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

Optimal. Leaf size=132 \[ -\frac {12 c \sqrt {a+b x+c x^2}}{d^3 \left (b^2-4 a c\right )^2 (b+2 c x)^2}-\frac {2}{d^3 \left (b^2-4 a c\right ) (b+2 c x)^2 \sqrt {a+b x+c x^2}}-\frac {6 \sqrt {c} \tan ^{-1}\left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}\right )}{d^3 \left (b^2-4 a c\right )^{5/2}} \]

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Rubi [A]  time = 0.08, antiderivative size = 132, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {687, 693, 688, 205} \begin {gather*} -\frac {12 c \sqrt {a+b x+c x^2}}{d^3 \left (b^2-4 a c\right )^2 (b+2 c x)^2}-\frac {2}{d^3 \left (b^2-4 a c\right ) (b+2 c x)^2 \sqrt {a+b x+c x^2}}-\frac {6 \sqrt {c} \tan ^{-1}\left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}\right )}{d^3 \left (b^2-4 a c\right )^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-2/((b^2 - 4*a*c)*d^3*(b + 2*c*x)^2*Sqrt[a + b*x + c*x^2]) - (12*c*Sqrt[a + b*x + c*x^2])/((b^2 - 4*a*c)^2*d^3
*(b + 2*c*x)^2) - (6*Sqrt[c]*ArcTan[(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])/Sqrt[b^2 - 4*a*c]])/((b^2 - 4*a*c)^(5/2)
*d^3)

Rule 205

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

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 688

Int[1/(((d_) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[4*c, Subst[Int[1/(b^2*e
- 4*a*c*e + 4*c*e*x^2), x], x, Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0]
&& EqQ[2*c*d - b*e, 0]

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])

Rubi steps

\begin {align*} \int \frac {1}{(b d+2 c d x)^3 \left (a+b x+c x^2\right )^{3/2}} \, dx &=-\frac {2}{\left (b^2-4 a c\right ) d^3 (b+2 c x)^2 \sqrt {a+b x+c x^2}}-\frac {(12 c) \int \frac {1}{(b d+2 c d x)^3 \sqrt {a+b x+c x^2}} \, dx}{b^2-4 a c}\\ &=-\frac {2}{\left (b^2-4 a c\right ) d^3 (b+2 c x)^2 \sqrt {a+b x+c x^2}}-\frac {12 c \sqrt {a+b x+c x^2}}{\left (b^2-4 a c\right )^2 d^3 (b+2 c x)^2}-\frac {(6 c) \int \frac {1}{(b d+2 c d x) \sqrt {a+b x+c x^2}} \, dx}{\left (b^2-4 a c\right )^2 d^2}\\ &=-\frac {2}{\left (b^2-4 a c\right ) d^3 (b+2 c x)^2 \sqrt {a+b x+c x^2}}-\frac {12 c \sqrt {a+b x+c x^2}}{\left (b^2-4 a c\right )^2 d^3 (b+2 c x)^2}-\frac {\left (24 c^2\right ) \operatorname {Subst}\left (\int \frac {1}{2 b^2 c d-8 a c^2 d+8 c^2 d x^2} \, dx,x,\sqrt {a+b x+c x^2}\right )}{\left (b^2-4 a c\right )^2 d^2}\\ &=-\frac {2}{\left (b^2-4 a c\right ) d^3 (b+2 c x)^2 \sqrt {a+b x+c x^2}}-\frac {12 c \sqrt {a+b x+c x^2}}{\left (b^2-4 a c\right )^2 d^3 (b+2 c x)^2}-\frac {6 \sqrt {c} \tan ^{-1}\left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{5/2} d^3}\\ \end {align*}

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Mathematica [C]  time = 0.03, size = 60, normalized size = 0.45 \begin {gather*} -\frac {2 \, _2F_1\left (-\frac {1}{2},2;\frac {1}{2};\frac {4 c (a+x (b+c x))}{4 a c-b^2}\right )}{d^3 \left (b^2-4 a c\right )^2 \sqrt {a+x (b+c x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [B]  time = 14.95, size = 3934, normalized size = 29.80 \begin {gather*} \text {Result too large to show} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(6291456*c^12*x^15 - 6291456*c^(23/2)*x^14*Sqrt[a + b*x + c*x^2] + 2*Sqrt[c]*Sqrt[a + b*x + c*x^2]*(-28*a^3*b^
8 - 80*a^2*b^9*x - 72*a*b^10*x^2 - 20*b^11*x^3) + 2*(2*a^3*b^9 + 5*a^2*b^10*x + 4*a*b^11*x^2 + b^12*x^3) + 2*c
^(3/2)*Sqrt[a + b*x + c*x^2]*(-616*a^4*b^6 - 3464*a^3*b^7*x - 6672*a^2*b^8*x^2 - 5272*a*b^9*x^3 - 1440*b^10*x^
4) + 2*c*(172*a^4*b^7 + 778*a^3*b^8*x + 1254*a^2*b^9*x^2 + 854*a*b^10*x^3 + 206*b^11*x^4) + 2*c^(5/2)*Sqrt[a +
b*x + c*x^2]*(-2464*a^5*b^4 - 25760*a^4*b^5*x - 95592*a^3*b^6*x^2 - 159264*a^2*b^7*x^3 - 120576*a*b^8*x^4 - 3
3384*b^9*x^5) + 2*c^2*(1456*a^5*b^5 + 12040*a^4*b^6*x + 36100*a^3*b^7*x^2 + 49974*a^2*b^8*x^3 + 32272*a*b^9*x^
4 + 7806*b^10*x^5) + 2*c^(7/2)*Sqrt[a + b*x + c*x^2]*(-2944*a^6*b^2 - 51584*a^5*b^3*x - 344352*a^4*b^4*x^2 - 1
092480*a^3*b^5*x^3 - 1735872*a^2*b^6*x^4 - 1321344*a*b^7*x^5 - 379632*b^8*x^6) + 2*c^3*(3136*a^6*b^3 + 44128*a
^5*b^4*x + 234640*a^4*b^5*x^2 + 599928*a^3*b^6*x^3 + 785472*a^2*b^7*x^4 + 504696*a*b^8*x^5 + 125280*b^9*x^6) +
2*c^(9/2)*Sqrt[a + b*x + c*x^2]*(-512*a^7 - 24064*a^6*b*x - 324480*a^5*b^2*x^2 - 1991168*a^4*b^3*x^3 - 624947
2*a^3*b^4*x^4 - 10206720*a^2*b^5*x^5 - 8157952*a*b^6*x^6 - 2494976*b^7*x^7) + 2*c^4*(1792*a^7*b + 45184*a^6*b^
2*x + 425152*a^5*b^3*x^2 + 1962528*a^4*b^4*x^3 + 4814976*a^3*b^5*x^4 + 6343872*a^2*b^6*x^5 + 4207872*a*b^7*x^6
+ 1095600*b^8*x^7) + 2*c^(11/2)*Sqrt[a + b*x + c*x^2]*(-38400*a^6*x^2 - 741376*a^5*b*x^3 - 5411840*a^4*b^2*x^
4 - 19187712*a^3*b^3*x^5 - 34764800*a^2*b^4*x^6 - 30588928*a*b^5*x^7 - 10263552*b^6*x^8) + 2*c^5*(6144*a^7*x +
167168*a^6*b*x^2 + 1651840*a^5*b^2*x^3 + 8023552*a^4*b^3*x^4 + 20888576*a^3*b^4*x^5 + 29391872*a^2*b^5*x^6 +
20903168*a*b^6*x^7 + 5851648*b^7*x^8) + 2*c^(13/2)*Sqrt[a + b*x + c*x^2]*(-540672*a^5*x^4 - 6764544*a^4*b*x^5
- 31901696*a^3*b^2*x^6 - 70258688*a^2*b^3*x^7 - 72019968*a*b^4*x^8 - 27510784*b^5*x^9) + 2*c^6*(167936*a^6*x^3
+ 2680832*a^5*b*x^4 + 16639488*a^4*b^2*x^5 + 51172352*a^3*b^3*x^6 + 82116096*a^2*b^4*x^7 + 65363968*a*b^5*x^8
+ 20254208*b^6*x^9) + 2*c^(15/2)*Sqrt[a + b*x + c*x^2]*(-3125248*a^4*x^6 - 26951680*a^3*b*x^7 - 82796544*a^2*
b^2*x^8 - 106807296*a*b^3*x^9 - 48754688*b^4*x^10) + 2*c^7*(1509376*a^5*x^5 + 16732160*a^4*b*x^6 + 70461440*a^
3*b^2*x^7 + 140189696*a^2*b^3*x^8 + 131688448*a*b^4*x^9 + 46809088*b^5*x^10) + 2*c^(17/2)*Sqrt[a + b*x + c*x^2
]*(-9043968*a^3*x^8 - 52428800*a^2*b*x^9 - 96698368*a*b^2*x^10 - 56623104*b^3*x^11) + 2*c^8*(6459392*a^4*x^7 +
50708480*a^3*b*x^8 + 142745600*a^2*b^2*x^9 + 170426368*a*b^3*x^10 + 72847360*b^4*x^11) + 2*c^(19/2)*Sqrt[a +
b*x + c*x^2]*(-13762560*a^2*x^10 - 48758784*a*b*x^11 - 41418752*b^2*x^12) + 2*c^9*(14811136*a^3*x^9 + 79495168
*a^2*b*x^10 + 136740864*a*b^2*x^11 + 75366400*b^3*x^12) + 2*c^(21/2)*Sqrt[a + b*x + c*x^2]*(-10485760*a*x^12 -
17301504*b*x^13) + 2*c^10*(18612224*a^2*x^11 + 61865984*a*b*x^12 + 49676288*b^2*x^13) + 2*c^11*(12058624*a*x^
13 + 18874368*b*x^14))/(33554432*a^2*c^(29/2)*d^3*x^16 - 33554432*a^2*c^14*d^3*x^15*Sqrt[a + b*x + c*x^2] + d^
3*Sqrt[a + b*x + c*x^2]*(-2*a^3*b^13 - 5*a^2*b^14*x - 4*a*b^15*x^2 - b^16*x^3) + c*d^3*Sqrt[a + b*x + c*x^2]*(
-152*a^4*b^11 - 724*a^3*b^12*x - 1204*a^2*b^13*x^2 - 836*a*b^14*x^3 - 204*b^15*x^4) + Sqrt[c]*d^3*(28*a^4*b^12
+ 108*a^3*b^13*x + 152*a^2*b^14*x^2 + 92*a*b^15*x^3 + 20*b^16*x^4) + c^2*d^3*Sqrt[a + b*x + c*x^2]*(192*a^5*b
^9 - 4160*a^4*b^10*x - 22328*a^3*b^11*x^2 - 39140*a^2*b^12*x^3 - 28576*a*b^13*x^4 - 7404*b^14*x^5) + c^(3/2)*d
^3*(336*a^5*b^10 + 2944*a^4*b^11*x + 8428*a^3*b^12*x^2 + 10800*a^2*b^13*x^3 + 6400*a*b^14*x^4 + 1420*b^15*x^5)
+ c^3*d^3*Sqrt[a + b*x + c*x^2]*(5376*a^6*b^7 + 46464*a^5*b^8*x + 80416*a^4*b^9*x^2 - 104688*a^3*b^10*x^3 - 4
08896*a^2*b^11*x^4 - 376848*a*b^12*x^5 - 111680*b^13*x^6) + c^(5/2)*d^3*(-2688*a^6*b^8 - 9024*a^5*b^9*x + 2137
6*a^4*b^10*x^2 + 127008*a^3*b^11*x^3 + 199808*a^2*b^12*x^4 + 132624*a*b^13*x^5 + 32104*b^14*x^6) + c^4*d^3*Sqr
t[a + b*x + c*x^2]*(-10752*a^7*b^5 + 13056*a^6*b^6*x + 592128*a^5*b^7*x^2 + 1901952*a^4*b^8*x^3 + 1572608*a^3*
b^9*x^4 - 1284288*a^2*b^10*x^5 - 2441216*a*b^11*x^6 - 913440*b^12*x^7) + c^(7/2)*d^3*(-1536*a^7*b^6 - 79872*a^
6*b^7*x - 414528*a^5*b^8*x^2 - 626304*a^4*b^9*x^3 + 218352*a^3*b^10*x^4 + 1412480*a^2*b^11*x^5 + 1261312*a*b^1
2*x^6 + 357008*b^13*x^7) + c^5*d^3*Sqrt[a + b*x + c*x^2]*(-14336*a^8*b^3 - 312320*a^7*b^4*x - 1331200*a^6*b^5*
x^2 + 470528*a^5*b^6*x^3 + 11174912*a^4*b^7*x^4 + 18502912*a^3*b^8*x^5 + 5460992*a^2*b^9*x^6 - 7994624*a*b^10*
x^7 - 4584448*b^11*x^8) + c^(9/2)*d^3*(19456*a^8*b^4 + 181248*a^7*b^5*x + 8192*a^6*b^6*x^2 - 3076096*a^5*b^7*x
^3 - 8503488*a^4*b^8*x^4 - 6916608*a^3*b^9*x^5 + 2683136*a^2*b^10*x^6 + 6354560*a*b^11*x^7 + 2331296*b^12*x^8)
+ c^6*d^3*Sqrt[a + b*x + c*x^2]*(-106496*a^8*b^2*x - 2267136*a^7*b^3*x^2 - 13005824*a^6*b^4*x^3 - 22962176*a^
5*b^5*x^4 + 13364224*a^4*b^6*x^5 + 76220416*a^3*b^7*x^6 + 57850368*a^2*b^8*x^7 - 8142848*a*b^9*x^8 - 15055872*
b^10*x^9) + c^(11/2)*d^3*(4096*a^9*b^2 + 262144*a^8*b^3*x + 2638848*a^7*b^4*x^2 + 7888896*a^6*b^5*x^3 + 71168*
a^5*b^6*x^4 - 36258816*a^4*b^7*x^5 - 56214528*a^3*b^8*x^6 - 18703104*a^2*b^9*x^7 + 16508160*a*b^10*x^8 + 97792
00*b^11*x^9) + c^7*d^3*Sqrt[a + b*x + c*x^2]*(-253952*a^8*b*x^2 - 6860800*a^7*b^2*x^3 - 50479104*a^6*b^3*x^4 -
138719232*a^5*b^4*x^5 - 109715456*a^4*b^5*x^6 + 121671680*a^3*b^6*x^7 + 204898304*a^2*b^7*x^8 + 34525184*a*b^
8*x^9 - 33390592*b^9*x^10) + c^(13/2)*d^3*(16384*a^9*b*x + 1114112*a^8*b^2*x^2 + 13312000*a^7*b^3*x^3 + 561254
40*a^6*b^4*x^4 + 84168704*a^5*b^5*x^5 - 25155584*a^4*b^6*x^6 - 183726080*a^3*b^7*x^7 - 136103424*a^2*b^8*x^8 +
10051584*a*b^9*x^9 + 27643904*b^10*x^10) + c^8*d^3*Sqrt[a + b*x + c*x^2]*(-196608*a^8*x^3 - 9371648*a^7*b*x^4
- 96944128*a^6*b^2*x^5 - 367951872*a^5*b^3*x^6 - 536485888*a^4*b^4*x^7 - 97550336*a^3*b^5*x^8 + 367853568*a^2
*b^6*x^9 + 159711232*a*b^7*x^10 - 50421760*b^8*x^11) + c^(15/2)*d^3*(16384*a^9*x^2 + 1933312*a^8*b*x^3 + 31240
192*a^7*b^2*x^4 + 174194688*a^6*b^3*x^5 + 401866752*a^5*b^4*x^6 + 297906176*a^4*b^5*x^7 - 230699008*a^3*b^6*x^
8 - 398909440*a^2*b^7*x^9 - 72998912*a*b^8*x^10 + 53858304*b^9*x^11) + c^9*d^3*Sqrt[a + b*x + c*x^2]*(-4784128
*a^7*x^5 - 91619328*a^6*b*x^6 - 512753664*a^5*b^2*x^7 - 1102053376*a^4*b^3*x^8 - 752287744*a^3*b^4*x^9 + 28219
8016*a^2*b^5*x^10 + 319029248*a*b^6*x^11 - 51118080*b^7*x^12) + c^(17/2)*d^3*(1196032*a^8*x^4 + 34865152*a^7*b
*x^5 + 279052288*a^6*b^2*x^6 + 896303104*a^5*b^3*x^7 + 1172938752*a^4*b^4*x^8 + 247496704*a^3*b^5*x^9 - 612286
464*a^2*b^6*x^10 - 266272768*a*b^7*x^11 + 72523776*b^8*x^12) + c^10*d^3*Sqrt[a + b*x + c*x^2]*(-33947648*a^6*x
^7 - 365428736*a^5*b*x^8 - 1205600256*a^4*b^2*x^9 - 1375731712*a^3*b^3*x^10 - 141950976*a^2*b^4*x^11 + 3711959
04*a*b^5*x^12 - 33292288*b^6*x^13) + c^(19/2)*d^3*(14942208*a^7*x^6 + 225378304*a^6*b*x^7 + 1078591488*a^5*b^2
*x^8 + 2077097984*a^4*b^3*x^9 + 1336147968*a^3*b^4*x^10 - 391315456*a^2*b^5*x^11 - 463994880*a*b^6*x^12 + 6632
2432*b^7*x^13) + c^11*d^3*Sqrt[a + b*x + c*x^2]*(-104857600*a^5*x^9 - 685768704*a^4*b*x^10 - 1269825536*a^3*b^
2*x^11 - 515899392*a^2*b^3*x^12 + 258998272*a*b^4*x^13 - 12582912*b^5*x^14) + c^(21/2)*d^3*(72482816*a^6*x^8 +
674758656*a^5*b*x^9 + 2004090880*a^4*b^2*x^10 + 2123890688*a^3*b^3*x^11 + 244187136*a^2*b^4*x^12 - 483393536*
a*b^5*x^13 + 39321600*b^6*x^14) + c^13*d^3*Sqrt[a + b*x + c*x^2]*(-117440512*a^3*x^13 - 201326592*a^2*b*x^14 +
16777216*a*b^2*x^15) + c^12*d^3*Sqrt[a + b*x + c*x^2]*(-159383552*a^4*x^11 - 603979776*a^3*b*x^12 - 473956352
*a^2*b^2*x^13 + 100663296*a*b^3*x^14 - 2097152*b^4*x^15) + c^(23/2)*d^3*(171966464*a^5*x^10 + 1019215872*a^4*b
*x^11 + 1752170496*a^3*b^2*x^12 + 683671552*a^2*b^3*x^13 - 306184192*a*b^4*x^14 + 13631488*b^5*x^15) + c^(27/2
)*d^3*(134217728*a^3*x^14 + 218103808*a^2*b*x^15 - 16777216*a*b^2*x^16) + c^(25/2)*d^3*(213909504*a^4*x^12 + 7
54974720*a^3*b*x^13 + 562036736*a^2*b^2*x^14 - 109051904*a*b^3*x^15 + 2097152*b^4*x^16)) + (12*Sqrt[c]*ArcTan[
b/Sqrt[b^2 - 4*a*c] + (2*c*x)/Sqrt[b^2 - 4*a*c] - (2*Sqrt[c]*Sqrt[a + b*x + c*x^2])/Sqrt[b^2 - 4*a*c]])/((b^2
- 4*a*c)^(5/2)*d^3)

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fricas [B]  time = 1.25, size = 702, normalized size = 5.32 \begin {gather*} \left [\frac {3 \, {\left (4 \, c^{3} x^{4} + 8 \, b c^{2} x^{3} + a b^{2} + {\left (5 \, b^{2} c + 4 \, a c^{2}\right )} x^{2} + {\left (b^{3} + 4 \, a b c\right )} x\right )} \sqrt {-\frac {c}{b^{2} - 4 \, a c}} \log \left (-\frac {4 \, c^{2} x^{2} + 4 \, b c x - b^{2} + 8 \, a c - 4 \, \sqrt {c x^{2} + b x + a} {\left (b^{2} - 4 \, a c\right )} \sqrt {-\frac {c}{b^{2} - 4 \, a c}}}{4 \, c^{2} x^{2} + 4 \, b c x + b^{2}}\right ) - 2 \, {\left (6 \, c^{2} x^{2} + 6 \, b c x + b^{2} + 2 \, a c\right )} \sqrt {c x^{2} + b x + a}}{4 \, {\left (b^{4} c^{3} - 8 \, a b^{2} c^{4} + 16 \, a^{2} c^{5}\right )} d^{3} x^{4} + 8 \, {\left (b^{5} c^{2} - 8 \, a b^{3} c^{3} + 16 \, a^{2} b c^{4}\right )} d^{3} x^{3} + {\left (5 \, b^{6} c - 36 \, a b^{4} c^{2} + 48 \, a^{2} b^{2} c^{3} + 64 \, a^{3} c^{4}\right )} d^{3} x^{2} + {\left (b^{7} - 4 \, a b^{5} c - 16 \, a^{2} b^{3} c^{2} + 64 \, a^{3} b c^{3}\right )} d^{3} x + {\left (a b^{6} - 8 \, a^{2} b^{4} c + 16 \, a^{3} b^{2} c^{2}\right )} d^{3}}, -\frac {2 \, {\left (3 \, {\left (4 \, c^{3} x^{4} + 8 \, b c^{2} x^{3} + a b^{2} + {\left (5 \, b^{2} c + 4 \, a c^{2}\right )} x^{2} + {\left (b^{3} + 4 \, a b c\right )} x\right )} \sqrt {\frac {c}{b^{2} - 4 \, a c}} \arctan \left (-\frac {\sqrt {c x^{2} + b x + a} {\left (b^{2} - 4 \, a c\right )} \sqrt {\frac {c}{b^{2} - 4 \, a c}}}{2 \, {\left (c^{2} x^{2} + b c x + a c\right )}}\right ) + {\left (6 \, c^{2} x^{2} + 6 \, b c x + b^{2} + 2 \, a c\right )} \sqrt {c x^{2} + b x + a}\right )}}{4 \, {\left (b^{4} c^{3} - 8 \, a b^{2} c^{4} + 16 \, a^{2} c^{5}\right )} d^{3} x^{4} + 8 \, {\left (b^{5} c^{2} - 8 \, a b^{3} c^{3} + 16 \, a^{2} b c^{4}\right )} d^{3} x^{3} + {\left (5 \, b^{6} c - 36 \, a b^{4} c^{2} + 48 \, a^{2} b^{2} c^{3} + 64 \, a^{3} c^{4}\right )} d^{3} x^{2} + {\left (b^{7} - 4 \, a b^{5} c - 16 \, a^{2} b^{3} c^{2} + 64 \, a^{3} b c^{3}\right )} d^{3} x + {\left (a b^{6} - 8 \, a^{2} b^{4} c + 16 \, a^{3} b^{2} c^{2}\right )} d^{3}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[(3*(4*c^3*x^4 + 8*b*c^2*x^3 + a*b^2 + (5*b^2*c + 4*a*c^2)*x^2 + (b^3 + 4*a*b*c)*x)*sqrt(-c/(b^2 - 4*a*c))*log
(-(4*c^2*x^2 + 4*b*c*x - b^2 + 8*a*c - 4*sqrt(c*x^2 + b*x + a)*(b^2 - 4*a*c)*sqrt(-c/(b^2 - 4*a*c)))/(4*c^2*x^
2 + 4*b*c*x + b^2)) - 2*(6*c^2*x^2 + 6*b*c*x + b^2 + 2*a*c)*sqrt(c*x^2 + b*x + a))/(4*(b^4*c^3 - 8*a*b^2*c^4 +
16*a^2*c^5)*d^3*x^4 + 8*(b^5*c^2 - 8*a*b^3*c^3 + 16*a^2*b*c^4)*d^3*x^3 + (5*b^6*c - 36*a*b^4*c^2 + 48*a^2*b^2
*c^3 + 64*a^3*c^4)*d^3*x^2 + (b^7 - 4*a*b^5*c - 16*a^2*b^3*c^2 + 64*a^3*b*c^3)*d^3*x + (a*b^6 - 8*a^2*b^4*c +
16*a^3*b^2*c^2)*d^3), -2*(3*(4*c^3*x^4 + 8*b*c^2*x^3 + a*b^2 + (5*b^2*c + 4*a*c^2)*x^2 + (b^3 + 4*a*b*c)*x)*sq
rt(c/(b^2 - 4*a*c))*arctan(-1/2*sqrt(c*x^2 + b*x + a)*(b^2 - 4*a*c)*sqrt(c/(b^2 - 4*a*c))/(c^2*x^2 + b*c*x + a
*c)) + (6*c^2*x^2 + 6*b*c*x + b^2 + 2*a*c)*sqrt(c*x^2 + b*x + a))/(4*(b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5)*d^3*
x^4 + 8*(b^5*c^2 - 8*a*b^3*c^3 + 16*a^2*b*c^4)*d^3*x^3 + (5*b^6*c - 36*a*b^4*c^2 + 48*a^2*b^2*c^3 + 64*a^3*c^4
)*d^3*x^2 + (b^7 - 4*a*b^5*c - 16*a^2*b^3*c^2 + 64*a^3*b*c^3)*d^3*x + (a*b^6 - 8*a^2*b^4*c + 16*a^3*b^2*c^2)*d
^3)]

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Unab
le to divide, perhaps due to rounding error%%%{%%%{128,[5]%%%},[6,2,3,0]%%%}+%%%{%%%{-64,[4]%%%},[6,1,3,2]%%%}
+%%%{%%%{8,[3]%%%},[6,0,3,4]%%%}+%%%{%%{[%%%{-384,[4]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[5,2,3,1]%%%}+%%%{%%{[%%%
{192,[3]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[5,1,3,3]%%%}+%%%{%%{[%%%{-24,[2]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[5,0,
3,5]%%%}+%%%{%%%{-384,[5]%%%},[4,3,3,0]%%%}+%%%{%%%{768,[4]%%%},[4,2,3,2]%%%}+%%%{%%%{-312,[3]%%%},[4,1,3,4]%%
%}+%%%{%%%{36,[2]%%%},[4,0,3,6]%%%}+%%%{%%{[%%%{768,[4]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[3,3,3,1]%%%}+%%%{%%{[%
%%{-896,[3]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[3,2,3,3]%%%}+%%%{%%{[%%%{304,[2]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[3
,1,3,5]%%%}+%%%{%%{[%%%{-32,[1]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[3,0,3,7]%%%}+%%%{%%%{384,[5]%%%},[2,4,3,0]%%%}
+%%%{%%%{-960,[4]%%%},[2,3,3,2]%%%}+%%%{%%%{696,[3]%%%},[2,2,3,4]%%%}+%%%{%%%{-192,[2]%%%},[2,1,3,6]%%%}+%%%{%
%%{18,[1]%%%},[2,0,3,8]%%%}+%%%{%%{[%%%{-384,[4]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[1,4,3,1]%%%}+%%%{%%{[%%%{576,
[3]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[1,3,3,3]%%%}+%%%{%%{[%%%{-312,[2]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[1,2,3,5]
%%%}+%%%{%%{[%%%{72,[1]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[1,1,3,7]%%%}+%%%{%%{[-6,0]:[1,0,%%%{-1,[1]%%%}]%%},[1,
0,3,9]%%%}+%%%{%%%{-128,[5]%%%},[0,5,3,0]%%%}+%%%{%%%{256,[4]%%%},[0,4,3,2]%%%}+%%%{%%%{-200,[3]%%%},[0,3,3,4]
%%%}+%%%{%%%{76,[2]%%%},[0,2,3,6]%%%}+%%%{%%%{-14,[1]%%%},[0,1,3,8]%%%}+%%%{1,[0,0,3,10]%%%} / %%%{%%{poly1[%%
%{-8,[4]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[6,0,0,0]%%%}+%%%{%%%{24,[4]%%%},[5,0,0,1]%%%}+%%%{%%{[%%%{24,[4]%%%},
0]:[1,0,%%%{-1,[1]%%%}]%%},[4,1,0,0]%%%}+%%%{%%{poly1[%%%{-36,[3]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[4,0,0,2]%%%}
+%%%{%%%{-48,[4]%%%},[3,1,0,1]%%%}+%%%{%%%{32,[3]%%%},[3,0,0,3]%%%}+%%%{%%{[%%%{-24,[4]%%%},0]:[1,0,%%%{-1,[1]
%%%}]%%},[2,2,0,0]%%%}+%%%{%%{[%%%{48,[3]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[2,1,0,2]%%%}+%%%{%%{poly1[%%%{-18,[2
]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[2,0,0,4]%%%}+%%%{%%%{24,[4]%%%},[1,2,0,1]%%%}+%%%{%%%{-24,[3]%%%},[1,1,0,3]%
%%}+%%%{%%%{6,[2]%%%},[1,0,0,5]%%%}+%%%{%%{[%%%{8,[4]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[0,3,0,0]%%%}+%%%{%%{[%%%
{-12,[3]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[0,2,0,2]%%%}+%%%{%%{[%%%{6,[2]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[0,1,0,
4]%%%}+%%%{%%{poly1[%%%{-1,[1]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[0,0,0,6]%%%} Error: Bad Argument Value

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maple [A]  time = 0.05, size = 218, normalized size = 1.65 \begin {gather*} \frac {6 \ln \left (\frac {\frac {4 a c -b^{2}}{2 c}+\frac {\sqrt {\frac {4 a c -b^{2}}{c}}\, \sqrt {4 \left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{c}}}{2}}{x +\frac {b}{2 c}}\right )}{\left (4 a c -b^{2}\right )^{2} \sqrt {\frac {4 a c -b^{2}}{c}}\, d^{3}}-\frac {3}{\left (4 a c -b^{2}\right )^{2} \sqrt {\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}}\, d^{3}}-\frac {1}{4 \left (4 a c -b^{2}\right ) \left (x +\frac {b}{2 c}\right )^{2} \sqrt {\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}}\, c^{2} d^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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(1/(2*c*d*x+b*d)^3/(c*x^2+b*x+a)^(3/2),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 [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{{\left (b\,d+2\,c\,d\,x\right )}^3\,{\left (c\,x^2+b\,x+a\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {\int \frac {1}{a b^{3} \sqrt {a + b x + c x^{2}} + 6 a b^{2} c x \sqrt {a + b x + c x^{2}} + 12 a b c^{2} x^{2} \sqrt {a + b x + c x^{2}} + 8 a c^{3} x^{3} \sqrt {a + b x + c x^{2}} + b^{4} x \sqrt {a + b x + c x^{2}} + 7 b^{3} c x^{2} \sqrt {a + b x + c x^{2}} + 18 b^{2} c^{2} x^{3} \sqrt {a + b x + c x^{2}} + 20 b c^{3} x^{4} \sqrt {a + b x + c x^{2}} + 8 c^{4} x^{5} \sqrt {a + b x + c x^{2}}}\, dx}{d^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/(a*b**3*sqrt(a + b*x + c*x**2) + 6*a*b**2*c*x*sqrt(a + b*x + c*x**2) + 12*a*b*c**2*x**2*sqrt(a + b*
x + c*x**2) + 8*a*c**3*x**3*sqrt(a + b*x + c*x**2) + b**4*x*sqrt(a + b*x + c*x**2) + 7*b**3*c*x**2*sqrt(a + b*
x + c*x**2) + 18*b**2*c**2*x**3*sqrt(a + b*x + c*x**2) + 20*b*c**3*x**4*sqrt(a + b*x + c*x**2) + 8*c**4*x**5*s
qrt(a + b*x + c*x**2)), x)/d**3

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