### 3.1296 $$\int \frac{(b d+2 c d x)^{13/2}}{(a+b x+c x^2)^2} \, dx$$

Optimal. Leaf size=181 $\frac{44}{3} c d^5 \left (b^2-4 a c\right ) (b d+2 c d x)^{3/2}+22 c d^{13/2} \left (b^2-4 a c\right )^{7/4} \tan ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )-22 c d^{13/2} \left (b^2-4 a c\right )^{7/4} \tanh ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )-\frac{d (b d+2 c d x)^{11/2}}{a+b x+c x^2}+\frac{44}{7} c d^3 (b d+2 c d x)^{7/2}$

[Out]

(44*c*(b^2 - 4*a*c)*d^5*(b*d + 2*c*d*x)^(3/2))/3 + (44*c*d^3*(b*d + 2*c*d*x)^(7/2))/7 - (d*(b*d + 2*c*d*x)^(11
/2))/(a + b*x + c*x^2) + 22*c*(b^2 - 4*a*c)^(7/4)*d^(13/2)*ArcTan[Sqrt[d*(b + 2*c*x)]/((b^2 - 4*a*c)^(1/4)*Sqr
t[d])] - 22*c*(b^2 - 4*a*c)^(7/4)*d^(13/2)*ArcTanh[Sqrt[d*(b + 2*c*x)]/((b^2 - 4*a*c)^(1/4)*Sqrt[d])]

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Rubi [A]  time = 0.159613, antiderivative size = 181, normalized size of antiderivative = 1., 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, 298, 203, 206} $\frac{44}{3} c d^5 \left (b^2-4 a c\right ) (b d+2 c d x)^{3/2}+22 c d^{13/2} \left (b^2-4 a c\right )^{7/4} \tan ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )-22 c d^{13/2} \left (b^2-4 a c\right )^{7/4} \tanh ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )-\frac{d (b d+2 c d x)^{11/2}}{a+b x+c x^2}+\frac{44}{7} c d^3 (b d+2 c d x)^{7/2}$

Antiderivative was successfully veriﬁed.

[In]

Int[(b*d + 2*c*d*x)^(13/2)/(a + b*x + c*x^2)^2,x]

[Out]

(44*c*(b^2 - 4*a*c)*d^5*(b*d + 2*c*d*x)^(3/2))/3 + (44*c*d^3*(b*d + 2*c*d*x)^(7/2))/7 - (d*(b*d + 2*c*d*x)^(11
/2))/(a + b*x + c*x^2) + 22*c*(b^2 - 4*a*c)^(7/4)*d^(13/2)*ArcTan[Sqrt[d*(b + 2*c*x)]/((b^2 - 4*a*c)^(1/4)*Sqr
t[d])] - 22*c*(b^2 - 4*a*c)^(7/4)*d^(13/2)*ArcTanh[Sqrt[d*(b + 2*c*x)]/((b^2 - 4*a*c)^(1/4)*Sqrt[d])]

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]

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 298

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

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])

Rubi steps

\begin{align*} \int \frac{(b d+2 c d x)^{13/2}}{\left (a+b x+c x^2\right )^2} \, dx &=-\frac{d (b d+2 c d x)^{11/2}}{a+b x+c x^2}+\left (11 c d^2\right ) \int \frac{(b d+2 c d x)^{9/2}}{a+b x+c x^2} \, dx\\ &=\frac{44}{7} c d^3 (b d+2 c d x)^{7/2}-\frac{d (b d+2 c d x)^{11/2}}{a+b x+c x^2}+\left (11 c \left (b^2-4 a c\right ) d^4\right ) \int \frac{(b d+2 c d x)^{5/2}}{a+b x+c x^2} \, dx\\ &=\frac{44}{3} c \left (b^2-4 a c\right ) d^5 (b d+2 c d x)^{3/2}+\frac{44}{7} c d^3 (b d+2 c d x)^{7/2}-\frac{d (b d+2 c d x)^{11/2}}{a+b x+c x^2}+\left (11 c \left (b^2-4 a c\right )^2 d^6\right ) \int \frac{\sqrt{b d+2 c d x}}{a+b x+c x^2} \, dx\\ &=\frac{44}{3} c \left (b^2-4 a c\right ) d^5 (b d+2 c d x)^{3/2}+\frac{44}{7} c d^3 (b d+2 c d x)^{7/2}-\frac{d (b d+2 c d x)^{11/2}}{a+b x+c x^2}+\frac{1}{2} \left (11 \left (b^2-4 a c\right )^2 d^5\right ) \operatorname{Subst}\left (\int \frac{\sqrt{x}}{a-\frac{b^2}{4 c}+\frac{x^2}{4 c d^2}} \, dx,x,b d+2 c d x\right )\\ &=\frac{44}{3} c \left (b^2-4 a c\right ) d^5 (b d+2 c d x)^{3/2}+\frac{44}{7} c d^3 (b d+2 c d x)^{7/2}-\frac{d (b d+2 c d x)^{11/2}}{a+b x+c x^2}+\left (11 \left (b^2-4 a c\right )^2 d^5\right ) \operatorname{Subst}\left (\int \frac{x^2}{a-\frac{b^2}{4 c}+\frac{x^4}{4 c d^2}} \, dx,x,\sqrt{d (b+2 c x)}\right )\\ &=\frac{44}{3} c \left (b^2-4 a c\right ) d^5 (b d+2 c d x)^{3/2}+\frac{44}{7} c d^3 (b d+2 c d x)^{7/2}-\frac{d (b d+2 c d x)^{11/2}}{a+b x+c x^2}-\left (22 c \left (b^2-4 a c\right )^2 d^7\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 (22 c \left (b^2-4 a c\right )^2 d^7\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 )\\ &=\frac{44}{3} c \left (b^2-4 a c\right ) d^5 (b d+2 c d x)^{3/2}+\frac{44}{7} c d^3 (b d+2 c d x)^{7/2}-\frac{d (b d+2 c d x)^{11/2}}{a+b x+c x^2}+22 c \left (b^2-4 a c\right )^{7/4} d^{13/2} \tan ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )-22 c \left (b^2-4 a c\right )^{7/4} d^{13/2} \tanh ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\\ \end{align*}

Mathematica [C]  time = 0.183084, size = 171, normalized size = 0.94 $\frac{4 d^5 (d (b+2 c x))^{3/2} \left (308 c \left (4 a^2 c+a \left (-b^2+4 b c x+4 c^2 x^2\right )-b^2 x (b+c x)\right ) \, _2F_1\left (\frac{3}{4},2;\frac{7}{4};\frac{(b+2 c x)^2}{b^2-4 a c}\right )+16 c^2 \left (-77 a^2-11 a c x^2+3 c^2 x^4\right )+4 b^2 c \left (143 a+29 c x^2\right )+16 b c^2 x \left (6 c x^2-11 a\right )+68 b^3 c x-63 b^4\right )}{21 (a+x (b+c x))}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(b*d + 2*c*d*x)^(13/2)/(a + b*x + c*x^2)^2,x]

[Out]

(4*d^5*(d*(b + 2*c*x))^(3/2)*(-63*b^4 + 68*b^3*c*x + 16*b*c^2*x*(-11*a + 6*c*x^2) + 4*b^2*c*(143*a + 29*c*x^2)
+ 16*c^2*(-77*a^2 - 11*a*c*x^2 + 3*c^2*x^4) + 308*c*(4*a^2*c - b^2*x*(b + c*x) + a*(-b^2 + 4*b*c*x + 4*c^2*x^
2))*Hypergeometric2F1[3/4, 2, 7/4, (b + 2*c*x)^2/(b^2 - 4*a*c)]))/(21*(a + x*(b + c*x)))

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Maple [B]  time = 0.202, size = 1090, normalized size = 6. \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((2*c*d*x+b*d)^(13/2)/(c*x^2+b*x+a)^2,x)

[Out]

16/7*c*d^3*(2*c*d*x+b*d)^(7/2)-128/3*c^2*d^5*(2*c*d*x+b*d)^(3/2)*a+32/3*c*d^5*(2*c*d*x+b*d)^(3/2)*b^2-64*c^3*d
^7*(2*c*d*x+b*d)^(3/2)/(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)^(3/2)/(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)^(3/2)/(4*c^2*d^2*x^2+4*b*c*d^2*x+4*a*c*d^2)*b^4+88*c^3*d^7
*2^(1/2)/(4*a*c*d^2-b^2*d^2)^(1/4)*a^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)))+176*c^3*d^7*2^(1/2)/(4*a*c*d^2-b^2*d^2)^(1/4)*a^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)-176*c^3*d^7*2^(1/2)/(4*a*c*d^2-b^2*d^2)^(1/4)*a^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)-44*c^2*d^7*2^(1/2)/(4*a*c*d^2-b^2*d^2)^(1/4)*a*b^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)))-88*c^2*d^7*2^(1/2)/(4*a*c*d^2-b^2*d^2)^(1/4)*a*b^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)+88*c^2*d^7*2^(1/2)/(4*a*c*d^2-b^2*d^2)^(1/4)*a*b^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)+11/2*c*d^7*2^(1/2)/(4*a*c*d^2-b^2*d^2)^(1/4)*b^4*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)))+11*c*d^7*2^(1/2)/(4*a*c*d^2-b^2*d^2
)^(1/4)*b^4*arctan(2^(1/2)/(4*a*c*d^2-b^2*d^2)^(1/4)*(2*c*d*x+b*d)^(1/2)+1)-11*c*d^7*2^(1/2)/(4*a*c*d^2-b^2*d^
2)^(1/4)*b^4*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., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*d*x+b*d)^(13/2)/(c*x^2+b*x+a)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.30889, size = 3710, normalized size = 20.5 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*d*x+b*d)^(13/2)/(c*x^2+b*x+a)^2,x, algorithm="fricas")

[Out]

-1/21*(924*((b^14*c^4 - 28*a*b^12*c^5 + 336*a^2*b^10*c^6 - 2240*a^3*b^8*c^7 + 8960*a^4*b^6*c^8 - 21504*a^5*b^4
*c^9 + 28672*a^6*b^2*c^10 - 16384*a^7*c^11)*d^26)^(1/4)*(c*x^2 + b*x + a)*arctan(-(((b^14*c^4 - 28*a*b^12*c^5
+ 336*a^2*b^10*c^6 - 2240*a^3*b^8*c^7 + 8960*a^4*b^6*c^8 - 21504*a^5*b^4*c^9 + 28672*a^6*b^2*c^10 - 16384*a^7*
c^11)*d^26)^(1/4)*(b^10*c^3 - 20*a*b^8*c^4 + 160*a^2*b^6*c^5 - 640*a^3*b^4*c^6 + 1280*a^4*b^2*c^7 - 1024*a^5*c
^8)*sqrt(2*c*d*x + b*d)*d^19 + sqrt(2*(b^20*c^7 - 40*a*b^18*c^8 + 720*a^2*b^16*c^9 - 7680*a^3*b^14*c^10 + 5376
0*a^4*b^12*c^11 - 258048*a^5*b^10*c^12 + 860160*a^6*b^8*c^13 - 1966080*a^7*b^6*c^14 + 2949120*a^8*b^4*c^15 - 2
621440*a^9*b^2*c^16 + 1048576*a^10*c^17)*d^39*x + (b^21*c^6 - 40*a*b^19*c^7 + 720*a^2*b^17*c^8 - 7680*a^3*b^15
*c^9 + 53760*a^4*b^13*c^10 - 258048*a^5*b^11*c^11 + 860160*a^6*b^9*c^12 - 1966080*a^7*b^7*c^13 + 2949120*a^8*b
^5*c^14 - 2621440*a^9*b^3*c^15 + 1048576*a^10*b*c^16)*d^39 + sqrt((b^14*c^4 - 28*a*b^12*c^5 + 336*a^2*b^10*c^6
- 2240*a^3*b^8*c^7 + 8960*a^4*b^6*c^8 - 21504*a^5*b^4*c^9 + 28672*a^6*b^2*c^10 - 16384*a^7*c^11)*d^26)*(b^14*
c^4 - 28*a*b^12*c^5 + 336*a^2*b^10*c^6 - 2240*a^3*b^8*c^7 + 8960*a^4*b^6*c^8 - 21504*a^5*b^4*c^9 + 28672*a^6*b
^2*c^10 - 16384*a^7*c^11)*d^26)*((b^14*c^4 - 28*a*b^12*c^5 + 336*a^2*b^10*c^6 - 2240*a^3*b^8*c^7 + 8960*a^4*b^
6*c^8 - 21504*a^5*b^4*c^9 + 28672*a^6*b^2*c^10 - 16384*a^7*c^11)*d^26)^(1/4))/((b^14*c^4 - 28*a*b^12*c^5 + 336
*a^2*b^10*c^6 - 2240*a^3*b^8*c^7 + 8960*a^4*b^6*c^8 - 21504*a^5*b^4*c^9 + 28672*a^6*b^2*c^10 - 16384*a^7*c^11)
*d^26)) - 231*((b^14*c^4 - 28*a*b^12*c^5 + 336*a^2*b^10*c^6 - 2240*a^3*b^8*c^7 + 8960*a^4*b^6*c^8 - 21504*a^5*
b^4*c^9 + 28672*a^6*b^2*c^10 - 16384*a^7*c^11)*d^26)^(1/4)*(c*x^2 + b*x + a)*log(-1331*(b^10*c^3 - 20*a*b^8*c^
4 + 160*a^2*b^6*c^5 - 640*a^3*b^4*c^6 + 1280*a^4*b^2*c^7 - 1024*a^5*c^8)*sqrt(2*c*d*x + b*d)*d^19 + 1331*((b^1
4*c^4 - 28*a*b^12*c^5 + 336*a^2*b^10*c^6 - 2240*a^3*b^8*c^7 + 8960*a^4*b^6*c^8 - 21504*a^5*b^4*c^9 + 28672*a^6
*b^2*c^10 - 16384*a^7*c^11)*d^26)^(3/4)) + 231*((b^14*c^4 - 28*a*b^12*c^5 + 336*a^2*b^10*c^6 - 2240*a^3*b^8*c^
7 + 8960*a^4*b^6*c^8 - 21504*a^5*b^4*c^9 + 28672*a^6*b^2*c^10 - 16384*a^7*c^11)*d^26)^(1/4)*(c*x^2 + b*x + a)*
log(-1331*(b^10*c^3 - 20*a*b^8*c^4 + 160*a^2*b^6*c^5 - 640*a^3*b^4*c^6 + 1280*a^4*b^2*c^7 - 1024*a^5*c^8)*sqrt
(2*c*d*x + b*d)*d^19 - 1331*((b^14*c^4 - 28*a*b^12*c^5 + 336*a^2*b^10*c^6 - 2240*a^3*b^8*c^7 + 8960*a^4*b^6*c^
8 - 21504*a^5*b^4*c^9 + 28672*a^6*b^2*c^10 - 16384*a^7*c^11)*d^26)^(3/4)) - (384*c^5*d^6*x^5 + 960*b*c^4*d^6*x
^4 + 32*(41*b^2*c^3 - 44*a*c^4)*d^6*x^3 + 48*(21*b^3*c^2 - 44*a*b*c^3)*d^6*x^2 + 2*(115*b^4*c + 88*a*b^2*c^2 -
1232*a^2*c^3)*d^6*x - (21*b^5 - 440*a*b^3*c + 1232*a^2*b*c^2)*d^6)*sqrt(2*c*d*x + b*d))/(c*x^2 + b*x + a)

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*d*x+b*d)**(13/2)/(c*x**2+b*x+a)**2,x)

[Out]

Timed out

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Giac [B]  time = 1.31144, size = 767, normalized size = 4.24 \begin{align*} \frac{32}{3} \,{\left (2 \, c d x + b d\right )}^{\frac{3}{2}} b^{2} c d^{5} - \frac{128}{3} \,{\left (2 \, c d x + b d\right )}^{\frac{3}{2}} a c^{2} d^{5} + \frac{16}{7} \,{\left (2 \, c d x + b d\right )}^{\frac{7}{2}} c d^{3} + \frac{11}{2} \, \sqrt{2}{\left (b^{2} c d^{5} - 4 \, a c^{2} d^{5}\right )}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{3}{4}} \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{11}{2} \, \sqrt{2}{\left (b^{2} c d^{5} - 4 \, a c^{2} d^{5}\right )}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{3}{4}} \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 ) - 11 \,{\left (\sqrt{2} b^{2} c d^{5} - 4 \, \sqrt{2} a c^{2} d^{5}\right )}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{3}{4}} \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 ) - 11 \,{\left (\sqrt{2} b^{2} c d^{5} - 4 \, \sqrt{2} a c^{2} d^{5}\right )}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{3}{4}} \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{4 \,{\left ({\left (2 \, c d x + b d\right )}^{\frac{3}{2}} b^{4} c d^{7} - 8 \,{\left (2 \, c d x + b d\right )}^{\frac{3}{2}} a b^{2} c^{2} d^{7} + 16 \,{\left (2 \, c d x + b d\right )}^{\frac{3}{2}} 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{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*d*x+b*d)^(13/2)/(c*x^2+b*x+a)^2,x, algorithm="giac")

[Out]

32/3*(2*c*d*x + b*d)^(3/2)*b^2*c*d^5 - 128/3*(2*c*d*x + b*d)^(3/2)*a*c^2*d^5 + 16/7*(2*c*d*x + b*d)^(7/2)*c*d^
3 + 11/2*sqrt(2)*(b^2*c*d^5 - 4*a*c^2*d^5)*(-b^2*d^2 + 4*a*c*d^2)^(3/4)*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)) - 11/2*sqrt(2)*(b^2*c*d^5 - 4*a*c^2*d^5)*
(-b^2*d^2 + 4*a*c*d^2)^(3/4)*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) + sq
rt(-b^2*d^2 + 4*a*c*d^2)) - 11*(sqrt(2)*b^2*c*d^5 - 4*sqrt(2)*a*c^2*d^5)*(-b^2*d^2 + 4*a*c*d^2)^(3/4)*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)) - 11*(
sqrt(2)*b^2*c*d^5 - 4*sqrt(2)*a*c^2*d^5)*(-b^2*d^2 + 4*a*c*d^2)^(3/4)*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)) + 4*((2*c*d*x + b*d)^(3/2)*b^4*c*d^7
- 8*(2*c*d*x + b*d)^(3/2)*a*b^2*c^2*d^7 + 16*(2*c*d*x + b*d)^(3/2)*a^2*c^3*d^7)/(b^2*d^2 - 4*a*c*d^2 - (2*c*d*
x + b*d)^2)