3.13.85 \(\int \frac {(b d+2 c d x)^{11/2}}{a+b x+c x^2} \, dx\) [1285]

Optimal. Leaf size=175 \[ 4 \left (b^2-4 a c\right )^2 d^5 \sqrt {b d+2 c d x}+\frac {4}{5} \left (b^2-4 a c\right ) d^3 (b d+2 c d x)^{5/2}+\frac {4}{9} d (b d+2 c d x)^{9/2}-2 \left (b^2-4 a c\right )^{9/4} d^{11/2} \tan ^{-1}\left (\frac {\sqrt {d (b+2 c x)}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )-2 \left (b^2-4 a c\right )^{9/4} d^{11/2} \tanh ^{-1}\left (\frac {\sqrt {d (b+2 c x)}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right ) \]

[Out]

4/5*(-4*a*c+b^2)*d^3*(2*c*d*x+b*d)^(5/2)+4/9*d*(2*c*d*x+b*d)^(9/2)-2*(-4*a*c+b^2)^(9/4)*d^(11/2)*arctan((d*(2*
c*x+b))^(1/2)/(-4*a*c+b^2)^(1/4)/d^(1/2))-2*(-4*a*c+b^2)^(9/4)*d^(11/2)*arctanh((d*(2*c*x+b))^(1/2)/(-4*a*c+b^
2)^(1/4)/d^(1/2))+4*(-4*a*c+b^2)^2*d^5*(2*c*d*x+b*d)^(1/2)

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Rubi [A]
time = 0.14, antiderivative size = 175, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {706, 708, 335, 218, 212, 209} \begin {gather*} -2 d^{11/2} \left (b^2-4 a c\right )^{9/4} \text {ArcTan}\left (\frac {\sqrt {d (b+2 c x)}}{\sqrt {d} \sqrt [4]{b^2-4 a c}}\right )-2 d^{11/2} \left (b^2-4 a c\right )^{9/4} \tanh ^{-1}\left (\frac {\sqrt {d (b+2 c x)}}{\sqrt {d} \sqrt [4]{b^2-4 a c}}\right )+4 d^5 \left (b^2-4 a c\right )^2 \sqrt {b d+2 c d x}+\frac {4}{5} d^3 \left (b^2-4 a c\right ) (b d+2 c d x)^{5/2}+\frac {4}{9} d (b d+2 c d x)^{9/2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(b*d + 2*c*d*x)^(11/2)/(a + b*x + c*x^2),x]

[Out]

4*(b^2 - 4*a*c)^2*d^5*Sqrt[b*d + 2*c*d*x] + (4*(b^2 - 4*a*c)*d^3*(b*d + 2*c*d*x)^(5/2))/5 + (4*d*(b*d + 2*c*d*
x)^(9/2))/9 - 2*(b^2 - 4*a*c)^(9/4)*d^(11/2)*ArcTan[Sqrt[d*(b + 2*c*x)]/((b^2 - 4*a*c)^(1/4)*Sqrt[d])] - 2*(b^
2 - 4*a*c)^(9/4)*d^(11/2)*ArcTanh[Sqrt[d*(b + 2*c*x)]/((b^2 - 4*a*c)^(1/4)*Sqrt[d])]

Rule 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

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 218

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]},
Dist[r/(2*a), Int[1/(r - s*x^2), x], x] + Dist[r/(2*a), Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] &&  !Gt
Q[a/b, 0]

Rule 335

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/c, Subst[I
nt[x^(k*(m + 1) - 1)*(a + b*(x^(k*n)/c^n))^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0]
&& FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]

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])

Rule 708

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

Rubi steps

\begin {align*} \int \frac {(b d+2 c d x)^{11/2}}{a+b x+c x^2} \, dx &=\frac {4}{9} d (b d+2 c d x)^{9/2}+\left (\left (b^2-4 a c\right ) d^2\right ) \int \frac {(b d+2 c d x)^{7/2}}{a+b x+c x^2} \, dx\\ &=\frac {4}{5} \left (b^2-4 a c\right ) d^3 (b d+2 c d x)^{5/2}+\frac {4}{9} d (b d+2 c d x)^{9/2}+\left (\left (b^2-4 a c\right )^2 d^4\right ) \int \frac {(b d+2 c d x)^{3/2}}{a+b x+c x^2} \, dx\\ &=4 \left (b^2-4 a c\right )^2 d^5 \sqrt {b d+2 c d x}+\frac {4}{5} \left (b^2-4 a c\right ) d^3 (b d+2 c d x)^{5/2}+\frac {4}{9} d (b d+2 c d x)^{9/2}+\left (\left (b^2-4 a c\right )^3 d^6\right ) \int \frac {1}{\sqrt {b d+2 c d x} \left (a+b x+c x^2\right )} \, dx\\ &=4 \left (b^2-4 a c\right )^2 d^5 \sqrt {b d+2 c d x}+\frac {4}{5} \left (b^2-4 a c\right ) d^3 (b d+2 c d x)^{5/2}+\frac {4}{9} d (b d+2 c d x)^{9/2}+\frac {\left (\left (b^2-4 a c\right )^3 d^5\right ) \text {Subst}\left (\int \frac {1}{\sqrt {x} \left (a-\frac {b^2}{4 c}+\frac {x^2}{4 c d^2}\right )} \, dx,x,b d+2 c d x\right )}{2 c}\\ &=4 \left (b^2-4 a c\right )^2 d^5 \sqrt {b d+2 c d x}+\frac {4}{5} \left (b^2-4 a c\right ) d^3 (b d+2 c d x)^{5/2}+\frac {4}{9} d (b d+2 c d x)^{9/2}+\frac {\left (\left (b^2-4 a c\right )^3 d^5\right ) \text {Subst}\left (\int \frac {1}{a-\frac {b^2}{4 c}+\frac {x^4}{4 c d^2}} \, dx,x,\sqrt {d (b+2 c x)}\right )}{c}\\ &=4 \left (b^2-4 a c\right )^2 d^5 \sqrt {b d+2 c d x}+\frac {4}{5} \left (b^2-4 a c\right ) d^3 (b d+2 c d x)^{5/2}+\frac {4}{9} d (b d+2 c d x)^{9/2}-\left (2 \left (b^2-4 a c\right )^{5/2} d^6\right ) \text {Subst}\left (\int \frac {1}{\sqrt {b^2-4 a c} d-x^2} \, dx,x,\sqrt {d (b+2 c x)}\right )-\left (2 \left (b^2-4 a c\right )^{5/2} d^6\right ) \text {Subst}\left (\int \frac {1}{\sqrt {b^2-4 a c} d+x^2} \, dx,x,\sqrt {d (b+2 c x)}\right )\\ &=4 \left (b^2-4 a c\right )^2 d^5 \sqrt {b d+2 c d x}+\frac {4}{5} \left (b^2-4 a c\right ) d^3 (b d+2 c d x)^{5/2}+\frac {4}{9} d (b d+2 c d x)^{9/2}-2 \left (b^2-4 a c\right )^{9/4} d^{11/2} \tan ^{-1}\left (\frac {\sqrt {d (b+2 c x)}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )-2 \left (b^2-4 a c\right )^{9/4} d^{11/2} \tanh ^{-1}\left (\frac {\sqrt {d (b+2 c x)}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 0.35, size = 254, normalized size = 1.45 \begin {gather*} \frac {\left (\frac {1}{45}-\frac {i}{45}\right ) d (d (b+2 c x))^{9/2} \left ((2+2 i) \sqrt {b+2 c x} \left (45 b^4-360 a b^2 c+720 a^2 c^2+9 b^2 (b+2 c x)^2-36 a c (b+2 c x)^2+5 (b+2 c x)^4\right )+45 \left (b^2-4 a c\right )^{9/4} \tan ^{-1}\left (1-\frac {(1+i) \sqrt {b+2 c x}}{\sqrt [4]{b^2-4 a c}}\right )-45 \left (b^2-4 a c\right )^{9/4} \tan ^{-1}\left (1+\frac {(1+i) \sqrt {b+2 c x}}{\sqrt [4]{b^2-4 a c}}\right )-45 \left (b^2-4 a c\right )^{9/4} \tanh ^{-1}\left (\frac {(1+i) \sqrt [4]{b^2-4 a c} \sqrt {b+2 c x}}{\sqrt {b^2-4 a c}+i (b+2 c x)}\right )\right )}{(b+2 c x)^{9/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(b*d + 2*c*d*x)^(11/2)/(a + b*x + c*x^2),x]

[Out]

((1/45 - I/45)*d*(d*(b + 2*c*x))^(9/2)*((2 + 2*I)*Sqrt[b + 2*c*x]*(45*b^4 - 360*a*b^2*c + 720*a^2*c^2 + 9*b^2*
(b + 2*c*x)^2 - 36*a*c*(b + 2*c*x)^2 + 5*(b + 2*c*x)^4) + 45*(b^2 - 4*a*c)^(9/4)*ArcTan[1 - ((1 + I)*Sqrt[b +
2*c*x])/(b^2 - 4*a*c)^(1/4)] - 45*(b^2 - 4*a*c)^(9/4)*ArcTan[1 + ((1 + I)*Sqrt[b + 2*c*x])/(b^2 - 4*a*c)^(1/4)
] - 45*(b^2 - 4*a*c)^(9/4)*ArcTanh[((1 + I)*(b^2 - 4*a*c)^(1/4)*Sqrt[b + 2*c*x])/(Sqrt[b^2 - 4*a*c] + I*(b + 2
*c*x))]))/(b + 2*c*x)^(9/2)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(376\) vs. \(2(145)=290\).
time = 0.74, size = 377, normalized size = 2.15

method	result	size
derivativedivides	\(4 d \left (16 a^{2} c^{2} d^{4} \sqrt {2 c d x +b d}-8 a \,b^{2} c \,d^{4} \sqrt {2 c d x +b d}-\frac {4 a c \,d^{2} \left (2 c d x +b d \right )^{\frac {5}{2}}}{5}+b^{4} d^{4} \sqrt {2 c d x +b d}+\frac {b^{2} d^{2} \left (2 c d x +b d \right )^{\frac {5}{2}}}{5}+\frac {\left (2 c d x +b d \right )^{\frac {9}{2}}}{9}-\frac {d^{6} \left (64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right ) \sqrt {2}\, \left (\ln \left (\frac {2 c d x +b d +\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}} \sqrt {2 c d x +b d}\, \sqrt {2}+\sqrt {4 a c \,d^{2}-b^{2} d^{2}}}{2 c d x +b d -\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}} \sqrt {2 c d x +b d}\, \sqrt {2}+\sqrt {4 a c \,d^{2}-b^{2} d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {2 c d x +b d}}{\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {2 c d x +b d}}{\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {3}{4}}}\right )\)	\(377\)
default	\(4 d \left (16 a^{2} c^{2} d^{4} \sqrt {2 c d x +b d}-8 a \,b^{2} c \,d^{4} \sqrt {2 c d x +b d}-\frac {4 a c \,d^{2} \left (2 c d x +b d \right )^{\frac {5}{2}}}{5}+b^{4} d^{4} \sqrt {2 c d x +b d}+\frac {b^{2} d^{2} \left (2 c d x +b d \right )^{\frac {5}{2}}}{5}+\frac {\left (2 c d x +b d \right )^{\frac {9}{2}}}{9}-\frac {d^{6} \left (64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right ) \sqrt {2}\, \left (\ln \left (\frac {2 c d x +b d +\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}} \sqrt {2 c d x +b d}\, \sqrt {2}+\sqrt {4 a c \,d^{2}-b^{2} d^{2}}}{2 c d x +b d -\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}} \sqrt {2 c d x +b d}\, \sqrt {2}+\sqrt {4 a c \,d^{2}-b^{2} d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {2 c d x +b d}}{\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {2 c d x +b d}}{\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {3}{4}}}\right )\)	\(377\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*c*d*x+b*d)^(11/2)/(c*x^2+b*x+a),x,method=_RETURNVERBOSE)

[Out]

4*d*(16*a^2*c^2*d^4*(2*c*d*x+b*d)^(1/2)-8*a*b^2*c*d^4*(2*c*d*x+b*d)^(1/2)-4/5*a*c*d^2*(2*c*d*x+b*d)^(5/2)+b^4*
d^4*(2*c*d*x+b*d)^(1/2)+1/5*b^2*d^2*(2*c*d*x+b*d)^(5/2)+1/9*(2*c*d*x+b*d)^(9/2)-1/8*d^6*(64*a^3*c^3-48*a^2*b^2
*c^2+12*a*b^4*c-b^6)/(4*a*c*d^2-b^2*d^2)^(3/4)*2^(1/2)*(ln((2*c*d*x+b*d+(4*a*c*d^2-b^2*d^2)^(1/4)*(2*c*d*x+b*d
)^(1/2)*2^(1/2)+(4*a*c*d^2-b^2*d^2)^(1/2))/(2*c*d*x+b*d-(4*a*c*d^2-b^2*d^2)^(1/4)*(2*c*d*x+b*d)^(1/2)*2^(1/2)+
(4*a*c*d^2-b^2*d^2)^(1/2)))+2*arctan(2^(1/2)/(4*a*c*d^2-b^2*d^2)^(1/4)*(2*c*d*x+b*d)^(1/2)+1)-2*arctan(-2^(1/2
)/(4*a*c*d^2-b^2*d^2)^(1/4)*(2*c*d*x+b*d)^(1/2)+1)))

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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)^(11/2)/(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 [B] Leaf count of result is larger than twice the leaf count of optimal. 1241 vs. \(2 (145) = 290\).
time = 3.68, size = 1241, normalized size = 7.09 \begin {gather*} \frac {4}{45} \, {\left (80 \, c^{4} d^{5} x^{4} + 160 \, b c^{3} d^{5} x^{3} + 12 \, {\left (13 \, b^{2} c^{2} - 12 \, a c^{3}\right )} d^{5} x^{2} + 4 \, {\left (19 \, b^{3} c - 36 \, a b c^{2}\right )} d^{5} x + {\left (59 \, b^{4} - 396 \, a b^{2} c + 720 \, a^{2} c^{2}\right )} d^{5}\right )} \sqrt {2 \, c d x + b d} - 4 \, \left ({\left (b^{18} - 36 \, a b^{16} c + 576 \, a^{2} b^{14} c^{2} - 5376 \, a^{3} b^{12} c^{3} + 32256 \, a^{4} b^{10} c^{4} - 129024 \, a^{5} b^{8} c^{5} + 344064 \, a^{6} b^{6} c^{6} - 589824 \, a^{7} b^{4} c^{7} + 589824 \, a^{8} b^{2} c^{8} - 262144 \, a^{9} c^{9}\right )} d^{22}\right )^{\frac {1}{4}} \arctan \left (\frac {\left ({\left (b^{18} - 36 \, a b^{16} c + 576 \, a^{2} b^{14} c^{2} - 5376 \, a^{3} b^{12} c^{3} + 32256 \, a^{4} b^{10} c^{4} - 129024 \, a^{5} b^{8} c^{5} + 344064 \, a^{6} b^{6} c^{6} - 589824 \, a^{7} b^{4} c^{7} + 589824 \, a^{8} b^{2} c^{8} - 262144 \, a^{9} c^{9}\right )} d^{22}\right )^{\frac {3}{4}} {\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} \sqrt {2 \, c d x + b d} d^{5} - \left ({\left (b^{18} - 36 \, a b^{16} c + 576 \, a^{2} b^{14} c^{2} - 5376 \, a^{3} b^{12} c^{3} + 32256 \, a^{4} b^{10} c^{4} - 129024 \, a^{5} b^{8} c^{5} + 344064 \, a^{6} b^{6} c^{6} - 589824 \, a^{7} b^{4} c^{7} + 589824 \, a^{8} b^{2} c^{8} - 262144 \, a^{9} c^{9}\right )} d^{22}\right )^{\frac {3}{4}} \sqrt {2 \, {\left (b^{8} c - 16 \, a b^{6} c^{2} + 96 \, a^{2} b^{4} c^{3} - 256 \, a^{3} b^{2} c^{4} + 256 \, a^{4} c^{5}\right )} d^{11} x + {\left (b^{9} - 16 \, a b^{7} c + 96 \, a^{2} b^{5} c^{2} - 256 \, a^{3} b^{3} c^{3} + 256 \, a^{4} b c^{4}\right )} d^{11} + \sqrt {{\left (b^{18} - 36 \, a b^{16} c + 576 \, a^{2} b^{14} c^{2} - 5376 \, a^{3} b^{12} c^{3} + 32256 \, a^{4} b^{10} c^{4} - 129024 \, a^{5} b^{8} c^{5} + 344064 \, a^{6} b^{6} c^{6} - 589824 \, a^{7} b^{4} c^{7} + 589824 \, a^{8} b^{2} c^{8} - 262144 \, a^{9} c^{9}\right )} d^{22}}}}{{\left (b^{18} - 36 \, a b^{16} c + 576 \, a^{2} b^{14} c^{2} - 5376 \, a^{3} b^{12} c^{3} + 32256 \, a^{4} b^{10} c^{4} - 129024 \, a^{5} b^{8} c^{5} + 344064 \, a^{6} b^{6} c^{6} - 589824 \, a^{7} b^{4} c^{7} + 589824 \, a^{8} b^{2} c^{8} - 262144 \, a^{9} c^{9}\right )} d^{22}}\right ) - \left ({\left (b^{18} - 36 \, a b^{16} c + 576 \, a^{2} b^{14} c^{2} - 5376 \, a^{3} b^{12} c^{3} + 32256 \, a^{4} b^{10} c^{4} - 129024 \, a^{5} b^{8} c^{5} + 344064 \, a^{6} b^{6} c^{6} - 589824 \, a^{7} b^{4} c^{7} + 589824 \, a^{8} b^{2} c^{8} - 262144 \, a^{9} c^{9}\right )} d^{22}\right )^{\frac {1}{4}} \log \left ({\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} \sqrt {2 \, c d x + b d} d^{5} + \left ({\left (b^{18} - 36 \, a b^{16} c + 576 \, a^{2} b^{14} c^{2} - 5376 \, a^{3} b^{12} c^{3} + 32256 \, a^{4} b^{10} c^{4} - 129024 \, a^{5} b^{8} c^{5} + 344064 \, a^{6} b^{6} c^{6} - 589824 \, a^{7} b^{4} c^{7} + 589824 \, a^{8} b^{2} c^{8} - 262144 \, a^{9} c^{9}\right )} d^{22}\right )^{\frac {1}{4}}\right ) + \left ({\left (b^{18} - 36 \, a b^{16} c + 576 \, a^{2} b^{14} c^{2} - 5376 \, a^{3} b^{12} c^{3} + 32256 \, a^{4} b^{10} c^{4} - 129024 \, a^{5} b^{8} c^{5} + 344064 \, a^{6} b^{6} c^{6} - 589824 \, a^{7} b^{4} c^{7} + 589824 \, a^{8} b^{2} c^{8} - 262144 \, a^{9} c^{9}\right )} d^{22}\right )^{\frac {1}{4}} \log \left ({\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} \sqrt {2 \, c d x + b d} d^{5} - \left ({\left (b^{18} - 36 \, a b^{16} c + 576 \, a^{2} b^{14} c^{2} - 5376 \, a^{3} b^{12} c^{3} + 32256 \, a^{4} b^{10} c^{4} - 129024 \, a^{5} b^{8} c^{5} + 344064 \, a^{6} b^{6} c^{6} - 589824 \, a^{7} b^{4} c^{7} + 589824 \, a^{8} b^{2} c^{8} - 262144 \, a^{9} c^{9}\right )} d^{22}\right )^{\frac {1}{4}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*d*x+b*d)^(11/2)/(c*x^2+b*x+a),x, algorithm="fricas")

[Out]

4/45*(80*c^4*d^5*x^4 + 160*b*c^3*d^5*x^3 + 12*(13*b^2*c^2 - 12*a*c^3)*d^5*x^2 + 4*(19*b^3*c - 36*a*b*c^2)*d^5*
x + (59*b^4 - 396*a*b^2*c + 720*a^2*c^2)*d^5)*sqrt(2*c*d*x + b*d) - 4*((b^18 - 36*a*b^16*c + 576*a^2*b^14*c^2
- 5376*a^3*b^12*c^3 + 32256*a^4*b^10*c^4 - 129024*a^5*b^8*c^5 + 344064*a^6*b^6*c^6 - 589824*a^7*b^4*c^7 + 5898
24*a^8*b^2*c^8 - 262144*a^9*c^9)*d^22)^(1/4)*arctan((((b^18 - 36*a*b^16*c + 576*a^2*b^14*c^2 - 5376*a^3*b^12*c
^3 + 32256*a^4*b^10*c^4 - 129024*a^5*b^8*c^5 + 344064*a^6*b^6*c^6 - 589824*a^7*b^4*c^7 + 589824*a^8*b^2*c^8 -
262144*a^9*c^9)*d^22)^(3/4)*(b^4 - 8*a*b^2*c + 16*a^2*c^2)*sqrt(2*c*d*x + b*d)*d^5 - ((b^18 - 36*a*b^16*c + 57
6*a^2*b^14*c^2 - 5376*a^3*b^12*c^3 + 32256*a^4*b^10*c^4 - 129024*a^5*b^8*c^5 + 344064*a^6*b^6*c^6 - 589824*a^7
*b^4*c^7 + 589824*a^8*b^2*c^8 - 262144*a^9*c^9)*d^22)^(3/4)*sqrt(2*(b^8*c - 16*a*b^6*c^2 + 96*a^2*b^4*c^3 - 25
6*a^3*b^2*c^4 + 256*a^4*c^5)*d^11*x + (b^9 - 16*a*b^7*c + 96*a^2*b^5*c^2 - 256*a^3*b^3*c^3 + 256*a^4*b*c^4)*d^
11 + sqrt((b^18 - 36*a*b^16*c + 576*a^2*b^14*c^2 - 5376*a^3*b^12*c^3 + 32256*a^4*b^10*c^4 - 129024*a^5*b^8*c^5
+ 344064*a^6*b^6*c^6 - 589824*a^7*b^4*c^7 + 589824*a^8*b^2*c^8 - 262144*a^9*c^9)*d^22)))/((b^18 - 36*a*b^16*c
+ 576*a^2*b^14*c^2 - 5376*a^3*b^12*c^3 + 32256*a^4*b^10*c^4 - 129024*a^5*b^8*c^5 + 344064*a^6*b^6*c^6 - 58982
4*a^7*b^4*c^7 + 589824*a^8*b^2*c^8 - 262144*a^9*c^9)*d^22)) - ((b^18 - 36*a*b^16*c + 576*a^2*b^14*c^2 - 5376*a
^3*b^12*c^3 + 32256*a^4*b^10*c^4 - 129024*a^5*b^8*c^5 + 344064*a^6*b^6*c^6 - 589824*a^7*b^4*c^7 + 589824*a^8*b
^2*c^8 - 262144*a^9*c^9)*d^22)^(1/4)*log((b^4 - 8*a*b^2*c + 16*a^2*c^2)*sqrt(2*c*d*x + b*d)*d^5 + ((b^18 - 36*
a*b^16*c + 576*a^2*b^14*c^2 - 5376*a^3*b^12*c^3 + 32256*a^4*b^10*c^4 - 129024*a^5*b^8*c^5 + 344064*a^6*b^6*c^6
- 589824*a^7*b^4*c^7 + 589824*a^8*b^2*c^8 - 262144*a^9*c^9)*d^22)^(1/4)) + ((b^18 - 36*a*b^16*c + 576*a^2*b^1
4*c^2 - 5376*a^3*b^12*c^3 + 32256*a^4*b^10*c^4 - 129024*a^5*b^8*c^5 + 344064*a^6*b^6*c^6 - 589824*a^7*b^4*c^7
+ 589824*a^8*b^2*c^8 - 262144*a^9*c^9)*d^22)^(1/4)*log((b^4 - 8*a*b^2*c + 16*a^2*c^2)*sqrt(2*c*d*x + b*d)*d^5
- ((b^18 - 36*a*b^16*c + 576*a^2*b^14*c^2 - 5376*a^3*b^12*c^3 + 32256*a^4*b^10*c^4 - 129024*a^5*b^8*c^5 + 3440
64*a^6*b^6*c^6 - 589824*a^7*b^4*c^7 + 589824*a^8*b^2*c^8 - 262144*a^9*c^9)*d^22)^(1/4))

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*d*x+b*d)**(11/2)/(c*x**2+b*x+a),x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 733 vs. \(2 (145) = 290\).
time = 1.80, size = 733, normalized size = 4.19 \begin {gather*} 4 \, \sqrt {2 \, c d x + b d} b^{4} d^{5} - 32 \, \sqrt {2 \, c d x + b d} a b^{2} c d^{5} + 64 \, \sqrt {2 \, c d x + b d} a^{2} c^{2} d^{5} + \frac {4}{5} \, {\left (2 \, c d x + b d\right )}^{\frac {5}{2}} b^{2} d^{3} - \frac {16}{5} \, {\left (2 \, c d x + b d\right )}^{\frac {5}{2}} a c d^{3} + \frac {4}{9} \, {\left (2 \, c d x + b d\right )}^{\frac {9}{2}} d - {\left (\sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} b^{4} d^{5} - 8 \, \sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} a b^{2} c d^{5} + 16 \, \sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} a^{2} c^{2} d^{5}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} + 2 \, \sqrt {2 \, c d x + b d}\right )}}{2 \, {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}}}\right ) - {\left (\sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} b^{4} d^{5} - 8 \, \sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} a b^{2} c d^{5} + 16 \, \sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} a^{2} c^{2} d^{5}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} - 2 \, \sqrt {2 \, c d x + b d}\right )}}{2 \, {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}}}\right ) - \frac {1}{2} \, {\left (\sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} b^{4} d^{5} - 8 \, \sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} a b^{2} c d^{5} + 16 \, \sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} a^{2} c^{2} d^{5}\right )} \log \left (2 \, c d x + b d + \sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} \sqrt {2 \, c d x + b d} + \sqrt {-b^{2} d^{2} + 4 \, a c d^{2}}\right ) + \frac {1}{2} \, {\left (\sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} b^{4} d^{5} - 8 \, \sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} a b^{2} c d^{5} + 16 \, \sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} a^{2} c^{2} d^{5}\right )} \log \left (2 \, c d x + b d - \sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} \sqrt {2 \, c d x + b d} + \sqrt {-b^{2} d^{2} + 4 \, a c d^{2}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*d*x+b*d)^(11/2)/(c*x^2+b*x+a),x, algorithm="giac")

[Out]

4*sqrt(2*c*d*x + b*d)*b^4*d^5 - 32*sqrt(2*c*d*x + b*d)*a*b^2*c*d^5 + 64*sqrt(2*c*d*x + b*d)*a^2*c^2*d^5 + 4/5*
(2*c*d*x + b*d)^(5/2)*b^2*d^3 - 16/5*(2*c*d*x + b*d)^(5/2)*a*c*d^3 + 4/9*(2*c*d*x + b*d)^(9/2)*d - (sqrt(2)*(-
b^2*d^2 + 4*a*c*d^2)^(1/4)*b^4*d^5 - 8*sqrt(2)*(-b^2*d^2 + 4*a*c*d^2)^(1/4)*a*b^2*c*d^5 + 16*sqrt(2)*(-b^2*d^2
+ 4*a*c*d^2)^(1/4)*a^2*c^2*d^5)*arctan(1/2*sqrt(2)*(sqrt(2)*(-b^2*d^2 + 4*a*c*d^2)^(1/4) + 2*sqrt(2*c*d*x + b
*d))/(-b^2*d^2 + 4*a*c*d^2)^(1/4)) - (sqrt(2)*(-b^2*d^2 + 4*a*c*d^2)^(1/4)*b^4*d^5 - 8*sqrt(2)*(-b^2*d^2 + 4*a
*c*d^2)^(1/4)*a*b^2*c*d^5 + 16*sqrt(2)*(-b^2*d^2 + 4*a*c*d^2)^(1/4)*a^2*c^2*d^5)*arctan(-1/2*sqrt(2)*(sqrt(2)*
(-b^2*d^2 + 4*a*c*d^2)^(1/4) - 2*sqrt(2*c*d*x + b*d))/(-b^2*d^2 + 4*a*c*d^2)^(1/4)) - 1/2*(sqrt(2)*(-b^2*d^2 +
4*a*c*d^2)^(1/4)*b^4*d^5 - 8*sqrt(2)*(-b^2*d^2 + 4*a*c*d^2)^(1/4)*a*b^2*c*d^5 + 16*sqrt(2)*(-b^2*d^2 + 4*a*c*
d^2)^(1/4)*a^2*c^2*d^5)*log(2*c*d*x + b*d + sqrt(2)*(-b^2*d^2 + 4*a*c*d^2)^(1/4)*sqrt(2*c*d*x + b*d) + sqrt(-b
^2*d^2 + 4*a*c*d^2)) + 1/2*(sqrt(2)*(-b^2*d^2 + 4*a*c*d^2)^(1/4)*b^4*d^5 - 8*sqrt(2)*(-b^2*d^2 + 4*a*c*d^2)^(1
/4)*a*b^2*c*d^5 + 16*sqrt(2)*(-b^2*d^2 + 4*a*c*d^2)^(1/4)*a^2*c^2*d^5)*log(2*c*d*x + b*d - sqrt(2)*(-b^2*d^2 +
4*a*c*d^2)^(1/4)*sqrt(2*c*d*x + b*d) + sqrt(-b^2*d^2 + 4*a*c*d^2))

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Mupad [B]
time = 0.65, size = 240, normalized size = 1.37 \begin {gather*} \frac {4\,d\,{\left (b\,d+2\,c\,d\,x\right )}^{9/2}}{9}-\frac {4\,d^3\,{\left (b\,d+2\,c\,d\,x\right )}^{5/2}\,\left (4\,a\,c-b^2\right )}{5}+4\,d^5\,\sqrt {b\,d+2\,c\,d\,x}\,{\left (4\,a\,c-b^2\right )}^2-2\,d^{11/2}\,\mathrm {atan}\left (\frac {b^4\,\sqrt {b\,d+2\,c\,d\,x}+16\,a^2\,c^2\,\sqrt {b\,d+2\,c\,d\,x}-8\,a\,b^2\,c\,\sqrt {b\,d+2\,c\,d\,x}}{\sqrt {d}\,{\left (b^2-4\,a\,c\right )}^{9/4}}\right )\,{\left (b^2-4\,a\,c\right )}^{9/4}+d^{11/2}\,\mathrm {atan}\left (\frac {b^4\,\sqrt {b\,d+2\,c\,d\,x}\,1{}\mathrm {i}+a^2\,c^2\,\sqrt {b\,d+2\,c\,d\,x}\,16{}\mathrm {i}-a\,b^2\,c\,\sqrt {b\,d+2\,c\,d\,x}\,8{}\mathrm {i}}{\sqrt {d}\,{\left (b^2-4\,a\,c\right )}^{9/4}}\right )\,{\left (b^2-4\,a\,c\right )}^{9/4}\,2{}\mathrm {i} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*d + 2*c*d*x)^(11/2)/(a + b*x + c*x^2),x)

[Out]

(4*d*(b*d + 2*c*d*x)^(9/2))/9 - (4*d^3*(b*d + 2*c*d*x)^(5/2)*(4*a*c - b^2))/5 + 4*d^5*(b*d + 2*c*d*x)^(1/2)*(4
*a*c - b^2)^2 - 2*d^(11/2)*atan((b^4*(b*d + 2*c*d*x)^(1/2) + 16*a^2*c^2*(b*d + 2*c*d*x)^(1/2) - 8*a*b^2*c*(b*d
+ 2*c*d*x)^(1/2))/(d^(1/2)*(b^2 - 4*a*c)^(9/4)))*(b^2 - 4*a*c)^(9/4) + d^(11/2)*atan((b^4*(b*d + 2*c*d*x)^(1/
2)*1i + a^2*c^2*(b*d + 2*c*d*x)^(1/2)*16i - a*b^2*c*(b*d + 2*c*d*x)^(1/2)*8i)/(d^(1/2)*(b^2 - 4*a*c)^(9/4)))*(
b^2 - 4*a*c)^(9/4)*2i

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