3.11.44 \(\int \frac {(a+b x+c x^2)^{5/2}}{(b d+2 c d x)^{11}} \, dx\)

Optimal. Leaf size=239 \[ \frac {3 \tan ^{-1}\left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}\right )}{16384 c^{7/2} d^{11} \left (b^2-4 a c\right )^{5/2}}+\frac {3 \sqrt {a+b x+c x^2}}{8192 c^3 d^{11} \left (b^2-4 a c\right )^2 (b+2 c x)^2}+\frac {\sqrt {a+b x+c x^2}}{4096 c^3 d^{11} \left (b^2-4 a c\right ) (b+2 c x)^4}-\frac {\sqrt {a+b x+c x^2}}{1024 c^3 d^{11} (b+2 c x)^6}-\frac {\left (a+b x+c x^2\right )^{3/2}}{128 c^2 d^{11} (b+2 c x)^8}-\frac {\left (a+b x+c x^2\right )^{5/2}}{20 c d^{11} (b+2 c x)^{10}} \]

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Rubi [A]  time = 0.18, antiderivative size = 239, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {684, 693, 688, 205} \begin {gather*} \frac {3 \sqrt {a+b x+c x^2}}{8192 c^3 d^{11} \left (b^2-4 a c\right )^2 (b+2 c x)^2}+\frac {\sqrt {a+b x+c x^2}}{4096 c^3 d^{11} \left (b^2-4 a c\right ) (b+2 c x)^4}+\frac {3 \tan ^{-1}\left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}\right )}{16384 c^{7/2} d^{11} \left (b^2-4 a c\right )^{5/2}}-\frac {\left (a+b x+c x^2\right )^{3/2}}{128 c^2 d^{11} (b+2 c x)^8}-\frac {\sqrt {a+b x+c x^2}}{1024 c^3 d^{11} (b+2 c x)^6}-\frac {\left (a+b x+c x^2\right )^{5/2}}{20 c d^{11} (b+2 c x)^{10}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x + c*x^2)^(5/2)/(b*d + 2*c*d*x)^11,x]

[Out]

-Sqrt[a + b*x + c*x^2]/(1024*c^3*d^11*(b + 2*c*x)^6) + Sqrt[a + b*x + c*x^2]/(4096*c^3*(b^2 - 4*a*c)*d^11*(b +
2*c*x)^4) + (3*Sqrt[a + b*x + c*x^2])/(8192*c^3*(b^2 - 4*a*c)^2*d^11*(b + 2*c*x)^2) - (a + b*x + c*x^2)^(3/2)
/(128*c^2*d^11*(b + 2*c*x)^8) - (a + b*x + c*x^2)^(5/2)/(20*c*d^11*(b + 2*c*x)^10) + (3*ArcTan[(2*Sqrt[c]*Sqrt
[a + b*x + c*x^2])/Sqrt[b^2 - 4*a*c]])/(16384*c^(7/2)*(b^2 - 4*a*c)^(5/2)*d^11)

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 684

Int[((d_) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[((d + e*x)^(m + 1)*(
a + b*x + c*x^2)^p)/(e*(m + 1)), x] - Dist[(b*p)/(d*e*(m + 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] &&
GtQ[p, 0] && LtQ[m, -1] &&  !(IntegerQ[m/2] && LtQ[m + 2*p + 3, 0]) && 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 {\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^{11}} \, dx &=-\frac {\left (a+b x+c x^2\right )^{5/2}}{20 c d^{11} (b+2 c x)^{10}}+\frac {\int \frac {\left (a+b x+c x^2\right )^{3/2}}{(b d+2 c d x)^9} \, dx}{8 c d^2}\\ &=-\frac {\left (a+b x+c x^2\right )^{3/2}}{128 c^2 d^{11} (b+2 c x)^8}-\frac {\left (a+b x+c x^2\right )^{5/2}}{20 c d^{11} (b+2 c x)^{10}}+\frac {3 \int \frac {\sqrt {a+b x+c x^2}}{(b d+2 c d x)^7} \, dx}{256 c^2 d^4}\\ &=-\frac {\sqrt {a+b x+c x^2}}{1024 c^3 d^{11} (b+2 c x)^6}-\frac {\left (a+b x+c x^2\right )^{3/2}}{128 c^2 d^{11} (b+2 c x)^8}-\frac {\left (a+b x+c x^2\right )^{5/2}}{20 c d^{11} (b+2 c x)^{10}}+\frac {\int \frac {1}{(b d+2 c d x)^5 \sqrt {a+b x+c x^2}} \, dx}{2048 c^3 d^6}\\ &=-\frac {\sqrt {a+b x+c x^2}}{1024 c^3 d^{11} (b+2 c x)^6}+\frac {\sqrt {a+b x+c x^2}}{4096 c^3 \left (b^2-4 a c\right ) d^{11} (b+2 c x)^4}-\frac {\left (a+b x+c x^2\right )^{3/2}}{128 c^2 d^{11} (b+2 c x)^8}-\frac {\left (a+b x+c x^2\right )^{5/2}}{20 c d^{11} (b+2 c x)^{10}}+\frac {3 \int \frac {1}{(b d+2 c d x)^3 \sqrt {a+b x+c x^2}} \, dx}{8192 c^3 \left (b^2-4 a c\right ) d^8}\\ &=-\frac {\sqrt {a+b x+c x^2}}{1024 c^3 d^{11} (b+2 c x)^6}+\frac {\sqrt {a+b x+c x^2}}{4096 c^3 \left (b^2-4 a c\right ) d^{11} (b+2 c x)^4}+\frac {3 \sqrt {a+b x+c x^2}}{8192 c^3 \left (b^2-4 a c\right )^2 d^{11} (b+2 c x)^2}-\frac {\left (a+b x+c x^2\right )^{3/2}}{128 c^2 d^{11} (b+2 c x)^8}-\frac {\left (a+b x+c x^2\right )^{5/2}}{20 c d^{11} (b+2 c x)^{10}}+\frac {3 \int \frac {1}{(b d+2 c d x) \sqrt {a+b x+c x^2}} \, dx}{16384 c^3 \left (b^2-4 a c\right )^2 d^{10}}\\ &=-\frac {\sqrt {a+b x+c x^2}}{1024 c^3 d^{11} (b+2 c x)^6}+\frac {\sqrt {a+b x+c x^2}}{4096 c^3 \left (b^2-4 a c\right ) d^{11} (b+2 c x)^4}+\frac {3 \sqrt {a+b x+c x^2}}{8192 c^3 \left (b^2-4 a c\right )^2 d^{11} (b+2 c x)^2}-\frac {\left (a+b x+c x^2\right )^{3/2}}{128 c^2 d^{11} (b+2 c x)^8}-\frac {\left (a+b x+c x^2\right )^{5/2}}{20 c d^{11} (b+2 c x)^{10}}+\frac {3 \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 )}{4096 c^2 \left (b^2-4 a c\right )^2 d^{10}}\\ &=-\frac {\sqrt {a+b x+c x^2}}{1024 c^3 d^{11} (b+2 c x)^6}+\frac {\sqrt {a+b x+c x^2}}{4096 c^3 \left (b^2-4 a c\right ) d^{11} (b+2 c x)^4}+\frac {3 \sqrt {a+b x+c x^2}}{8192 c^3 \left (b^2-4 a c\right )^2 d^{11} (b+2 c x)^2}-\frac {\left (a+b x+c x^2\right )^{3/2}}{128 c^2 d^{11} (b+2 c x)^8}-\frac {\left (a+b x+c x^2\right )^{5/2}}{20 c d^{11} (b+2 c x)^{10}}+\frac {3 \tan ^{-1}\left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}\right )}{16384 c^{7/2} \left (b^2-4 a c\right )^{5/2} d^{11}}\\ \end {align*}

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Mathematica [C]  time = 0.04, size = 62, normalized size = 0.26 \begin {gather*} \frac {2 (a+x (b+c x))^{7/2} \, _2F_1\left (\frac {7}{2},6;\frac {9}{2};\frac {4 c (a+x (b+c x))}{4 a c-b^2}\right )}{7 d^{11} \left (b^2-4 a c\right )^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x + c*x^2)^(5/2)/(b*d + 2*c*d*x)^11,x]

[Out]

(2*(a + x*(b + c*x))^(7/2)*Hypergeometric2F1[7/2, 6, 9/2, (4*c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)])/(7*(b^2 - 4
*a*c)^6*d^11)

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IntegrateAlgebraic [F]  time = 180.02, size = 0, normalized size = 0.00 \begin {gather*} \text {\$Aborted} \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(a + b*x + c*x^2)^(5/2)/(b*d + 2*c*d*x)^11,x]

[Out]

$Aborted

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fricas [B]  time = 43.70, size = 2114, normalized size = 8.85

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)^(5/2)/(2*c*d*x+b*d)^11,x, algorithm="fricas")

[Out]

[-1/163840*(15*(1024*c^10*x^10 + 5120*b*c^9*x^9 + 11520*b^2*c^8*x^8 + 15360*b^3*c^7*x^7 + 13440*b^4*c^6*x^6 +
8064*b^5*c^5*x^5 + 3360*b^6*c^4*x^4 + 960*b^7*c^3*x^3 + 180*b^8*c^2*x^2 + 20*b^9*c*x + b^10)*sqrt(-b^2*c + 4*a
*c^2)*log(-(4*c^2*x^2 + 4*b*c*x - b^2 + 8*a*c - 4*sqrt(-b^2*c + 4*a*c^2)*sqrt(c*x^2 + b*x + a))/(4*c^2*x^2 + 4
*b*c*x + b^2)) + 4*(15*b^10*c - 20*a*b^8*c^2 - 32*a^2*b^6*c^3 - 11776*a^3*b^4*c^4 + 77824*a^4*b^2*c^5 - 131072
*a^5*c^6 - 3840*(b^2*c^9 - 4*a*c^10)*x^8 - 15360*(b^3*c^8 - 4*a*b*c^9)*x^7 - 640*(43*b^4*c^7 - 176*a*b^2*c^8 +
16*a^2*c^9)*x^6 - 1920*(15*b^5*c^6 - 64*a*b^3*c^7 + 16*a^2*b*c^8)*x^5 - 128*(119*b^6*c^5 - 303*a*b^4*c^6 - 11
88*a^2*b^2*c^7 + 1984*a^3*c^8)*x^4 - 128*(3*b^7*c^4 + 434*a*b^5*c^5 - 2776*a^2*b^3*c^6 + 3968*a^3*b*c^7)*x^3 +
24*(97*b^8*c^3 - 1600*a*b^6*c^4 + 6128*a^2*b^4*c^5 - 1536*a^3*b^2*c^6 - 14336*a^4*c^7)*x^2 + 8*(35*b^9*c^2 -
48*a*b^7*c^3 - 4464*a^2*b^5*c^4 + 27136*a^3*b^3*c^5 - 43008*a^4*b*c^6)*x)*sqrt(c*x^2 + b*x + a))/(1024*(b^6*c^
14 - 12*a*b^4*c^15 + 48*a^2*b^2*c^16 - 64*a^3*c^17)*d^11*x^10 + 5120*(b^7*c^13 - 12*a*b^5*c^14 + 48*a^2*b^3*c^
15 - 64*a^3*b*c^16)*d^11*x^9 + 11520*(b^8*c^12 - 12*a*b^6*c^13 + 48*a^2*b^4*c^14 - 64*a^3*b^2*c^15)*d^11*x^8 +
15360*(b^9*c^11 - 12*a*b^7*c^12 + 48*a^2*b^5*c^13 - 64*a^3*b^3*c^14)*d^11*x^7 + 13440*(b^10*c^10 - 12*a*b^8*c
^11 + 48*a^2*b^6*c^12 - 64*a^3*b^4*c^13)*d^11*x^6 + 8064*(b^11*c^9 - 12*a*b^9*c^10 + 48*a^2*b^7*c^11 - 64*a^3*
b^5*c^12)*d^11*x^5 + 3360*(b^12*c^8 - 12*a*b^10*c^9 + 48*a^2*b^8*c^10 - 64*a^3*b^6*c^11)*d^11*x^4 + 960*(b^13*
c^7 - 12*a*b^11*c^8 + 48*a^2*b^9*c^9 - 64*a^3*b^7*c^10)*d^11*x^3 + 180*(b^14*c^6 - 12*a*b^12*c^7 + 48*a^2*b^10
*c^8 - 64*a^3*b^8*c^9)*d^11*x^2 + 20*(b^15*c^5 - 12*a*b^13*c^6 + 48*a^2*b^11*c^7 - 64*a^3*b^9*c^8)*d^11*x + (b
^16*c^4 - 12*a*b^14*c^5 + 48*a^2*b^12*c^6 - 64*a^3*b^10*c^7)*d^11), -1/81920*(15*(1024*c^10*x^10 + 5120*b*c^9*
x^9 + 11520*b^2*c^8*x^8 + 15360*b^3*c^7*x^7 + 13440*b^4*c^6*x^6 + 8064*b^5*c^5*x^5 + 3360*b^6*c^4*x^4 + 960*b^
7*c^3*x^3 + 180*b^8*c^2*x^2 + 20*b^9*c*x + b^10)*sqrt(b^2*c - 4*a*c^2)*arctan(1/2*sqrt(b^2*c - 4*a*c^2)*sqrt(c
*x^2 + b*x + a)/(c^2*x^2 + b*c*x + a*c)) + 2*(15*b^10*c - 20*a*b^8*c^2 - 32*a^2*b^6*c^3 - 11776*a^3*b^4*c^4 +
77824*a^4*b^2*c^5 - 131072*a^5*c^6 - 3840*(b^2*c^9 - 4*a*c^10)*x^8 - 15360*(b^3*c^8 - 4*a*b*c^9)*x^7 - 640*(43
*b^4*c^7 - 176*a*b^2*c^8 + 16*a^2*c^9)*x^6 - 1920*(15*b^5*c^6 - 64*a*b^3*c^7 + 16*a^2*b*c^8)*x^5 - 128*(119*b^
6*c^5 - 303*a*b^4*c^6 - 1188*a^2*b^2*c^7 + 1984*a^3*c^8)*x^4 - 128*(3*b^7*c^4 + 434*a*b^5*c^5 - 2776*a^2*b^3*c
^6 + 3968*a^3*b*c^7)*x^3 + 24*(97*b^8*c^3 - 1600*a*b^6*c^4 + 6128*a^2*b^4*c^5 - 1536*a^3*b^2*c^6 - 14336*a^4*c
^7)*x^2 + 8*(35*b^9*c^2 - 48*a*b^7*c^3 - 4464*a^2*b^5*c^4 + 27136*a^3*b^3*c^5 - 43008*a^4*b*c^6)*x)*sqrt(c*x^2
+ b*x + a))/(1024*(b^6*c^14 - 12*a*b^4*c^15 + 48*a^2*b^2*c^16 - 64*a^3*c^17)*d^11*x^10 + 5120*(b^7*c^13 - 12*
a*b^5*c^14 + 48*a^2*b^3*c^15 - 64*a^3*b*c^16)*d^11*x^9 + 11520*(b^8*c^12 - 12*a*b^6*c^13 + 48*a^2*b^4*c^14 - 6
4*a^3*b^2*c^15)*d^11*x^8 + 15360*(b^9*c^11 - 12*a*b^7*c^12 + 48*a^2*b^5*c^13 - 64*a^3*b^3*c^14)*d^11*x^7 + 134
40*(b^10*c^10 - 12*a*b^8*c^11 + 48*a^2*b^6*c^12 - 64*a^3*b^4*c^13)*d^11*x^6 + 8064*(b^11*c^9 - 12*a*b^9*c^10 +
48*a^2*b^7*c^11 - 64*a^3*b^5*c^12)*d^11*x^5 + 3360*(b^12*c^8 - 12*a*b^10*c^9 + 48*a^2*b^8*c^10 - 64*a^3*b^6*c
^11)*d^11*x^4 + 960*(b^13*c^7 - 12*a*b^11*c^8 + 48*a^2*b^9*c^9 - 64*a^3*b^7*c^10)*d^11*x^3 + 180*(b^14*c^6 - 1
2*a*b^12*c^7 + 48*a^2*b^10*c^8 - 64*a^3*b^8*c^9)*d^11*x^2 + 20*(b^15*c^5 - 12*a*b^13*c^6 + 48*a^2*b^11*c^7 - 6
4*a^3*b^9*c^8)*d^11*x + (b^16*c^4 - 12*a*b^14*c^5 + 48*a^2*b^12*c^6 - 64*a^3*b^10*c^7)*d^11)]

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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((c*x^2+b*x+a)^(5/2)/(2*c*d*x+b*d)^11,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%%%{%%%{2048,[11]%%%},[22,11,0,0]%%%}+%%%{%%{[%%%{-22528,[10]%%%},0
]:[1,0,%%%{-1,[1]%%%}]%%},[21,11,1,0]%%%}+%%%{%%%{123904,[10]%%%},[20,11,2,0]%%%}+%%%{%%%{-22528,[11]%%%},[20,
11,0,1]%%%}+%%%{%%{[%%%{-450560,[9]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[19,11,3,0]%%%}+%%%{%%{[%%%{225280,[10]%%%}
,0]:[1,0,%%%{-1,[1]%%%}]%%},[19,11,1,1]%%%}+%%%{%%%{1210880,[9]%%%},[18,11,4,0]%%%}+%%%{%%%{-1126400,[10]%%%},
[18,11,2,1]%%%}+%%%{%%%{112640,[11]%%%},[18,11,0,2]%%%}+%%%{%%{[%%%{-2551296,[8]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%
},[17,11,5,0]%%%}+%%%{%%{[%%%{3717120,[9]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[17,11,3,1]%%%}+%%%{%%{[%%%{-1013760,
[10]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[17,11,1,2]%%%}+%%%{%%%{4367616,[8]%%%},[16,11,6,0]%%%}+%%%{%%%{-9039360,[
9]%%%},[16,11,4,1]%%%}+%%%{%%%{4561920,[10]%%%},[16,11,2,2]%%%}+%%%{%%%{-337920,[11]%%%},[16,11,0,3]%%%}+%%%{%
%{[%%%{-6217728,[7]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[15,11,7,0]%%%}+%%%{%%{[%%%{17166336,[8]%%%},0]:[1,0,%%%{-1
,[1]%%%}]%%},[15,11,5,1]%%%}+%%%{%%{[%%%{-13516800,[9]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[15,11,3,2]%%%}+%%%{%%{[
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5,0,17,0]%%%}+%%%{%%%{-536448,[9]%%%},[5,0,15,1]%%%}+%%%{%%%{3104640,[10]%%%},[5,0,13,2]%%%}+%%%{%%%{-10053120
,[11]%%%},[5,0,11,3]%%%}+%%%{%%%{19810560,[12]%%%},[5,0,9,4]%%%}+%%%{%%%{-24127488,[13]%%%},[5,0,7,5]%%%}+%%%{
%%%{17504256,[14]%%%},[5,0,5,6]%%%}+%%%{%%%{-6758400,[15]%%%},[5,0,3,7]%%%}+%%%{%%%{1013760,[16]%%%},[5,0,1,8]
%%%}+%%%{%%{poly1[%%%{-9460,[7]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[4,0,18,0]%%%}+%%%{%%{[%%%{141240,[8]%%%},0]:[1
,0,%%%{-1,[1]%%%}]%%},[4,0,16,1]%%%}+%%%{%%{poly1[%%%{-918720,[9]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[4,0,14,2]%%%
}+%%%{%%{[%%%{3400320,[10]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[4,0,12,3]%%%}+%%%{%%{poly1[%%%{-7835520,[11]%%%},0]
:[1,0,%%%{-1,[1]%%%}]%%},[4,0,10,4]%%%}+%%%{%%{[%%%{11531520,[12]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[4,0,8,5]%%%}
+%%%{%%{poly1[%%%{-10644480,[13]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[4,0,6,6]%%%}+%%%{%%{[%%%{5744640,[14]%%%},0]:
[1,0,%%%{-1,[1]%%%}]%%},[4,0,4,7]%%%}+%%%{%%{poly1[%%%{-1520640,[15]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[4,0,2,8]%
%%}+%%%{%%{[%%%{112640,[16]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[4,0,0,9]%%%}+%%%{%%%{1760,[7]%%%},[3,0,19,0]%%%}+%
%%{%%%{-29040,[8]%%%},[3,0,17,1]%%%}+%%%{%%%{211200,[9]%%%},[3,0,15,2]%%%}+%%%{%%%{-887040,[10]%%%},[3,0,13,3]
%%%}+%%%{%%%{2365440,[11]%%%},[3,0,11,4]%%%}+%%%{%%%{-4139520,[12]%%%},[3,0,9,5]%%%}+%%%{%%%{4730880,[13]%%%},
[3,0,7,6]%%%}+%%%{%%%{-3379200,[14]%%%},[3,0,5,7]%%%}+%%%{%%%{1351680,[15]%%%},[3,0,3,8]%%%}+%%%{%%%{-225280,[
16]%%%},[3,0,1,9]%%%}+%%%{%%{poly1[%%%{-242,[6]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[2,0,20,0]%%%}+%%%{%%{[%%%{4400
,[7]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[2,0,18,1]%%%}+%%%{%%{poly1[%%%{-35640,[8]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},
[2,0,16,2]%%%}+%%%{%%{[%%%{168960,[9]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[2,0,14,3]%%%}+%%%{%%{poly1[%%%{-517440,[
10]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[2,0,12,4]%%%}+%%%{%%{[%%%{1064448,[11]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[2,0
,10,5]%%%}+%%%{%%{poly1[%%%{-1478400,[12]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[2,0,8,6]%%%}+%%%{%%{[%%%{1351680,[13
]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[2,0,6,7]%%%}+%%%{%%{poly1[%%%{-760320,[14]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[2
,0,4,8]%%%}+%%%{%%{[%%%{225280,[15]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[2,0,2,9]%%%}+%%%{%%{[%%%{-22528,[16]%%%},0
]:[1,0,%%%{-1,[1]%%%}]%%},[2,0,0,10]%%%}+%%%{%%%{22,[6]%%%},[1,0,21,0]%%%}+%%%{%%%{-440,[7]%%%},[1,0,19,1]%%%}
+%%%{%%%{3960,[8]%%%},[1,0,17,2]%%%}+%%%{%%%{-21120,[9]%%%},[1,0,15,3]%%%}+%%%{%%%{73920,[10]%%%},[1,0,13,4]%%
%}+%%%{%%%{-177408,[11]%%%},[1,0,11,5]%%%}+%%%{%%%{295680,[12]%%%},[1,0,9,6]%%%}+%%%{%%%{-337920,[13]%%%},[1,0
,7,7]%%%}+%%%{%%%{253440,[14]%%%},[1,0,5,8]%%%}+%%%{%%%{-112640,[15]%%%},[1,0,3,9]%%%}+%%%{%%%{22528,[16]%%%},
[1,0,1,10]%%%}+%%%{%%{poly1[%%%{-1,[5]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[0,0,22,0]%%%}+%%%{%%{[%%%{22,[6]%%%},0]
:[1,0,%%%{-1,[1]%%%}]%%},[0,0,20,1]%%%}+%%%{%%{poly1[%%%{-220,[7]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[0,0,18,2]%%%
}+%%%{%%{[%%%{1320,[8]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[0,0,16,3]%%%}+%%%{%%{poly1[%%%{-5280,[9]%%%},0]:[1,0,%%
%{-1,[1]%%%}]%%},[0,0,14,4]%%%}+%%%{%%{[%%%{14784,[10]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[0,0,12,5]%%%}+%%%{%%{po
ly1[%%%{-29568,[11]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[0,0,10,6]%%%}+%%%{%%{[%%%{42240,[12]%%%},0]:[1,0,%%%{-1,[1
]%%%}]%%},[0,0,8,7]%%%}+%%%{%%{poly1[%%%{-42240,[13]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[0,0,6,8]%%%}+%%%{%%{[%%%{
28160,[14]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[0,0,4,9]%%%}+%%%{%%{[%%%{-11264,[15]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%}
,[0,0,2,10]%%%}+%%%{%%{[%%%{2048,[16]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[0,0,0,11]%%%} Error: Bad Argument Value

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maple [B]  time = 0.13, size = 1080, normalized size = 4.52 \begin {gather*} -\frac {3 a^{3} \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 )}{256 \left (4 a c -b^{2}\right )^{5} \sqrt {\frac {4 a c -b^{2}}{c}}\, c \,d^{11}}+\frac {9 a^{2} b^{2} \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 )}{1024 \left (4 a c -b^{2}\right )^{5} \sqrt {\frac {4 a c -b^{2}}{c}}\, c^{2} d^{11}}-\frac {9 a \,b^{4} \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 )}{4096 \left (4 a c -b^{2}\right )^{5} \sqrt {\frac {4 a c -b^{2}}{c}}\, c^{3} d^{11}}+\frac {3 b^{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 )}{16384 \left (4 a c -b^{2}\right )^{5} \sqrt {\frac {4 a c -b^{2}}{c}}\, c^{4} d^{11}}+\frac {3 \sqrt {4 \left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{c}}\, a^{2}}{1024 \left (4 a c -b^{2}\right )^{5} c \,d^{11}}-\frac {3 \sqrt {4 \left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{c}}\, a \,b^{2}}{2048 \left (4 a c -b^{2}\right )^{5} c^{2} d^{11}}+\frac {3 \sqrt {4 \left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{c}}\, b^{4}}{16384 \left (4 a c -b^{2}\right )^{5} c^{3} d^{11}}+\frac {\left (\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}\right )^{\frac {3}{2}} a}{512 \left (4 a c -b^{2}\right )^{5} c \,d^{11}}-\frac {\left (\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}\right )^{\frac {3}{2}} b^{2}}{2048 \left (4 a c -b^{2}\right )^{5} c^{2} d^{11}}+\frac {3 \left (\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}\right )^{\frac {5}{2}}}{2560 \left (4 a c -b^{2}\right )^{5} c \,d^{11}}-\frac {3 \left (\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}\right )^{\frac {7}{2}}}{2560 \left (4 a c -b^{2}\right )^{5} \left (x +\frac {b}{2 c}\right )^{2} c^{2} d^{11}}-\frac {\left (\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}\right )^{\frac {7}{2}}}{5120 \left (4 a c -b^{2}\right )^{4} \left (x +\frac {b}{2 c}\right )^{4} c^{4} d^{11}}-\frac {\left (\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}\right )^{\frac {7}{2}}}{5120 \left (4 a c -b^{2}\right )^{3} \left (x +\frac {b}{2 c}\right )^{6} c^{6} d^{11}}+\frac {3 \left (\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}\right )^{\frac {7}{2}}}{10240 \left (4 a c -b^{2}\right )^{2} \left (x +\frac {b}{2 c}\right )^{8} c^{8} d^{11}}-\frac {\left (\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}\right )^{\frac {7}{2}}}{5120 \left (4 a c -b^{2}\right ) \left (x +\frac {b}{2 c}\right )^{10} c^{10} d^{11}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2+b*x+a)^(5/2)/(2*c*d*x+b*d)^11,x)

[Out]

-1/5120/d^11/c^10/(4*a*c-b^2)/(x+1/2*b/c)^10*((x+1/2*b/c)^2*c+1/4*(4*a*c-b^2)/c)^(7/2)+3/10240/d^11/c^8/(4*a*c
-b^2)^2/(x+1/2*b/c)^8*((x+1/2*b/c)^2*c+1/4*(4*a*c-b^2)/c)^(7/2)-1/5120/d^11/c^6/(4*a*c-b^2)^3/(x+1/2*b/c)^6*((
x+1/2*b/c)^2*c+1/4*(4*a*c-b^2)/c)^(7/2)-1/5120/d^11/c^4/(4*a*c-b^2)^4/(x+1/2*b/c)^4*((x+1/2*b/c)^2*c+1/4*(4*a*
c-b^2)/c)^(7/2)-3/2560/d^11/c^2/(4*a*c-b^2)^5/(x+1/2*b/c)^2*((x+1/2*b/c)^2*c+1/4*(4*a*c-b^2)/c)^(7/2)+3/2560/d
^11/c/(4*a*c-b^2)^5*((x+1/2*b/c)^2*c+1/4*(4*a*c-b^2)/c)^(5/2)+1/512/d^11/c/(4*a*c-b^2)^5*((x+1/2*b/c)^2*c+1/4*
(4*a*c-b^2)/c)^(3/2)*a-1/2048/d^11/c^2/(4*a*c-b^2)^5*((x+1/2*b/c)^2*c+1/4*(4*a*c-b^2)/c)^(3/2)*b^2+3/1024/d^11
/c/(4*a*c-b^2)^5*(4*(x+1/2*b/c)^2*c+(4*a*c-b^2)/c)^(1/2)*a^2-3/2048/d^11/c^2/(4*a*c-b^2)^5*(4*(x+1/2*b/c)^2*c+
(4*a*c-b^2)/c)^(1/2)*a*b^2+3/16384/d^11/c^3/(4*a*c-b^2)^5*(4*(x+1/2*b/c)^2*c+(4*a*c-b^2)/c)^(1/2)*b^4-3/256/d^
11/c/(4*a*c-b^2)^5/((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))*a^3+9/1024/d^11/c^2/(4*a*c-b^2)^5/((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))*a^2*b^2-9/4096/d^11/c^3/(4*a*
c-b^2)^5/((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))*a*b^4+3/16384/d^11/c^4/(4*a*c-b^2)^5/((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))*b^6

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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((c*x^2+b*x+a)^(5/2)/(2*c*d*x+b*d)^11,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 zero or nonzero?

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (c\,x^2+b\,x+a\right )}^{5/2}}{{\left (b\,d+2\,c\,d\,x\right )}^{11}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x + c*x^2)^(5/2)/(b*d + 2*c*d*x)^11,x)

[Out]

int((a + b*x + c*x^2)^(5/2)/(b*d + 2*c*d*x)^11, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {\int \frac {a^{2} \sqrt {a + b x + c x^{2}}}{b^{11} + 22 b^{10} c x + 220 b^{9} c^{2} x^{2} + 1320 b^{8} c^{3} x^{3} + 5280 b^{7} c^{4} x^{4} + 14784 b^{6} c^{5} x^{5} + 29568 b^{5} c^{6} x^{6} + 42240 b^{4} c^{7} x^{7} + 42240 b^{3} c^{8} x^{8} + 28160 b^{2} c^{9} x^{9} + 11264 b c^{10} x^{10} + 2048 c^{11} x^{11}}\, dx + \int \frac {b^{2} x^{2} \sqrt {a + b x + c x^{2}}}{b^{11} + 22 b^{10} c x + 220 b^{9} c^{2} x^{2} + 1320 b^{8} c^{3} x^{3} + 5280 b^{7} c^{4} x^{4} + 14784 b^{6} c^{5} x^{5} + 29568 b^{5} c^{6} x^{6} + 42240 b^{4} c^{7} x^{7} + 42240 b^{3} c^{8} x^{8} + 28160 b^{2} c^{9} x^{9} + 11264 b c^{10} x^{10} + 2048 c^{11} x^{11}}\, dx + \int \frac {c^{2} x^{4} \sqrt {a + b x + c x^{2}}}{b^{11} + 22 b^{10} c x + 220 b^{9} c^{2} x^{2} + 1320 b^{8} c^{3} x^{3} + 5280 b^{7} c^{4} x^{4} + 14784 b^{6} c^{5} x^{5} + 29568 b^{5} c^{6} x^{6} + 42240 b^{4} c^{7} x^{7} + 42240 b^{3} c^{8} x^{8} + 28160 b^{2} c^{9} x^{9} + 11264 b c^{10} x^{10} + 2048 c^{11} x^{11}}\, dx + \int \frac {2 a b x \sqrt {a + b x + c x^{2}}}{b^{11} + 22 b^{10} c x + 220 b^{9} c^{2} x^{2} + 1320 b^{8} c^{3} x^{3} + 5280 b^{7} c^{4} x^{4} + 14784 b^{6} c^{5} x^{5} + 29568 b^{5} c^{6} x^{6} + 42240 b^{4} c^{7} x^{7} + 42240 b^{3} c^{8} x^{8} + 28160 b^{2} c^{9} x^{9} + 11264 b c^{10} x^{10} + 2048 c^{11} x^{11}}\, dx + \int \frac {2 a c x^{2} \sqrt {a + b x + c x^{2}}}{b^{11} + 22 b^{10} c x + 220 b^{9} c^{2} x^{2} + 1320 b^{8} c^{3} x^{3} + 5280 b^{7} c^{4} x^{4} + 14784 b^{6} c^{5} x^{5} + 29568 b^{5} c^{6} x^{6} + 42240 b^{4} c^{7} x^{7} + 42240 b^{3} c^{8} x^{8} + 28160 b^{2} c^{9} x^{9} + 11264 b c^{10} x^{10} + 2048 c^{11} x^{11}}\, dx + \int \frac {2 b c x^{3} \sqrt {a + b x + c x^{2}}}{b^{11} + 22 b^{10} c x + 220 b^{9} c^{2} x^{2} + 1320 b^{8} c^{3} x^{3} + 5280 b^{7} c^{4} x^{4} + 14784 b^{6} c^{5} x^{5} + 29568 b^{5} c^{6} x^{6} + 42240 b^{4} c^{7} x^{7} + 42240 b^{3} c^{8} x^{8} + 28160 b^{2} c^{9} x^{9} + 11264 b c^{10} x^{10} + 2048 c^{11} x^{11}}\, dx}{d^{11}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2+b*x+a)**(5/2)/(2*c*d*x+b*d)**11,x)

[Out]

(Integral(a**2*sqrt(a + b*x + c*x**2)/(b**11 + 22*b**10*c*x + 220*b**9*c**2*x**2 + 1320*b**8*c**3*x**3 + 5280*
b**7*c**4*x**4 + 14784*b**6*c**5*x**5 + 29568*b**5*c**6*x**6 + 42240*b**4*c**7*x**7 + 42240*b**3*c**8*x**8 + 2
8160*b**2*c**9*x**9 + 11264*b*c**10*x**10 + 2048*c**11*x**11), x) + Integral(b**2*x**2*sqrt(a + b*x + c*x**2)/
(b**11 + 22*b**10*c*x + 220*b**9*c**2*x**2 + 1320*b**8*c**3*x**3 + 5280*b**7*c**4*x**4 + 14784*b**6*c**5*x**5
+ 29568*b**5*c**6*x**6 + 42240*b**4*c**7*x**7 + 42240*b**3*c**8*x**8 + 28160*b**2*c**9*x**9 + 11264*b*c**10*x*
*10 + 2048*c**11*x**11), x) + Integral(c**2*x**4*sqrt(a + b*x + c*x**2)/(b**11 + 22*b**10*c*x + 220*b**9*c**2*
x**2 + 1320*b**8*c**3*x**3 + 5280*b**7*c**4*x**4 + 14784*b**6*c**5*x**5 + 29568*b**5*c**6*x**6 + 42240*b**4*c*
*7*x**7 + 42240*b**3*c**8*x**8 + 28160*b**2*c**9*x**9 + 11264*b*c**10*x**10 + 2048*c**11*x**11), x) + Integral
(2*a*b*x*sqrt(a + b*x + c*x**2)/(b**11 + 22*b**10*c*x + 220*b**9*c**2*x**2 + 1320*b**8*c**3*x**3 + 5280*b**7*c
**4*x**4 + 14784*b**6*c**5*x**5 + 29568*b**5*c**6*x**6 + 42240*b**4*c**7*x**7 + 42240*b**3*c**8*x**8 + 28160*b
**2*c**9*x**9 + 11264*b*c**10*x**10 + 2048*c**11*x**11), x) + Integral(2*a*c*x**2*sqrt(a + b*x + c*x**2)/(b**1
1 + 22*b**10*c*x + 220*b**9*c**2*x**2 + 1320*b**8*c**3*x**3 + 5280*b**7*c**4*x**4 + 14784*b**6*c**5*x**5 + 295
68*b**5*c**6*x**6 + 42240*b**4*c**7*x**7 + 42240*b**3*c**8*x**8 + 28160*b**2*c**9*x**9 + 11264*b*c**10*x**10 +
2048*c**11*x**11), x) + Integral(2*b*c*x**3*sqrt(a + b*x + c*x**2)/(b**11 + 22*b**10*c*x + 220*b**9*c**2*x**2
+ 1320*b**8*c**3*x**3 + 5280*b**7*c**4*x**4 + 14784*b**6*c**5*x**5 + 29568*b**5*c**6*x**6 + 42240*b**4*c**7*x
**7 + 42240*b**3*c**8*x**8 + 28160*b**2*c**9*x**9 + 11264*b*c**10*x**10 + 2048*c**11*x**11), x))/d**11

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