### 3.1314 $$\int \frac{(b d+2 c d x)^{3/2}}{(a+b x+c x^2)^3} \, dx$$

Optimal. Leaf size=178 $\frac{3 c^2 d^{3/2} \tan ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{7/4}}+\frac{3 c^2 d^{3/2} \tanh ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{7/4}}-\frac{c d \sqrt{b d+2 c d x}}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac{d \sqrt{b d+2 c d x}}{2 \left (a+b x+c x^2\right )^2}$

[Out]

-(d*Sqrt[b*d + 2*c*d*x])/(2*(a + b*x + c*x^2)^2) - (c*d*Sqrt[b*d + 2*c*d*x])/(2*(b^2 - 4*a*c)*(a + b*x + c*x^2
)) + (3*c^2*d^(3/2)*ArcTan[Sqrt[d*(b + 2*c*x)]/((b^2 - 4*a*c)^(1/4)*Sqrt[d])])/(b^2 - 4*a*c)^(7/4) + (3*c^2*d^
(3/2)*ArcTanh[Sqrt[d*(b + 2*c*x)]/((b^2 - 4*a*c)^(1/4)*Sqrt[d])])/(b^2 - 4*a*c)^(7/4)

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Rubi [A]  time = 0.128863, antiderivative size = 178, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 26, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.269, Rules used = {686, 687, 694, 329, 212, 206, 203} $\frac{3 c^2 d^{3/2} \tan ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{7/4}}+\frac{3 c^2 d^{3/2} \tanh ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{7/4}}-\frac{c d \sqrt{b d+2 c d x}}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac{d \sqrt{b d+2 c d x}}{2 \left (a+b x+c x^2\right )^2}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-(d*Sqrt[b*d + 2*c*d*x])/(2*(a + b*x + c*x^2)^2) - (c*d*Sqrt[b*d + 2*c*d*x])/(2*(b^2 - 4*a*c)*(a + b*x + c*x^2
)) + (3*c^2*d^(3/2)*ArcTan[Sqrt[d*(b + 2*c*x)]/((b^2 - 4*a*c)^(1/4)*Sqrt[d])])/(b^2 - 4*a*c)^(7/4) + (3*c^2*d^
(3/2)*ArcTanh[Sqrt[d*(b + 2*c*x)]/((b^2 - 4*a*c)^(1/4)*Sqrt[d])])/(b^2 - 4*a*c)^(7/4)

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 687

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

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 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 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 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])

Rubi steps

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

Mathematica [A]  time = 0.378265, size = 171, normalized size = 0.96 $\frac{(d (b+2 c x))^{3/2} \left (\frac{8 c (a+x (b+c x)) \left (-\left (b^2-4 a c\right )^{3/4} \sqrt{b+2 c x}+6 c (a+x (b+c x)) \tan ^{-1}\left (\frac{\sqrt{b+2 c x}}{\sqrt [4]{b^2-4 a c}}\right )+6 c (a+x (b+c x)) \tanh ^{-1}\left (\frac{\sqrt{b+2 c x}}{\sqrt [4]{b^2-4 a c}}\right )\right )}{\left (b^2-4 a c\right )^{7/4}}-8 \sqrt{b+2 c x}\right )}{16 (b+2 c x)^{3/2} (a+x (b+c x))^2}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

((d*(b + 2*c*x))^(3/2)*(-8*Sqrt[b + 2*c*x] + (8*c*(a + x*(b + c*x))*(-((b^2 - 4*a*c)^(3/4)*Sqrt[b + 2*c*x]) +
6*c*(a + x*(b + c*x))*ArcTan[Sqrt[b + 2*c*x]/(b^2 - 4*a*c)^(1/4)] + 6*c*(a + x*(b + c*x))*ArcTanh[Sqrt[b + 2*c
*x]/(b^2 - 4*a*c)^(1/4)]))/(b^2 - 4*a*c)^(7/4)))/(16*(b + 2*c*x)^(3/2)*(a + x*(b + c*x))^2)

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Maple [B]  time = 0.2, size = 431, normalized size = 2.4 \begin{align*} 2\,{\frac{{c}^{2}{d}^{3} \left ( 2\,cdx+bd \right ) ^{5/2}}{ \left ( 4\,{c}^{2}{d}^{2}{x}^{2}+4\,bc{d}^{2}x+4\,ac{d}^{2} \right ) ^{2} \left ( 4\,ac-{b}^{2} \right ) }}-6\,{\frac{{c}^{2}{d}^{5}\sqrt{2\,cdx+bd}}{ \left ( 4\,{c}^{2}{d}^{2}{x}^{2}+4\,bc{d}^{2}x+4\,ac{d}^{2} \right ) ^{2}}}+{\frac{3\,{c}^{2}{d}^{3}\sqrt{2}}{16\,ac-4\,{b}^{2}}\ln \left ({ \left ( 2\,cdx+bd+\sqrt [4]{4\,ac{d}^{2}-{b}^{2}{d}^{2}}\sqrt{2\,cdx+bd}\sqrt{2}+\sqrt{4\,ac{d}^{2}-{b}^{2}{d}^{2}} \right ) \left ( 2\,cdx+bd-\sqrt [4]{4\,ac{d}^{2}-{b}^{2}{d}^{2}}\sqrt{2\,cdx+bd}\sqrt{2}+\sqrt{4\,ac{d}^{2}-{b}^{2}{d}^{2}} \right ) ^{-1}} \right ) \left ( 4\,ac{d}^{2}-{b}^{2}{d}^{2} \right ) ^{-{\frac{3}{4}}}}+{\frac{3\,{c}^{2}{d}^{3}\sqrt{2}}{8\,ac-2\,{b}^{2}}\arctan \left ({\sqrt{2}\sqrt{2\,cdx+bd}{\frac{1}{\sqrt [4]{4\,ac{d}^{2}-{b}^{2}{d}^{2}}}}}+1 \right ) \left ( 4\,ac{d}^{2}-{b}^{2}{d}^{2} \right ) ^{-{\frac{3}{4}}}}-{\frac{3\,{c}^{2}{d}^{3}\sqrt{2}}{8\,ac-2\,{b}^{2}}\arctan \left ( -{\sqrt{2}\sqrt{2\,cdx+bd}{\frac{1}{\sqrt [4]{4\,ac{d}^{2}-{b}^{2}{d}^{2}}}}}+1 \right ) \left ( 4\,ac{d}^{2}-{b}^{2}{d}^{2} \right ) ^{-{\frac{3}{4}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.07073, size = 3089, normalized size = 17.35 \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)^(3/2)/(c*x^2+b*x+a)^3,x, algorithm="fricas")

[Out]

1/2*(12*(c^8*d^6/(b^14 - 28*a*b^12*c + 336*a^2*b^10*c^2 - 2240*a^3*b^8*c^3 + 8960*a^4*b^6*c^4 - 21504*a^5*b^4*
c^5 + 28672*a^6*b^2*c^6 - 16384*a^7*c^7))^(1/4)*((b^2*c^2 - 4*a*c^3)*x^4 + a^2*b^2 - 4*a^3*c + 2*(b^3*c - 4*a*
b*c^2)*x^3 + (b^4 - 2*a*b^2*c - 8*a^2*c^2)*x^2 + 2*(a*b^3 - 4*a^2*b*c)*x)*arctan(((c^8*d^6/(b^14 - 28*a*b^12*c
+ 336*a^2*b^10*c^2 - 2240*a^3*b^8*c^3 + 8960*a^4*b^6*c^4 - 21504*a^5*b^4*c^5 + 28672*a^6*b^2*c^6 - 16384*a^7*
c^7))^(3/4)*(b^10*c^2 - 20*a*b^8*c^3 + 160*a^2*b^6*c^4 - 640*a^3*b^4*c^5 + 1280*a^4*b^2*c^6 - 1024*a^5*c^7)*sq
rt(2*c*d*x + b*d)*d - (c^8*d^6/(b^14 - 28*a*b^12*c + 336*a^2*b^10*c^2 - 2240*a^3*b^8*c^3 + 8960*a^4*b^6*c^4 -
21504*a^5*b^4*c^5 + 28672*a^6*b^2*c^6 - 16384*a^7*c^7))^(3/4)*(b^10 - 20*a*b^8*c + 160*a^2*b^6*c^2 - 640*a^3*b
^4*c^3 + 1280*a^4*b^2*c^4 - 1024*a^5*c^5)*sqrt(2*c^5*d^3*x + b*c^4*d^3 + sqrt(c^8*d^6/(b^14 - 28*a*b^12*c + 33
6*a^2*b^10*c^2 - 2240*a^3*b^8*c^3 + 8960*a^4*b^6*c^4 - 21504*a^5*b^4*c^5 + 28672*a^6*b^2*c^6 - 16384*a^7*c^7))
*(b^8 - 16*a*b^6*c + 96*a^2*b^4*c^2 - 256*a^3*b^2*c^3 + 256*a^4*c^4)))/(c^8*d^6)) + 3*(c^8*d^6/(b^14 - 28*a*b^
12*c + 336*a^2*b^10*c^2 - 2240*a^3*b^8*c^3 + 8960*a^4*b^6*c^4 - 21504*a^5*b^4*c^5 + 28672*a^6*b^2*c^6 - 16384*
a^7*c^7))^(1/4)*((b^2*c^2 - 4*a*c^3)*x^4 + a^2*b^2 - 4*a^3*c + 2*(b^3*c - 4*a*b*c^2)*x^3 + (b^4 - 2*a*b^2*c -
8*a^2*c^2)*x^2 + 2*(a*b^3 - 4*a^2*b*c)*x)*log(3*sqrt(2*c*d*x + b*d)*c^2*d + 3*(c^8*d^6/(b^14 - 28*a*b^12*c + 3
36*a^2*b^10*c^2 - 2240*a^3*b^8*c^3 + 8960*a^4*b^6*c^4 - 21504*a^5*b^4*c^5 + 28672*a^6*b^2*c^6 - 16384*a^7*c^7)
)^(1/4)*(b^4 - 8*a*b^2*c + 16*a^2*c^2)) - 3*(c^8*d^6/(b^14 - 28*a*b^12*c + 336*a^2*b^10*c^2 - 2240*a^3*b^8*c^3
+ 8960*a^4*b^6*c^4 - 21504*a^5*b^4*c^5 + 28672*a^6*b^2*c^6 - 16384*a^7*c^7))^(1/4)*((b^2*c^2 - 4*a*c^3)*x^4 +
a^2*b^2 - 4*a^3*c + 2*(b^3*c - 4*a*b*c^2)*x^3 + (b^4 - 2*a*b^2*c - 8*a^2*c^2)*x^2 + 2*(a*b^3 - 4*a^2*b*c)*x)*
log(3*sqrt(2*c*d*x + b*d)*c^2*d - 3*(c^8*d^6/(b^14 - 28*a*b^12*c + 336*a^2*b^10*c^2 - 2240*a^3*b^8*c^3 + 8960*
a^4*b^6*c^4 - 21504*a^5*b^4*c^5 + 28672*a^6*b^2*c^6 - 16384*a^7*c^7))^(1/4)*(b^4 - 8*a*b^2*c + 16*a^2*c^2)) -
(c^2*d*x^2 + b*c*d*x + (b^2 - 3*a*c)*d)*sqrt(2*c*d*x + b*d))/((b^2*c^2 - 4*a*c^3)*x^4 + a^2*b^2 - 4*a^3*c + 2*
(b^3*c - 4*a*b*c^2)*x^3 + (b^4 - 2*a*b^2*c - 8*a^2*c^2)*x^2 + 2*(a*b^3 - 4*a^2*b*c)*x)

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

[Out]

Timed out

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Giac [B]  time = 1.23079, size = 765, normalized size = 4.3 \begin{align*} \frac{3 \,{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} c^{2} d \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 )}{\sqrt{2} b^{4} - 8 \, \sqrt{2} a b^{2} c + 16 \, \sqrt{2} a^{2} c^{2}} + \frac{3 \,{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} c^{2} d \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 )}{\sqrt{2} b^{4} - 8 \, \sqrt{2} a b^{2} c + 16 \, \sqrt{2} a^{2} c^{2}} + \frac{3 \,{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} c^{2} d \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 )}{2 \,{\left (\sqrt{2} b^{4} - 8 \, \sqrt{2} a b^{2} c + 16 \, \sqrt{2} a^{2} c^{2}\right )}} - \frac{3 \,{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} c^{2} d \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 )}{2 \,{\left (\sqrt{2} b^{4} - 8 \, \sqrt{2} a b^{2} c + 16 \, \sqrt{2} a^{2} c^{2}\right )}} - \frac{2 \,{\left (3 \, \sqrt{2 \, c d x + b d} b^{2} c^{2} d^{5} - 12 \, \sqrt{2 \, c d x + b d} a c^{3} d^{5} +{\left (2 \, c d x + b d\right )}^{\frac{5}{2}} c^{2} d^{3}\right )}}{{\left (b^{2} d^{2} - 4 \, a c d^{2} -{\left (2 \, c d x + b d\right )}^{2}\right )}^{2}{\left (b^{2} - 4 \, a c\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*(-b^2*d^2 + 4*a*c*d^2)^(1/4)*c^2*d*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^4 - 8*sqrt(2)*a*b^2*c + 16*sqrt(2)*a^2*c^2) + 3*(-b^2*d^2 +
4*a*c*d^2)^(1/4)*c^2*d*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^4 - 8*sqrt(2)*a*b^2*c + 16*sqrt(2)*a^2*c^2) + 3/2*(-b^2*d^2 + 4*a*c*d^2)^
(1/4)*c^2*d*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))/(sqrt(2)*b^4 - 8*sqrt(2)*a*b^2*c + 16*sqrt(2)*a^2*c^2) - 3/2*(-b^2*d^2 + 4*a*c*d^2)^(1/4)*c^2*d*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))/(sqrt(2)
*b^4 - 8*sqrt(2)*a*b^2*c + 16*sqrt(2)*a^2*c^2) - 2*(3*sqrt(2*c*d*x + b*d)*b^2*c^2*d^5 - 12*sqrt(2*c*d*x + b*d)
*a*c^3*d^5 + (2*c*d*x + b*d)^(5/2)*c^2*d^3)/((b^2*d^2 - 4*a*c*d^2 - (2*c*d*x + b*d)^2)^2*(b^2 - 4*a*c))