3.1318 \(\int \frac {1}{(b d+2 c d x)^{5/2} (a+b x+c x^2)^3} \, dx\)
Optimal. Leaf size=225 \[ -\frac {77 c^2 \tan ^{-1}\left (\frac {\sqrt {d (b+2 c x)}}{\sqrt {d} \sqrt [4]{b^2-4 a c}}\right )}{d^{5/2} \left (b^2-4 a c\right )^{15/4}}-\frac {77 c^2 \tanh ^{-1}\left (\frac {\sqrt {d (b+2 c x)}}{\sqrt {d} \sqrt [4]{b^2-4 a c}}\right )}{d^{5/2} \left (b^2-4 a c\right )^{15/4}}+\frac {154 c^2}{3 d \left (b^2-4 a c\right )^3 (b d+2 c d x)^{3/2}}+\frac {11 c}{2 d \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right ) (b d+2 c d x)^{3/2}}-\frac {1}{2 d \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2 (b d+2 c d x)^{3/2}} \]
[Out]
154/3*c^2/(-4*a*c+b^2)^3/d/(2*c*d*x+b*d)^(3/2)-1/2/(-4*a*c+b^2)/d/(2*c*d*x+b*d)^(3/2)/(c*x^2+b*x+a)^2+11/2*c/(
-4*a*c+b^2)^2/d/(2*c*d*x+b*d)^(3/2)/(c*x^2+b*x+a)-77*c^2*arctan((d*(2*c*x+b))^(1/2)/(-4*a*c+b^2)^(1/4)/d^(1/2)
)/(-4*a*c+b^2)^(15/4)/d^(5/2)-77*c^2*arctanh((d*(2*c*x+b))^(1/2)/(-4*a*c+b^2)^(1/4)/d^(1/2))/(-4*a*c+b^2)^(15/
4)/d^(5/2)
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Rubi [A] time = 0.19, antiderivative size = 225, 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 =
{687, 693, 694, 329, 212, 206, 203} \[ -\frac {77 c^2 \tan ^{-1}\left (\frac {\sqrt {d (b+2 c x)}}{\sqrt {d} \sqrt [4]{b^2-4 a c}}\right )}{d^{5/2} \left (b^2-4 a c\right )^{15/4}}-\frac {77 c^2 \tanh ^{-1}\left (\frac {\sqrt {d (b+2 c x)}}{\sqrt {d} \sqrt [4]{b^2-4 a c}}\right )}{d^{5/2} \left (b^2-4 a c\right )^{15/4}}+\frac {154 c^2}{3 d \left (b^2-4 a c\right )^3 (b d+2 c d x)^{3/2}}+\frac {11 c}{2 d \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right ) (b d+2 c d x)^{3/2}}-\frac {1}{2 d \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2 (b d+2 c d x)^{3/2}} \]
Antiderivative was successfully verified.
[In]
Int[1/((b*d + 2*c*d*x)^(5/2)*(a + b*x + c*x^2)^3),x]
[Out]
(154*c^2)/(3*(b^2 - 4*a*c)^3*d*(b*d + 2*c*d*x)^(3/2)) - 1/(2*(b^2 - 4*a*c)*d*(b*d + 2*c*d*x)^(3/2)*(a + b*x +
c*x^2)^2) + (11*c)/(2*(b^2 - 4*a*c)^2*d*(b*d + 2*c*d*x)^(3/2)*(a + b*x + c*x^2)) - (77*c^2*ArcTan[Sqrt[d*(b +
2*c*x)]/((b^2 - 4*a*c)^(1/4)*Sqrt[d])])/((b^2 - 4*a*c)^(15/4)*d^(5/2)) - (77*c^2*ArcTanh[Sqrt[d*(b + 2*c*x)]/(
(b^2 - 4*a*c)^(1/4)*Sqrt[d])])/((b^2 - 4*a*c)^(15/4)*d^(5/2))
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 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]
Rubi steps
\begin {align*} \int \frac {1}{(b d+2 c d x)^{5/2} \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)^{3/2} \left (a+b x+c x^2\right )^2}-\frac {(11 c) \int \frac {1}{(b d+2 c d x)^{5/2} \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)^{3/2} \left (a+b x+c x^2\right )^2}+\frac {11 c}{2 \left (b^2-4 a c\right )^2 d (b d+2 c d x)^{3/2} \left (a+b x+c x^2\right )}+\frac {\left (77 c^2\right ) \int \frac {1}{(b d+2 c d x)^{5/2} \left (a+b x+c x^2\right )} \, dx}{2 \left (b^2-4 a c\right )^2}\\ &=\frac {154 c^2}{3 \left (b^2-4 a c\right )^3 d (b d+2 c d x)^{3/2}}-\frac {1}{2 \left (b^2-4 a c\right ) d (b d+2 c d x)^{3/2} \left (a+b x+c x^2\right )^2}+\frac {11 c}{2 \left (b^2-4 a c\right )^2 d (b d+2 c d x)^{3/2} \left (a+b x+c x^2\right )}+\frac {\left (77 c^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 )^3 d^2}\\ &=\frac {154 c^2}{3 \left (b^2-4 a c\right )^3 d (b d+2 c d x)^{3/2}}-\frac {1}{2 \left (b^2-4 a c\right ) d (b d+2 c d x)^{3/2} \left (a+b x+c x^2\right )^2}+\frac {11 c}{2 \left (b^2-4 a c\right )^2 d (b d+2 c d x)^{3/2} \left (a+b x+c x^2\right )}+\frac {(77 c) \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 )^3 d^3}\\ &=\frac {154 c^2}{3 \left (b^2-4 a c\right )^3 d (b d+2 c d x)^{3/2}}-\frac {1}{2 \left (b^2-4 a c\right ) d (b d+2 c d x)^{3/2} \left (a+b x+c x^2\right )^2}+\frac {11 c}{2 \left (b^2-4 a c\right )^2 d (b d+2 c d x)^{3/2} \left (a+b x+c x^2\right )}+\frac {(77 c) \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 )^3 d^3}\\ &=\frac {154 c^2}{3 \left (b^2-4 a c\right )^3 d (b d+2 c d x)^{3/2}}-\frac {1}{2 \left (b^2-4 a c\right ) d (b d+2 c d x)^{3/2} \left (a+b x+c x^2\right )^2}+\frac {11 c}{2 \left (b^2-4 a c\right )^2 d (b d+2 c d x)^{3/2} \left (a+b x+c x^2\right )}-\frac {\left (77 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 )^{7/2} d^2}-\frac {\left (77 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 )^{7/2} d^2}\\ &=\frac {154 c^2}{3 \left (b^2-4 a c\right )^3 d (b d+2 c d x)^{3/2}}-\frac {1}{2 \left (b^2-4 a c\right ) d (b d+2 c d x)^{3/2} \left (a+b x+c x^2\right )^2}+\frac {11 c}{2 \left (b^2-4 a c\right )^2 d (b d+2 c d x)^{3/2} \left (a+b x+c x^2\right )}-\frac {77 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 )^{15/4} d^{5/2}}-\frac {77 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 )^{15/4} d^{5/2}}\\ \end {align*}
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Mathematica [C] time = 0.07, size = 59, normalized size = 0.26 \[ \frac {64 c^2 \, _2F_1\left (-\frac {3}{4},3;\frac {1}{4};\frac {(b+2 c x)^2}{b^2-4 a c}\right )}{3 d \left (b^2-4 a c\right )^3 (d (b+2 c x))^{3/2}} \]
Antiderivative was successfully verified.
[In]
Integrate[1/((b*d + 2*c*d*x)^(5/2)*(a + b*x + c*x^2)^3),x]
[Out]
(64*c^2*Hypergeometric2F1[-3/4, 3, 1/4, (b + 2*c*x)^2/(b^2 - 4*a*c)])/(3*(b^2 - 4*a*c)^3*d*(d*(b + 2*c*x))^(3/
2))
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fricas [B] time = 0.99, size = 3401, normalized size = 15.12 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(2*c*d*x+b*d)^(5/2)/(c*x^2+b*x+a)^3,x, algorithm="fricas")
[Out]
-1/6*(924*(4*(b^6*c^4 - 12*a*b^4*c^5 + 48*a^2*b^2*c^6 - 64*a^3*c^7)*d^3*x^6 + 12*(b^7*c^3 - 12*a*b^5*c^4 + 48*
a^2*b^3*c^5 - 64*a^3*b*c^6)*d^3*x^5 + (13*b^8*c^2 - 148*a*b^6*c^3 + 528*a^2*b^4*c^4 - 448*a^3*b^2*c^5 - 512*a^
4*c^6)*d^3*x^4 + 2*(3*b^9*c - 28*a*b^7*c^2 + 48*a^2*b^5*c^3 + 192*a^3*b^3*c^4 - 512*a^4*b*c^5)*d^3*x^3 + (b^10
- 2*a*b^8*c - 68*a^2*b^6*c^2 + 368*a^3*b^4*c^3 - 448*a^4*b^2*c^4 - 256*a^5*c^5)*d^3*x^2 + 2*(a*b^9 - 10*a^2*b
^7*c + 24*a^3*b^5*c^2 + 32*a^4*b^3*c^3 - 128*a^5*b*c^4)*d^3*x + (a^2*b^8 - 12*a^3*b^6*c + 48*a^4*b^4*c^2 - 64*
a^5*b^2*c^3)*d^3)*(c^8/((b^30 - 60*a*b^28*c + 1680*a^2*b^26*c^2 - 29120*a^3*b^24*c^3 + 349440*a^4*b^22*c^4 - 3
075072*a^5*b^20*c^5 + 20500480*a^6*b^18*c^6 - 105431040*a^7*b^16*c^7 + 421724160*a^8*b^14*c^8 - 1312030720*a^9
*b^12*c^9 + 3148873728*a^10*b^10*c^10 - 5725224960*a^11*b^8*c^11 + 7633633280*a^12*b^6*c^12 - 7046430720*a^13*
b^4*c^13 + 4026531840*a^14*b^2*c^14 - 1073741824*a^15*c^15)*d^10))^(1/4)*arctan(-((b^22 - 44*a*b^20*c + 880*a^
2*b^18*c^2 - 10560*a^3*b^16*c^3 + 84480*a^4*b^14*c^4 - 473088*a^5*b^12*c^5 + 1892352*a^6*b^10*c^6 - 5406720*a^
7*b^8*c^7 + 10813440*a^8*b^6*c^8 - 14417920*a^9*b^4*c^9 + 11534336*a^10*b^2*c^10 - 4194304*a^11*c^11)*sqrt((b^
16 - 32*a*b^14*c + 448*a^2*b^12*c^2 - 3584*a^3*b^10*c^3 + 17920*a^4*b^8*c^4 - 57344*a^5*b^6*c^5 + 114688*a^6*b
^4*c^6 - 131072*a^7*b^2*c^7 + 65536*a^8*c^8)*d^6*sqrt(c^8/((b^30 - 60*a*b^28*c + 1680*a^2*b^26*c^2 - 29120*a^3
*b^24*c^3 + 349440*a^4*b^22*c^4 - 3075072*a^5*b^20*c^5 + 20500480*a^6*b^18*c^6 - 105431040*a^7*b^16*c^7 + 4217
24160*a^8*b^14*c^8 - 1312030720*a^9*b^12*c^9 + 3148873728*a^10*b^10*c^10 - 5725224960*a^11*b^8*c^11 + 76336332
80*a^12*b^6*c^12 - 7046430720*a^13*b^4*c^13 + 4026531840*a^14*b^2*c^14 - 1073741824*a^15*c^15)*d^10)) + 2*c^5*
d*x + b*c^4*d)*d^7*(c^8/((b^30 - 60*a*b^28*c + 1680*a^2*b^26*c^2 - 29120*a^3*b^24*c^3 + 349440*a^4*b^22*c^4 -
3075072*a^5*b^20*c^5 + 20500480*a^6*b^18*c^6 - 105431040*a^7*b^16*c^7 + 421724160*a^8*b^14*c^8 - 1312030720*a^
9*b^12*c^9 + 3148873728*a^10*b^10*c^10 - 5725224960*a^11*b^8*c^11 + 7633633280*a^12*b^6*c^12 - 7046430720*a^13
*b^4*c^13 + 4026531840*a^14*b^2*c^14 - 1073741824*a^15*c^15)*d^10))^(3/4) - (b^22*c^2 - 44*a*b^20*c^3 + 880*a^
2*b^18*c^4 - 10560*a^3*b^16*c^5 + 84480*a^4*b^14*c^6 - 473088*a^5*b^12*c^7 + 1892352*a^6*b^10*c^8 - 5406720*a^
7*b^8*c^9 + 10813440*a^8*b^6*c^10 - 14417920*a^9*b^4*c^11 + 11534336*a^10*b^2*c^12 - 4194304*a^11*c^13)*sqrt(2
*c*d*x + b*d)*d^7*(c^8/((b^30 - 60*a*b^28*c + 1680*a^2*b^26*c^2 - 29120*a^3*b^24*c^3 + 349440*a^4*b^22*c^4 - 3
075072*a^5*b^20*c^5 + 20500480*a^6*b^18*c^6 - 105431040*a^7*b^16*c^7 + 421724160*a^8*b^14*c^8 - 1312030720*a^9
*b^12*c^9 + 3148873728*a^10*b^10*c^10 - 5725224960*a^11*b^8*c^11 + 7633633280*a^12*b^6*c^12 - 7046430720*a^13*
b^4*c^13 + 4026531840*a^14*b^2*c^14 - 1073741824*a^15*c^15)*d^10))^(3/4))/c^8) + 231*(4*(b^6*c^4 - 12*a*b^4*c^
5 + 48*a^2*b^2*c^6 - 64*a^3*c^7)*d^3*x^6 + 12*(b^7*c^3 - 12*a*b^5*c^4 + 48*a^2*b^3*c^5 - 64*a^3*b*c^6)*d^3*x^5
+ (13*b^8*c^2 - 148*a*b^6*c^3 + 528*a^2*b^4*c^4 - 448*a^3*b^2*c^5 - 512*a^4*c^6)*d^3*x^4 + 2*(3*b^9*c - 28*a*
b^7*c^2 + 48*a^2*b^5*c^3 + 192*a^3*b^3*c^4 - 512*a^4*b*c^5)*d^3*x^3 + (b^10 - 2*a*b^8*c - 68*a^2*b^6*c^2 + 368
*a^3*b^4*c^3 - 448*a^4*b^2*c^4 - 256*a^5*c^5)*d^3*x^2 + 2*(a*b^9 - 10*a^2*b^7*c + 24*a^3*b^5*c^2 + 32*a^4*b^3*
c^3 - 128*a^5*b*c^4)*d^3*x + (a^2*b^8 - 12*a^3*b^6*c + 48*a^4*b^4*c^2 - 64*a^5*b^2*c^3)*d^3)*(c^8/((b^30 - 60*
a*b^28*c + 1680*a^2*b^26*c^2 - 29120*a^3*b^24*c^3 + 349440*a^4*b^22*c^4 - 3075072*a^5*b^20*c^5 + 20500480*a^6*
b^18*c^6 - 105431040*a^7*b^16*c^7 + 421724160*a^8*b^14*c^8 - 1312030720*a^9*b^12*c^9 + 3148873728*a^10*b^10*c^
10 - 5725224960*a^11*b^8*c^11 + 7633633280*a^12*b^6*c^12 - 7046430720*a^13*b^4*c^13 + 4026531840*a^14*b^2*c^14
- 1073741824*a^15*c^15)*d^10))^(1/4)*log(77*(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^30 - 60*a*b^28*c + 1680*a^2*b^26*c^2 - 29120*a^3*b^24*c^3 + 349440*a^4*b^22*c^4 - 3075072*a^5*
b^20*c^5 + 20500480*a^6*b^18*c^6 - 105431040*a^7*b^16*c^7 + 421724160*a^8*b^14*c^8 - 1312030720*a^9*b^12*c^9 +
3148873728*a^10*b^10*c^10 - 5725224960*a^11*b^8*c^11 + 7633633280*a^12*b^6*c^12 - 7046430720*a^13*b^4*c^13 +
4026531840*a^14*b^2*c^14 - 1073741824*a^15*c^15)*d^10))^(1/4) + 77*sqrt(2*c*d*x + b*d)*c^2) - 231*(4*(b^6*c^4
- 12*a*b^4*c^5 + 48*a^2*b^2*c^6 - 64*a^3*c^7)*d^3*x^6 + 12*(b^7*c^3 - 12*a*b^5*c^4 + 48*a^2*b^3*c^5 - 64*a^3*b
*c^6)*d^3*x^5 + (13*b^8*c^2 - 148*a*b^6*c^3 + 528*a^2*b^4*c^4 - 448*a^3*b^2*c^5 - 512*a^4*c^6)*d^3*x^4 + 2*(3*
b^9*c - 28*a*b^7*c^2 + 48*a^2*b^5*c^3 + 192*a^3*b^3*c^4 - 512*a^4*b*c^5)*d^3*x^3 + (b^10 - 2*a*b^8*c - 68*a^2*
b^6*c^2 + 368*a^3*b^4*c^3 - 448*a^4*b^2*c^4 - 256*a^5*c^5)*d^3*x^2 + 2*(a*b^9 - 10*a^2*b^7*c + 24*a^3*b^5*c^2
+ 32*a^4*b^3*c^3 - 128*a^5*b*c^4)*d^3*x + (a^2*b^8 - 12*a^3*b^6*c + 48*a^4*b^4*c^2 - 64*a^5*b^2*c^3)*d^3)*(c^8
/((b^30 - 60*a*b^28*c + 1680*a^2*b^26*c^2 - 29120*a^3*b^24*c^3 + 349440*a^4*b^22*c^4 - 3075072*a^5*b^20*c^5 +
20500480*a^6*b^18*c^6 - 105431040*a^7*b^16*c^7 + 421724160*a^8*b^14*c^8 - 1312030720*a^9*b^12*c^9 + 3148873728
*a^10*b^10*c^10 - 5725224960*a^11*b^8*c^11 + 7633633280*a^12*b^6*c^12 - 7046430720*a^13*b^4*c^13 + 4026531840*
a^14*b^2*c^14 - 1073741824*a^15*c^15)*d^10))^(1/4)*log(-77*(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^30 - 60*a*b^28*c + 1680*a^2*b^26*c^2 - 29120*a^3*b^24*c^3 + 349440*a^4*b^22*c^4
- 3075072*a^5*b^20*c^5 + 20500480*a^6*b^18*c^6 - 105431040*a^7*b^16*c^7 + 421724160*a^8*b^14*c^8 - 1312030720*
a^9*b^12*c^9 + 3148873728*a^10*b^10*c^10 - 5725224960*a^11*b^8*c^11 + 7633633280*a^12*b^6*c^12 - 7046430720*a^
13*b^4*c^13 + 4026531840*a^14*b^2*c^14 - 1073741824*a^15*c^15)*d^10))^(1/4) + 77*sqrt(2*c*d*x + b*d)*c^2) - (3
08*c^4*x^4 + 616*b*c^3*x^3 - 3*b^4 + 57*a*b^2*c + 128*a^2*c^2 + 11*(31*b^2*c^2 + 44*a*c^3)*x^2 + 11*(3*b^3*c +
44*a*b*c^2)*x)*sqrt(2*c*d*x + b*d))/(4*(b^6*c^4 - 12*a*b^4*c^5 + 48*a^2*b^2*c^6 - 64*a^3*c^7)*d^3*x^6 + 12*(b
^7*c^3 - 12*a*b^5*c^4 + 48*a^2*b^3*c^5 - 64*a^3*b*c^6)*d^3*x^5 + (13*b^8*c^2 - 148*a*b^6*c^3 + 528*a^2*b^4*c^4
- 448*a^3*b^2*c^5 - 512*a^4*c^6)*d^3*x^4 + 2*(3*b^9*c - 28*a*b^7*c^2 + 48*a^2*b^5*c^3 + 192*a^3*b^3*c^4 - 512
*a^4*b*c^5)*d^3*x^3 + (b^10 - 2*a*b^8*c - 68*a^2*b^6*c^2 + 368*a^3*b^4*c^3 - 448*a^4*b^2*c^4 - 256*a^5*c^5)*d^
3*x^2 + 2*(a*b^9 - 10*a^2*b^7*c + 24*a^3*b^5*c^2 + 32*a^4*b^3*c^3 - 128*a^5*b*c^4)*d^3*x + (a^2*b^8 - 12*a^3*b
^6*c + 48*a^4*b^4*c^2 - 64*a^5*b^2*c^3)*d^3)
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giac [B] time = 0.35, size = 813, normalized size = 3.61 \[ -\frac {77 \, {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} c^{2} \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^{8} d^{3} - 16 \, \sqrt {2} a b^{6} c d^{3} + 96 \, \sqrt {2} a^{2} b^{4} c^{2} d^{3} - 256 \, \sqrt {2} a^{3} b^{2} c^{3} d^{3} + 256 \, \sqrt {2} a^{4} c^{4} d^{3}} - \frac {77 \, {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} c^{2} \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^{8} d^{3} - 16 \, \sqrt {2} a b^{6} c d^{3} + 96 \, \sqrt {2} a^{2} b^{4} c^{2} d^{3} - 256 \, \sqrt {2} a^{3} b^{2} c^{3} d^{3} + 256 \, \sqrt {2} a^{4} c^{4} d^{3}} - \frac {77 \, {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} c^{2} \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^{8} d^{3} - 16 \, \sqrt {2} a b^{6} c d^{3} + 96 \, \sqrt {2} a^{2} b^{4} c^{2} d^{3} - 256 \, \sqrt {2} a^{3} b^{2} c^{3} d^{3} + 256 \, \sqrt {2} a^{4} c^{4} d^{3}\right )}} + \frac {77 \, {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} c^{2} \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^{8} d^{3} - 16 \, \sqrt {2} a b^{6} c d^{3} + 96 \, \sqrt {2} a^{2} b^{4} c^{2} d^{3} - 256 \, \sqrt {2} a^{3} b^{2} c^{3} d^{3} + 256 \, \sqrt {2} a^{4} c^{4} d^{3}\right )}} + \frac {64 \, c^{2}}{3 \, {\left (b^{6} d - 12 \, a b^{4} c d + 48 \, a^{2} b^{2} c^{2} d - 64 \, a^{3} c^{3} d\right )} {\left (2 \, c d x + b d\right )}^{\frac {3}{2}}} - \frac {2 \, {\left (19 \, \sqrt {2 \, c d x + b d} b^{2} c^{2} d^{2} - 76 \, \sqrt {2 \, c d x + b d} a c^{3} d^{2} - 15 \, {\left (2 \, c d x + b d\right )}^{\frac {5}{2}} c^{2}\right )}}{{\left (b^{6} d - 12 \, a b^{4} c d + 48 \, a^{2} b^{2} c^{2} d - 64 \, a^{3} c^{3} d\right )} {\left (b^{2} d^{2} - 4 \, a c d^{2} - {\left (2 \, c d x + b d\right )}^{2}\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(2*c*d*x+b*d)^(5/2)/(c*x^2+b*x+a)^3,x, algorithm="giac")
[Out]
-77*(-b^2*d^2 + 4*a*c*d^2)^(1/4)*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^8*d^3 - 16*sqrt(2)*a*b^6*c*d^3 + 96*sqrt(2)*a^2*b^4*c^2*d^3
- 256*sqrt(2)*a^3*b^2*c^3*d^3 + 256*sqrt(2)*a^4*c^4*d^3) - 77*(-b^2*d^2 + 4*a*c*d^2)^(1/4)*c^2*arctan(-1/2*sqr
t(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^8
*d^3 - 16*sqrt(2)*a*b^6*c*d^3 + 96*sqrt(2)*a^2*b^4*c^2*d^3 - 256*sqrt(2)*a^3*b^2*c^3*d^3 + 256*sqrt(2)*a^4*c^4
*d^3) - 77/2*(-b^2*d^2 + 4*a*c*d^2)^(1/4)*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^8*d^3 - 16*sqrt(2)*a*b^6*c*d^3 + 96*sqrt(2)*a^2*b^4*c^2*
d^3 - 256*sqrt(2)*a^3*b^2*c^3*d^3 + 256*sqrt(2)*a^4*c^4*d^3) + 77/2*(-b^2*d^2 + 4*a*c*d^2)^(1/4)*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^8
*d^3 - 16*sqrt(2)*a*b^6*c*d^3 + 96*sqrt(2)*a^2*b^4*c^2*d^3 - 256*sqrt(2)*a^3*b^2*c^3*d^3 + 256*sqrt(2)*a^4*c^4
*d^3) + 64/3*c^2/((b^6*d - 12*a*b^4*c*d + 48*a^2*b^2*c^2*d - 64*a^3*c^3*d)*(2*c*d*x + b*d)^(3/2)) - 2*(19*sqrt
(2*c*d*x + b*d)*b^2*c^2*d^2 - 76*sqrt(2*c*d*x + b*d)*a*c^3*d^2 - 15*(2*c*d*x + b*d)^(5/2)*c^2)/((b^6*d - 12*a*
b^4*c*d + 48*a^2*b^2*c^2*d - 64*a^3*c^3*d)*(b^2*d^2 - 4*a*c*d^2 - (2*c*d*x + b*d)^2)^2)
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maple [B] time = 0.08, size = 534, normalized size = 2.37 \[ -\frac {152 \sqrt {2 c d x +b d}\, a \,c^{3} d}{\left (4 a c -b^{2}\right )^{3} \left (4 c^{2} d^{2} x^{2}+4 b c \,d^{2} x +4 a c \,d^{2}\right )^{2}}+\frac {38 \sqrt {2 c d x +b d}\, b^{2} c^{2} d}{\left (4 a c -b^{2}\right )^{3} \left (4 c^{2} d^{2} x^{2}+4 b c \,d^{2} x +4 a c \,d^{2}\right )^{2}}+\frac {77 \sqrt {2}\, c^{2} \arctan \left (-\frac {\sqrt {2}\, \sqrt {2 c d x +b d}}{\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}}}+1\right )}{2 \left (4 a c -b^{2}\right )^{3} \left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {3}{4}} d}-\frac {77 \sqrt {2}\, c^{2} \arctan \left (\frac {\sqrt {2}\, \sqrt {2 c d x +b d}}{\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}}}+1\right )}{2 \left (4 a c -b^{2}\right )^{3} \left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {3}{4}} d}-\frac {77 \sqrt {2}\, c^{2} \ln \left (\frac {2 c d x +b d +\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}} \sqrt {2 c d x +b d}\, \sqrt {2}+\sqrt {4 a c \,d^{2}-b^{2} d^{2}}}{2 c d x +b d -\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}} \sqrt {2 c d x +b d}\, \sqrt {2}+\sqrt {4 a c \,d^{2}-b^{2} d^{2}}}\right )}{4 \left (4 a c -b^{2}\right )^{3} \left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {3}{4}} d}-\frac {30 \left (2 c d x +b d \right )^{\frac {5}{2}} c^{2}}{\left (4 a c -b^{2}\right )^{3} \left (4 c^{2} d^{2} x^{2}+4 b c \,d^{2} x +4 a c \,d^{2}\right )^{2} d}-\frac {64 c^{2}}{3 \left (4 a c -b^{2}\right )^{3} \left (2 c d x +b d \right )^{\frac {3}{2}} d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(2*c*d*x+b*d)^(5/2)/(c*x^2+b*x+a)^3,x)
[Out]
-30*c^2/d/(4*a*c-b^2)^3/(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)^(5/2)-152*c^3*d/(4*a*c-b^2)^3/(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)*a+38*c^2*d/(4*a*c-b^2)^3/(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)*b^2-77/4*c^2/d/(4*a*c-b^2)^3/(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)))-77/2*c^2/d/(4*a*c-b^2)^3/(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)+77/2*c^2/d/(4*a*c-b^2)^3/(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)-64/3*c^2/d/(4*a
*c-b^2)^3/(2*c*d*x+b*d)^(3/2)
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(2*c*d*x+b*d)^(5/2)/(c*x^2+b*x+a)^3,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 = 1.11, size = 3227, normalized size = 14.34 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/((b*d + 2*c*d*x)^(5/2)*(a + b*x + c*x^2)^3),x)
[Out]
- ((64*c^2*d^3)/(3*(4*a*c - b^2)) - (154*c^2*(b*d + 2*c*d*x)^4)/(3*(b^6*d - 64*a^3*c^3*d + 48*a^2*b^2*c^2*d -
12*a*b^4*c*d)) + (242*c^2*d*(b*d + 2*c*d*x)^2)/(3*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)))/((b*d + 2*c*d*x)^(3/2)*(b^4
*d^4 + 16*a^2*c^2*d^4 - 8*a*b^2*c*d^4) - (b*d + 2*c*d*x)^(7/2)*(2*b^2*d^2 - 8*a*c*d^2) + (b*d + 2*c*d*x)^(11/2
)) - (c^2*atan(((c^2*((b*d + 2*c*d*x)^(1/2)*(24868028416*a^9*c^13*d^3 - 94864*b^18*c^4*d^3 + 3415104*a*b^16*c^
5*d^3 - 54641664*a^2*b^14*c^6*d^3 + 509988864*a^3*b^12*c^7*d^3 - 3059933184*a^4*b^10*c^8*d^3 + 12239732736*a^5
*b^8*c^9*d^3 - 32639287296*a^6*b^6*c^10*d^3 + 55953063936*a^7*b^4*c^11*d^3 - 55953063936*a^8*b^2*c^12*d^3) - (
77*c^2*(165356240896*a^13*c^15*d^6 - 2464*b^26*c^2*d^6 + 128128*a*b^24*c^3*d^6 - 3075072*a^2*b^22*c^4*d^6 + 45
101056*a^3*b^20*c^5*d^6 - 451010560*a^4*b^18*c^6*d^6 + 3247276032*a^5*b^16*c^7*d^6 - 17318805504*a^6*b^14*c^8*
d^6 + 69275222016*a^7*b^12*c^9*d^6 - 207825666048*a^8*b^10*c^10*d^6 + 461834813440*a^9*b^8*c^11*d^6 - 73893570
1504*a^10*b^6*c^12*d^6 + 806111674368*a^11*b^4*c^13*d^6 - 537407782912*a^12*b^2*c^14*d^6))/(2*d^(5/2)*(b^2 - 4
*a*c)^(15/4)))*77i)/(2*d^(5/2)*(b^2 - 4*a*c)^(15/4)) + (c^2*((b*d + 2*c*d*x)^(1/2)*(24868028416*a^9*c^13*d^3 -
94864*b^18*c^4*d^3 + 3415104*a*b^16*c^5*d^3 - 54641664*a^2*b^14*c^6*d^3 + 509988864*a^3*b^12*c^7*d^3 - 305993
3184*a^4*b^10*c^8*d^3 + 12239732736*a^5*b^8*c^9*d^3 - 32639287296*a^6*b^6*c^10*d^3 + 55953063936*a^7*b^4*c^11*
d^3 - 55953063936*a^8*b^2*c^12*d^3) + (77*c^2*(165356240896*a^13*c^15*d^6 - 2464*b^26*c^2*d^6 + 128128*a*b^24*
c^3*d^6 - 3075072*a^2*b^22*c^4*d^6 + 45101056*a^3*b^20*c^5*d^6 - 451010560*a^4*b^18*c^6*d^6 + 3247276032*a^5*b
^16*c^7*d^6 - 17318805504*a^6*b^14*c^8*d^6 + 69275222016*a^7*b^12*c^9*d^6 - 207825666048*a^8*b^10*c^10*d^6 + 4
61834813440*a^9*b^8*c^11*d^6 - 738935701504*a^10*b^6*c^12*d^6 + 806111674368*a^11*b^4*c^13*d^6 - 537407782912*
a^12*b^2*c^14*d^6))/(2*d^(5/2)*(b^2 - 4*a*c)^(15/4)))*77i)/(2*d^(5/2)*(b^2 - 4*a*c)^(15/4)))/((77*c^2*((b*d +
2*c*d*x)^(1/2)*(24868028416*a^9*c^13*d^3 - 94864*b^18*c^4*d^3 + 3415104*a*b^16*c^5*d^3 - 54641664*a^2*b^14*c^6
*d^3 + 509988864*a^3*b^12*c^7*d^3 - 3059933184*a^4*b^10*c^8*d^3 + 12239732736*a^5*b^8*c^9*d^3 - 32639287296*a^
6*b^6*c^10*d^3 + 55953063936*a^7*b^4*c^11*d^3 - 55953063936*a^8*b^2*c^12*d^3) - (77*c^2*(165356240896*a^13*c^1
5*d^6 - 2464*b^26*c^2*d^6 + 128128*a*b^24*c^3*d^6 - 3075072*a^2*b^22*c^4*d^6 + 45101056*a^3*b^20*c^5*d^6 - 451
010560*a^4*b^18*c^6*d^6 + 3247276032*a^5*b^16*c^7*d^6 - 17318805504*a^6*b^14*c^8*d^6 + 69275222016*a^7*b^12*c^
9*d^6 - 207825666048*a^8*b^10*c^10*d^6 + 461834813440*a^9*b^8*c^11*d^6 - 738935701504*a^10*b^6*c^12*d^6 + 8061
11674368*a^11*b^4*c^13*d^6 - 537407782912*a^12*b^2*c^14*d^6))/(2*d^(5/2)*(b^2 - 4*a*c)^(15/4))))/(2*d^(5/2)*(b
^2 - 4*a*c)^(15/4)) - (77*c^2*((b*d + 2*c*d*x)^(1/2)*(24868028416*a^9*c^13*d^3 - 94864*b^18*c^4*d^3 + 3415104*
a*b^16*c^5*d^3 - 54641664*a^2*b^14*c^6*d^3 + 509988864*a^3*b^12*c^7*d^3 - 3059933184*a^4*b^10*c^8*d^3 + 122397
32736*a^5*b^8*c^9*d^3 - 32639287296*a^6*b^6*c^10*d^3 + 55953063936*a^7*b^4*c^11*d^3 - 55953063936*a^8*b^2*c^12
*d^3) + (77*c^2*(165356240896*a^13*c^15*d^6 - 2464*b^26*c^2*d^6 + 128128*a*b^24*c^3*d^6 - 3075072*a^2*b^22*c^4
*d^6 + 45101056*a^3*b^20*c^5*d^6 - 451010560*a^4*b^18*c^6*d^6 + 3247276032*a^5*b^16*c^7*d^6 - 17318805504*a^6*
b^14*c^8*d^6 + 69275222016*a^7*b^12*c^9*d^6 - 207825666048*a^8*b^10*c^10*d^6 + 461834813440*a^9*b^8*c^11*d^6 -
738935701504*a^10*b^6*c^12*d^6 + 806111674368*a^11*b^4*c^13*d^6 - 537407782912*a^12*b^2*c^14*d^6))/(2*d^(5/2)
*(b^2 - 4*a*c)^(15/4))))/(2*d^(5/2)*(b^2 - 4*a*c)^(15/4))))*77i)/(d^(5/2)*(b^2 - 4*a*c)^(15/4)) - (77*c^2*atan
(((77*c^2*((b*d + 2*c*d*x)^(1/2)*(24868028416*a^9*c^13*d^3 - 94864*b^18*c^4*d^3 + 3415104*a*b^16*c^5*d^3 - 546
41664*a^2*b^14*c^6*d^3 + 509988864*a^3*b^12*c^7*d^3 - 3059933184*a^4*b^10*c^8*d^3 + 12239732736*a^5*b^8*c^9*d^
3 - 32639287296*a^6*b^6*c^10*d^3 + 55953063936*a^7*b^4*c^11*d^3 - 55953063936*a^8*b^2*c^12*d^3) - (c^2*(165356
240896*a^13*c^15*d^6 - 2464*b^26*c^2*d^6 + 128128*a*b^24*c^3*d^6 - 3075072*a^2*b^22*c^4*d^6 + 45101056*a^3*b^2
0*c^5*d^6 - 451010560*a^4*b^18*c^6*d^6 + 3247276032*a^5*b^16*c^7*d^6 - 17318805504*a^6*b^14*c^8*d^6 + 69275222
016*a^7*b^12*c^9*d^6 - 207825666048*a^8*b^10*c^10*d^6 + 461834813440*a^9*b^8*c^11*d^6 - 738935701504*a^10*b^6*
c^12*d^6 + 806111674368*a^11*b^4*c^13*d^6 - 537407782912*a^12*b^2*c^14*d^6)*77i)/(2*d^(5/2)*(b^2 - 4*a*c)^(15/
4))))/(2*d^(5/2)*(b^2 - 4*a*c)^(15/4)) + (77*c^2*((b*d + 2*c*d*x)^(1/2)*(24868028416*a^9*c^13*d^3 - 94864*b^18
*c^4*d^3 + 3415104*a*b^16*c^5*d^3 - 54641664*a^2*b^14*c^6*d^3 + 509988864*a^3*b^12*c^7*d^3 - 3059933184*a^4*b^
10*c^8*d^3 + 12239732736*a^5*b^8*c^9*d^3 - 32639287296*a^6*b^6*c^10*d^3 + 55953063936*a^7*b^4*c^11*d^3 - 55953
063936*a^8*b^2*c^12*d^3) + (c^2*(165356240896*a^13*c^15*d^6 - 2464*b^26*c^2*d^6 + 128128*a*b^24*c^3*d^6 - 3075
072*a^2*b^22*c^4*d^6 + 45101056*a^3*b^20*c^5*d^6 - 451010560*a^4*b^18*c^6*d^6 + 3247276032*a^5*b^16*c^7*d^6 -
17318805504*a^6*b^14*c^8*d^6 + 69275222016*a^7*b^12*c^9*d^6 - 207825666048*a^8*b^10*c^10*d^6 + 461834813440*a^
9*b^8*c^11*d^6 - 738935701504*a^10*b^6*c^12*d^6 + 806111674368*a^11*b^4*c^13*d^6 - 537407782912*a^12*b^2*c^14*
d^6)*77i)/(2*d^(5/2)*(b^2 - 4*a*c)^(15/4))))/(2*d^(5/2)*(b^2 - 4*a*c)^(15/4)))/((c^2*((b*d + 2*c*d*x)^(1/2)*(2
4868028416*a^9*c^13*d^3 - 94864*b^18*c^4*d^3 + 3415104*a*b^16*c^5*d^3 - 54641664*a^2*b^14*c^6*d^3 + 509988864*
a^3*b^12*c^7*d^3 - 3059933184*a^4*b^10*c^8*d^3 + 12239732736*a^5*b^8*c^9*d^3 - 32639287296*a^6*b^6*c^10*d^3 +
55953063936*a^7*b^4*c^11*d^3 - 55953063936*a^8*b^2*c^12*d^3) - (c^2*(165356240896*a^13*c^15*d^6 - 2464*b^26*c^
2*d^6 + 128128*a*b^24*c^3*d^6 - 3075072*a^2*b^22*c^4*d^6 + 45101056*a^3*b^20*c^5*d^6 - 451010560*a^4*b^18*c^6*
d^6 + 3247276032*a^5*b^16*c^7*d^6 - 17318805504*a^6*b^14*c^8*d^6 + 69275222016*a^7*b^12*c^9*d^6 - 207825666048
*a^8*b^10*c^10*d^6 + 461834813440*a^9*b^8*c^11*d^6 - 738935701504*a^10*b^6*c^12*d^6 + 806111674368*a^11*b^4*c^
13*d^6 - 537407782912*a^12*b^2*c^14*d^6)*77i)/(2*d^(5/2)*(b^2 - 4*a*c)^(15/4)))*77i)/(2*d^(5/2)*(b^2 - 4*a*c)^
(15/4)) - (c^2*((b*d + 2*c*d*x)^(1/2)*(24868028416*a^9*c^13*d^3 - 94864*b^18*c^4*d^3 + 3415104*a*b^16*c^5*d^3
- 54641664*a^2*b^14*c^6*d^3 + 509988864*a^3*b^12*c^7*d^3 - 3059933184*a^4*b^10*c^8*d^3 + 12239732736*a^5*b^8*c
^9*d^3 - 32639287296*a^6*b^6*c^10*d^3 + 55953063936*a^7*b^4*c^11*d^3 - 55953063936*a^8*b^2*c^12*d^3) + (c^2*(1
65356240896*a^13*c^15*d^6 - 2464*b^26*c^2*d^6 + 128128*a*b^24*c^3*d^6 - 3075072*a^2*b^22*c^4*d^6 + 45101056*a^
3*b^20*c^5*d^6 - 451010560*a^4*b^18*c^6*d^6 + 3247276032*a^5*b^16*c^7*d^6 - 17318805504*a^6*b^14*c^8*d^6 + 692
75222016*a^7*b^12*c^9*d^6 - 207825666048*a^8*b^10*c^10*d^6 + 461834813440*a^9*b^8*c^11*d^6 - 738935701504*a^10
*b^6*c^12*d^6 + 806111674368*a^11*b^4*c^13*d^6 - 537407782912*a^12*b^2*c^14*d^6)*77i)/(2*d^(5/2)*(b^2 - 4*a*c)
^(15/4)))*77i)/(2*d^(5/2)*(b^2 - 4*a*c)^(15/4)))))/(d^(5/2)*(b^2 - 4*a*c)^(15/4))
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(2*c*d*x+b*d)**(5/2)/(c*x**2+b*x+a)**3,x)
[Out]
Timed out
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