3.18.64 \(\int (a+b x) (d+e x)^8 (a^2+2 a b x+b^2 x^2)^{5/2} \, dx\)

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

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Rubi [A]  time = 0.59, 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} (d+e x)^{15}}{15 e^7 (a+b x)}-\frac {3 b^5 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{14} (b d-a e)}{7 e^7 (a+b x)}+\frac {15 b^4 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{13} (b d-a e)^2}{13 e^7 (a+b x)}-\frac {5 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{12} (b d-a e)^3}{3 e^7 (a+b x)}+\frac {15 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{11} (b d-a e)^4}{11 e^7 (a+b x)}-\frac {3 b \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{10} (b d-a e)^5}{5 e^7 (a+b x)}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^9 (b d-a e)^6}{9 e^7 (a+b x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)*(d + e*x)^8*(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]

[Out]

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

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 (a+b x) (d+e x)^8 \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int (a+b x) \left (a b+b^2 x\right )^5 (d+e x)^8 \, 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 (a+b x)^6 (d+e x)^8 \, 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 (d+e x)^8}{e^6}-\frac {6 b (b d-a e)^5 (d+e x)^9}{e^6}+\frac {15 b^2 (b d-a e)^4 (d+e x)^{10}}{e^6}-\frac {20 b^3 (b d-a e)^3 (d+e x)^{11}}{e^6}+\frac {15 b^4 (b d-a e)^2 (d+e x)^{12}}{e^6}-\frac {6 b^5 (b d-a e) (d+e x)^{13}}{e^6}+\frac {b^6 (d+e x)^{14}}{e^6}\right ) \, dx}{a b+b^2 x}\\ &=\frac {(b d-a e)^6 (d+e x)^9 \sqrt {a^2+2 a b x+b^2 x^2}}{9 e^7 (a+b x)}-\frac {3 b (b d-a e)^5 (d+e x)^{10} \sqrt {a^2+2 a b x+b^2 x^2}}{5 e^7 (a+b x)}+\frac {15 b^2 (b d-a e)^4 (d+e x)^{11} \sqrt {a^2+2 a b x+b^2 x^2}}{11 e^7 (a+b x)}-\frac {5 b^3 (b d-a e)^3 (d+e x)^{12} \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^7 (a+b x)}+\frac {15 b^4 (b d-a e)^2 (d+e x)^{13} \sqrt {a^2+2 a b x+b^2 x^2}}{13 e^7 (a+b x)}-\frac {3 b^5 (b d-a e) (d+e x)^{14} \sqrt {a^2+2 a b x+b^2 x^2}}{7 e^7 (a+b x)}+\frac {b^6 (d+e x)^{15} \sqrt {a^2+2 a b x+b^2 x^2}}{15 e^7 (a+b x)}\\ \end {align*}

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Mathematica [A]  time = 0.22, size = 679, normalized size = 1.88 \begin {gather*} \frac {x \sqrt {(a+b x)^2} \left (5005 a^6 \left (9 d^8+36 d^7 e x+84 d^6 e^2 x^2+126 d^5 e^3 x^3+126 d^4 e^4 x^4+84 d^3 e^5 x^5+36 d^2 e^6 x^6+9 d e^7 x^7+e^8 x^8\right )+3003 a^5 b x \left (45 d^8+240 d^7 e x+630 d^6 e^2 x^2+1008 d^5 e^3 x^3+1050 d^4 e^4 x^4+720 d^3 e^5 x^5+315 d^2 e^6 x^6+80 d e^7 x^7+9 e^8 x^8\right )+1365 a^4 b^2 x^2 \left (165 d^8+990 d^7 e x+2772 d^6 e^2 x^2+4620 d^5 e^3 x^3+4950 d^4 e^4 x^4+3465 d^3 e^5 x^5+1540 d^2 e^6 x^6+396 d e^7 x^7+45 e^8 x^8\right )+455 a^3 b^3 x^3 \left (495 d^8+3168 d^7 e x+9240 d^6 e^2 x^2+15840 d^5 e^3 x^3+17325 d^4 e^4 x^4+12320 d^3 e^5 x^5+5544 d^2 e^6 x^6+1440 d e^7 x^7+165 e^8 x^8\right )+105 a^2 b^4 x^4 \left (1287 d^8+8580 d^7 e x+25740 d^6 e^2 x^2+45045 d^5 e^3 x^3+50050 d^4 e^4 x^4+36036 d^3 e^5 x^5+16380 d^2 e^6 x^6+4290 d e^7 x^7+495 e^8 x^8\right )+15 a b^5 x^5 \left (3003 d^8+20592 d^7 e x+63063 d^6 e^2 x^2+112112 d^5 e^3 x^3+126126 d^4 e^4 x^4+91728 d^3 e^5 x^5+42042 d^2 e^6 x^6+11088 d e^7 x^7+1287 e^8 x^8\right )+b^6 x^6 \left (6435 d^8+45045 d^7 e x+140140 d^6 e^2 x^2+252252 d^5 e^3 x^3+286650 d^4 e^4 x^4+210210 d^3 e^5 x^5+97020 d^2 e^6 x^6+25740 d e^7 x^7+3003 e^8 x^8\right )\right )}{45045 (a+b x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)*(d + e*x)^8*(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]

[Out]

(x*Sqrt[(a + b*x)^2]*(5005*a^6*(9*d^8 + 36*d^7*e*x + 84*d^6*e^2*x^2 + 126*d^5*e^3*x^3 + 126*d^4*e^4*x^4 + 84*d
^3*e^5*x^5 + 36*d^2*e^6*x^6 + 9*d*e^7*x^7 + e^8*x^8) + 3003*a^5*b*x*(45*d^8 + 240*d^7*e*x + 630*d^6*e^2*x^2 +
1008*d^5*e^3*x^3 + 1050*d^4*e^4*x^4 + 720*d^3*e^5*x^5 + 315*d^2*e^6*x^6 + 80*d*e^7*x^7 + 9*e^8*x^8) + 1365*a^4
*b^2*x^2*(165*d^8 + 990*d^7*e*x + 2772*d^6*e^2*x^2 + 4620*d^5*e^3*x^3 + 4950*d^4*e^4*x^4 + 3465*d^3*e^5*x^5 +
1540*d^2*e^6*x^6 + 396*d*e^7*x^7 + 45*e^8*x^8) + 455*a^3*b^3*x^3*(495*d^8 + 3168*d^7*e*x + 9240*d^6*e^2*x^2 +
15840*d^5*e^3*x^3 + 17325*d^4*e^4*x^4 + 12320*d^3*e^5*x^5 + 5544*d^2*e^6*x^6 + 1440*d*e^7*x^7 + 165*e^8*x^8) +
105*a^2*b^4*x^4*(1287*d^8 + 8580*d^7*e*x + 25740*d^6*e^2*x^2 + 45045*d^5*e^3*x^3 + 50050*d^4*e^4*x^4 + 36036*
d^3*e^5*x^5 + 16380*d^2*e^6*x^6 + 4290*d*e^7*x^7 + 495*e^8*x^8) + 15*a*b^5*x^5*(3003*d^8 + 20592*d^7*e*x + 630
63*d^6*e^2*x^2 + 112112*d^5*e^3*x^3 + 126126*d^4*e^4*x^4 + 91728*d^3*e^5*x^5 + 42042*d^2*e^6*x^6 + 11088*d*e^7
*x^7 + 1287*e^8*x^8) + b^6*x^6*(6435*d^8 + 45045*d^7*e*x + 140140*d^6*e^2*x^2 + 252252*d^5*e^3*x^3 + 286650*d^
4*e^4*x^4 + 210210*d^3*e^5*x^5 + 97020*d^2*e^6*x^6 + 25740*d*e^7*x^7 + 3003*e^8*x^8)))/(45045*(a + b*x))

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IntegrateAlgebraic [F]  time = 7.88, size = 0, normalized size = 0.00 \begin {gather*} \int (a+b x) (d+e x)^8 \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(a + b*x)*(d + e*x)^8*(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]

[Out]

Defer[IntegrateAlgebraic][(a + b*x)*(d + e*x)^8*(a^2 + 2*a*b*x + b^2*x^2)^(5/2), x]

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fricas [B]  time = 0.44, size = 797, normalized size = 2.20 \begin {gather*} \frac {1}{15} \, b^{6} e^{8} x^{15} + a^{6} d^{8} x + \frac {1}{7} \, {\left (4 \, b^{6} d e^{7} + 3 \, a b^{5} e^{8}\right )} x^{14} + \frac {1}{13} \, {\left (28 \, b^{6} d^{2} e^{6} + 48 \, a b^{5} d e^{7} + 15 \, a^{2} b^{4} e^{8}\right )} x^{13} + \frac {1}{3} \, {\left (14 \, b^{6} d^{3} e^{5} + 42 \, a b^{5} d^{2} e^{6} + 30 \, a^{2} b^{4} d e^{7} + 5 \, a^{3} b^{3} e^{8}\right )} x^{12} + \frac {1}{11} \, {\left (70 \, b^{6} d^{4} e^{4} + 336 \, a b^{5} d^{3} e^{5} + 420 \, a^{2} b^{4} d^{2} e^{6} + 160 \, a^{3} b^{3} d e^{7} + 15 \, a^{4} b^{2} e^{8}\right )} x^{11} + \frac {1}{5} \, {\left (28 \, b^{6} d^{5} e^{3} + 210 \, a b^{5} d^{4} e^{4} + 420 \, a^{2} b^{4} d^{3} e^{5} + 280 \, a^{3} b^{3} d^{2} e^{6} + 60 \, a^{4} b^{2} d e^{7} + 3 \, a^{5} b e^{8}\right )} x^{10} + \frac {1}{9} \, {\left (28 \, b^{6} d^{6} e^{2} + 336 \, a b^{5} d^{5} e^{3} + 1050 \, a^{2} b^{4} d^{4} e^{4} + 1120 \, a^{3} b^{3} d^{3} e^{5} + 420 \, a^{4} b^{2} d^{2} e^{6} + 48 \, a^{5} b d e^{7} + a^{6} e^{8}\right )} x^{9} + {\left (b^{6} d^{7} e + 21 \, a b^{5} d^{6} e^{2} + 105 \, a^{2} b^{4} d^{5} e^{3} + 175 \, a^{3} b^{3} d^{4} e^{4} + 105 \, a^{4} b^{2} d^{3} e^{5} + 21 \, a^{5} b d^{2} e^{6} + a^{6} d e^{7}\right )} x^{8} + \frac {1}{7} \, {\left (b^{6} d^{8} + 48 \, a b^{5} d^{7} e + 420 \, a^{2} b^{4} d^{6} e^{2} + 1120 \, a^{3} b^{3} d^{5} e^{3} + 1050 \, a^{4} b^{2} d^{4} e^{4} + 336 \, a^{5} b d^{3} e^{5} + 28 \, a^{6} d^{2} e^{6}\right )} x^{7} + \frac {1}{3} \, {\left (3 \, a b^{5} d^{8} + 60 \, a^{2} b^{4} d^{7} e + 280 \, a^{3} b^{3} d^{6} e^{2} + 420 \, a^{4} b^{2} d^{5} e^{3} + 210 \, a^{5} b d^{4} e^{4} + 28 \, a^{6} d^{3} e^{5}\right )} x^{6} + \frac {1}{5} \, {\left (15 \, a^{2} b^{4} d^{8} + 160 \, a^{3} b^{3} d^{7} e + 420 \, a^{4} b^{2} d^{6} e^{2} + 336 \, a^{5} b d^{5} e^{3} + 70 \, a^{6} d^{4} e^{4}\right )} x^{5} + {\left (5 \, a^{3} b^{3} d^{8} + 30 \, a^{4} b^{2} d^{7} e + 42 \, a^{5} b d^{6} e^{2} + 14 \, a^{6} d^{5} e^{3}\right )} x^{4} + \frac {1}{3} \, {\left (15 \, a^{4} b^{2} d^{8} + 48 \, a^{5} b d^{7} e + 28 \, a^{6} d^{6} e^{2}\right )} x^{3} + {\left (3 \, a^{5} b d^{8} + 4 \, a^{6} d^{7} e\right )} x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(e*x+d)^8*(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="fricas")

[Out]

1/15*b^6*e^8*x^15 + a^6*d^8*x + 1/7*(4*b^6*d*e^7 + 3*a*b^5*e^8)*x^14 + 1/13*(28*b^6*d^2*e^6 + 48*a*b^5*d*e^7 +
15*a^2*b^4*e^8)*x^13 + 1/3*(14*b^6*d^3*e^5 + 42*a*b^5*d^2*e^6 + 30*a^2*b^4*d*e^7 + 5*a^3*b^3*e^8)*x^12 + 1/11
*(70*b^6*d^4*e^4 + 336*a*b^5*d^3*e^5 + 420*a^2*b^4*d^2*e^6 + 160*a^3*b^3*d*e^7 + 15*a^4*b^2*e^8)*x^11 + 1/5*(2
8*b^6*d^5*e^3 + 210*a*b^5*d^4*e^4 + 420*a^2*b^4*d^3*e^5 + 280*a^3*b^3*d^2*e^6 + 60*a^4*b^2*d*e^7 + 3*a^5*b*e^8
)*x^10 + 1/9*(28*b^6*d^6*e^2 + 336*a*b^5*d^5*e^3 + 1050*a^2*b^4*d^4*e^4 + 1120*a^3*b^3*d^3*e^5 + 420*a^4*b^2*d
^2*e^6 + 48*a^5*b*d*e^7 + a^6*e^8)*x^9 + (b^6*d^7*e + 21*a*b^5*d^6*e^2 + 105*a^2*b^4*d^5*e^3 + 175*a^3*b^3*d^4
*e^4 + 105*a^4*b^2*d^3*e^5 + 21*a^5*b*d^2*e^6 + a^6*d*e^7)*x^8 + 1/7*(b^6*d^8 + 48*a*b^5*d^7*e + 420*a^2*b^4*d
^6*e^2 + 1120*a^3*b^3*d^5*e^3 + 1050*a^4*b^2*d^4*e^4 + 336*a^5*b*d^3*e^5 + 28*a^6*d^2*e^6)*x^7 + 1/3*(3*a*b^5*
d^8 + 60*a^2*b^4*d^7*e + 280*a^3*b^3*d^6*e^2 + 420*a^4*b^2*d^5*e^3 + 210*a^5*b*d^4*e^4 + 28*a^6*d^3*e^5)*x^6 +
1/5*(15*a^2*b^4*d^8 + 160*a^3*b^3*d^7*e + 420*a^4*b^2*d^6*e^2 + 336*a^5*b*d^5*e^3 + 70*a^6*d^4*e^4)*x^5 + (5*
a^3*b^3*d^8 + 30*a^4*b^2*d^7*e + 42*a^5*b*d^6*e^2 + 14*a^6*d^5*e^3)*x^4 + 1/3*(15*a^4*b^2*d^8 + 48*a^5*b*d^7*e
+ 28*a^6*d^6*e^2)*x^3 + (3*a^5*b*d^8 + 4*a^6*d^7*e)*x^2

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giac [B]  time = 0.24, size = 1242, normalized size = 3.43

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(e*x+d)^8*(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="giac")

[Out]

1/15*b^6*x^15*e^8*sgn(b*x + a) + 4/7*b^6*d*x^14*e^7*sgn(b*x + a) + 28/13*b^6*d^2*x^13*e^6*sgn(b*x + a) + 14/3*
b^6*d^3*x^12*e^5*sgn(b*x + a) + 70/11*b^6*d^4*x^11*e^4*sgn(b*x + a) + 28/5*b^6*d^5*x^10*e^3*sgn(b*x + a) + 28/
9*b^6*d^6*x^9*e^2*sgn(b*x + a) + b^6*d^7*x^8*e*sgn(b*x + a) + 1/7*b^6*d^8*x^7*sgn(b*x + a) + 3/7*a*b^5*x^14*e^
8*sgn(b*x + a) + 48/13*a*b^5*d*x^13*e^7*sgn(b*x + a) + 14*a*b^5*d^2*x^12*e^6*sgn(b*x + a) + 336/11*a*b^5*d^3*x
^11*e^5*sgn(b*x + a) + 42*a*b^5*d^4*x^10*e^4*sgn(b*x + a) + 112/3*a*b^5*d^5*x^9*e^3*sgn(b*x + a) + 21*a*b^5*d^
6*x^8*e^2*sgn(b*x + a) + 48/7*a*b^5*d^7*x^7*e*sgn(b*x + a) + a*b^5*d^8*x^6*sgn(b*x + a) + 15/13*a^2*b^4*x^13*e
^8*sgn(b*x + a) + 10*a^2*b^4*d*x^12*e^7*sgn(b*x + a) + 420/11*a^2*b^4*d^2*x^11*e^6*sgn(b*x + a) + 84*a^2*b^4*d
^3*x^10*e^5*sgn(b*x + a) + 350/3*a^2*b^4*d^4*x^9*e^4*sgn(b*x + a) + 105*a^2*b^4*d^5*x^8*e^3*sgn(b*x + a) + 60*
a^2*b^4*d^6*x^7*e^2*sgn(b*x + a) + 20*a^2*b^4*d^7*x^6*e*sgn(b*x + a) + 3*a^2*b^4*d^8*x^5*sgn(b*x + a) + 5/3*a^
3*b^3*x^12*e^8*sgn(b*x + a) + 160/11*a^3*b^3*d*x^11*e^7*sgn(b*x + a) + 56*a^3*b^3*d^2*x^10*e^6*sgn(b*x + a) +
1120/9*a^3*b^3*d^3*x^9*e^5*sgn(b*x + a) + 175*a^3*b^3*d^4*x^8*e^4*sgn(b*x + a) + 160*a^3*b^3*d^5*x^7*e^3*sgn(b
*x + a) + 280/3*a^3*b^3*d^6*x^6*e^2*sgn(b*x + a) + 32*a^3*b^3*d^7*x^5*e*sgn(b*x + a) + 5*a^3*b^3*d^8*x^4*sgn(b
*x + a) + 15/11*a^4*b^2*x^11*e^8*sgn(b*x + a) + 12*a^4*b^2*d*x^10*e^7*sgn(b*x + a) + 140/3*a^4*b^2*d^2*x^9*e^6
*sgn(b*x + a) + 105*a^4*b^2*d^3*x^8*e^5*sgn(b*x + a) + 150*a^4*b^2*d^4*x^7*e^4*sgn(b*x + a) + 140*a^4*b^2*d^5*
x^6*e^3*sgn(b*x + a) + 84*a^4*b^2*d^6*x^5*e^2*sgn(b*x + a) + 30*a^4*b^2*d^7*x^4*e*sgn(b*x + a) + 5*a^4*b^2*d^8
*x^3*sgn(b*x + a) + 3/5*a^5*b*x^10*e^8*sgn(b*x + a) + 16/3*a^5*b*d*x^9*e^7*sgn(b*x + a) + 21*a^5*b*d^2*x^8*e^6
*sgn(b*x + a) + 48*a^5*b*d^3*x^7*e^5*sgn(b*x + a) + 70*a^5*b*d^4*x^6*e^4*sgn(b*x + a) + 336/5*a^5*b*d^5*x^5*e^
3*sgn(b*x + a) + 42*a^5*b*d^6*x^4*e^2*sgn(b*x + a) + 16*a^5*b*d^7*x^3*e*sgn(b*x + a) + 3*a^5*b*d^8*x^2*sgn(b*x
+ a) + 1/9*a^6*x^9*e^8*sgn(b*x + a) + a^6*d*x^8*e^7*sgn(b*x + a) + 4*a^6*d^2*x^7*e^6*sgn(b*x + a) + 28/3*a^6*
d^3*x^6*e^5*sgn(b*x + a) + 14*a^6*d^4*x^5*e^4*sgn(b*x + a) + 14*a^6*d^5*x^4*e^3*sgn(b*x + a) + 28/3*a^6*d^6*x^
3*e^2*sgn(b*x + a) + 4*a^6*d^7*x^2*e*sgn(b*x + a) + a^6*d^8*x*sgn(b*x + a)

________________________________________________________________________________________

maple [B]  time = 0.05, size = 925, normalized size = 2.56 \begin {gather*} \frac {\left (3003 b^{6} e^{8} x^{14}+19305 x^{13} a \,b^{5} e^{8}+25740 x^{13} b^{6} d \,e^{7}+51975 x^{12} a^{2} b^{4} e^{8}+166320 x^{12} a \,b^{5} d \,e^{7}+97020 x^{12} b^{6} d^{2} e^{6}+75075 x^{11} a^{3} b^{3} e^{8}+450450 x^{11} a^{2} b^{4} d \,e^{7}+630630 x^{11} a \,b^{5} d^{2} e^{6}+210210 x^{11} b^{6} d^{3} e^{5}+61425 x^{10} a^{4} b^{2} e^{8}+655200 x^{10} a^{3} b^{3} d \,e^{7}+1719900 x^{10} a^{2} b^{4} d^{2} e^{6}+1375920 x^{10} a \,b^{5} d^{3} e^{5}+286650 x^{10} b^{6} d^{4} e^{4}+27027 x^{9} a^{5} b \,e^{8}+540540 x^{9} a^{4} b^{2} d \,e^{7}+2522520 x^{9} a^{3} b^{3} d^{2} e^{6}+3783780 x^{9} a^{2} b^{4} d^{3} e^{5}+1891890 x^{9} a \,b^{5} d^{4} e^{4}+252252 x^{9} b^{6} d^{5} e^{3}+5005 x^{8} a^{6} e^{8}+240240 x^{8} a^{5} b d \,e^{7}+2102100 x^{8} a^{4} b^{2} d^{2} e^{6}+5605600 x^{8} a^{3} b^{3} d^{3} e^{5}+5255250 x^{8} a^{2} b^{4} d^{4} e^{4}+1681680 x^{8} a \,b^{5} d^{5} e^{3}+140140 x^{8} b^{6} d^{6} e^{2}+45045 a^{6} d \,e^{7} x^{7}+945945 a^{5} b \,d^{2} e^{6} x^{7}+4729725 a^{4} b^{2} d^{3} e^{5} x^{7}+7882875 a^{3} b^{3} d^{4} e^{4} x^{7}+4729725 a^{2} b^{4} d^{5} e^{3} x^{7}+945945 a \,b^{5} d^{6} e^{2} x^{7}+45045 b^{6} d^{7} e \,x^{7}+180180 x^{6} a^{6} d^{2} e^{6}+2162160 x^{6} a^{5} b \,d^{3} e^{5}+6756750 x^{6} a^{4} b^{2} d^{4} e^{4}+7207200 x^{6} a^{3} b^{3} d^{5} e^{3}+2702700 x^{6} a^{2} b^{4} d^{6} e^{2}+308880 x^{6} a \,b^{5} d^{7} e +6435 x^{6} b^{6} d^{8}+420420 x^{5} a^{6} d^{3} e^{5}+3153150 x^{5} a^{5} b \,d^{4} e^{4}+6306300 x^{5} a^{4} b^{2} d^{5} e^{3}+4204200 x^{5} a^{3} b^{3} d^{6} e^{2}+900900 x^{5} a^{2} b^{4} d^{7} e +45045 x^{5} a \,b^{5} d^{8}+630630 x^{4} a^{6} d^{4} e^{4}+3027024 x^{4} a^{5} b \,d^{5} e^{3}+3783780 x^{4} a^{4} b^{2} d^{6} e^{2}+1441440 x^{4} a^{3} b^{3} d^{7} e +135135 x^{4} a^{2} b^{4} d^{8}+630630 a^{6} d^{5} e^{3} x^{3}+1891890 a^{5} b \,d^{6} e^{2} x^{3}+1351350 a^{4} b^{2} d^{7} e \,x^{3}+225225 a^{3} b^{3} d^{8} x^{3}+420420 x^{2} a^{6} d^{6} e^{2}+720720 x^{2} a^{5} b \,d^{7} e +225225 x^{2} a^{4} b^{2} d^{8}+180180 a^{6} d^{7} e x +135135 a^{5} b \,d^{8} x +45045 a^{6} d^{8}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}} x}{45045 \left (b x +a \right )^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)*(e*x+d)^8*(b^2*x^2+2*a*b*x+a^2)^(5/2),x)

[Out]

1/45045*x*(3003*b^6*e^8*x^14+19305*a*b^5*e^8*x^13+25740*b^6*d*e^7*x^13+51975*a^2*b^4*e^8*x^12+166320*a*b^5*d*e
^7*x^12+97020*b^6*d^2*e^6*x^12+75075*a^3*b^3*e^8*x^11+450450*a^2*b^4*d*e^7*x^11+630630*a*b^5*d^2*e^6*x^11+2102
10*b^6*d^3*e^5*x^11+61425*a^4*b^2*e^8*x^10+655200*a^3*b^3*d*e^7*x^10+1719900*a^2*b^4*d^2*e^6*x^10+1375920*a*b^
5*d^3*e^5*x^10+286650*b^6*d^4*e^4*x^10+27027*a^5*b*e^8*x^9+540540*a^4*b^2*d*e^7*x^9+2522520*a^3*b^3*d^2*e^6*x^
9+3783780*a^2*b^4*d^3*e^5*x^9+1891890*a*b^5*d^4*e^4*x^9+252252*b^6*d^5*e^3*x^9+5005*a^6*e^8*x^8+240240*a^5*b*d
*e^7*x^8+2102100*a^4*b^2*d^2*e^6*x^8+5605600*a^3*b^3*d^3*e^5*x^8+5255250*a^2*b^4*d^4*e^4*x^8+1681680*a*b^5*d^5
*e^3*x^8+140140*b^6*d^6*e^2*x^8+45045*a^6*d*e^7*x^7+945945*a^5*b*d^2*e^6*x^7+4729725*a^4*b^2*d^3*e^5*x^7+78828
75*a^3*b^3*d^4*e^4*x^7+4729725*a^2*b^4*d^5*e^3*x^7+945945*a*b^5*d^6*e^2*x^7+45045*b^6*d^7*e*x^7+180180*a^6*d^2
*e^6*x^6+2162160*a^5*b*d^3*e^5*x^6+6756750*a^4*b^2*d^4*e^4*x^6+7207200*a^3*b^3*d^5*e^3*x^6+2702700*a^2*b^4*d^6
*e^2*x^6+308880*a*b^5*d^7*e*x^6+6435*b^6*d^8*x^6+420420*a^6*d^3*e^5*x^5+3153150*a^5*b*d^4*e^4*x^5+6306300*a^4*
b^2*d^5*e^3*x^5+4204200*a^3*b^3*d^6*e^2*x^5+900900*a^2*b^4*d^7*e*x^5+45045*a*b^5*d^8*x^5+630630*a^6*d^4*e^4*x^
4+3027024*a^5*b*d^5*e^3*x^4+3783780*a^4*b^2*d^6*e^2*x^4+1441440*a^3*b^3*d^7*e*x^4+135135*a^2*b^4*d^8*x^4+63063
0*a^6*d^5*e^3*x^3+1891890*a^5*b*d^6*e^2*x^3+1351350*a^4*b^2*d^7*e*x^3+225225*a^3*b^3*d^8*x^3+420420*a^6*d^6*e^
2*x^2+720720*a^5*b*d^7*e*x^2+225225*a^4*b^2*d^8*x^2+180180*a^6*d^7*e*x+135135*a^5*b*d^8*x+45045*a^6*d^8)*((b*x
+a)^2)^(5/2)/(b*x+a)^5

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maxima [B]  time = 0.82, size = 2653, normalized size = 7.33

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(e*x+d)^8*(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="maxima")

[Out]

1/15*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*e^8*x^8/b - 23/210*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a*e^8*x^7/b^2 + 53/390
*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a^2*e^8*x^6/b^3 - 59/390*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a^3*e^8*x^5/b^4 + 13
7/858*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a^4*e^8*x^4/b^5 - 703/4290*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a^5*e^8*x^3/b
^6 + 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a*d^8*x - 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^9*e^8*x/b^8 + 237/143
0*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a^6*e^8*x^2/b^7 + 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^2*d^8/b - 1/6*(b^2*x
^2 + 2*a*b*x + a^2)^(5/2)*a^10*e^8/b^9 - 119/715*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a^7*e^8*x/b^8 + 834/5005*(b^2
*x^2 + 2*a*b*x + a^2)^(7/2)*a^8*e^8/b^9 + 1/14*(8*b*d*e^7 + a*e^8)*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*x^7/b^2 - 3
/26*(8*b*d*e^7 + a*e^8)*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a*x^6/b^3 + 4/13*(7*b*d^2*e^6 + 2*a*d*e^7)*(b^2*x^2 +
2*a*b*x + a^2)^(7/2)*x^6/b^2 + 11/78*(8*b*d*e^7 + a*e^8)*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a^2*x^5/b^4 - 19/39*(
7*b*d^2*e^6 + 2*a*d*e^7)*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a*x^5/b^3 + 7/3*(2*b*d^3*e^5 + a*d^2*e^6)*(b^2*x^2 +
2*a*b*x + a^2)^(7/2)*x^5/b^2 - 133/858*(8*b*d*e^7 + a*e^8)*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a^3*x^4/b^5 + 251/4
29*(7*b*d^2*e^6 + 2*a*d*e^7)*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a^2*x^4/b^4 - 119/33*(2*b*d^3*e^5 + a*d^2*e^6)*(b
^2*x^2 + 2*a*b*x + a^2)^(7/2)*a*x^4/b^3 + 14/11*(5*b*d^4*e^4 + 4*a*d^3*e^5)*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*x^
4/b^2 + 139/858*(8*b*d*e^7 + a*e^8)*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a^4*x^3/b^6 - 272/429*(7*b*d^2*e^6 + 2*a*d
*e^7)*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a^3*x^3/b^5 + 140/33*(2*b*d^3*e^5 + a*d^2*e^6)*(b^2*x^2 + 2*a*b*x + a^2)
^(7/2)*a^2*x^3/b^4 - 21/11*(5*b*d^4*e^4 + 4*a*d^3*e^5)*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a*x^3/b^3 + 7/5*(4*b*d^
5*e^3 + 5*a*d^4*e^4)*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*x^3/b^2 + 1/6*(8*b*d*e^7 + a*e^8)*(b^2*x^2 + 2*a*b*x + a^
2)^(5/2)*a^8*x/b^8 - 2/3*(7*b*d^2*e^6 + 2*a*d*e^7)*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^7*x/b^7 + 14/3*(2*b*d^3*e
^5 + a*d^2*e^6)*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^6*x/b^6 - 7/3*(5*b*d^4*e^4 + 4*a*d^3*e^5)*(b^2*x^2 + 2*a*b*x
+ a^2)^(5/2)*a^5*x/b^5 + 7/3*(4*b*d^5*e^3 + 5*a*d^4*e^4)*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^4*x/b^4 - 14/3*(b*
d^6*e^2 + 2*a*d^5*e^3)*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^3*x/b^3 + 2/3*(2*b*d^7*e + 7*a*d^6*e^2)*(b^2*x^2 + 2*
a*b*x + a^2)^(5/2)*a^2*x/b^2 - 1/6*(b*d^8 + 8*a*d^7*e)*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a*x/b - 425/2574*(8*b*d
*e^7 + a*e^8)*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a^5*x^2/b^7 + 844/1287*(7*b*d^2*e^6 + 2*a*d*e^7)*(b^2*x^2 + 2*a*
b*x + a^2)^(7/2)*a^4*x^2/b^6 - 448/99*(2*b*d^3*e^5 + a*d^2*e^6)*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a^3*x^2/b^5 +
217/99*(5*b*d^4*e^4 + 4*a*d^3*e^5)*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a^2*x^2/b^4 - 91/45*(4*b*d^5*e^3 + 5*a*d^4*
e^4)*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a*x^2/b^3 + 28/9*(b*d^6*e^2 + 2*a*d^5*e^3)*(b^2*x^2 + 2*a*b*x + a^2)^(7/2
)*x^2/b^2 + 1/6*(8*b*d*e^7 + a*e^8)*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^9/b^9 - 2/3*(7*b*d^2*e^6 + 2*a*d*e^7)*(b
^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^8/b^8 + 14/3*(2*b*d^3*e^5 + a*d^2*e^6)*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^7/b^7
- 7/3*(5*b*d^4*e^4 + 4*a*d^3*e^5)*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^6/b^6 + 7/3*(4*b*d^5*e^3 + 5*a*d^4*e^4)*(
b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^5/b^5 - 14/3*(b*d^6*e^2 + 2*a*d^5*e^3)*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^4/b^
4 + 2/3*(2*b*d^7*e + 7*a*d^6*e^2)*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^3/b^3 - 1/6*(b*d^8 + 8*a*d^7*e)*(b^2*x^2 +
2*a*b*x + a^2)^(5/2)*a^2/b^2 + 214/1287*(8*b*d*e^7 + a*e^8)*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a^6*x/b^8 - 1709/
2574*(7*b*d^2*e^6 + 2*a*d*e^7)*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a^5*x/b^7 + 917/198*(2*b*d^3*e^5 + a*d^2*e^6)*(
b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a^4*x/b^6 - 455/198*(5*b*d^4*e^4 + 4*a*d^3*e^5)*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)
*a^3*x/b^5 + 203/90*(4*b*d^5*e^3 + 5*a*d^4*e^4)*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a^2*x/b^4 - 77/18*(b*d^6*e^2 +
2*a*d^5*e^3)*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a*x/b^3 + 1/2*(2*b*d^7*e + 7*a*d^6*e^2)*(b^2*x^2 + 2*a*b*x + a^2
)^(7/2)*x/b^2 - 1501/9009*(8*b*d*e^7 + a*e^8)*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a^7/b^9 + 1715/2574*(7*b*d^2*e^6
+ 2*a*d*e^7)*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a^6/b^8 - 923/198*(2*b*d^3*e^5 + a*d^2*e^6)*(b^2*x^2 + 2*a*b*x +
a^2)^(7/2)*a^5/b^7 + 461/198*(5*b*d^4*e^4 + 4*a*d^3*e^5)*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a^4/b^6 - 209/90*(4*
b*d^5*e^3 + 5*a*d^4*e^4)*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a^3/b^5 + 83/18*(b*d^6*e^2 + 2*a*d^5*e^3)*(b^2*x^2 +
2*a*b*x + a^2)^(7/2)*a^2/b^4 - 9/14*(2*b*d^7*e + 7*a*d^6*e^2)*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a/b^3 + 1/7*(b*d
^8 + 8*a*d^7*e)*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)/b^2

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \left (a+b\,x\right )\,{\left (d+e\,x\right )}^8\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)*(d + e*x)^8*(a^2 + b^2*x^2 + 2*a*b*x)^(5/2),x)

[Out]

int((a + b*x)*(d + e*x)^8*(a^2 + b^2*x^2 + 2*a*b*x)^(5/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a + b x\right ) \left (d + e x\right )^{8} \left (\left (a + b x\right )^{2}\right )^{\frac {5}{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(e*x+d)**8*(b**2*x**2+2*a*b*x+a**2)**(5/2),x)

[Out]

Integral((a + b*x)*(d + e*x)**8*((a + b*x)**2)**(5/2), x)

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