3.18.88 \(\int \frac {(a+b x) (a^2+2 a b x+b^2 x^2)^{5/2}}{(d+e x)^{16}} \, dx\)

Optimal. Leaf size=362 \[ -\frac {15 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^4}{13 e^7 (a+b x) (d+e x)^{13}}+\frac {3 b \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^5}{7 e^7 (a+b x) (d+e x)^{14}}-\frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^6}{15 e^7 (a+b x) (d+e x)^{15}}-\frac {b^6 \sqrt {a^2+2 a b x+b^2 x^2}}{9 e^7 (a+b x) (d+e x)^9}+\frac {3 b^5 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}{5 e^7 (a+b x) (d+e x)^{10}}-\frac {15 b^4 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2}{11 e^7 (a+b x) (d+e x)^{11}}+\frac {5 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3}{3 e^7 (a+b x) (d+e x)^{12}} \]

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Rubi [A]  time = 0.20, antiderivative size = 362, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {770, 21, 43} \begin {gather*} -\frac {b^6 \sqrt {a^2+2 a b x+b^2 x^2}}{9 e^7 (a+b x) (d+e x)^9}+\frac {3 b^5 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}{5 e^7 (a+b x) (d+e x)^{10}}-\frac {15 b^4 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2}{11 e^7 (a+b x) (d+e x)^{11}}+\frac {5 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3}{3 e^7 (a+b x) (d+e x)^{12}}-\frac {15 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^4}{13 e^7 (a+b x) (d+e x)^{13}}+\frac {3 b \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^5}{7 e^7 (a+b x) (d+e x)^{14}}-\frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^6}{15 e^7 (a+b x) (d+e x)^{15}} \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)^16,x]

[Out]

-((b*d - a*e)^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(15*e^7*(a + b*x)*(d + e*x)^15) + (3*b*(b*d - a*e)^5*Sqrt[a^2 +
2*a*b*x + b^2*x^2])/(7*e^7*(a + b*x)*(d + e*x)^14) - (15*b^2*(b*d - a*e)^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(13
*e^7*(a + b*x)*(d + e*x)^13) + (5*b^3*(b*d - a*e)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*e^7*(a + b*x)*(d + e*x)^
12) - (15*b^4*(b*d - a*e)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(11*e^7*(a + b*x)*(d + e*x)^11) + (3*b^5*(b*d - a*e
)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5*e^7*(a + b*x)*(d + e*x)^10) - (b^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(9*e^7*(
a + b*x)*(d + e*x)^9)

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 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

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)^{16}} \, 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)^{16}} \, 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)^{16}} \, dx}{a b+b^2 x}\\ &=\frac {\left (b \sqrt {a^2+2 a b x+b^2 x^2}\right ) \int \left (\frac {(-b d+a e)^6}{e^6 (d+e x)^{16}}-\frac {6 b (b d-a e)^5}{e^6 (d+e x)^{15}}+\frac {15 b^2 (b d-a e)^4}{e^6 (d+e x)^{14}}-\frac {20 b^3 (b d-a e)^3}{e^6 (d+e x)^{13}}+\frac {15 b^4 (b d-a e)^2}{e^6 (d+e x)^{12}}-\frac {6 b^5 (b d-a e)}{e^6 (d+e x)^{11}}+\frac {b^6}{e^6 (d+e x)^{10}}\right ) \, dx}{a b+b^2 x}\\ &=-\frac {(b d-a e)^6 \sqrt {a^2+2 a b x+b^2 x^2}}{15 e^7 (a+b x) (d+e x)^{15}}+\frac {3 b (b d-a e)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{7 e^7 (a+b x) (d+e x)^{14}}-\frac {15 b^2 (b d-a e)^4 \sqrt {a^2+2 a b x+b^2 x^2}}{13 e^7 (a+b x) (d+e x)^{13}}+\frac {5 b^3 (b d-a e)^3 \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^7 (a+b x) (d+e x)^{12}}-\frac {15 b^4 (b d-a e)^2 \sqrt {a^2+2 a b x+b^2 x^2}}{11 e^7 (a+b x) (d+e x)^{11}}+\frac {3 b^5 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}{5 e^7 (a+b x) (d+e x)^{10}}-\frac {b^6 \sqrt {a^2+2 a b x+b^2 x^2}}{9 e^7 (a+b x) (d+e x)^9}\\ \end {align*}

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Mathematica [A]  time = 0.12, size = 295, normalized size = 0.81 \begin {gather*} -\frac {\sqrt {(a+b x)^2} \left (3003 a^6 e^6+1287 a^5 b e^5 (d+15 e x)+495 a^4 b^2 e^4 \left (d^2+15 d e x+105 e^2 x^2\right )+165 a^3 b^3 e^3 \left (d^3+15 d^2 e x+105 d e^2 x^2+455 e^3 x^3\right )+45 a^2 b^4 e^2 \left (d^4+15 d^3 e x+105 d^2 e^2 x^2+455 d e^3 x^3+1365 e^4 x^4\right )+9 a b^5 e \left (d^5+15 d^4 e x+105 d^3 e^2 x^2+455 d^2 e^3 x^3+1365 d e^4 x^4+3003 e^5 x^5\right )+b^6 \left (d^6+15 d^5 e x+105 d^4 e^2 x^2+455 d^3 e^3 x^3+1365 d^2 e^4 x^4+3003 d e^5 x^5+5005 e^6 x^6\right )\right )}{45045 e^7 (a+b x) (d+e x)^{15}} \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)^16,x]

[Out]

-1/45045*(Sqrt[(a + b*x)^2]*(3003*a^6*e^6 + 1287*a^5*b*e^5*(d + 15*e*x) + 495*a^4*b^2*e^4*(d^2 + 15*d*e*x + 10
5*e^2*x^2) + 165*a^3*b^3*e^3*(d^3 + 15*d^2*e*x + 105*d*e^2*x^2 + 455*e^3*x^3) + 45*a^2*b^4*e^2*(d^4 + 15*d^3*e
*x + 105*d^2*e^2*x^2 + 455*d*e^3*x^3 + 1365*e^4*x^4) + 9*a*b^5*e*(d^5 + 15*d^4*e*x + 105*d^3*e^2*x^2 + 455*d^2
*e^3*x^3 + 1365*d*e^4*x^4 + 3003*e^5*x^5) + b^6*(d^6 + 15*d^5*e*x + 105*d^4*e^2*x^2 + 455*d^3*e^3*x^3 + 1365*d
^2*e^4*x^4 + 3003*d*e^5*x^5 + 5005*e^6*x^6)))/(e^7*(a + b*x)*(d + e*x)^15)

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IntegrateAlgebraic [F]  time = 180.24, 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)^16,x]

[Out]

$Aborted

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fricas [A]  time = 0.44, size = 507, normalized size = 1.40 \begin {gather*} -\frac {5005 \, b^{6} e^{6} x^{6} + b^{6} d^{6} + 9 \, a b^{5} d^{5} e + 45 \, a^{2} b^{4} d^{4} e^{2} + 165 \, a^{3} b^{3} d^{3} e^{3} + 495 \, a^{4} b^{2} d^{2} e^{4} + 1287 \, a^{5} b d e^{5} + 3003 \, a^{6} e^{6} + 3003 \, {\left (b^{6} d e^{5} + 9 \, a b^{5} e^{6}\right )} x^{5} + 1365 \, {\left (b^{6} d^{2} e^{4} + 9 \, a b^{5} d e^{5} + 45 \, a^{2} b^{4} e^{6}\right )} x^{4} + 455 \, {\left (b^{6} d^{3} e^{3} + 9 \, a b^{5} d^{2} e^{4} + 45 \, a^{2} b^{4} d e^{5} + 165 \, a^{3} b^{3} e^{6}\right )} x^{3} + 105 \, {\left (b^{6} d^{4} e^{2} + 9 \, a b^{5} d^{3} e^{3} + 45 \, a^{2} b^{4} d^{2} e^{4} + 165 \, a^{3} b^{3} d e^{5} + 495 \, a^{4} b^{2} e^{6}\right )} x^{2} + 15 \, {\left (b^{6} d^{5} e + 9 \, a b^{5} d^{4} e^{2} + 45 \, a^{2} b^{4} d^{3} e^{3} + 165 \, a^{3} b^{3} d^{2} e^{4} + 495 \, a^{4} b^{2} d e^{5} + 1287 \, a^{5} b e^{6}\right )} x}{45045 \, {\left (e^{22} x^{15} + 15 \, d e^{21} x^{14} + 105 \, d^{2} e^{20} x^{13} + 455 \, d^{3} e^{19} x^{12} + 1365 \, d^{4} e^{18} x^{11} + 3003 \, d^{5} e^{17} x^{10} + 5005 \, d^{6} e^{16} x^{9} + 6435 \, d^{7} e^{15} x^{8} + 6435 \, d^{8} e^{14} x^{7} + 5005 \, d^{9} e^{13} x^{6} + 3003 \, d^{10} e^{12} x^{5} + 1365 \, d^{11} e^{11} x^{4} + 455 \, d^{12} e^{10} x^{3} + 105 \, d^{13} e^{9} x^{2} + 15 \, d^{14} e^{8} x + d^{15} 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)^16,x, algorithm="fricas")

[Out]

-1/45045*(5005*b^6*e^6*x^6 + b^6*d^6 + 9*a*b^5*d^5*e + 45*a^2*b^4*d^4*e^2 + 165*a^3*b^3*d^3*e^3 + 495*a^4*b^2*
d^2*e^4 + 1287*a^5*b*d*e^5 + 3003*a^6*e^6 + 3003*(b^6*d*e^5 + 9*a*b^5*e^6)*x^5 + 1365*(b^6*d^2*e^4 + 9*a*b^5*d
*e^5 + 45*a^2*b^4*e^6)*x^4 + 455*(b^6*d^3*e^3 + 9*a*b^5*d^2*e^4 + 45*a^2*b^4*d*e^5 + 165*a^3*b^3*e^6)*x^3 + 10
5*(b^6*d^4*e^2 + 9*a*b^5*d^3*e^3 + 45*a^2*b^4*d^2*e^4 + 165*a^3*b^3*d*e^5 + 495*a^4*b^2*e^6)*x^2 + 15*(b^6*d^5
*e + 9*a*b^5*d^4*e^2 + 45*a^2*b^4*d^3*e^3 + 165*a^3*b^3*d^2*e^4 + 495*a^4*b^2*d*e^5 + 1287*a^5*b*e^6)*x)/(e^22
*x^15 + 15*d*e^21*x^14 + 105*d^2*e^20*x^13 + 455*d^3*e^19*x^12 + 1365*d^4*e^18*x^11 + 3003*d^5*e^17*x^10 + 500
5*d^6*e^16*x^9 + 6435*d^7*e^15*x^8 + 6435*d^8*e^14*x^7 + 5005*d^9*e^13*x^6 + 3003*d^10*e^12*x^5 + 1365*d^11*e^
11*x^4 + 455*d^12*e^10*x^3 + 105*d^13*e^9*x^2 + 15*d^14*e^8*x + d^15*e^7)

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giac [A]  time = 0.19, size = 520, normalized size = 1.44 \begin {gather*} -\frac {{\left (5005 \, b^{6} x^{6} e^{6} \mathrm {sgn}\left (b x + a\right ) + 3003 \, b^{6} d x^{5} e^{5} \mathrm {sgn}\left (b x + a\right ) + 1365 \, b^{6} d^{2} x^{4} e^{4} \mathrm {sgn}\left (b x + a\right ) + 455 \, b^{6} d^{3} x^{3} e^{3} \mathrm {sgn}\left (b x + a\right ) + 105 \, b^{6} d^{4} x^{2} e^{2} \mathrm {sgn}\left (b x + a\right ) + 15 \, b^{6} d^{5} x e \mathrm {sgn}\left (b x + a\right ) + b^{6} d^{6} \mathrm {sgn}\left (b x + a\right ) + 27027 \, a b^{5} x^{5} e^{6} \mathrm {sgn}\left (b x + a\right ) + 12285 \, a b^{5} d x^{4} e^{5} \mathrm {sgn}\left (b x + a\right ) + 4095 \, a b^{5} d^{2} x^{3} e^{4} \mathrm {sgn}\left (b x + a\right ) + 945 \, a b^{5} d^{3} x^{2} e^{3} \mathrm {sgn}\left (b x + a\right ) + 135 \, a b^{5} d^{4} x e^{2} \mathrm {sgn}\left (b x + a\right ) + 9 \, a b^{5} d^{5} e \mathrm {sgn}\left (b x + a\right ) + 61425 \, a^{2} b^{4} x^{4} e^{6} \mathrm {sgn}\left (b x + a\right ) + 20475 \, a^{2} b^{4} d x^{3} e^{5} \mathrm {sgn}\left (b x + a\right ) + 4725 \, a^{2} b^{4} d^{2} x^{2} e^{4} \mathrm {sgn}\left (b x + a\right ) + 675 \, a^{2} b^{4} d^{3} x e^{3} \mathrm {sgn}\left (b x + a\right ) + 45 \, a^{2} b^{4} d^{4} e^{2} \mathrm {sgn}\left (b x + a\right ) + 75075 \, a^{3} b^{3} x^{3} e^{6} \mathrm {sgn}\left (b x + a\right ) + 17325 \, a^{3} b^{3} d x^{2} e^{5} \mathrm {sgn}\left (b x + a\right ) + 2475 \, a^{3} b^{3} d^{2} x e^{4} \mathrm {sgn}\left (b x + a\right ) + 165 \, a^{3} b^{3} d^{3} e^{3} \mathrm {sgn}\left (b x + a\right ) + 51975 \, a^{4} b^{2} x^{2} e^{6} \mathrm {sgn}\left (b x + a\right ) + 7425 \, a^{4} b^{2} d x e^{5} \mathrm {sgn}\left (b x + a\right ) + 495 \, a^{4} b^{2} d^{2} e^{4} \mathrm {sgn}\left (b x + a\right ) + 19305 \, a^{5} b x e^{6} \mathrm {sgn}\left (b x + a\right ) + 1287 \, a^{5} b d e^{5} \mathrm {sgn}\left (b x + a\right ) + 3003 \, a^{6} e^{6} \mathrm {sgn}\left (b x + a\right )\right )} e^{\left (-7\right )}}{45045 \, {\left (x e + d\right )}^{15}} \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)^16,x, algorithm="giac")

[Out]

-1/45045*(5005*b^6*x^6*e^6*sgn(b*x + a) + 3003*b^6*d*x^5*e^5*sgn(b*x + a) + 1365*b^6*d^2*x^4*e^4*sgn(b*x + a)
+ 455*b^6*d^3*x^3*e^3*sgn(b*x + a) + 105*b^6*d^4*x^2*e^2*sgn(b*x + a) + 15*b^6*d^5*x*e*sgn(b*x + a) + b^6*d^6*
sgn(b*x + a) + 27027*a*b^5*x^5*e^6*sgn(b*x + a) + 12285*a*b^5*d*x^4*e^5*sgn(b*x + a) + 4095*a*b^5*d^2*x^3*e^4*
sgn(b*x + a) + 945*a*b^5*d^3*x^2*e^3*sgn(b*x + a) + 135*a*b^5*d^4*x*e^2*sgn(b*x + a) + 9*a*b^5*d^5*e*sgn(b*x +
a) + 61425*a^2*b^4*x^4*e^6*sgn(b*x + a) + 20475*a^2*b^4*d*x^3*e^5*sgn(b*x + a) + 4725*a^2*b^4*d^2*x^2*e^4*sgn
(b*x + a) + 675*a^2*b^4*d^3*x*e^3*sgn(b*x + a) + 45*a^2*b^4*d^4*e^2*sgn(b*x + a) + 75075*a^3*b^3*x^3*e^6*sgn(b
*x + a) + 17325*a^3*b^3*d*x^2*e^5*sgn(b*x + a) + 2475*a^3*b^3*d^2*x*e^4*sgn(b*x + a) + 165*a^3*b^3*d^3*e^3*sgn
(b*x + a) + 51975*a^4*b^2*x^2*e^6*sgn(b*x + a) + 7425*a^4*b^2*d*x*e^5*sgn(b*x + a) + 495*a^4*b^2*d^2*e^4*sgn(b
*x + a) + 19305*a^5*b*x*e^6*sgn(b*x + a) + 1287*a^5*b*d*e^5*sgn(b*x + a) + 3003*a^6*e^6*sgn(b*x + a))*e^(-7)/(
x*e + d)^15

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maple [A]  time = 0.07, size = 392, normalized size = 1.08 \begin {gather*} -\frac {\left (5005 b^{6} e^{6} x^{6}+27027 a \,b^{5} e^{6} x^{5}+3003 b^{6} d \,e^{5} x^{5}+61425 a^{2} b^{4} e^{6} x^{4}+12285 a \,b^{5} d \,e^{5} x^{4}+1365 b^{6} d^{2} e^{4} x^{4}+75075 a^{3} b^{3} e^{6} x^{3}+20475 a^{2} b^{4} d \,e^{5} x^{3}+4095 a \,b^{5} d^{2} e^{4} x^{3}+455 b^{6} d^{3} e^{3} x^{3}+51975 a^{4} b^{2} e^{6} x^{2}+17325 a^{3} b^{3} d \,e^{5} x^{2}+4725 a^{2} b^{4} d^{2} e^{4} x^{2}+945 a \,b^{5} d^{3} e^{3} x^{2}+105 b^{6} d^{4} e^{2} x^{2}+19305 a^{5} b \,e^{6} x +7425 a^{4} b^{2} d \,e^{5} x +2475 a^{3} b^{3} d^{2} e^{4} x +675 a^{2} b^{4} d^{3} e^{3} x +135 a \,b^{5} d^{4} e^{2} x +15 b^{6} d^{5} e x +3003 a^{6} e^{6}+1287 a^{5} b d \,e^{5}+495 a^{4} b^{2} d^{2} e^{4}+165 a^{3} b^{3} d^{3} e^{3}+45 a^{2} b^{4} d^{4} e^{2}+9 a \,b^{5} d^{5} e +b^{6} d^{6}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{45045 \left (e x +d \right )^{15} \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)^16,x)

[Out]

-1/45045/e^7*(5005*b^6*e^6*x^6+27027*a*b^5*e^6*x^5+3003*b^6*d*e^5*x^5+61425*a^2*b^4*e^6*x^4+12285*a*b^5*d*e^5*
x^4+1365*b^6*d^2*e^4*x^4+75075*a^3*b^3*e^6*x^3+20475*a^2*b^4*d*e^5*x^3+4095*a*b^5*d^2*e^4*x^3+455*b^6*d^3*e^3*
x^3+51975*a^4*b^2*e^6*x^2+17325*a^3*b^3*d*e^5*x^2+4725*a^2*b^4*d^2*e^4*x^2+945*a*b^5*d^3*e^3*x^2+105*b^6*d^4*e
^2*x^2+19305*a^5*b*e^6*x+7425*a^4*b^2*d*e^5*x+2475*a^3*b^3*d^2*e^4*x+675*a^2*b^4*d^3*e^3*x+135*a*b^5*d^4*e^2*x
+15*b^6*d^5*e*x+3003*a^6*e^6+1287*a^5*b*d*e^5+495*a^4*b^2*d^2*e^4+165*a^3*b^3*d^3*e^3+45*a^2*b^4*d^4*e^2+9*a*b
^5*d^5*e+b^6*d^6)*((b*x+a)^2)^(5/2)/(e*x+d)^15/(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)^16,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 = 2.45, size = 1010, normalized size = 2.79 \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}{14\,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}{14\,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}{14\,e^7}-\frac {d\,\left (\frac {d\,\left (\frac {b^6\,d}{14\,e^3}-\frac {b^5\,\left (6\,a\,e-b\,d\right )}{14\,e^3}\right )}{e}+\frac {b^4\,\left (15\,a^2\,e^2-6\,a\,b\,d\,e+b^2\,d^2\right )}{14\,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 )}^{14}}-\frac {\left (\frac {15\,a^2\,b^4\,e^2-24\,a\,b^5\,d\,e+10\,b^6\,d^2}{11\,e^7}+\frac {d\,\left (\frac {b^6\,d}{11\,e^6}-\frac {2\,b^5\,\left (3\,a\,e-2\,b\,d\right )}{11\,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 )}^{11}}-\frac {\left (\frac {a^6}{15\,e}-\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (\frac {2\,a\,b^5}{5\,e}-\frac {b^6\,d}{15\,e^2}\right )}{e}-\frac {a^2\,b^4}{e}\right )}{e}+\frac {4\,a^3\,b^3}{3\,e}\right )}{e}-\frac {a^4\,b^2}{e}\right )}{e}+\frac {2\,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 )}^{15}}-\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}{13\,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}{13\,e^7}+\frac {d\,\left (\frac {d\,\left (\frac {b^6\,d}{13\,e^4}-\frac {2\,b^5\,\left (3\,a\,e-b\,d\right )}{13\,e^4}\right )}{e}+\frac {3\,b^4\,\left (5\,a^2\,e^2-4\,a\,b\,d\,e+b^2\,d^2\right )}{13\,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 )}^{13}}+\frac {\left (\frac {5\,b^6\,d-6\,a\,b^5\,e}{10\,e^7}+\frac {b^6\,d}{10\,e^7}\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 {-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}{12\,e^7}+\frac {d\,\left (\frac {d\,\left (\frac {b^6\,d}{12\,e^5}-\frac {b^5\,\left (2\,a\,e-b\,d\right )}{4\,e^5}\right )}{e}+\frac {b^4\,\left (5\,a^2\,e^2-6\,a\,b\,d\,e+2\,b^2\,d^2\right )}{4\,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 )}^{12}}-\frac {b^6\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{9\,e^7\,\left (a+b\,x\right )\,{\left (d+e\,x\right )}^9} \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)^16,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)/(14*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)/(14*e^7) - (d*(
(20*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)/(14*e^7) - (d*((d*((b^6*d)/(14*e^3) - (b^5
*(6*a*e - b*d))/(14*e^3)))/e + (b^4*(15*a^2*e^2 + b^2*d^2 - 6*a*b*d*e))/(14*e^4)))/e))/e))/e)*(a^2 + b^2*x^2 +
2*a*b*x)^(1/2))/((a + b*x)*(d + e*x)^14) - (((10*b^6*d^2 + 15*a^2*b^4*e^2 - 24*a*b^5*d*e)/(11*e^7) + (d*((b^6
*d)/(11*e^6) - (2*b^5*(3*a*e - 2*b*d))/(11*e^6)))/e)*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/((a + b*x)*(d + e*x)^11)
- ((a^6/(15*e) - (d*((d*((d*((d*((d*((2*a*b^5)/(5*e) - (b^6*d)/(15*e^2)))/e - (a^2*b^4)/e))/e + (4*a^3*b^3)/(
3*e)))/e - (a^4*b^2)/e))/e + (2*a^5*b)/(5*e)))/e)*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/((a + b*x)*(d + e*x)^15) -
(((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)/(13*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)/(13*e^7) + (d*((d*((b^6*d)/(13*e^4) - (2*b^5*(3*
a*e - b*d))/(13*e^4)))/e + (3*b^4*(5*a^2*e^2 + b^2*d^2 - 4*a*b*d*e))/(13*e^5)))/e))/e)*(a^2 + b^2*x^2 + 2*a*b*
x)^(1/2))/((a + b*x)*(d + e*x)^13) + (((5*b^6*d - 6*a*b^5*e)/(10*e^7) + (b^6*d)/(10*e^7))*(a^2 + b^2*x^2 + 2*a
*b*x)^(1/2))/((a + b*x)*(d + e*x)^10) + (((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)/(1
2*e^7) + (d*((d*((b^6*d)/(12*e^5) - (b^5*(2*a*e - b*d))/(4*e^5)))/e + (b^4*(5*a^2*e^2 + 2*b^2*d^2 - 6*a*b*d*e)
)/(4*e^6)))/e)*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/((a + b*x)*(d + e*x)^12) - (b^6*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2
))/(9*e^7*(a + b*x)*(d + e*x)^9)

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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)**16,x)

[Out]

Timed out

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