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

Optimal. Leaf size=200 \[ \frac {b^2 \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^6}{120 (d+e x)^8 (b d-a e)^3}+\frac {b \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^6}{30 (d+e x)^9 (b d-a e)^2}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^6}{10 (d+e x)^{10} (b d-a e)}+\frac {b^3 \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^6}{840 (d+e x)^7 (b d-a e)^4} \]

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Rubi [A]  time = 0.09, antiderivative size = 200, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.121, Rules used = {770, 21, 45, 37} \begin {gather*} \frac {b^3 \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^6}{840 (d+e x)^7 (b d-a e)^4}+\frac {b^2 \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^6}{120 (d+e x)^8 (b d-a e)^3}+\frac {b \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^6}{30 (d+e x)^9 (b d-a e)^2}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^6}{10 (d+e x)^{10} (b d-a e)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((a + b*x)^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(10*(b*d - a*e)*(d + e*x)^10) + (b*(a + b*x)^6*Sqrt[a^2 + 2*a*b*x
+ b^2*x^2])/(30*(b*d - a*e)^2*(d + e*x)^9) + (b^2*(a + b*x)^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(120*(b*d - a*e)^
3*(d + e*x)^8) + (b^3*(a + b*x)^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(840*(b*d - a*e)^4*(d + e*x)^7)

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*Simplify[m + n + 2])/((b*c - a*d)*(m + 1)), Int[(a + b*x)^Simplify[m +
1]*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[Simplify[m + n + 2], 0] &&
NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (
SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 1])

Rule 770

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dis
t[(a + b*x + c*x^2)^FracPart[p]/(c^IntPart[p]*(b/2 + c*x)^(2*FracPart[p])), Int[(d + e*x)^m*(f + g*x)*(b/2 + c
*x)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && EqQ[b^2 - 4*a*c, 0]

Rubi steps

\begin {align*} \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{11}} \, dx &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {(a+b x) \left (a b+b^2 x\right )^5}{(d+e x)^{11}} \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=\frac {\left (b \sqrt {a^2+2 a b x+b^2 x^2}\right ) \int \frac {(a+b x)^6}{(d+e x)^{11}} \, dx}{a b+b^2 x}\\ &=\frac {(a+b x)^6 \sqrt {a^2+2 a b x+b^2 x^2}}{10 (b d-a e) (d+e x)^{10}}+\frac {\left (3 b^2 \sqrt {a^2+2 a b x+b^2 x^2}\right ) \int \frac {(a+b x)^6}{(d+e x)^{10}} \, dx}{10 (b d-a e) \left (a b+b^2 x\right )}\\ &=\frac {(a+b x)^6 \sqrt {a^2+2 a b x+b^2 x^2}}{10 (b d-a e) (d+e x)^{10}}+\frac {b (a+b x)^6 \sqrt {a^2+2 a b x+b^2 x^2}}{30 (b d-a e)^2 (d+e x)^9}+\frac {\left (b^3 \sqrt {a^2+2 a b x+b^2 x^2}\right ) \int \frac {(a+b x)^6}{(d+e x)^9} \, dx}{15 (b d-a e)^2 \left (a b+b^2 x\right )}\\ &=\frac {(a+b x)^6 \sqrt {a^2+2 a b x+b^2 x^2}}{10 (b d-a e) (d+e x)^{10}}+\frac {b (a+b x)^6 \sqrt {a^2+2 a b x+b^2 x^2}}{30 (b d-a e)^2 (d+e x)^9}+\frac {b^2 (a+b x)^6 \sqrt {a^2+2 a b x+b^2 x^2}}{120 (b d-a e)^3 (d+e x)^8}+\frac {\left (b^4 \sqrt {a^2+2 a b x+b^2 x^2}\right ) \int \frac {(a+b x)^6}{(d+e x)^8} \, dx}{120 (b d-a e)^3 \left (a b+b^2 x\right )}\\ &=\frac {(a+b x)^6 \sqrt {a^2+2 a b x+b^2 x^2}}{10 (b d-a e) (d+e x)^{10}}+\frac {b (a+b x)^6 \sqrt {a^2+2 a b x+b^2 x^2}}{30 (b d-a e)^2 (d+e x)^9}+\frac {b^2 (a+b x)^6 \sqrt {a^2+2 a b x+b^2 x^2}}{120 (b d-a e)^3 (d+e x)^8}+\frac {b^3 (a+b x)^6 \sqrt {a^2+2 a b x+b^2 x^2}}{840 (b d-a e)^4 (d+e x)^7}\\ \end {align*}

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Mathematica [A]  time = 0.11, size = 295, normalized size = 1.48 \begin {gather*} -\frac {\sqrt {(a+b x)^2} \left (84 a^6 e^6+56 a^5 b e^5 (d+10 e x)+35 a^4 b^2 e^4 \left (d^2+10 d e x+45 e^2 x^2\right )+20 a^3 b^3 e^3 \left (d^3+10 d^2 e x+45 d e^2 x^2+120 e^3 x^3\right )+10 a^2 b^4 e^2 \left (d^4+10 d^3 e x+45 d^2 e^2 x^2+120 d e^3 x^3+210 e^4 x^4\right )+4 a b^5 e \left (d^5+10 d^4 e x+45 d^3 e^2 x^2+120 d^2 e^3 x^3+210 d e^4 x^4+252 e^5 x^5\right )+b^6 \left (d^6+10 d^5 e x+45 d^4 e^2 x^2+120 d^3 e^3 x^3+210 d^2 e^4 x^4+252 d e^5 x^5+210 e^6 x^6\right )\right )}{840 e^7 (a+b x) (d+e x)^{10}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/840*(Sqrt[(a + b*x)^2]*(84*a^6*e^6 + 56*a^5*b*e^5*(d + 10*e*x) + 35*a^4*b^2*e^4*(d^2 + 10*d*e*x + 45*e^2*x^
2) + 20*a^3*b^3*e^3*(d^3 + 10*d^2*e*x + 45*d*e^2*x^2 + 120*e^3*x^3) + 10*a^2*b^4*e^2*(d^4 + 10*d^3*e*x + 45*d^
2*e^2*x^2 + 120*d*e^3*x^3 + 210*e^4*x^4) + 4*a*b^5*e*(d^5 + 10*d^4*e*x + 45*d^3*e^2*x^2 + 120*d^2*e^3*x^3 + 21
0*d*e^4*x^4 + 252*e^5*x^5) + b^6*(d^6 + 10*d^5*e*x + 45*d^4*e^2*x^2 + 120*d^3*e^3*x^3 + 210*d^2*e^4*x^4 + 252*
d*e^5*x^5 + 210*e^6*x^6)))/(e^7*(a + b*x)*(d + e*x)^10)

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IntegrateAlgebraic [F]  time = 180.06, size = 0, normalized size = 0.00 \begin {gather*} \text {\$Aborted} \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

$Aborted

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fricas [B]  time = 0.45, size = 452, normalized size = 2.26 \begin {gather*} -\frac {210 \, b^{6} e^{6} x^{6} + b^{6} d^{6} + 4 \, a b^{5} d^{5} e + 10 \, a^{2} b^{4} d^{4} e^{2} + 20 \, a^{3} b^{3} d^{3} e^{3} + 35 \, a^{4} b^{2} d^{2} e^{4} + 56 \, a^{5} b d e^{5} + 84 \, a^{6} e^{6} + 252 \, {\left (b^{6} d e^{5} + 4 \, a b^{5} e^{6}\right )} x^{5} + 210 \, {\left (b^{6} d^{2} e^{4} + 4 \, a b^{5} d e^{5} + 10 \, a^{2} b^{4} e^{6}\right )} x^{4} + 120 \, {\left (b^{6} d^{3} e^{3} + 4 \, a b^{5} d^{2} e^{4} + 10 \, a^{2} b^{4} d e^{5} + 20 \, a^{3} b^{3} e^{6}\right )} x^{3} + 45 \, {\left (b^{6} d^{4} e^{2} + 4 \, a b^{5} d^{3} e^{3} + 10 \, a^{2} b^{4} d^{2} e^{4} + 20 \, a^{3} b^{3} d e^{5} + 35 \, a^{4} b^{2} e^{6}\right )} x^{2} + 10 \, {\left (b^{6} d^{5} e + 4 \, a b^{5} d^{4} e^{2} + 10 \, a^{2} b^{4} d^{3} e^{3} + 20 \, a^{3} b^{3} d^{2} e^{4} + 35 \, a^{4} b^{2} d e^{5} + 56 \, a^{5} b e^{6}\right )} x}{840 \, {\left (e^{17} x^{10} + 10 \, d e^{16} x^{9} + 45 \, d^{2} e^{15} x^{8} + 120 \, d^{3} e^{14} x^{7} + 210 \, d^{4} e^{13} x^{6} + 252 \, d^{5} e^{12} x^{5} + 210 \, d^{6} e^{11} x^{4} + 120 \, d^{7} e^{10} x^{3} + 45 \, d^{8} e^{9} x^{2} + 10 \, d^{9} e^{8} x + d^{10} e^{7}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/840*(210*b^6*e^6*x^6 + b^6*d^6 + 4*a*b^5*d^5*e + 10*a^2*b^4*d^4*e^2 + 20*a^3*b^3*d^3*e^3 + 35*a^4*b^2*d^2*e
^4 + 56*a^5*b*d*e^5 + 84*a^6*e^6 + 252*(b^6*d*e^5 + 4*a*b^5*e^6)*x^5 + 210*(b^6*d^2*e^4 + 4*a*b^5*d*e^5 + 10*a
^2*b^4*e^6)*x^4 + 120*(b^6*d^3*e^3 + 4*a*b^5*d^2*e^4 + 10*a^2*b^4*d*e^5 + 20*a^3*b^3*e^6)*x^3 + 45*(b^6*d^4*e^
2 + 4*a*b^5*d^3*e^3 + 10*a^2*b^4*d^2*e^4 + 20*a^3*b^3*d*e^5 + 35*a^4*b^2*e^6)*x^2 + 10*(b^6*d^5*e + 4*a*b^5*d^
4*e^2 + 10*a^2*b^4*d^3*e^3 + 20*a^3*b^3*d^2*e^4 + 35*a^4*b^2*d*e^5 + 56*a^5*b*e^6)*x)/(e^17*x^10 + 10*d*e^16*x
^9 + 45*d^2*e^15*x^8 + 120*d^3*e^14*x^7 + 210*d^4*e^13*x^6 + 252*d^5*e^12*x^5 + 210*d^6*e^11*x^4 + 120*d^7*e^1
0*x^3 + 45*d^8*e^9*x^2 + 10*d^9*e^8*x + d^10*e^7)

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giac [B]  time = 0.20, size = 520, normalized size = 2.60 \begin {gather*} -\frac {{\left (210 \, b^{6} x^{6} e^{6} \mathrm {sgn}\left (b x + a\right ) + 252 \, b^{6} d x^{5} e^{5} \mathrm {sgn}\left (b x + a\right ) + 210 \, b^{6} d^{2} x^{4} e^{4} \mathrm {sgn}\left (b x + a\right ) + 120 \, b^{6} d^{3} x^{3} e^{3} \mathrm {sgn}\left (b x + a\right ) + 45 \, b^{6} d^{4} x^{2} e^{2} \mathrm {sgn}\left (b x + a\right ) + 10 \, b^{6} d^{5} x e \mathrm {sgn}\left (b x + a\right ) + b^{6} d^{6} \mathrm {sgn}\left (b x + a\right ) + 1008 \, a b^{5} x^{5} e^{6} \mathrm {sgn}\left (b x + a\right ) + 840 \, a b^{5} d x^{4} e^{5} \mathrm {sgn}\left (b x + a\right ) + 480 \, a b^{5} d^{2} x^{3} e^{4} \mathrm {sgn}\left (b x + a\right ) + 180 \, a b^{5} d^{3} x^{2} e^{3} \mathrm {sgn}\left (b x + a\right ) + 40 \, a b^{5} d^{4} x e^{2} \mathrm {sgn}\left (b x + a\right ) + 4 \, a b^{5} d^{5} e \mathrm {sgn}\left (b x + a\right ) + 2100 \, a^{2} b^{4} x^{4} e^{6} \mathrm {sgn}\left (b x + a\right ) + 1200 \, a^{2} b^{4} d x^{3} e^{5} \mathrm {sgn}\left (b x + a\right ) + 450 \, a^{2} b^{4} d^{2} x^{2} e^{4} \mathrm {sgn}\left (b x + a\right ) + 100 \, a^{2} b^{4} d^{3} x e^{3} \mathrm {sgn}\left (b x + a\right ) + 10 \, a^{2} b^{4} d^{4} e^{2} \mathrm {sgn}\left (b x + a\right ) + 2400 \, a^{3} b^{3} x^{3} e^{6} \mathrm {sgn}\left (b x + a\right ) + 900 \, a^{3} b^{3} d x^{2} e^{5} \mathrm {sgn}\left (b x + a\right ) + 200 \, a^{3} b^{3} d^{2} x e^{4} \mathrm {sgn}\left (b x + a\right ) + 20 \, a^{3} b^{3} d^{3} e^{3} \mathrm {sgn}\left (b x + a\right ) + 1575 \, a^{4} b^{2} x^{2} e^{6} \mathrm {sgn}\left (b x + a\right ) + 350 \, a^{4} b^{2} d x e^{5} \mathrm {sgn}\left (b x + a\right ) + 35 \, a^{4} b^{2} d^{2} e^{4} \mathrm {sgn}\left (b x + a\right ) + 560 \, a^{5} b x e^{6} \mathrm {sgn}\left (b x + a\right ) + 56 \, a^{5} b d e^{5} \mathrm {sgn}\left (b x + a\right ) + 84 \, a^{6} e^{6} \mathrm {sgn}\left (b x + a\right )\right )} e^{\left (-7\right )}}{840 \, {\left (x e + d\right )}^{10}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/840*(210*b^6*x^6*e^6*sgn(b*x + a) + 252*b^6*d*x^5*e^5*sgn(b*x + a) + 210*b^6*d^2*x^4*e^4*sgn(b*x + a) + 120
*b^6*d^3*x^3*e^3*sgn(b*x + a) + 45*b^6*d^4*x^2*e^2*sgn(b*x + a) + 10*b^6*d^5*x*e*sgn(b*x + a) + b^6*d^6*sgn(b*
x + a) + 1008*a*b^5*x^5*e^6*sgn(b*x + a) + 840*a*b^5*d*x^4*e^5*sgn(b*x + a) + 480*a*b^5*d^2*x^3*e^4*sgn(b*x +
a) + 180*a*b^5*d^3*x^2*e^3*sgn(b*x + a) + 40*a*b^5*d^4*x*e^2*sgn(b*x + a) + 4*a*b^5*d^5*e*sgn(b*x + a) + 2100*
a^2*b^4*x^4*e^6*sgn(b*x + a) + 1200*a^2*b^4*d*x^3*e^5*sgn(b*x + a) + 450*a^2*b^4*d^2*x^2*e^4*sgn(b*x + a) + 10
0*a^2*b^4*d^3*x*e^3*sgn(b*x + a) + 10*a^2*b^4*d^4*e^2*sgn(b*x + a) + 2400*a^3*b^3*x^3*e^6*sgn(b*x + a) + 900*a
^3*b^3*d*x^2*e^5*sgn(b*x + a) + 200*a^3*b^3*d^2*x*e^4*sgn(b*x + a) + 20*a^3*b^3*d^3*e^3*sgn(b*x + a) + 1575*a^
4*b^2*x^2*e^6*sgn(b*x + a) + 350*a^4*b^2*d*x*e^5*sgn(b*x + a) + 35*a^4*b^2*d^2*e^4*sgn(b*x + a) + 560*a^5*b*x*
e^6*sgn(b*x + a) + 56*a^5*b*d*e^5*sgn(b*x + a) + 84*a^6*e^6*sgn(b*x + a))*e^(-7)/(x*e + d)^10

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maple [B]  time = 0.06, size = 392, normalized size = 1.96 \begin {gather*} -\frac {\left (210 b^{6} e^{6} x^{6}+1008 a \,b^{5} e^{6} x^{5}+252 b^{6} d \,e^{5} x^{5}+2100 a^{2} b^{4} e^{6} x^{4}+840 a \,b^{5} d \,e^{5} x^{4}+210 b^{6} d^{2} e^{4} x^{4}+2400 a^{3} b^{3} e^{6} x^{3}+1200 a^{2} b^{4} d \,e^{5} x^{3}+480 a \,b^{5} d^{2} e^{4} x^{3}+120 b^{6} d^{3} e^{3} x^{3}+1575 a^{4} b^{2} e^{6} x^{2}+900 a^{3} b^{3} d \,e^{5} x^{2}+450 a^{2} b^{4} d^{2} e^{4} x^{2}+180 a \,b^{5} d^{3} e^{3} x^{2}+45 b^{6} d^{4} e^{2} x^{2}+560 a^{5} b \,e^{6} x +350 a^{4} b^{2} d \,e^{5} x +200 a^{3} b^{3} d^{2} e^{4} x +100 a^{2} b^{4} d^{3} e^{3} x +40 a \,b^{5} d^{4} e^{2} x +10 b^{6} d^{5} e x +84 a^{6} e^{6}+56 a^{5} b d \,e^{5}+35 a^{4} b^{2} d^{2} e^{4}+20 a^{3} b^{3} d^{3} e^{3}+10 a^{2} b^{4} d^{4} e^{2}+4 a \,b^{5} d^{5} e +b^{6} d^{6}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{840 \left (e x +d \right )^{10} \left (b x +a \right )^{5} e^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/840/e^7*(210*b^6*e^6*x^6+1008*a*b^5*e^6*x^5+252*b^6*d*e^5*x^5+2100*a^2*b^4*e^6*x^4+840*a*b^5*d*e^5*x^4+210*
b^6*d^2*e^4*x^4+2400*a^3*b^3*e^6*x^3+1200*a^2*b^4*d*e^5*x^3+480*a*b^5*d^2*e^4*x^3+120*b^6*d^3*e^3*x^3+1575*a^4
*b^2*e^6*x^2+900*a^3*b^3*d*e^5*x^2+450*a^2*b^4*d^2*e^4*x^2+180*a*b^5*d^3*e^3*x^2+45*b^6*d^4*e^2*x^2+560*a^5*b*
e^6*x+350*a^4*b^2*d*e^5*x+200*a^3*b^3*d^2*e^4*x+100*a^2*b^4*d^3*e^3*x+40*a*b^5*d^4*e^2*x+10*b^6*d^5*e*x+84*a^6
*e^6+56*a^5*b*d*e^5+35*a^4*b^2*d^2*e^4+20*a^3*b^3*d^3*e^3+10*a^2*b^4*d^4*e^2+4*a*b^5*d^5*e+b^6*d^6)*((b*x+a)^2
)^(5/2)/(e*x+d)^10/(b*x+a)^5

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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((b*x+a)*(b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+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(a*e-b*d>0)', see `assume?` for
more details)Is a*e-b*d zero or nonzero?

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mupad [B]  time = 2.53, size = 1010, normalized size = 5.05 \begin {gather*} \frac {\left (\frac {-6\,a^5\,b\,e^5+15\,a^4\,b^2\,d\,e^4-20\,a^3\,b^3\,d^2\,e^3+15\,a^2\,b^4\,d^3\,e^2-6\,a\,b^5\,d^4\,e+b^6\,d^5}{9\,e^7}+\frac {d\,\left (\frac {15\,a^4\,b^2\,e^5-20\,a^3\,b^3\,d\,e^4+15\,a^2\,b^4\,d^2\,e^3-6\,a\,b^5\,d^3\,e^2+b^6\,d^4\,e}{9\,e^7}-\frac {d\,\left (\frac {20\,a^3\,b^3\,e^5-15\,a^2\,b^4\,d\,e^4+6\,a\,b^5\,d^2\,e^3-b^6\,d^3\,e^2}{9\,e^7}-\frac {d\,\left (\frac {d\,\left (\frac {b^6\,d}{9\,e^3}-\frac {b^5\,\left (6\,a\,e-b\,d\right )}{9\,e^3}\right )}{e}+\frac {b^4\,\left (15\,a^2\,e^2-6\,a\,b\,d\,e+b^2\,d^2\right )}{9\,e^4}\right )}{e}\right )}{e}\right )}{e}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^9}-\frac {\left (\frac {15\,a^2\,b^4\,e^2-24\,a\,b^5\,d\,e+10\,b^6\,d^2}{6\,e^7}+\frac {d\,\left (\frac {b^6\,d}{6\,e^6}-\frac {b^5\,\left (3\,a\,e-2\,b\,d\right )}{3\,e^6}\right )}{e}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^6}-\frac {\left (\frac {a^6}{10\,e}-\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (\frac {3\,a\,b^5}{5\,e}-\frac {b^6\,d}{10\,e^2}\right )}{e}-\frac {3\,a^2\,b^4}{2\,e}\right )}{e}+\frac {2\,a^3\,b^3}{e}\right )}{e}-\frac {3\,a^4\,b^2}{2\,e}\right )}{e}+\frac {3\,a^5\,b}{5\,e}\right )}{e}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^{10}}-\frac {\left (\frac {15\,a^4\,b^2\,e^4-40\,a^3\,b^3\,d\,e^3+45\,a^2\,b^4\,d^2\,e^2-24\,a\,b^5\,d^3\,e+5\,b^6\,d^4}{8\,e^7}+\frac {d\,\left (\frac {-20\,a^3\,b^3\,e^4+30\,a^2\,b^4\,d\,e^3-18\,a\,b^5\,d^2\,e^2+4\,b^6\,d^3\,e}{8\,e^7}+\frac {d\,\left (\frac {d\,\left (\frac {b^6\,d}{8\,e^4}-\frac {b^5\,\left (3\,a\,e-b\,d\right )}{4\,e^4}\right )}{e}+\frac {3\,b^4\,\left (5\,a^2\,e^2-4\,a\,b\,d\,e+b^2\,d^2\right )}{8\,e^5}\right )}{e}\right )}{e}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^8}+\frac {\left (\frac {5\,b^6\,d-6\,a\,b^5\,e}{5\,e^7}+\frac {b^6\,d}{5\,e^7}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^5}+\frac {\left (\frac {-20\,a^3\,b^3\,e^3+45\,a^2\,b^4\,d\,e^2-36\,a\,b^5\,d^2\,e+10\,b^6\,d^3}{7\,e^7}+\frac {d\,\left (\frac {d\,\left (\frac {b^6\,d}{7\,e^5}-\frac {3\,b^5\,\left (2\,a\,e-b\,d\right )}{7\,e^5}\right )}{e}+\frac {3\,b^4\,\left (5\,a^2\,e^2-6\,a\,b\,d\,e+2\,b^2\,d^2\right )}{7\,e^6}\right )}{e}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^7}-\frac {b^6\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{4\,e^7\,\left (a+b\,x\right )\,{\left (d+e\,x\right )}^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(((b^6*d^5 - 6*a^5*b*e^5 + 15*a^4*b^2*d*e^4 + 15*a^2*b^4*d^3*e^2 - 20*a^3*b^3*d^2*e^3 - 6*a*b^5*d^4*e)/(9*e^7)
+ (d*((b^6*d^4*e + 15*a^4*b^2*e^5 - 6*a*b^5*d^3*e^2 - 20*a^3*b^3*d*e^4 + 15*a^2*b^4*d^2*e^3)/(9*e^7) - (d*((2
0*a^3*b^3*e^5 - b^6*d^3*e^2 + 6*a*b^5*d^2*e^3 - 15*a^2*b^4*d*e^4)/(9*e^7) - (d*((d*((b^6*d)/(9*e^3) - (b^5*(6*
a*e - b*d))/(9*e^3)))/e + (b^4*(15*a^2*e^2 + b^2*d^2 - 6*a*b*d*e))/(9*e^4)))/e))/e))/e)*(a^2 + b^2*x^2 + 2*a*b
*x)^(1/2))/((a + b*x)*(d + e*x)^9) - (((10*b^6*d^2 + 15*a^2*b^4*e^2 - 24*a*b^5*d*e)/(6*e^7) + (d*((b^6*d)/(6*e
^6) - (b^5*(3*a*e - 2*b*d))/(3*e^6)))/e)*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/((a + b*x)*(d + e*x)^6) - ((a^6/(10*
e) - (d*((d*((d*((d*((d*((3*a*b^5)/(5*e) - (b^6*d)/(10*e^2)))/e - (3*a^2*b^4)/(2*e)))/e + (2*a^3*b^3)/e))/e -
(3*a^4*b^2)/(2*e)))/e + (3*a^5*b)/(5*e)))/e)*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/((a + b*x)*(d + e*x)^10) - (((5*
b^6*d^4 + 15*a^4*b^2*e^4 - 40*a^3*b^3*d*e^3 + 45*a^2*b^4*d^2*e^2 - 24*a*b^5*d^3*e)/(8*e^7) + (d*((4*b^6*d^3*e
- 20*a^3*b^3*e^4 - 18*a*b^5*d^2*e^2 + 30*a^2*b^4*d*e^3)/(8*e^7) + (d*((d*((b^6*d)/(8*e^4) - (b^5*(3*a*e - b*d)
)/(4*e^4)))/e + (3*b^4*(5*a^2*e^2 + b^2*d^2 - 4*a*b*d*e))/(8*e^5)))/e))/e)*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/((
a + b*x)*(d + e*x)^8) + (((5*b^6*d - 6*a*b^5*e)/(5*e^7) + (b^6*d)/(5*e^7))*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/((
a + b*x)*(d + e*x)^5) + (((10*b^6*d^3 - 20*a^3*b^3*e^3 + 45*a^2*b^4*d*e^2 - 36*a*b^5*d^2*e)/(7*e^7) + (d*((d*(
(b^6*d)/(7*e^5) - (3*b^5*(2*a*e - b*d))/(7*e^5)))/e + (3*b^4*(5*a^2*e^2 + 2*b^2*d^2 - 6*a*b*d*e))/(7*e^6)))/e)
*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/((a + b*x)*(d + e*x)^7) - (b^6*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/(4*e^7*(a +
b*x)*(d + e*x)^4)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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