### 3.1319 $$\int \frac{1}{(b d+2 c d x)^{7/2} (a+b x+c x^2)^3} \, dx$$

Optimal. Leaf size=256 $\frac{234 c^2}{d^3 \left (b^2-4 a c\right )^4 \sqrt{b d+2 c d x}}+\frac{117 c^2 \tan ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )}{d^{7/2} \left (b^2-4 a c\right )^{17/4}}-\frac{117 c^2 \tanh ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )}{d^{7/2} \left (b^2-4 a c\right )^{17/4}}+\frac{234 c^2}{5 d \left (b^2-4 a c\right )^3 (b d+2 c d x)^{5/2}}+\frac{13 c}{2 d \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right ) (b d+2 c d x)^{5/2}}-\frac{1}{2 d \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2 (b d+2 c d x)^{5/2}}$

[Out]

(234*c^2)/(5*(b^2 - 4*a*c)^3*d*(b*d + 2*c*d*x)^(5/2)) + (234*c^2)/((b^2 - 4*a*c)^4*d^3*Sqrt[b*d + 2*c*d*x]) -
1/(2*(b^2 - 4*a*c)*d*(b*d + 2*c*d*x)^(5/2)*(a + b*x + c*x^2)^2) + (13*c)/(2*(b^2 - 4*a*c)^2*d*(b*d + 2*c*d*x)^
(5/2)*(a + b*x + c*x^2)) + (117*c^2*ArcTan[Sqrt[d*(b + 2*c*x)]/((b^2 - 4*a*c)^(1/4)*Sqrt[d])])/((b^2 - 4*a*c)^
(17/4)*d^(7/2)) - (117*c^2*ArcTanh[Sqrt[d*(b + 2*c*x)]/((b^2 - 4*a*c)^(1/4)*Sqrt[d])])/((b^2 - 4*a*c)^(17/4)*d
^(7/2))

________________________________________________________________________________________

Rubi [A]  time = 0.227809, antiderivative size = 256, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 26, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.269, Rules used = {687, 693, 694, 329, 298, 203, 206} $\frac{234 c^2}{d^3 \left (b^2-4 a c\right )^4 \sqrt{b d+2 c d x}}+\frac{117 c^2 \tan ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )}{d^{7/2} \left (b^2-4 a c\right )^{17/4}}-\frac{117 c^2 \tanh ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )}{d^{7/2} \left (b^2-4 a c\right )^{17/4}}+\frac{234 c^2}{5 d \left (b^2-4 a c\right )^3 (b d+2 c d x)^{5/2}}+\frac{13 c}{2 d \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right ) (b d+2 c d x)^{5/2}}-\frac{1}{2 d \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2 (b d+2 c d x)^{5/2}}$

Antiderivative was successfully veriﬁed.

[In]

Int[1/((b*d + 2*c*d*x)^(7/2)*(a + b*x + c*x^2)^3),x]

[Out]

(234*c^2)/(5*(b^2 - 4*a*c)^3*d*(b*d + 2*c*d*x)^(5/2)) + (234*c^2)/((b^2 - 4*a*c)^4*d^3*Sqrt[b*d + 2*c*d*x]) -
1/(2*(b^2 - 4*a*c)*d*(b*d + 2*c*d*x)^(5/2)*(a + b*x + c*x^2)^2) + (13*c)/(2*(b^2 - 4*a*c)^2*d*(b*d + 2*c*d*x)^
(5/2)*(a + b*x + c*x^2)) + (117*c^2*ArcTan[Sqrt[d*(b + 2*c*x)]/((b^2 - 4*a*c)^(1/4)*Sqrt[d])])/((b^2 - 4*a*c)^
(17/4)*d^(7/2)) - (117*c^2*ArcTanh[Sqrt[d*(b + 2*c*x)]/((b^2 - 4*a*c)^(1/4)*Sqrt[d])])/((b^2 - 4*a*c)^(17/4)*d
^(7/2))

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 693

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

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{1}{(b d+2 c d x)^{7/2} \left (a+b x+c x^2\right )^3} \, dx &=-\frac{1}{2 \left (b^2-4 a c\right ) d (b d+2 c d x)^{5/2} \left (a+b x+c x^2\right )^2}-\frac{(13 c) \int \frac{1}{(b d+2 c d x)^{7/2} \left (a+b x+c x^2\right )^2} \, dx}{2 \left (b^2-4 a c\right )}\\ &=-\frac{1}{2 \left (b^2-4 a c\right ) d (b d+2 c d x)^{5/2} \left (a+b x+c x^2\right )^2}+\frac{13 c}{2 \left (b^2-4 a c\right )^2 d (b d+2 c d x)^{5/2} \left (a+b x+c x^2\right )}+\frac{\left (117 c^2\right ) \int \frac{1}{(b d+2 c d x)^{7/2} \left (a+b x+c x^2\right )} \, dx}{2 \left (b^2-4 a c\right )^2}\\ &=\frac{234 c^2}{5 \left (b^2-4 a c\right )^3 d (b d+2 c d x)^{5/2}}-\frac{1}{2 \left (b^2-4 a c\right ) d (b d+2 c d x)^{5/2} \left (a+b x+c x^2\right )^2}+\frac{13 c}{2 \left (b^2-4 a c\right )^2 d (b d+2 c d x)^{5/2} \left (a+b x+c x^2\right )}+\frac{\left (117 c^2\right ) \int \frac{1}{(b d+2 c d x)^{3/2} \left (a+b x+c x^2\right )} \, dx}{2 \left (b^2-4 a c\right )^3 d^2}\\ &=\frac{234 c^2}{5 \left (b^2-4 a c\right )^3 d (b d+2 c d x)^{5/2}}+\frac{234 c^2}{\left (b^2-4 a c\right )^4 d^3 \sqrt{b d+2 c d x}}-\frac{1}{2 \left (b^2-4 a c\right ) d (b d+2 c d x)^{5/2} \left (a+b x+c x^2\right )^2}+\frac{13 c}{2 \left (b^2-4 a c\right )^2 d (b d+2 c d x)^{5/2} \left (a+b x+c x^2\right )}+\frac{\left (117 c^2\right ) \int \frac{\sqrt{b d+2 c d x}}{a+b x+c x^2} \, dx}{2 \left (b^2-4 a c\right )^4 d^4}\\ &=\frac{234 c^2}{5 \left (b^2-4 a c\right )^3 d (b d+2 c d x)^{5/2}}+\frac{234 c^2}{\left (b^2-4 a c\right )^4 d^3 \sqrt{b d+2 c d x}}-\frac{1}{2 \left (b^2-4 a c\right ) d (b d+2 c d x)^{5/2} \left (a+b x+c x^2\right )^2}+\frac{13 c}{2 \left (b^2-4 a c\right )^2 d (b d+2 c d x)^{5/2} \left (a+b x+c x^2\right )}+\frac{(117 c) \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 )}{4 \left (b^2-4 a c\right )^4 d^5}\\ &=\frac{234 c^2}{5 \left (b^2-4 a c\right )^3 d (b d+2 c d x)^{5/2}}+\frac{234 c^2}{\left (b^2-4 a c\right )^4 d^3 \sqrt{b d+2 c d x}}-\frac{1}{2 \left (b^2-4 a c\right ) d (b d+2 c d x)^{5/2} \left (a+b x+c x^2\right )^2}+\frac{13 c}{2 \left (b^2-4 a c\right )^2 d (b d+2 c d x)^{5/2} \left (a+b x+c x^2\right )}+\frac{(117 c) \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 )}{2 \left (b^2-4 a c\right )^4 d^5}\\ &=\frac{234 c^2}{5 \left (b^2-4 a c\right )^3 d (b d+2 c d x)^{5/2}}+\frac{234 c^2}{\left (b^2-4 a c\right )^4 d^3 \sqrt{b d+2 c d x}}-\frac{1}{2 \left (b^2-4 a c\right ) d (b d+2 c d x)^{5/2} \left (a+b x+c x^2\right )^2}+\frac{13 c}{2 \left (b^2-4 a c\right )^2 d (b d+2 c d x)^{5/2} \left (a+b x+c x^2\right )}-\frac{\left (117 c^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 )^4 d^3}+\frac{\left (117 c^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 )^4 d^3}\\ &=\frac{234 c^2}{5 \left (b^2-4 a c\right )^3 d (b d+2 c d x)^{5/2}}+\frac{234 c^2}{\left (b^2-4 a c\right )^4 d^3 \sqrt{b d+2 c d x}}-\frac{1}{2 \left (b^2-4 a c\right ) d (b d+2 c d x)^{5/2} \left (a+b x+c x^2\right )^2}+\frac{13 c}{2 \left (b^2-4 a c\right )^2 d (b d+2 c d x)^{5/2} \left (a+b x+c x^2\right )}+\frac{117 c^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 )^{17/4} d^{7/2}}-\frac{117 c^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 )^{17/4} d^{7/2}}\\ \end{align*}

Mathematica [C]  time = 0.0903296, size = 59, normalized size = 0.23 $\frac{64 c^2 \, _2F_1\left (-\frac{5}{4},3;-\frac{1}{4};\frac{(b+2 c x)^2}{b^2-4 a c}\right )}{5 d \left (b^2-4 a c\right )^3 (d (b+2 c x))^{5/2}}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[1/((b*d + 2*c*d*x)^(7/2)*(a + b*x + c*x^2)^3),x]

[Out]

(64*c^2*Hypergeometric2F1[-5/4, 3, -1/4, (b + 2*c*x)^2/(b^2 - 4*a*c)])/(5*(b^2 - 4*a*c)^3*d*(d*(b + 2*c*x))^(5
/2))

________________________________________________________________________________________

Maple [B]  time = 0.21, size = 569, normalized size = 2.2 \begin{align*} -{\frac{64\,{c}^{2}}{5\,d \left ( 4\,ac-{b}^{2} \right ) ^{3}} \left ( 2\,cdx+bd \right ) ^{-{\frac{5}{2}}}}+192\,{\frac{{c}^{2}}{{d}^{3} \left ( 4\,ac-{b}^{2} \right ) ^{4}\sqrt{2\,cdx+bd}}}+42\,{\frac{{c}^{2} \left ( 2\,cdx+bd \right ) ^{7/2}}{{d}^{3} \left ( 4\,ac-{b}^{2} \right ) ^{4} \left ( 4\,{c}^{2}{d}^{2}{x}^{2}+4\,bc{d}^{2}x+4\,ac{d}^{2} \right ) ^{2}}}+200\,{\frac{{c}^{3} \left ( 2\,cdx+bd \right ) ^{3/2}a}{d \left ( 4\,ac-{b}^{2} \right ) ^{4} \left ( 4\,{c}^{2}{d}^{2}{x}^{2}+4\,bc{d}^{2}x+4\,ac{d}^{2} \right ) ^{2}}}-50\,{\frac{{c}^{2} \left ( 2\,cdx+bd \right ) ^{3/2}{b}^{2}}{d \left ( 4\,ac-{b}^{2} \right ) ^{4} \left ( 4\,{c}^{2}{d}^{2}{x}^{2}+4\,bc{d}^{2}x+4\,ac{d}^{2} \right ) ^{2}}}+{\frac{117\,{c}^{2}\sqrt{2}}{4\,{d}^{3} \left ( 4\,ac-{b}^{2} \right ) ^{4}}\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 ){\frac{1}{\sqrt [4]{4\,ac{d}^{2}-{b}^{2}{d}^{2}}}}}+{\frac{117\,{c}^{2}\sqrt{2}}{2\,{d}^{3} \left ( 4\,ac-{b}^{2} \right ) ^{4}}\arctan \left ({\sqrt{2}\sqrt{2\,cdx+bd}{\frac{1}{\sqrt [4]{4\,ac{d}^{2}-{b}^{2}{d}^{2}}}}}+1 \right ){\frac{1}{\sqrt [4]{4\,ac{d}^{2}-{b}^{2}{d}^{2}}}}}-{\frac{117\,{c}^{2}\sqrt{2}}{2\,{d}^{3} \left ( 4\,ac-{b}^{2} \right ) ^{4}}\arctan \left ( -{\sqrt{2}\sqrt{2\,cdx+bd}{\frac{1}{\sqrt [4]{4\,ac{d}^{2}-{b}^{2}{d}^{2}}}}}+1 \right ){\frac{1}{\sqrt [4]{4\,ac{d}^{2}-{b}^{2}{d}^{2}}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(1/(2*c*d*x+b*d)^(7/2)/(c*x^2+b*x+a)^3,x)

[Out]

-64/5*c^2/d/(4*a*c-b^2)^3/(2*c*d*x+b*d)^(5/2)+192*c^2/d^3/(4*a*c-b^2)^4/(2*c*d*x+b*d)^(1/2)+42*c^2/d^3/(4*a*c-
b^2)^4/(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)^(7/2)+200*c^3/d/(4*a*c-b^2)^4/(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)^(3/2)*a-50*c^2/d/(4*a*c-b^2)^4/(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)^(3/2)*b^2+117/4*c^2/d^3/(4*a*c-b^2)^4*2^(1/2)/(4*a*c*d^2-b^2*d^2)^(1/4)*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)))+117/2*c^2/d^3/(4*a*c-b^2)^4*2^(1/2)/(4*a*c*d^2-b^2*d^2)^
(1/4)*arctan(2^(1/2)/(4*a*c*d^2-b^2*d^2)^(1/4)*(2*c*d*x+b*d)^(1/2)+1)-117/2*c^2/d^3/(4*a*c-b^2)^4*2^(1/2)/(4*a
*c*d^2-b^2*d^2)^(1/4)*arctan(-2^(1/2)/(4*a*c*d^2-b^2*d^2)^(1/4)*(2*c*d*x+b*d)^(1/2)+1)

________________________________________________________________________________________

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(1/(2*c*d*x+b*d)^(7/2)/(c*x^2+b*x+a)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [B]  time = 2.70787, size = 11348, normalized size = 44.33 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(2*c*d*x+b*d)^(7/2)/(c*x^2+b*x+a)^3,x, algorithm="fricas")

[Out]

-1/10*(2340*(8*(b^8*c^5 - 16*a*b^6*c^6 + 96*a^2*b^4*c^7 - 256*a^3*b^2*c^8 + 256*a^4*c^9)*d^4*x^7 + 28*(b^9*c^4
- 16*a*b^7*c^5 + 96*a^2*b^5*c^6 - 256*a^3*b^3*c^7 + 256*a^4*b*c^8)*d^4*x^6 + 2*(19*b^10*c^3 - 296*a*b^8*c^4 +
1696*a^2*b^6*c^5 - 4096*a^3*b^4*c^6 + 2816*a^4*b^2*c^7 + 2048*a^5*c^8)*d^4*x^5 + 5*(5*b^11*c^2 - 72*a*b^9*c^3
+ 352*a^2*b^7*c^4 - 512*a^3*b^5*c^5 - 768*a^4*b^3*c^6 + 2048*a^5*b*c^7)*d^4*x^4 + 4*(2*b^12*c - 23*a*b^10*c^2
+ 50*a^2*b^8*c^3 + 320*a^3*b^6*c^4 - 1600*a^4*b^4*c^5 + 1792*a^5*b^2*c^6 + 512*a^6*c^7)*d^4*x^3 + (b^13 - 2*a
*b^11*c - 116*a^2*b^9*c^2 + 896*a^3*b^7*c^3 - 2176*a^4*b^5*c^4 + 512*a^5*b^3*c^5 + 3072*a^6*b*c^6)*d^4*x^2 + 2
*(a*b^12 - 13*a^2*b^10*c + 48*a^3*b^8*c^2 + 32*a^4*b^6*c^3 - 512*a^5*b^4*c^4 + 768*a^6*b^2*c^5)*d^4*x + (a^2*b
^11 - 16*a^3*b^9*c + 96*a^4*b^7*c^2 - 256*a^5*b^5*c^3 + 256*a^6*b^3*c^4)*d^4)*(c^8/((b^34 - 68*a*b^32*c + 2176
*a^2*b^30*c^2 - 43520*a^3*b^28*c^3 + 609280*a^4*b^26*c^4 - 6336512*a^5*b^24*c^5 + 50692096*a^6*b^22*c^6 - 3186
36032*a^7*b^20*c^7 + 1593180160*a^8*b^18*c^8 - 6372720640*a^9*b^16*c^9 + 20392706048*a^10*b^14*c^10 - 51908706
304*a^11*b^12*c^11 + 103817412608*a^12*b^10*c^12 - 159719096320*a^13*b^8*c^13 + 182536110080*a^14*b^6*c^14 - 1
46028888064*a^15*b^4*c^15 + 73014444032*a^16*b^2*c^16 - 17179869184*a^17*c^17)*d^14))^(1/4)*arctan((sqrt(2*c^1
3*d*x + b*c^12*d + (b^18*c^8 - 36*a*b^16*c^9 + 576*a^2*b^14*c^10 - 5376*a^3*b^12*c^11 + 32256*a^4*b^10*c^12 -
129024*a^5*b^8*c^13 + 344064*a^6*b^6*c^14 - 589824*a^7*b^4*c^15 + 589824*a^8*b^2*c^16 - 262144*a^9*c^17)*d^8*s
qrt(c^8/((b^34 - 68*a*b^32*c + 2176*a^2*b^30*c^2 - 43520*a^3*b^28*c^3 + 609280*a^4*b^26*c^4 - 6336512*a^5*b^24
*c^5 + 50692096*a^6*b^22*c^6 - 318636032*a^7*b^20*c^7 + 1593180160*a^8*b^18*c^8 - 6372720640*a^9*b^16*c^9 + 20
392706048*a^10*b^14*c^10 - 51908706304*a^11*b^12*c^11 + 103817412608*a^12*b^10*c^12 - 159719096320*a^13*b^8*c^
13 + 182536110080*a^14*b^6*c^14 - 146028888064*a^15*b^4*c^15 + 73014444032*a^16*b^2*c^16 - 17179869184*a^17*c^
17)*d^14)))*(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)*d^3*(c^8/((b^34 - 68*a*b^32*c
+ 2176*a^2*b^30*c^2 - 43520*a^3*b^28*c^3 + 609280*a^4*b^26*c^4 - 6336512*a^5*b^24*c^5 + 50692096*a^6*b^22*c^6
- 318636032*a^7*b^20*c^7 + 1593180160*a^8*b^18*c^8 - 6372720640*a^9*b^16*c^9 + 20392706048*a^10*b^14*c^10 - 51
908706304*a^11*b^12*c^11 + 103817412608*a^12*b^10*c^12 - 159719096320*a^13*b^8*c^13 + 182536110080*a^14*b^6*c^
14 - 146028888064*a^15*b^4*c^15 + 73014444032*a^16*b^2*c^16 - 17179869184*a^17*c^17)*d^14))^(1/4) - (b^8*c^6 -
16*a*b^6*c^7 + 96*a^2*b^4*c^8 - 256*a^3*b^2*c^9 + 256*a^4*c^10)*sqrt(2*c*d*x + b*d)*d^3*(c^8/((b^34 - 68*a*b^
32*c + 2176*a^2*b^30*c^2 - 43520*a^3*b^28*c^3 + 609280*a^4*b^26*c^4 - 6336512*a^5*b^24*c^5 + 50692096*a^6*b^22
*c^6 - 318636032*a^7*b^20*c^7 + 1593180160*a^8*b^18*c^8 - 6372720640*a^9*b^16*c^9 + 20392706048*a^10*b^14*c^10
- 51908706304*a^11*b^12*c^11 + 103817412608*a^12*b^10*c^12 - 159719096320*a^13*b^8*c^13 + 182536110080*a^14*b
^6*c^14 - 146028888064*a^15*b^4*c^15 + 73014444032*a^16*b^2*c^16 - 17179869184*a^17*c^17)*d^14))^(1/4))/c^8) +
585*(8*(b^8*c^5 - 16*a*b^6*c^6 + 96*a^2*b^4*c^7 - 256*a^3*b^2*c^8 + 256*a^4*c^9)*d^4*x^7 + 28*(b^9*c^4 - 16*a
*b^7*c^5 + 96*a^2*b^5*c^6 - 256*a^3*b^3*c^7 + 256*a^4*b*c^8)*d^4*x^6 + 2*(19*b^10*c^3 - 296*a*b^8*c^4 + 1696*a
^2*b^6*c^5 - 4096*a^3*b^4*c^6 + 2816*a^4*b^2*c^7 + 2048*a^5*c^8)*d^4*x^5 + 5*(5*b^11*c^2 - 72*a*b^9*c^3 + 352*
a^2*b^7*c^4 - 512*a^3*b^5*c^5 - 768*a^4*b^3*c^6 + 2048*a^5*b*c^7)*d^4*x^4 + 4*(2*b^12*c - 23*a*b^10*c^2 + 50*a
^2*b^8*c^3 + 320*a^3*b^6*c^4 - 1600*a^4*b^4*c^5 + 1792*a^5*b^2*c^6 + 512*a^6*c^7)*d^4*x^3 + (b^13 - 2*a*b^11*c
- 116*a^2*b^9*c^2 + 896*a^3*b^7*c^3 - 2176*a^4*b^5*c^4 + 512*a^5*b^3*c^5 + 3072*a^6*b*c^6)*d^4*x^2 + 2*(a*b^1
2 - 13*a^2*b^10*c + 48*a^3*b^8*c^2 + 32*a^4*b^6*c^3 - 512*a^5*b^4*c^4 + 768*a^6*b^2*c^5)*d^4*x + (a^2*b^11 - 1
6*a^3*b^9*c + 96*a^4*b^7*c^2 - 256*a^5*b^5*c^3 + 256*a^6*b^3*c^4)*d^4)*(c^8/((b^34 - 68*a*b^32*c + 2176*a^2*b^
30*c^2 - 43520*a^3*b^28*c^3 + 609280*a^4*b^26*c^4 - 6336512*a^5*b^24*c^5 + 50692096*a^6*b^22*c^6 - 318636032*a
^7*b^20*c^7 + 1593180160*a^8*b^18*c^8 - 6372720640*a^9*b^16*c^9 + 20392706048*a^10*b^14*c^10 - 51908706304*a^1
1*b^12*c^11 + 103817412608*a^12*b^10*c^12 - 159719096320*a^13*b^8*c^13 + 182536110080*a^14*b^6*c^14 - 14602888
8064*a^15*b^4*c^15 + 73014444032*a^16*b^2*c^16 - 17179869184*a^17*c^17)*d^14))^(1/4)*log(1601613*(b^26 - 52*a*
b^24*c + 1248*a^2*b^22*c^2 - 18304*a^3*b^20*c^3 + 183040*a^4*b^18*c^4 - 1317888*a^5*b^16*c^5 + 7028736*a^6*b^1
4*c^6 - 28114944*a^7*b^12*c^7 + 84344832*a^8*b^10*c^8 - 187432960*a^9*b^8*c^9 + 299892736*a^10*b^6*c^10 - 3271
55712*a^11*b^4*c^11 + 218103808*a^12*b^2*c^12 - 67108864*a^13*c^13)*d^11*(c^8/((b^34 - 68*a*b^32*c + 2176*a^2*
b^30*c^2 - 43520*a^3*b^28*c^3 + 609280*a^4*b^26*c^4 - 6336512*a^5*b^24*c^5 + 50692096*a^6*b^22*c^6 - 318636032
*a^7*b^20*c^7 + 1593180160*a^8*b^18*c^8 - 6372720640*a^9*b^16*c^9 + 20392706048*a^10*b^14*c^10 - 51908706304*a
^11*b^12*c^11 + 103817412608*a^12*b^10*c^12 - 159719096320*a^13*b^8*c^13 + 182536110080*a^14*b^6*c^14 - 146028
888064*a^15*b^4*c^15 + 73014444032*a^16*b^2*c^16 - 17179869184*a^17*c^17)*d^14))^(3/4) + 1601613*sqrt(2*c*d*x
+ b*d)*c^6) - 585*(8*(b^8*c^5 - 16*a*b^6*c^6 + 96*a^2*b^4*c^7 - 256*a^3*b^2*c^8 + 256*a^4*c^9)*d^4*x^7 + 28*(b
^9*c^4 - 16*a*b^7*c^5 + 96*a^2*b^5*c^6 - 256*a^3*b^3*c^7 + 256*a^4*b*c^8)*d^4*x^6 + 2*(19*b^10*c^3 - 296*a*b^8
*c^4 + 1696*a^2*b^6*c^5 - 4096*a^3*b^4*c^6 + 2816*a^4*b^2*c^7 + 2048*a^5*c^8)*d^4*x^5 + 5*(5*b^11*c^2 - 72*a*b
^9*c^3 + 352*a^2*b^7*c^4 - 512*a^3*b^5*c^5 - 768*a^4*b^3*c^6 + 2048*a^5*b*c^7)*d^4*x^4 + 4*(2*b^12*c - 23*a*b^
10*c^2 + 50*a^2*b^8*c^3 + 320*a^3*b^6*c^4 - 1600*a^4*b^4*c^5 + 1792*a^5*b^2*c^6 + 512*a^6*c^7)*d^4*x^3 + (b^13
- 2*a*b^11*c - 116*a^2*b^9*c^2 + 896*a^3*b^7*c^3 - 2176*a^4*b^5*c^4 + 512*a^5*b^3*c^5 + 3072*a^6*b*c^6)*d^4*x
^2 + 2*(a*b^12 - 13*a^2*b^10*c + 48*a^3*b^8*c^2 + 32*a^4*b^6*c^3 - 512*a^5*b^4*c^4 + 768*a^6*b^2*c^5)*d^4*x +
(a^2*b^11 - 16*a^3*b^9*c + 96*a^4*b^7*c^2 - 256*a^5*b^5*c^3 + 256*a^6*b^3*c^4)*d^4)*(c^8/((b^34 - 68*a*b^32*c
+ 2176*a^2*b^30*c^2 - 43520*a^3*b^28*c^3 + 609280*a^4*b^26*c^4 - 6336512*a^5*b^24*c^5 + 50692096*a^6*b^22*c^6
- 318636032*a^7*b^20*c^7 + 1593180160*a^8*b^18*c^8 - 6372720640*a^9*b^16*c^9 + 20392706048*a^10*b^14*c^10 - 51
908706304*a^11*b^12*c^11 + 103817412608*a^12*b^10*c^12 - 159719096320*a^13*b^8*c^13 + 182536110080*a^14*b^6*c^
14 - 146028888064*a^15*b^4*c^15 + 73014444032*a^16*b^2*c^16 - 17179869184*a^17*c^17)*d^14))^(1/4)*log(-1601613
*(b^26 - 52*a*b^24*c + 1248*a^2*b^22*c^2 - 18304*a^3*b^20*c^3 + 183040*a^4*b^18*c^4 - 1317888*a^5*b^16*c^5 + 7
028736*a^6*b^14*c^6 - 28114944*a^7*b^12*c^7 + 84344832*a^8*b^10*c^8 - 187432960*a^9*b^8*c^9 + 299892736*a^10*b
^6*c^10 - 327155712*a^11*b^4*c^11 + 218103808*a^12*b^2*c^12 - 67108864*a^13*c^13)*d^11*(c^8/((b^34 - 68*a*b^32
*c + 2176*a^2*b^30*c^2 - 43520*a^3*b^28*c^3 + 609280*a^4*b^26*c^4 - 6336512*a^5*b^24*c^5 + 50692096*a^6*b^22*c
^6 - 318636032*a^7*b^20*c^7 + 1593180160*a^8*b^18*c^8 - 6372720640*a^9*b^16*c^9 + 20392706048*a^10*b^14*c^10 -
51908706304*a^11*b^12*c^11 + 103817412608*a^12*b^10*c^12 - 159719096320*a^13*b^8*c^13 + 182536110080*a^14*b^6
*c^14 - 146028888064*a^15*b^4*c^15 + 73014444032*a^16*b^2*c^16 - 17179869184*a^17*c^17)*d^14))^(3/4) + 1601613
*sqrt(2*c*d*x + b*d)*c^6) - (9360*c^6*x^6 + 28080*b*c^5*x^5 - 5*b^6 + 125*a*b^4*c + 2048*a^2*b^2*c^2 - 512*a^3
*c^3 + 2808*(11*b^2*c^4 + 6*a*c^5)*x^4 + 3744*(4*b^3*c^3 + 9*a*b*c^4)*x^3 + 13*(221*b^4*c^2 + 1688*a*b^2*c^3 +
512*a^2*c^4)*x^2 + 13*(5*b^5*c + 392*a*b^3*c^2 + 512*a^2*b*c^3)*x)*sqrt(2*c*d*x + b*d))/(8*(b^8*c^5 - 16*a*b^
6*c^6 + 96*a^2*b^4*c^7 - 256*a^3*b^2*c^8 + 256*a^4*c^9)*d^4*x^7 + 28*(b^9*c^4 - 16*a*b^7*c^5 + 96*a^2*b^5*c^6
- 256*a^3*b^3*c^7 + 256*a^4*b*c^8)*d^4*x^6 + 2*(19*b^10*c^3 - 296*a*b^8*c^4 + 1696*a^2*b^6*c^5 - 4096*a^3*b^4*
c^6 + 2816*a^4*b^2*c^7 + 2048*a^5*c^8)*d^4*x^5 + 5*(5*b^11*c^2 - 72*a*b^9*c^3 + 352*a^2*b^7*c^4 - 512*a^3*b^5*
c^5 - 768*a^4*b^3*c^6 + 2048*a^5*b*c^7)*d^4*x^4 + 4*(2*b^12*c - 23*a*b^10*c^2 + 50*a^2*b^8*c^3 + 320*a^3*b^6*c
^4 - 1600*a^4*b^4*c^5 + 1792*a^5*b^2*c^6 + 512*a^6*c^7)*d^4*x^3 + (b^13 - 2*a*b^11*c - 116*a^2*b^9*c^2 + 896*a
^3*b^7*c^3 - 2176*a^4*b^5*c^4 + 512*a^5*b^3*c^5 + 3072*a^6*b*c^6)*d^4*x^2 + 2*(a*b^12 - 13*a^2*b^10*c + 48*a^3
*b^8*c^2 + 32*a^4*b^6*c^3 - 512*a^5*b^4*c^4 + 768*a^6*b^2*c^5)*d^4*x + (a^2*b^11 - 16*a^3*b^9*c + 96*a^4*b^7*c
^2 - 256*a^5*b^5*c^3 + 256*a^6*b^3*c^4)*d^4)

________________________________________________________________________________________

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(1/(2*c*d*x+b*d)**(7/2)/(c*x**2+b*x+a)**3,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [B]  time = 1.32044, size = 1293, normalized size = 5.05 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(2*c*d*x+b*d)^(7/2)/(c*x^2+b*x+a)^3,x, algorithm="giac")

[Out]

-117*(-b^2*d^2 + 4*a*c*d^2)^(3/4)*c^2*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^10*d^5 - 20*sqrt(2)*a*b^8*c*d^5 + 160*sqrt(2)*a^2*b^6*c^2*d
^5 - 640*sqrt(2)*a^3*b^4*c^3*d^5 + 1280*sqrt(2)*a^4*b^2*c^4*d^5 - 1024*sqrt(2)*a^5*c^5*d^5) - 117*(-b^2*d^2 +
4*a*c*d^2)^(3/4)*c^2*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^10*d^5 - 20*sqrt(2)*a*b^8*c*d^5 + 160*sqrt(2)*a^2*b^6*c^2*d^5 - 640*sqrt(2)
*a^3*b^4*c^3*d^5 + 1280*sqrt(2)*a^4*b^2*c^4*d^5 - 1024*sqrt(2)*a^5*c^5*d^5) + 117/2*(-b^2*d^2 + 4*a*c*d^2)^(3/
4)*c^2*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^10*d^5 - 20*sqrt(2)*a*b^8*c*d^5 + 160*sqrt(2)*a^2*b^6*c^2*d^5 - 640*sqrt(2)*a^3*b^4*c^3*d^5 + 1
280*sqrt(2)*a^4*b^2*c^4*d^5 - 1024*sqrt(2)*a^5*c^5*d^5) - 117/2*(-b^2*d^2 + 4*a*c*d^2)^(3/4)*c^2*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^10*d^
5 - 20*sqrt(2)*a*b^8*c*d^5 + 160*sqrt(2)*a^2*b^6*c^2*d^5 - 640*sqrt(2)*a^3*b^4*c^3*d^5 + 1280*sqrt(2)*a^4*b^2*
c^4*d^5 - 1024*sqrt(2)*a^5*c^5*d^5) - 2*(25*(2*c*d*x + b*d)^(3/2)*b^2*c^2*d^2 - 100*(2*c*d*x + b*d)^(3/2)*a*c^
3*d^2 - 21*(2*c*d*x + b*d)^(7/2)*c^2)/((b^8*d^3 - 16*a*b^6*c*d^3 + 96*a^2*b^4*c^2*d^3 - 256*a^3*b^2*c^3*d^3 +
256*a^4*c^4*d^3)*(b^2*d^2 - 4*a*c*d^2 - (2*c*d*x + b*d)^2)^2) + 64/5*(b^2*c^2*d^2 - 4*a*c^3*d^2 + 15*(2*c*d*x
+ b*d)^2*c^2)/((b^8*d^3 - 16*a*b^6*c*d^3 + 96*a^2*b^4*c^2*d^3 - 256*a^3*b^2*c^3*d^3 + 256*a^4*c^4*d^3)*(2*c*d*
x + b*d)^(5/2))