3 \, b^{5} c^{3} d^{5} x - 36 \, a b^{3} c^{4} d^{5} x + 144 \, a^{2} b c^{5} d^{5} x}{192 \, c^{6} d^{6}} \]

[In]

integrate((c*x^2+b*x+a)^3/(2*c*d*x+b*d),x, algorithm="giac")

[Out]

-1/128*(b^6 - 12*a*b^4*c + 48*a^2*b^2*c^2 - 64*a^3*c^3)*log(abs(2*c*x + b))/(c^4*d) + 1/192*(16*c^8*d^5*x^6 +
48*b*c^7*d^5*x^5 + 42*b^2*c^6*d^5*x^4 + 72*a*c^7*d^5*x^4 + 4*b^3*c^5*d^5*x^3 + 144*a*b*c^6*d^5*x^3 - 3*b^4*c^4
*d^5*x^2 + 36*a*b^2*c^5*d^5*x^2 + 144*a^2*c^6*d^5*x^2 + 3*b^5*c^3*d^5*x - 36*a*b^3*c^4*d^5*x + 144*a^2*b*c^5*d
^5*x)/(c^6*d^6)

Mupad [B] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 304, normalized size of antiderivative = 3.04 \[ \int \frac {\left (a+b x+c x^2\right )^3}{b d+2 c d x} \, dx=x^2\,\left (\frac {b\,\left (\frac {b\,\left (\frac {3\,\left (b^2+a\,c\right )}{2\,d}-\frac {5\,b^2}{8\,d}\right )}{2\,c}-\frac {b^3+6\,a\,c\,b}{2\,c\,d}\right )}{4\,c}+\frac {3\,a\,\left (b^2+a\,c\right )}{4\,c\,d}\right )-x^3\,\left (\frac {b\,\left (\frac {3\,\left (b^2+a\,c\right )}{2\,d}-\frac {5\,b^2}{8\,d}\right )}{6\,c}-\frac {b^3+6\,a\,c\,b}{6\,c\,d}\right )-x\,\left (\frac {b\,\left (\frac {b\,\left (\frac {b\,\left (\frac {3\,\left (b^2+a\,c\right )}{2\,d}-\frac {5\,b^2}{8\,d}\right )}{2\,c}-\frac {b^3+6\,a\,c\,b}{2\,c\,d}\right )}{2\,c}+\frac {3\,a\,\left (b^2+a\,c\right )}{2\,c\,d}\right )}{2\,c}-\frac {3\,a^2\,b}{2\,c\,d}\right )+x^4\,\left (\frac {3\,\left (b^2+a\,c\right )}{8\,d}-\frac {5\,b^2}{32\,d}\right )+\frac {c^2\,x^6}{12\,d}+\frac {b\,c\,x^5}{4\,d}-\frac {\ln \left (b+2\,c\,x\right )\,\left (-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6\right )}{128\,c^4\,d} \]

[In]

int((a + b*x + c*x^2)^3/(b*d + 2*c*d*x),x)

[Out]

x^2*((b*((b*((3*(a*c + b^2))/(2*d) - (5*b^2)/(8*d)))/(2*c) - (b^3 + 6*a*b*c)/(2*c*d)))/(4*c) + (3*a*(a*c + b^2
))/(4*c*d)) - x^3*((b*((3*(a*c + b^2))/(2*d) - (5*b^2)/(8*d)))/(6*c) - (b^3 + 6*a*b*c)/(6*c*d)) - x*((b*((b*((
b*((3*(a*c + b^2))/(2*d) - (5*b^2)/(8*d)))/(2*c) - (b^3 + 6*a*b*c)/(2*c*d)))/(2*c) + (3*a*(a*c + b^2))/(2*c*d)
))/(2*c) - (3*a^2*b)/(2*c*d)) + x^4*((3*(a*c + b^2))/(8*d) - (5*b^2)/(32*d)) + (c^2*x^6)/(12*d) + (b*c*x^5)/(4
*d) - (log(b + 2*c*x)*(b^6 - 64*a^3*c^3 + 48*a^2*b^2*c^2 - 12*a*b^4*c))/(128*c^4*d)