3.12.9 \(\int \frac {(b d+2 c d x)^{11/2}}{(a+b x+c x^2)^2} \, dx\)

Optimal. Leaf size=179 \[ -18 c d^{11/2} \left (b^2-4 a c\right )^{5/4} \tan ^{-1}\left (\frac {\sqrt {d (b+2 c x)}}{\sqrt {d} \sqrt [4]{b^2-4 a c}}\right )-18 c d^{11/2} \left (b^2-4 a c\right )^{5/4} \tanh ^{-1}\left (\frac {\sqrt {d (b+2 c x)}}{\sqrt {d} \sqrt [4]{b^2-4 a c}}\right )+36 c d^5 \left (b^2-4 a c\right ) \sqrt {b d+2 c d x}-\frac {d (b d+2 c d x)^{9/2}}{a+b x+c x^2}+\frac {36}{5} c d^3 (b d+2 c d x)^{5/2} \]

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Rubi [A]  time = 0.15, antiderivative size = 179, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {686, 692, 694, 329, 212, 206, 203} \begin {gather*} 36 c d^5 \left (b^2-4 a c\right ) \sqrt {b d+2 c d x}-18 c d^{11/2} \left (b^2-4 a c\right )^{5/4} \tan ^{-1}\left (\frac {\sqrt {d (b+2 c x)}}{\sqrt {d} \sqrt [4]{b^2-4 a c}}\right )-18 c d^{11/2} \left (b^2-4 a c\right )^{5/4} \tanh ^{-1}\left (\frac {\sqrt {d (b+2 c x)}}{\sqrt {d} \sqrt [4]{b^2-4 a c}}\right )-\frac {d (b d+2 c d x)^{9/2}}{a+b x+c x^2}+\frac {36}{5} c d^3 (b d+2 c d x)^{5/2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

36*c*(b^2 - 4*a*c)*d^5*Sqrt[b*d + 2*c*d*x] + (36*c*d^3*(b*d + 2*c*d*x)^(5/2))/5 - (d*(b*d + 2*c*d*x)^(9/2))/(a
+ b*x + c*x^2) - 18*c*(b^2 - 4*a*c)^(5/4)*d^(11/2)*ArcTan[Sqrt[d*(b + 2*c*x)]/((b^2 - 4*a*c)^(1/4)*Sqrt[d])]
- 18*c*(b^2 - 4*a*c)^(5/4)*d^(11/2)*ArcTanh[Sqrt[d*(b + 2*c*x)]/((b^2 - 4*a*c)^(1/4)*Sqrt[d])]

Rule 203

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

Rule 206

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

Rule 212

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] &&
!GtQ[a/b, 0]

Rule 329

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 686

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

Rule 692

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 694

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}}{\left (a+b x+c x^2\right )^2} \, dx &=-\frac {d (b d+2 c d x)^{9/2}}{a+b x+c x^2}+\left (9 c d^2\right ) \int \frac {(b d+2 c d x)^{7/2}}{a+b x+c x^2} \, dx\\ &=\frac {36}{5} c d^3 (b d+2 c d x)^{5/2}-\frac {d (b d+2 c d x)^{9/2}}{a+b x+c x^2}+\left (9 c \left (b^2-4 a c\right ) d^4\right ) \int \frac {(b d+2 c d x)^{3/2}}{a+b x+c x^2} \, dx\\ &=36 c \left (b^2-4 a c\right ) d^5 \sqrt {b d+2 c d x}+\frac {36}{5} c d^3 (b d+2 c d x)^{5/2}-\frac {d (b d+2 c d x)^{9/2}}{a+b x+c x^2}+\left (9 c \left (b^2-4 a c\right )^2 d^6\right ) \int \frac {1}{\sqrt {b d+2 c d x} \left (a+b x+c x^2\right )} \, dx\\ &=36 c \left (b^2-4 a c\right ) d^5 \sqrt {b d+2 c d x}+\frac {36}{5} c d^3 (b d+2 c d x)^{5/2}-\frac {d (b d+2 c d x)^{9/2}}{a+b x+c x^2}+\frac {1}{2} \left (9 \left (b^2-4 a c\right )^2 d^5\right ) \operatorname {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 )\\ &=36 c \left (b^2-4 a c\right ) d^5 \sqrt {b d+2 c d x}+\frac {36}{5} c d^3 (b d+2 c d x)^{5/2}-\frac {d (b d+2 c d x)^{9/2}}{a+b x+c x^2}+\left (9 \left (b^2-4 a c\right )^2 d^5\right ) \operatorname {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 )\\ &=36 c \left (b^2-4 a c\right ) d^5 \sqrt {b d+2 c d x}+\frac {36}{5} c d^3 (b d+2 c d x)^{5/2}-\frac {d (b d+2 c d x)^{9/2}}{a+b x+c x^2}-\left (18 c \left (b^2-4 a c\right )^{3/2} d^6\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {b^2-4 a c} d-x^2} \, dx,x,\sqrt {d (b+2 c x)}\right )-\left (18 c \left (b^2-4 a c\right )^{3/2} d^6\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {b^2-4 a c} d+x^2} \, dx,x,\sqrt {d (b+2 c x)}\right )\\ &=36 c \left (b^2-4 a c\right ) d^5 \sqrt {b d+2 c d x}+\frac {36}{5} c d^3 (b d+2 c d x)^{5/2}-\frac {d (b d+2 c d x)^{9/2}}{a+b x+c x^2}-18 c \left (b^2-4 a c\right )^{5/4} d^{11/2} \tan ^{-1}\left (\frac {\sqrt {d (b+2 c x)}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )-18 c \left (b^2-4 a c\right )^{5/4} 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 [A]  time = 0.34, size = 167, normalized size = 0.93 \begin {gather*} -\frac {d (d (b+2 c x))^{9/2} \left (-3 \left (b^2-4 a c\right ) \left (-30 \left (b^2-4 a c\right ) \sqrt {b+2 c x}-60 c \sqrt [4]{b^2-4 a c} (a+x (b+c x)) \left (\tan ^{-1}\left (\frac {\sqrt {b+2 c x}}{\sqrt [4]{b^2-4 a c}}\right )+\tanh ^{-1}\left (\frac {\sqrt {b+2 c x}}{\sqrt [4]{b^2-4 a c}}\right )\right )+24 (b+2 c x)^{5/2}\right )-8 (b+2 c x)^{9/2}\right )}{10 (b+2 c x)^{9/2} (a+x (b+c x))} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/10*(d*(d*(b + 2*c*x))^(9/2)*(-8*(b + 2*c*x)^(9/2) - 3*(b^2 - 4*a*c)*(-30*(b^2 - 4*a*c)*Sqrt[b + 2*c*x] + 24
*(b + 2*c*x)^(5/2) - 60*c*(b^2 - 4*a*c)^(1/4)*(a + x*(b + c*x))*(ArcTan[Sqrt[b + 2*c*x]/(b^2 - 4*a*c)^(1/4)] +
ArcTanh[Sqrt[b + 2*c*x]/(b^2 - 4*a*c)^(1/4)]))))/((b + 2*c*x)^(9/2)*(a + x*(b + c*x)))

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IntegrateAlgebraic [C]  time = 0.84, size = 354, normalized size = 1.98 \begin {gather*} \frac {\sqrt {b d+2 c d x} \left (-720 a^2 c^2 d^5+216 a b^2 c d^5-576 a b c^2 d^5 x-576 a c^3 d^5 x^2-5 b^4 d^5+176 b^3 c d^5 x+240 b^2 c^2 d^5 x^2+128 b c^3 d^5 x^3+64 c^4 d^5 x^4\right )}{5 \left (a+b x+c x^2\right )}+(9-9 i) \left (b^2 c d^{11/2} \sqrt [4]{b^2-4 a c}-4 a c^2 d^{11/2} \sqrt [4]{b^2-4 a c}\right ) \tan ^{-1}\left (\frac {\sqrt {d} \left ((1-i) \sqrt {b^2-4 a c}+(-1-i) b-(2+2 i) c x\right )}{2 \sqrt [4]{b^2-4 a c} \sqrt {b d+2 c d x}}\right )-(9-9 i) \left (b^2 c d^{11/2} \sqrt [4]{b^2-4 a c}-4 a c^2 d^{11/2} \sqrt [4]{b^2-4 a c}\right ) \tanh ^{-1}\left (\frac {(1+i) \sqrt [4]{b^2-4 a c} \sqrt {b d+2 c d x}}{\sqrt {d} \left (\sqrt {b^2-4 a c}+i b+2 i c x\right )}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[b*d + 2*c*d*x]*(-5*b^4*d^5 + 216*a*b^2*c*d^5 - 720*a^2*c^2*d^5 + 176*b^3*c*d^5*x - 576*a*b*c^2*d^5*x + 2
40*b^2*c^2*d^5*x^2 - 576*a*c^3*d^5*x^2 + 128*b*c^3*d^5*x^3 + 64*c^4*d^5*x^4))/(5*(a + b*x + c*x^2)) + (9 - 9*I
)*(b^2*c*(b^2 - 4*a*c)^(1/4)*d^(11/2) - 4*a*c^2*(b^2 - 4*a*c)^(1/4)*d^(11/2))*ArcTan[(Sqrt[d]*((-1 - I)*b + (1
- I)*Sqrt[b^2 - 4*a*c] - (2 + 2*I)*c*x))/(2*(b^2 - 4*a*c)^(1/4)*Sqrt[b*d + 2*c*d*x])] - (9 - 9*I)*(b^2*c*(b^2
- 4*a*c)^(1/4)*d^(11/2) - 4*a*c^2*(b^2 - 4*a*c)^(1/4)*d^(11/2))*ArcTanh[((1 + I)*(b^2 - 4*a*c)^(1/4)*Sqrt[b*d
+ 2*c*d*x])/(Sqrt[d]*(I*b + Sqrt[b^2 - 4*a*c] + (2*I)*c*x))]

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fricas [B]  time = 0.45, size = 891, normalized size = 4.98 \begin {gather*} \frac {180 \, \left ({\left (b^{10} c^{4} - 20 \, a b^{8} c^{5} + 160 \, a^{2} b^{6} c^{6} - 640 \, a^{3} b^{4} c^{7} + 1280 \, a^{4} b^{2} c^{8} - 1024 \, a^{5} c^{9}\right )} d^{22}\right )^{\frac {1}{4}} {\left (c x^{2} + b x + a\right )} \arctan \left (-\frac {\left ({\left (b^{10} c^{4} - 20 \, a b^{8} c^{5} + 160 \, a^{2} b^{6} c^{6} - 640 \, a^{3} b^{4} c^{7} + 1280 \, a^{4} b^{2} c^{8} - 1024 \, a^{5} c^{9}\right )} d^{22}\right )^{\frac {3}{4}} {\left (b^{2} c - 4 \, a c^{2}\right )} \sqrt {2 \, c d x + b d} d^{5} + \left ({\left (b^{10} c^{4} - 20 \, a b^{8} c^{5} + 160 \, a^{2} b^{6} c^{6} - 640 \, a^{3} b^{4} c^{7} + 1280 \, a^{4} b^{2} c^{8} - 1024 \, a^{5} c^{9}\right )} d^{22}\right )^{\frac {3}{4}} \sqrt {2 \, {\left (b^{4} c^{3} - 8 \, a b^{2} c^{4} + 16 \, a^{2} c^{5}\right )} d^{11} x + {\left (b^{5} c^{2} - 8 \, a b^{3} c^{3} + 16 \, a^{2} b c^{4}\right )} d^{11} + \sqrt {{\left (b^{10} c^{4} - 20 \, a b^{8} c^{5} + 160 \, a^{2} b^{6} c^{6} - 640 \, a^{3} b^{4} c^{7} + 1280 \, a^{4} b^{2} c^{8} - 1024 \, a^{5} c^{9}\right )} d^{22}}}}{{\left (b^{10} c^{4} - 20 \, a b^{8} c^{5} + 160 \, a^{2} b^{6} c^{6} - 640 \, a^{3} b^{4} c^{7} + 1280 \, a^{4} b^{2} c^{8} - 1024 \, a^{5} c^{9}\right )} d^{22}}\right ) + 45 \, \left ({\left (b^{10} c^{4} - 20 \, a b^{8} c^{5} + 160 \, a^{2} b^{6} c^{6} - 640 \, a^{3} b^{4} c^{7} + 1280 \, a^{4} b^{2} c^{8} - 1024 \, a^{5} c^{9}\right )} d^{22}\right )^{\frac {1}{4}} {\left (c x^{2} + b x + a\right )} \log \left (-9 \, {\left (b^{2} c - 4 \, a c^{2}\right )} \sqrt {2 \, c d x + b d} d^{5} + 9 \, \left ({\left (b^{10} c^{4} - 20 \, a b^{8} c^{5} + 160 \, a^{2} b^{6} c^{6} - 640 \, a^{3} b^{4} c^{7} + 1280 \, a^{4} b^{2} c^{8} - 1024 \, a^{5} c^{9}\right )} d^{22}\right )^{\frac {1}{4}}\right ) - 45 \, \left ({\left (b^{10} c^{4} - 20 \, a b^{8} c^{5} + 160 \, a^{2} b^{6} c^{6} - 640 \, a^{3} b^{4} c^{7} + 1280 \, a^{4} b^{2} c^{8} - 1024 \, a^{5} c^{9}\right )} d^{22}\right )^{\frac {1}{4}} {\left (c x^{2} + b x + a\right )} \log \left (-9 \, {\left (b^{2} c - 4 \, a c^{2}\right )} \sqrt {2 \, c d x + b d} d^{5} - 9 \, \left ({\left (b^{10} c^{4} - 20 \, a b^{8} c^{5} + 160 \, a^{2} b^{6} c^{6} - 640 \, a^{3} b^{4} c^{7} + 1280 \, a^{4} b^{2} c^{8} - 1024 \, a^{5} c^{9}\right )} d^{22}\right )^{\frac {1}{4}}\right ) + {\left (64 \, c^{4} d^{5} x^{4} + 128 \, b c^{3} d^{5} x^{3} + 48 \, {\left (5 \, b^{2} c^{2} - 12 \, a c^{3}\right )} d^{5} x^{2} + 16 \, {\left (11 \, b^{3} c - 36 \, a b c^{2}\right )} d^{5} x - {\left (5 \, b^{4} - 216 \, a b^{2} c + 720 \, a^{2} c^{2}\right )} d^{5}\right )} \sqrt {2 \, c d x + b d}}{5 \, {\left (c x^{2} + b x + a\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)^2,x, algorithm="fricas")

[Out]

1/5*(180*((b^10*c^4 - 20*a*b^8*c^5 + 160*a^2*b^6*c^6 - 640*a^3*b^4*c^7 + 1280*a^4*b^2*c^8 - 1024*a^5*c^9)*d^22
)^(1/4)*(c*x^2 + b*x + a)*arctan(-(((b^10*c^4 - 20*a*b^8*c^5 + 160*a^2*b^6*c^6 - 640*a^3*b^4*c^7 + 1280*a^4*b^
2*c^8 - 1024*a^5*c^9)*d^22)^(3/4)*(b^2*c - 4*a*c^2)*sqrt(2*c*d*x + b*d)*d^5 + ((b^10*c^4 - 20*a*b^8*c^5 + 160*
a^2*b^6*c^6 - 640*a^3*b^4*c^7 + 1280*a^4*b^2*c^8 - 1024*a^5*c^9)*d^22)^(3/4)*sqrt(2*(b^4*c^3 - 8*a*b^2*c^4 + 1
6*a^2*c^5)*d^11*x + (b^5*c^2 - 8*a*b^3*c^3 + 16*a^2*b*c^4)*d^11 + sqrt((b^10*c^4 - 20*a*b^8*c^5 + 160*a^2*b^6*
c^6 - 640*a^3*b^4*c^7 + 1280*a^4*b^2*c^8 - 1024*a^5*c^9)*d^22)))/((b^10*c^4 - 20*a*b^8*c^5 + 160*a^2*b^6*c^6 -
640*a^3*b^4*c^7 + 1280*a^4*b^2*c^8 - 1024*a^5*c^9)*d^22)) + 45*((b^10*c^4 - 20*a*b^8*c^5 + 160*a^2*b^6*c^6 -
640*a^3*b^4*c^7 + 1280*a^4*b^2*c^8 - 1024*a^5*c^9)*d^22)^(1/4)*(c*x^2 + b*x + a)*log(-9*(b^2*c - 4*a*c^2)*sqrt
(2*c*d*x + b*d)*d^5 + 9*((b^10*c^4 - 20*a*b^8*c^5 + 160*a^2*b^6*c^6 - 640*a^3*b^4*c^7 + 1280*a^4*b^2*c^8 - 102
4*a^5*c^9)*d^22)^(1/4)) - 45*((b^10*c^4 - 20*a*b^8*c^5 + 160*a^2*b^6*c^6 - 640*a^3*b^4*c^7 + 1280*a^4*b^2*c^8
- 1024*a^5*c^9)*d^22)^(1/4)*(c*x^2 + b*x + a)*log(-9*(b^2*c - 4*a*c^2)*sqrt(2*c*d*x + b*d)*d^5 - 9*((b^10*c^4
- 20*a*b^8*c^5 + 160*a^2*b^6*c^6 - 640*a^3*b^4*c^7 + 1280*a^4*b^2*c^8 - 1024*a^5*c^9)*d^22)^(1/4)) + (64*c^4*d
^5*x^4 + 128*b*c^3*d^5*x^3 + 48*(5*b^2*c^2 - 12*a*c^3)*d^5*x^2 + 16*(11*b^3*c - 36*a*b*c^2)*d^5*x - (5*b^4 - 2
16*a*b^2*c + 720*a^2*c^2)*d^5)*sqrt(2*c*d*x + b*d))/(c*x^2 + b*x + a)

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giac [B]  time = 0.29, size = 646, normalized size = 3.61 \begin {gather*} 32 \, \sqrt {2 \, c d x + b d} b^{2} c d^{5} - 128 \, \sqrt {2 \, c d x + b d} a c^{2} d^{5} + \frac {16}{5} \, {\left (2 \, c d x + b d\right )}^{\frac {5}{2}} c d^{3} - 9 \, {\left (\sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} b^{2} c d^{5} - 4 \, \sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} a 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 ) - 9 \, {\left (\sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} b^{2} c d^{5} - 4 \, \sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} a 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 {9}{2} \, {\left (\sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} b^{2} c d^{5} - 4 \, \sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} a 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 {9}{2} \, {\left (\sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} b^{2} c d^{5} - 4 \, \sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} a 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 {4 \, {\left (\sqrt {2 \, c d x + b d} b^{4} c d^{7} - 8 \, \sqrt {2 \, c d x + b d} a b^{2} c^{2} d^{7} + 16 \, \sqrt {2 \, c d x + b d} a^{2} c^{3} d^{7}\right )}}{b^{2} d^{2} - 4 \, a c d^{2} - {\left (2 \, c d x + b d\right )}^{2}} \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)^2,x, algorithm="giac")

[Out]

32*sqrt(2*c*d*x + b*d)*b^2*c*d^5 - 128*sqrt(2*c*d*x + b*d)*a*c^2*d^5 + 16/5*(2*c*d*x + b*d)^(5/2)*c*d^3 - 9*(s
qrt(2)*(-b^2*d^2 + 4*a*c*d^2)^(1/4)*b^2*c*d^5 - 4*sqrt(2)*(-b^2*d^2 + 4*a*c*d^2)^(1/4)*a*c^2*d^5)*arctan(1/2*s
qrt(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)) - 9*(sqrt(
2)*(-b^2*d^2 + 4*a*c*d^2)^(1/4)*b^2*c*d^5 - 4*sqrt(2)*(-b^2*d^2 + 4*a*c*d^2)^(1/4)*a*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)) - 9/2*(sqrt(2
)*(-b^2*d^2 + 4*a*c*d^2)^(1/4)*b^2*c*d^5 - 4*sqrt(2)*(-b^2*d^2 + 4*a*c*d^2)^(1/4)*a*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)) + 9/2*(sqrt(2)*(-b^2
*d^2 + 4*a*c*d^2)^(1/4)*b^2*c*d^5 - 4*sqrt(2)*(-b^2*d^2 + 4*a*c*d^2)^(1/4)*a*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)) + 4*(sqrt(2*c*d*x + b*d)*b^
4*c*d^7 - 8*sqrt(2*c*d*x + b*d)*a*b^2*c^2*d^7 + 16*sqrt(2*c*d*x + b*d)*a^2*c^3*d^7)/(b^2*d^2 - 4*a*c*d^2 - (2*
c*d*x + b*d)^2)

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maple [B]  time = 0.06, size = 1090, normalized size = 6.09 \begin {gather*} -\frac {144 \sqrt {2}\, a^{2} c^{3} d^{7} \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 )}{\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {3}{4}}}+\frac {144 \sqrt {2}\, a^{2} c^{3} d^{7} \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 )}{\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {3}{4}}}+\frac {72 \sqrt {2}\, a^{2} c^{3} d^{7} \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 )}{\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {3}{4}}}+\frac {72 \sqrt {2}\, a \,b^{2} c^{2} d^{7} \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 )}{\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {3}{4}}}-\frac {72 \sqrt {2}\, a \,b^{2} c^{2} d^{7} \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 )}{\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {3}{4}}}-\frac {36 \sqrt {2}\, a \,b^{2} c^{2} d^{7} \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 )}{\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {3}{4}}}-\frac {9 \sqrt {2}\, b^{4} c \,d^{7} \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 )}{\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {3}{4}}}+\frac {9 \sqrt {2}\, b^{4} c \,d^{7} \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 )}{\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {3}{4}}}+\frac {9 \sqrt {2}\, b^{4} c \,d^{7} \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 \left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {3}{4}}}-\frac {64 \sqrt {2 c d x +b d}\, a^{2} c^{3} d^{7}}{4 c^{2} d^{2} x^{2}+4 b c \,d^{2} x +4 a c \,d^{2}}+\frac {32 \sqrt {2 c d x +b d}\, a \,b^{2} c^{2} d^{7}}{4 c^{2} d^{2} x^{2}+4 b c \,d^{2} x +4 a c \,d^{2}}-\frac {4 \sqrt {2 c d x +b d}\, b^{4} c \,d^{7}}{4 c^{2} d^{2} x^{2}+4 b c \,d^{2} x +4 a c \,d^{2}}-128 \sqrt {2 c d x +b d}\, a \,c^{2} d^{5}+32 \sqrt {2 c d x +b d}\, b^{2} c \,d^{5}+\frac {16 \left (2 c d x +b d \right )^{\frac {5}{2}} c \,d^{3}}{5} \end {gather*}

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)^2,x)

[Out]

16/5*c*d^3*(2*c*d*x+b*d)^(5/2)-128*c^2*d^5*a*(2*c*d*x+b*d)^(1/2)+32*c*d^5*b^2*(2*c*d*x+b*d)^(1/2)-64*c^3*d^7*(
2*c*d*x+b*d)^(1/2)/(4*c^2*d^2*x^2+4*b*c*d^2*x+4*a*c*d^2)*a^2+32*c^2*d^7*(2*c*d*x+b*d)^(1/2)/(4*c^2*d^2*x^2+4*b
*c*d^2*x+4*a*c*d^2)*a*b^2-4*c*d^7*(2*c*d*x+b*d)^(1/2)/(4*c^2*d^2*x^2+4*b*c*d^2*x+4*a*c*d^2)*b^4+144*c^3*d^7/(4
*a*c*d^2-b^2*d^2)^(3/4)*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)*a^2-72*c^2*d^7
/(4*a*c*d^2-b^2*d^2)^(3/4)*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)*a*b^2+9*c*d
^7/(4*a*c*d^2-b^2*d^2)^(3/4)*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)*b^4-144*c
^3*d^7/(4*a*c*d^2-b^2*d^2)^(3/4)*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)*a^2+
72*c^2*d^7/(4*a*c*d^2-b^2*d^2)^(3/4)*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)*
a*b^2-9*c*d^7/(4*a*c*d^2-b^2*d^2)^(3/4)*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)*b^4+72*c^3*d^7/(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)))*a^2-36*c^2*d^7/(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)))*a*b^2+9/2*c*d^7/(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)))*b^4

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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)^2,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 details)Is 4*a*c-b^2 positive, negative or zero?

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mupad [B]  time = 0.20, size = 834, normalized size = 4.66 \begin {gather*} \frac {16\,c\,d^3\,{\left (b\,d+2\,c\,d\,x\right )}^{5/2}}{5}-\frac {\sqrt {b\,d+2\,c\,d\,x}\,\left (64\,a^2\,c^3\,d^7-32\,a\,b^2\,c^2\,d^7+4\,b^4\,c\,d^7\right )}{{\left (b\,d+2\,c\,d\,x\right )}^2-b^2\,d^2+4\,a\,c\,d^2}-18\,c\,d^{11/2}\,\mathrm {atan}\left (\frac {9\,c\,d^{11/2}\,{\left (b^2-4\,a\,c\right )}^{5/4}\,\left (\sqrt {b\,d+2\,c\,d\,x}\,\left (1327104\,a^4\,c^6\,d^{14}-1327104\,a^3\,b^2\,c^5\,d^{14}+497664\,a^2\,b^4\,c^4\,d^{14}-82944\,a\,b^6\,c^3\,d^{14}+5184\,b^8\,c^2\,d^{14}\right )-c\,d^{11/2}\,{\left (b^2-4\,a\,c\right )}^{5/4}\,\left (-36864\,a^3\,c^4\,d^9+27648\,a^2\,b^2\,c^3\,d^9-6912\,a\,b^4\,c^2\,d^9+576\,b^6\,c\,d^9\right )\,9{}\mathrm {i}\right )+9\,c\,d^{11/2}\,{\left (b^2-4\,a\,c\right )}^{5/4}\,\left (\sqrt {b\,d+2\,c\,d\,x}\,\left (1327104\,a^4\,c^6\,d^{14}-1327104\,a^3\,b^2\,c^5\,d^{14}+497664\,a^2\,b^4\,c^4\,d^{14}-82944\,a\,b^6\,c^3\,d^{14}+5184\,b^8\,c^2\,d^{14}\right )+c\,d^{11/2}\,{\left (b^2-4\,a\,c\right )}^{5/4}\,\left (-36864\,a^3\,c^4\,d^9+27648\,a^2\,b^2\,c^3\,d^9-6912\,a\,b^4\,c^2\,d^9+576\,b^6\,c\,d^9\right )\,9{}\mathrm {i}\right )}{c\,d^{11/2}\,{\left (b^2-4\,a\,c\right )}^{5/4}\,\left (\sqrt {b\,d+2\,c\,d\,x}\,\left (1327104\,a^4\,c^6\,d^{14}-1327104\,a^3\,b^2\,c^5\,d^{14}+497664\,a^2\,b^4\,c^4\,d^{14}-82944\,a\,b^6\,c^3\,d^{14}+5184\,b^8\,c^2\,d^{14}\right )-c\,d^{11/2}\,{\left (b^2-4\,a\,c\right )}^{5/4}\,\left (-36864\,a^3\,c^4\,d^9+27648\,a^2\,b^2\,c^3\,d^9-6912\,a\,b^4\,c^2\,d^9+576\,b^6\,c\,d^9\right )\,9{}\mathrm {i}\right )\,9{}\mathrm {i}-c\,d^{11/2}\,{\left (b^2-4\,a\,c\right )}^{5/4}\,\left (\sqrt {b\,d+2\,c\,d\,x}\,\left (1327104\,a^4\,c^6\,d^{14}-1327104\,a^3\,b^2\,c^5\,d^{14}+497664\,a^2\,b^4\,c^4\,d^{14}-82944\,a\,b^6\,c^3\,d^{14}+5184\,b^8\,c^2\,d^{14}\right )+c\,d^{11/2}\,{\left (b^2-4\,a\,c\right )}^{5/4}\,\left (-36864\,a^3\,c^4\,d^9+27648\,a^2\,b^2\,c^3\,d^9-6912\,a\,b^4\,c^2\,d^9+576\,b^6\,c\,d^9\right )\,9{}\mathrm {i}\right )\,9{}\mathrm {i}}\right )\,{\left (b^2-4\,a\,c\right )}^{5/4}-32\,c\,d^5\,\sqrt {b\,d+2\,c\,d\,x}\,\left (4\,a\,c-b^2\right )+c\,d^{11/2}\,\mathrm {atan}\left (\frac {b^2\,\sqrt {b\,d+2\,c\,d\,x}\,1{}\mathrm {i}-a\,c\,\sqrt {b\,d+2\,c\,d\,x}\,4{}\mathrm {i}}{\sqrt {d}\,{\left (b^2-4\,a\,c\right )}^{5/4}}\right )\,{\left (b^2-4\,a\,c\right )}^{5/4}\,18{}\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)^2,x)

[Out]

(16*c*d^3*(b*d + 2*c*d*x)^(5/2))/5 - ((b*d + 2*c*d*x)^(1/2)*(4*b^4*c*d^7 + 64*a^2*c^3*d^7 - 32*a*b^2*c^2*d^7))
/((b*d + 2*c*d*x)^2 - b^2*d^2 + 4*a*c*d^2) - 18*c*d^(11/2)*atan((9*c*d^(11/2)*(b^2 - 4*a*c)^(5/4)*((b*d + 2*c*
d*x)^(1/2)*(1327104*a^4*c^6*d^14 + 5184*b^8*c^2*d^14 - 82944*a*b^6*c^3*d^14 + 497664*a^2*b^4*c^4*d^14 - 132710
4*a^3*b^2*c^5*d^14) - c*d^(11/2)*(b^2 - 4*a*c)^(5/4)*(576*b^6*c*d^9 - 36864*a^3*c^4*d^9 - 6912*a*b^4*c^2*d^9 +
27648*a^2*b^2*c^3*d^9)*9i) + 9*c*d^(11/2)*(b^2 - 4*a*c)^(5/4)*((b*d + 2*c*d*x)^(1/2)*(1327104*a^4*c^6*d^14 +
5184*b^8*c^2*d^14 - 82944*a*b^6*c^3*d^14 + 497664*a^2*b^4*c^4*d^14 - 1327104*a^3*b^2*c^5*d^14) + c*d^(11/2)*(b
^2 - 4*a*c)^(5/4)*(576*b^6*c*d^9 - 36864*a^3*c^4*d^9 - 6912*a*b^4*c^2*d^9 + 27648*a^2*b^2*c^3*d^9)*9i))/(c*d^(
11/2)*(b^2 - 4*a*c)^(5/4)*((b*d + 2*c*d*x)^(1/2)*(1327104*a^4*c^6*d^14 + 5184*b^8*c^2*d^14 - 82944*a*b^6*c^3*d
^14 + 497664*a^2*b^4*c^4*d^14 - 1327104*a^3*b^2*c^5*d^14) - c*d^(11/2)*(b^2 - 4*a*c)^(5/4)*(576*b^6*c*d^9 - 36
864*a^3*c^4*d^9 - 6912*a*b^4*c^2*d^9 + 27648*a^2*b^2*c^3*d^9)*9i)*9i - c*d^(11/2)*(b^2 - 4*a*c)^(5/4)*((b*d +
2*c*d*x)^(1/2)*(1327104*a^4*c^6*d^14 + 5184*b^8*c^2*d^14 - 82944*a*b^6*c^3*d^14 + 497664*a^2*b^4*c^4*d^14 - 13
27104*a^3*b^2*c^5*d^14) + c*d^(11/2)*(b^2 - 4*a*c)^(5/4)*(576*b^6*c*d^9 - 36864*a^3*c^4*d^9 - 6912*a*b^4*c^2*d
^9 + 27648*a^2*b^2*c^3*d^9)*9i)*9i))*(b^2 - 4*a*c)^(5/4) - 32*c*d^5*(b*d + 2*c*d*x)^(1/2)*(4*a*c - b^2) + c*d^
(11/2)*atan((b^2*(b*d + 2*c*d*x)^(1/2)*1i - a*c*(b*d + 2*c*d*x)^(1/2)*4i)/(d^(1/2)*(b^2 - 4*a*c)^(5/4)))*(b^2
- 4*a*c)^(5/4)*18i

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sympy [F(-1)]  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)**2,x)

[Out]

Timed out

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