3.12.56 \(\int \frac {(b d+2 c d x)^6}{a+b x+c x^2} \, dx\) [1156]

Optimal. Leaf size=97 \[ 2 \left (b^2-4 a c\right )^2 d^6 (b+2 c x)+\frac {2}{3} \left (b^2-4 a c\right ) d^6 (b+2 c x)^3+\frac {2}{5} d^6 (b+2 c x)^5-2 \left (b^2-4 a c\right )^{5/2} d^6 \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right ) \]

[Out]

2*(-4*a*c+b^2)^2*d^6*(2*c*x+b)+2/3*(-4*a*c+b^2)*d^6*(2*c*x+b)^3+2/5*d^6*(2*c*x+b)^5-2*(-4*a*c+b^2)^(5/2)*d^6*a
rctanh((2*c*x+b)/(-4*a*c+b^2)^(1/2))

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Rubi [A]
time = 0.05, antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {706, 632, 212} \begin {gather*} \frac {2}{3} d^6 \left (b^2-4 a c\right ) (b+2 c x)^3+2 d^6 \left (b^2-4 a c\right )^2 (b+2 c x)-2 d^6 \left (b^2-4 a c\right )^{5/2} \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )+\frac {2}{5} d^6 (b+2 c x)^5 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(b*d + 2*c*d*x)^6/(a + b*x + c*x^2),x]

[Out]

2*(b^2 - 4*a*c)^2*d^6*(b + 2*c*x) + (2*(b^2 - 4*a*c)*d^6*(b + 2*c*x)^3)/3 + (2*d^6*(b + 2*c*x)^5)/5 - 2*(b^2 -
4*a*c)^(5/2)*d^6*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]]

Rule 212

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] && (GtQ[a, 0] || LtQ[b, 0])

Rule 632

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 706

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

Rubi steps

\begin {align*} \int \frac {(b d+2 c d x)^6}{a+b x+c x^2} \, dx &=\frac {2}{5} d^6 (b+2 c x)^5+\left (\left (b^2-4 a c\right ) d^2\right ) \int \frac {(b d+2 c d x)^4}{a+b x+c x^2} \, dx\\ &=\frac {2}{3} \left (b^2-4 a c\right ) d^6 (b+2 c x)^3+\frac {2}{5} d^6 (b+2 c x)^5+\left (\left (b^2-4 a c\right )^2 d^4\right ) \int \frac {(b d+2 c d x)^2}{a+b x+c x^2} \, dx\\ &=2 \left (b^2-4 a c\right )^2 d^6 (b+2 c x)+\frac {2}{3} \left (b^2-4 a c\right ) d^6 (b+2 c x)^3+\frac {2}{5} d^6 (b+2 c x)^5+\left (\left (b^2-4 a c\right )^3 d^6\right ) \int \frac {1}{a+b x+c x^2} \, dx\\ &=2 \left (b^2-4 a c\right )^2 d^6 (b+2 c x)+\frac {2}{3} \left (b^2-4 a c\right ) d^6 (b+2 c x)^3+\frac {2}{5} d^6 (b+2 c x)^5-\left (2 \left (b^2-4 a c\right )^3 d^6\right ) \text {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c x\right )\\ &=2 \left (b^2-4 a c\right )^2 d^6 (b+2 c x)+\frac {2}{3} \left (b^2-4 a c\right ) d^6 (b+2 c x)^3+\frac {2}{5} d^6 (b+2 c x)^5-2 \left (b^2-4 a c\right )^{5/2} d^6 \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )\\ \end {align*}

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Mathematica [A]
time = 0.05, size = 120, normalized size = 1.24 \begin {gather*} d^6 \left (\frac {4}{15} c x \left (45 b^4+90 b^3 c x+120 b c^2 x \left (-a+c x^2\right )+20 b^2 c \left (-9 a+7 c x^2\right )+16 c^2 \left (15 a^2-5 a c x^2+3 c^2 x^4\right )\right )-2 \left (-b^2+4 a c\right )^{5/2} \tan ^{-1}\left (\frac {b+2 c x}{\sqrt {-b^2+4 a c}}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(b*d + 2*c*d*x)^6/(a + b*x + c*x^2),x]

[Out]

d^6*((4*c*x*(45*b^4 + 90*b^3*c*x + 120*b*c^2*x*(-a + c*x^2) + 20*b^2*c*(-9*a + 7*c*x^2) + 16*c^2*(15*a^2 - 5*a
*c*x^2 + 3*c^2*x^4)))/15 - 2*(-b^2 + 4*a*c)^(5/2)*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])

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Maple [A]
time = 0.72, size = 154, normalized size = 1.59

method	result	size
default	\(d^{6} \left (\frac {64 c^{5} x^{5}}{5}+32 b \,c^{4} x^{4}-\frac {64 a \,c^{4} x^{3}}{3}+\frac {112 b^{2} c^{3} x^{3}}{3}-32 a b \,c^{3} x^{2}+24 b^{3} c^{2} x^{2}+64 a^{2} c^{3} x -48 b^{2} c^{2} a x +12 c \,b^{4} x +\frac {2 \left (-64 a^{3} c^{3}+48 a^{2} b^{2} c^{2}-12 a \,b^{4} c +b^{6}\right ) \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}\right )\)	\(154\)
risch	\(\frac {64 c^{5} d^{6} x^{5}}{5}+32 c^{4} d^{6} x^{4} b -\frac {64 c^{4} d^{6} a \,x^{3}}{3}+\frac {112 c^{3} d^{6} b^{2} x^{3}}{3}-32 c^{3} d^{6} a b \,x^{2}+24 c^{2} d^{6} b^{3} x^{2}+64 c^{3} d^{6} a^{2} x -48 c^{2} d^{6} x a \,b^{2}+12 c \,d^{6} b^{4} x +\left (-4 a c +b^{2}\right )^{\frac {5}{2}} d^{6} \ln \left (-2 \left (-4 a c +b^{2}\right )^{\frac {5}{2}} c x -\left (-4 a c +b^{2}\right )^{\frac {5}{2}} b -64 a^{3} c^{3}+48 a^{2} b^{2} c^{2}-12 a \,b^{4} c +b^{6}\right )-\left (-4 a c +b^{2}\right )^{\frac {5}{2}} d^{6} \ln \left (2 \left (-4 a c +b^{2}\right )^{\frac {5}{2}} c x +\left (-4 a c +b^{2}\right )^{\frac {5}{2}} b -64 a^{3} c^{3}+48 a^{2} b^{2} c^{2}-12 a \,b^{4} c +b^{6}\right )\)	\(257\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*c*d*x+b*d)^6/(c*x^2+b*x+a),x,method=_RETURNVERBOSE)

[Out]

d^6*(64/5*c^5*x^5+32*b*c^4*x^4-64/3*a*c^4*x^3+112/3*b^2*c^3*x^3-32*a*b*c^3*x^2+24*b^3*c^2*x^2+64*a^2*c^3*x-48*
b^2*c^2*a*x+12*c*b^4*x+2*(-64*a^3*c^3+48*a^2*b^2*c^2-12*a*b^4*c+b^6)/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c
-b^2)^(1/2)))

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*d*x+b*d)^6/(c*x^2+b*x+a),x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*a*c-b^2>0)', see `assume?` f
or more deta

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Fricas [A]
time = 1.92, size = 356, normalized size = 3.67 \begin {gather*} \left [\frac {64}{5} \, c^{5} d^{6} x^{5} + 32 \, b c^{4} d^{6} x^{4} + \frac {16}{3} \, {\left (7 \, b^{2} c^{3} - 4 \, a c^{4}\right )} d^{6} x^{3} + 8 \, {\left (3 \, b^{3} c^{2} - 4 \, a b c^{3}\right )} d^{6} x^{2} + {\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} \sqrt {b^{2} - 4 \, a c} d^{6} \log \left (\frac {2 \, c^{2} x^{2} + 2 \, b c x + b^{2} - 2 \, a c - \sqrt {b^{2} - 4 \, a c} {\left (2 \, c x + b\right )}}{c x^{2} + b x + a}\right ) + 4 \, {\left (3 \, b^{4} c - 12 \, a b^{2} c^{2} + 16 \, a^{2} c^{3}\right )} d^{6} x, \frac {64}{5} \, c^{5} d^{6} x^{5} + 32 \, b c^{4} d^{6} x^{4} + \frac {16}{3} \, {\left (7 \, b^{2} c^{3} - 4 \, a c^{4}\right )} d^{6} x^{3} + 8 \, {\left (3 \, b^{3} c^{2} - 4 \, a b c^{3}\right )} d^{6} x^{2} - 2 \, {\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} \sqrt {-b^{2} + 4 \, a c} d^{6} \arctan \left (-\frac {\sqrt {-b^{2} + 4 \, a c} {\left (2 \, c x + b\right )}}{b^{2} - 4 \, a c}\right ) + 4 \, {\left (3 \, b^{4} c - 12 \, a b^{2} c^{2} + 16 \, a^{2} c^{3}\right )} d^{6} x\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*d*x+b*d)^6/(c*x^2+b*x+a),x, algorithm="fricas")

[Out]

[64/5*c^5*d^6*x^5 + 32*b*c^4*d^6*x^4 + 16/3*(7*b^2*c^3 - 4*a*c^4)*d^6*x^3 + 8*(3*b^3*c^2 - 4*a*b*c^3)*d^6*x^2
+ (b^4 - 8*a*b^2*c + 16*a^2*c^2)*sqrt(b^2 - 4*a*c)*d^6*log((2*c^2*x^2 + 2*b*c*x + b^2 - 2*a*c - sqrt(b^2 - 4*a
*c)*(2*c*x + b))/(c*x^2 + b*x + a)) + 4*(3*b^4*c - 12*a*b^2*c^2 + 16*a^2*c^3)*d^6*x, 64/5*c^5*d^6*x^5 + 32*b*c
^4*d^6*x^4 + 16/3*(7*b^2*c^3 - 4*a*c^4)*d^6*x^3 + 8*(3*b^3*c^2 - 4*a*b*c^3)*d^6*x^2 - 2*(b^4 - 8*a*b^2*c + 16*
a^2*c^2)*sqrt(-b^2 + 4*a*c)*d^6*arctan(-sqrt(-b^2 + 4*a*c)*(2*c*x + b)/(b^2 - 4*a*c)) + 4*(3*b^4*c - 12*a*b^2*
c^2 + 16*a^2*c^3)*d^6*x]

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 337 vs. \(2 (99) = 198\).
time = 0.41, size = 337, normalized size = 3.47 \begin {gather*} 32 b c^{4} d^{6} x^{4} + \frac {64 c^{5} d^{6} x^{5}}{5} + d^{6} \sqrt {- \left (4 a c - b^{2}\right )^{5}} \log {\left (x + \frac {16 a^{2} b c^{2} d^{6} - 8 a b^{3} c d^{6} + b^{5} d^{6} - d^{6} \sqrt {- \left (4 a c - b^{2}\right )^{5}}}{32 a^{2} c^{3} d^{6} - 16 a b^{2} c^{2} d^{6} + 2 b^{4} c d^{6}} \right )} - d^{6} \sqrt {- \left (4 a c - b^{2}\right )^{5}} \log {\left (x + \frac {16 a^{2} b c^{2} d^{6} - 8 a b^{3} c d^{6} + b^{5} d^{6} + d^{6} \sqrt {- \left (4 a c - b^{2}\right )^{5}}}{32 a^{2} c^{3} d^{6} - 16 a b^{2} c^{2} d^{6} + 2 b^{4} c d^{6}} \right )} + x^{3} \left (- \frac {64 a c^{4} d^{6}}{3} + \frac {112 b^{2} c^{3} d^{6}}{3}\right ) + x^{2} \left (- 32 a b c^{3} d^{6} + 24 b^{3} c^{2} d^{6}\right ) + x \left (64 a^{2} c^{3} d^{6} - 48 a b^{2} c^{2} d^{6} + 12 b^{4} c d^{6}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*d*x+b*d)**6/(c*x**2+b*x+a),x)

[Out]

32*b*c**4*d**6*x**4 + 64*c**5*d**6*x**5/5 + d**6*sqrt(-(4*a*c - b**2)**5)*log(x + (16*a**2*b*c**2*d**6 - 8*a*b
**3*c*d**6 + b**5*d**6 - d**6*sqrt(-(4*a*c - b**2)**5))/(32*a**2*c**3*d**6 - 16*a*b**2*c**2*d**6 + 2*b**4*c*d*
*6)) - d**6*sqrt(-(4*a*c - b**2)**5)*log(x + (16*a**2*b*c**2*d**6 - 8*a*b**3*c*d**6 + b**5*d**6 + d**6*sqrt(-(
4*a*c - b**2)**5))/(32*a**2*c**3*d**6 - 16*a*b**2*c**2*d**6 + 2*b**4*c*d**6)) + x**3*(-64*a*c**4*d**6/3 + 112*
b**2*c**3*d**6/3) + x**2*(-32*a*b*c**3*d**6 + 24*b**3*c**2*d**6) + x*(64*a**2*c**3*d**6 - 48*a*b**2*c**2*d**6
+ 12*b**4*c*d**6)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 197 vs. \(2 (89) = 178\).
time = 1.06, size = 197, normalized size = 2.03 \begin {gather*} \frac {2 \, {\left (b^{6} d^{6} - 12 \, a b^{4} c d^{6} + 48 \, a^{2} b^{2} c^{2} d^{6} - 64 \, a^{3} c^{3} d^{6}\right )} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{\sqrt {-b^{2} + 4 \, a c}} + \frac {4 \, {\left (48 \, c^{10} d^{6} x^{5} + 120 \, b c^{9} d^{6} x^{4} + 140 \, b^{2} c^{8} d^{6} x^{3} - 80 \, a c^{9} d^{6} x^{3} + 90 \, b^{3} c^{7} d^{6} x^{2} - 120 \, a b c^{8} d^{6} x^{2} + 45 \, b^{4} c^{6} d^{6} x - 180 \, a b^{2} c^{7} d^{6} x + 240 \, a^{2} c^{8} d^{6} x\right )}}{15 \, c^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*d*x+b*d)^6/(c*x^2+b*x+a),x, algorithm="giac")

[Out]

2*(b^6*d^6 - 12*a*b^4*c*d^6 + 48*a^2*b^2*c^2*d^6 - 64*a^3*c^3*d^6)*arctan((2*c*x + b)/sqrt(-b^2 + 4*a*c))/sqrt
(-b^2 + 4*a*c) + 4/15*(48*c^10*d^6*x^5 + 120*b*c^9*d^6*x^4 + 140*b^2*c^8*d^6*x^3 - 80*a*c^9*d^6*x^3 + 90*b^3*c
^7*d^6*x^2 - 120*a*b*c^8*d^6*x^2 + 45*b^4*c^6*d^6*x - 180*a*b^2*c^7*d^6*x + 240*a^2*c^8*d^6*x)/c^5

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Mupad [B]
time = 0.44, size = 296, normalized size = 3.05 \begin {gather*} x\,\left (60\,b^4\,c\,d^6-\frac {b\,\left (160\,b^3\,c^2\,d^6+\frac {b\,\left (64\,a\,c^4\,d^6-112\,b^2\,c^3\,d^6\right )}{c}-128\,a\,b\,c^3\,d^6\right )}{c}+\frac {a\,\left (64\,a\,c^4\,d^6-112\,b^2\,c^3\,d^6\right )}{c}\right )-x^3\,\left (\frac {64\,a\,c^4\,d^6}{3}-\frac {112\,b^2\,c^3\,d^6}{3}\right )+x^2\,\left (80\,b^3\,c^2\,d^6+\frac {b\,\left (64\,a\,c^4\,d^6-112\,b^2\,c^3\,d^6\right )}{2\,c}-64\,a\,b\,c^3\,d^6\right )+2\,d^6\,\mathrm {atan}\left (\frac {b\,d^6\,{\left (4\,a\,c-b^2\right )}^{5/2}+2\,c\,d^6\,x\,{\left (4\,a\,c-b^2\right )}^{5/2}}{-64\,a^3\,c^3\,d^6+48\,a^2\,b^2\,c^2\,d^6-12\,a\,b^4\,c\,d^6+b^6\,d^6}\right )\,{\left (4\,a\,c-b^2\right )}^{5/2}+\frac {64\,c^5\,d^6\,x^5}{5}+32\,b\,c^4\,d^6\,x^4 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*d + 2*c*d*x)^6/(a + b*x + c*x^2),x)

[Out]

x*(60*b^4*c*d^6 - (b*(160*b^3*c^2*d^6 + (b*(64*a*c^4*d^6 - 112*b^2*c^3*d^6))/c - 128*a*b*c^3*d^6))/c + (a*(64*
a*c^4*d^6 - 112*b^2*c^3*d^6))/c) - x^3*((64*a*c^4*d^6)/3 - (112*b^2*c^3*d^6)/3) + x^2*(80*b^3*c^2*d^6 + (b*(64
*a*c^4*d^6 - 112*b^2*c^3*d^6))/(2*c) - 64*a*b*c^3*d^6) + 2*d^6*atan((b*d^6*(4*a*c - b^2)^(5/2) + 2*c*d^6*x*(4*
a*c - b^2)^(5/2))/(b^6*d^6 - 64*a^3*c^3*d^6 + 48*a^2*b^2*c^2*d^6 - 12*a*b^4*c*d^6))*(4*a*c - b^2)^(5/2) + (64*
c^5*d^6*x^5)/5 + 32*b*c^4*d^6*x^4

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