3.13.32 \(\int \frac {(a+b x+c x^2)^{5/2}}{(b d+2 c d x)^{11}} \, dx\) [1232]

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

[Out]

-1/128*(c*x^2+b*x+a)^(3/2)/c^2/d^11/(2*c*x+b)^8-1/20*(c*x^2+b*x+a)^(5/2)/c/d^11/(2*c*x+b)^10+3/16384*arctan(2*
c^(1/2)*(c*x^2+b*x+a)^(1/2)/(-4*a*c+b^2)^(1/2))/c^(7/2)/(-4*a*c+b^2)^(5/2)/d^11-1/1024*(c*x^2+b*x+a)^(1/2)/c^3
/d^11/(2*c*x+b)^6+1/4096*(c*x^2+b*x+a)^(1/2)/c^3/(-4*a*c+b^2)/d^11/(2*c*x+b)^4+3/8192*(c*x^2+b*x+a)^(1/2)/c^3/
(-4*a*c+b^2)^2/d^11/(2*c*x+b)^2

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Rubi [A]
time = 0.12, 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 = {698, 707, 702, 211} \begin {gather*} \frac {3 \text {ArcTan}\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}} \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]

-1/1024*Sqrt[a + b*x + c*x^2]/(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]*Sq
rt[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 211

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

Rule 698

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 702

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 707

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 \text {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] Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
time = 10.03, 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))}{-b^2+4 a c}\right )}{7 \left (b^2-4 a c\right )^6 d^{11}} \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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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(609\) vs. \(2(207)=414\).
time = 0.70, size = 610, normalized size = 2.55

method	result	size
default	\(\frac {-\frac {2 c \left (\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}\right )^{\frac {7}{2}}}{5 \left (4 a c -b^{2}\right ) \left (x +\frac {b}{2 c}\right )^{10}}-\frac {6 c^{2} \left (-\frac {c \left (\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}\right )^{\frac {7}{2}}}{2 \left (4 a c -b^{2}\right ) \left (x +\frac {b}{2 c}\right )^{8}}-\frac {c^{2} \left (-\frac {2 c \left (\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}\right )^{\frac {7}{2}}}{3 \left (4 a c -b^{2}\right ) \left (x +\frac {b}{2 c}\right )^{6}}+\frac {2 c^{2} \left (-\frac {c \left (\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}\right )^{\frac {7}{2}}}{\left (4 a c -b^{2}\right ) \left (x +\frac {b}{2 c}\right )^{4}}+\frac {3 c^{2} \left (-\frac {2 c \left (\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}\right )^{\frac {7}{2}}}{\left (4 a c -b^{2}\right ) \left (x +\frac {b}{2 c}\right )^{2}}+\frac {10 c^{2} \left (\frac {\left (\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}\right )^{\frac {5}{2}}}{5}+\frac {\left (4 a c -b^{2}\right ) \left (\frac {\left (\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}\right )^{\frac {3}{2}}}{3}+\frac {\left (4 a c -b^{2}\right ) \left (\frac {\sqrt {4 \left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{c}}}{2}-\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {4 a c -b^{2}}{2 c}+\frac {\sqrt {\frac {4 a c -b^{2}}{c}}\, \sqrt {4 \left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{c}}}{2}}{x +\frac {b}{2 c}}\right )}{2 c \sqrt {\frac {4 a c -b^{2}}{c}}}\right )}{4 c}\right )}{4 c}\right )}{4 a c -b^{2}}\right )}{4 a c -b^{2}}\right )}{3 \left (4 a c -b^{2}\right )}\right )}{2 \left (4 a c -b^{2}\right )}\right )}{5 \left (4 a c -b^{2}\right )}}{2048 d^{11} c^{11}}\)	\(610\)

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,method=_RETURNVERBOSE)

[Out]

1/2048/d^11/c^11*(-2/5/(4*a*c-b^2)*c/(x+1/2*b/c)^10*((x+1/2*b/c)^2*c+1/4*(4*a*c-b^2)/c)^(7/2)-6/5*c^2/(4*a*c-b
^2)*(-1/2/(4*a*c-b^2)*c/(x+1/2*b/c)^8*((x+1/2*b/c)^2*c+1/4*(4*a*c-b^2)/c)^(7/2)-1/2*c^2/(4*a*c-b^2)*(-2/3/(4*a
*c-b^2)*c/(x+1/2*b/c)^6*((x+1/2*b/c)^2*c+1/4*(4*a*c-b^2)/c)^(7/2)+2/3*c^2/(4*a*c-b^2)*(-1/(4*a*c-b^2)*c/(x+1/2
*b/c)^4*((x+1/2*b/c)^2*c+1/4*(4*a*c-b^2)/c)^(7/2)+3*c^2/(4*a*c-b^2)*(-2/(4*a*c-b^2)*c/(x+1/2*b/c)^2*((x+1/2*b/
c)^2*c+1/4*(4*a*c-b^2)/c)^(7/2)+10*c^2/(4*a*c-b^2)*(1/5*((x+1/2*b/c)^2*c+1/4*(4*a*c-b^2)/c)^(5/2)+1/4*(4*a*c-b
^2)/c*(1/3*((x+1/2*b/c)^2*c+1/4*(4*a*c-b^2)/c)^(3/2)+1/4*(4*a*c-b^2)/c*(1/2*(4*(x+1/2*b/c)^2*c+(4*a*c-b^2)/c)^
(1/2)-1/2*(4*a*c-b^2)/c/((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((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 deta

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1042 vs. \(2 (207) = 414\).
time = 70.68, size = 2114, normalized size = 8.85 \begin {gather*} \text {Too large to display} \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="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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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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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,sageVARx):;OUTP
UT:Unable to divide, perhaps due to rounding error%%%{%%%{2048,[11]%%%},[22,11,0,0]%%%}+%%%{%%{[%%%{-22528,[10
]%%%},0]:[1

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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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