3.20.70 \(\int (a+b x) (d+e x)^5 (a^2+2 a b x+b^2 x^2)^{3/2} \, dx\) [1970]

Optimal. Leaf size=254 \[ \frac {(b d-a e)^4 (d+e x)^6 \sqrt {a^2+2 a b x+b^2 x^2}}{6 e^5 (a+b x)}-\frac {4 b (b d-a e)^3 (d+e x)^7 \sqrt {a^2+2 a b x+b^2 x^2}}{7 e^5 (a+b x)}+\frac {3 b^2 (b d-a e)^2 (d+e x)^8 \sqrt {a^2+2 a b x+b^2 x^2}}{4 e^5 (a+b x)}-\frac {4 b^3 (b d-a e) (d+e x)^9 \sqrt {a^2+2 a b x+b^2 x^2}}{9 e^5 (a+b x)}+\frac {b^4 (d+e x)^{10} \sqrt {a^2+2 a b x+b^2 x^2}}{10 e^5 (a+b x)} \]

[Out]

1/6*(-a*e+b*d)^4*(e*x+d)^6*((b*x+a)^2)^(1/2)/e^5/(b*x+a)-4/7*b*(-a*e+b*d)^3*(e*x+d)^7*((b*x+a)^2)^(1/2)/e^5/(b
*x+a)+3/4*b^2*(-a*e+b*d)^2*(e*x+d)^8*((b*x+a)^2)^(1/2)/e^5/(b*x+a)-4/9*b^3*(-a*e+b*d)*(e*x+d)^9*((b*x+a)^2)^(1
/2)/e^5/(b*x+a)+1/10*b^4*(e*x+d)^10*((b*x+a)^2)^(1/2)/e^5/(b*x+a)

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Rubi [A]
time = 0.19, antiderivative size = 254, 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 = {784, 21, 45} \begin {gather*} \frac {3 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^8 (b d-a e)^2}{4 e^5 (a+b x)}-\frac {4 b \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^7 (b d-a e)^3}{7 e^5 (a+b x)}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^6 (b d-a e)^4}{6 e^5 (a+b x)}+\frac {b^4 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{10}}{10 e^5 (a+b x)}-\frac {4 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^9 (b d-a e)}{9 e^5 (a+b x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)*(d + e*x)^5*(a^2 + 2*a*b*x + b^2*x^2)^(3/2),x]

[Out]

((b*d - a*e)^4*(d + e*x)^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(6*e^5*(a + b*x)) - (4*b*(b*d - a*e)^3*(d + e*x)^7*S
qrt[a^2 + 2*a*b*x + b^2*x^2])/(7*e^5*(a + b*x)) + (3*b^2*(b*d - a*e)^2*(d + e*x)^8*Sqrt[a^2 + 2*a*b*x + b^2*x^
2])/(4*e^5*(a + b*x)) - (4*b^3*(b*d - a*e)*(d + e*x)^9*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(9*e^5*(a + b*x)) + (b^4
*(d + e*x)^10*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(10*e^5*(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 45

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 784

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

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Mathematica [A]
time = 0.07, size = 322, normalized size = 1.27 \begin {gather*} \frac {x \sqrt {(a+b x)^2} \left (210 a^4 \left (6 d^5+15 d^4 e x+20 d^3 e^2 x^2+15 d^2 e^3 x^3+6 d e^4 x^4+e^5 x^5\right )+120 a^3 b x \left (21 d^5+70 d^4 e x+105 d^3 e^2 x^2+84 d^2 e^3 x^3+35 d e^4 x^4+6 e^5 x^5\right )+45 a^2 b^2 x^2 \left (56 d^5+210 d^4 e x+336 d^3 e^2 x^2+280 d^2 e^3 x^3+120 d e^4 x^4+21 e^5 x^5\right )+10 a b^3 x^3 \left (126 d^5+504 d^4 e x+840 d^3 e^2 x^2+720 d^2 e^3 x^3+315 d e^4 x^4+56 e^5 x^5\right )+b^4 x^4 \left (252 d^5+1050 d^4 e x+1800 d^3 e^2 x^2+1575 d^2 e^3 x^3+700 d e^4 x^4+126 e^5 x^5\right )\right )}{1260 (a+b x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)*(d + e*x)^5*(a^2 + 2*a*b*x + b^2*x^2)^(3/2),x]

[Out]

(x*Sqrt[(a + b*x)^2]*(210*a^4*(6*d^5 + 15*d^4*e*x + 20*d^3*e^2*x^2 + 15*d^2*e^3*x^3 + 6*d*e^4*x^4 + e^5*x^5) +
120*a^3*b*x*(21*d^5 + 70*d^4*e*x + 105*d^3*e^2*x^2 + 84*d^2*e^3*x^3 + 35*d*e^4*x^4 + 6*e^5*x^5) + 45*a^2*b^2*
x^2*(56*d^5 + 210*d^4*e*x + 336*d^3*e^2*x^2 + 280*d^2*e^3*x^3 + 120*d*e^4*x^4 + 21*e^5*x^5) + 10*a*b^3*x^3*(12
6*d^5 + 504*d^4*e*x + 840*d^3*e^2*x^2 + 720*d^2*e^3*x^3 + 315*d*e^4*x^4 + 56*e^5*x^5) + b^4*x^4*(252*d^5 + 105
0*d^4*e*x + 1800*d^3*e^2*x^2 + 1575*d^2*e^3*x^3 + 700*d*e^4*x^4 + 126*e^5*x^5)))/(1260*(a + b*x))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(413\) vs. \(2(189)=378\).
time = 0.09, size = 414, normalized size = 1.63

method	result	size
gosper	\(\frac {x \left (126 b^{4} e^{5} x^{9}+560 x^{8} a \,b^{3} e^{5}+700 x^{8} b^{4} d \,e^{4}+945 x^{7} a^{2} b^{2} e^{5}+3150 x^{7} a \,b^{3} d \,e^{4}+1575 x^{7} b^{4} d^{2} e^{3}+720 x^{6} a^{3} b \,e^{5}+5400 x^{6} a^{2} b^{2} d \,e^{4}+7200 x^{6} a \,b^{3} d^{2} e^{3}+1800 x^{6} b^{4} d^{3} e^{2}+210 x^{5} a^{4} e^{5}+4200 x^{5} a^{3} b d \,e^{4}+12600 x^{5} a^{2} b^{2} d^{2} e^{3}+8400 x^{5} a \,b^{3} d^{3} e^{2}+1050 x^{5} b^{4} d^{4} e +1260 x^{4} a^{4} d \,e^{4}+10080 x^{4} a^{3} b \,d^{2} e^{3}+15120 x^{4} a^{2} b^{2} d^{3} e^{2}+5040 x^{4} a \,b^{3} d^{4} e +252 x^{4} b^{4} d^{5}+3150 x^{3} a^{4} d^{2} e^{3}+12600 x^{3} a^{3} b \,d^{3} e^{2}+9450 x^{3} a^{2} b^{2} d^{4} e +1260 x^{3} a \,b^{3} d^{5}+4200 x^{2} a^{4} d^{3} e^{2}+8400 x^{2} a^{3} b \,d^{4} e +2520 x^{2} a^{2} b^{2} d^{5}+3150 x \,a^{4} d^{4} e +2520 x \,a^{3} b \,d^{5}+1260 a^{4} d^{5}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}{1260 \left (b x +a \right )^{3}}\)	\(414\)
default	\(\frac {x \left (126 b^{4} e^{5} x^{9}+560 x^{8} a \,b^{3} e^{5}+700 x^{8} b^{4} d \,e^{4}+945 x^{7} a^{2} b^{2} e^{5}+3150 x^{7} a \,b^{3} d \,e^{4}+1575 x^{7} b^{4} d^{2} e^{3}+720 x^{6} a^{3} b \,e^{5}+5400 x^{6} a^{2} b^{2} d \,e^{4}+7200 x^{6} a \,b^{3} d^{2} e^{3}+1800 x^{6} b^{4} d^{3} e^{2}+210 x^{5} a^{4} e^{5}+4200 x^{5} a^{3} b d \,e^{4}+12600 x^{5} a^{2} b^{2} d^{2} e^{3}+8400 x^{5} a \,b^{3} d^{3} e^{2}+1050 x^{5} b^{4} d^{4} e +1260 x^{4} a^{4} d \,e^{4}+10080 x^{4} a^{3} b \,d^{2} e^{3}+15120 x^{4} a^{2} b^{2} d^{3} e^{2}+5040 x^{4} a \,b^{3} d^{4} e +252 x^{4} b^{4} d^{5}+3150 x^{3} a^{4} d^{2} e^{3}+12600 x^{3} a^{3} b \,d^{3} e^{2}+9450 x^{3} a^{2} b^{2} d^{4} e +1260 x^{3} a \,b^{3} d^{5}+4200 x^{2} a^{4} d^{3} e^{2}+8400 x^{2} a^{3} b \,d^{4} e +2520 x^{2} a^{2} b^{2} d^{5}+3150 x \,a^{4} d^{4} e +2520 x \,a^{3} b \,d^{5}+1260 a^{4} d^{5}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}{1260 \left (b x +a \right )^{3}}\)	\(414\)
risch	\(\frac {\sqrt {\left (b x +a \right )^{2}}\, b^{4} e^{5} x^{10}}{10 b x +10 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (4 a \,b^{3} e^{5}+5 b^{4} d \,e^{4}\right ) x^{9}}{9 b x +9 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (6 a^{2} b^{2} e^{5}+20 a \,b^{3} d \,e^{4}+10 b^{4} d^{2} e^{3}\right ) x^{8}}{8 b x +8 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (4 a^{3} b \,e^{5}+30 a^{2} b^{2} d \,e^{4}+40 a \,b^{3} d^{2} e^{3}+10 b^{4} d^{3} e^{2}\right ) x^{7}}{7 b x +7 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (a^{4} e^{5}+20 a^{3} b d \,e^{4}+60 a^{2} b^{2} d^{2} e^{3}+40 a \,b^{3} d^{3} e^{2}+5 b^{4} d^{4} e \right ) x^{6}}{6 b x +6 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (5 a^{4} d \,e^{4}+40 a^{3} b \,d^{2} e^{3}+60 a^{2} b^{2} d^{3} e^{2}+20 a \,b^{3} d^{4} e +b^{4} d^{5}\right ) x^{5}}{5 b x +5 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (10 a^{4} d^{2} e^{3}+40 a^{3} b \,d^{3} e^{2}+30 a^{2} b^{2} d^{4} e +4 a \,b^{3} d^{5}\right ) x^{4}}{4 b x +4 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (10 a^{4} d^{3} e^{2}+20 a^{3} b \,d^{4} e +6 a^{2} b^{2} d^{5}\right ) x^{3}}{3 b x +3 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (5 a^{4} d^{4} e +4 a^{3} b \,d^{5}\right ) x^{2}}{2 b x +2 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, a^{4} d^{5} x}{b x +a}\)	\(521\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)*(e*x+d)^5*(b^2*x^2+2*a*b*x+a^2)^(3/2),x,method=_RETURNVERBOSE)

[Out]

1/1260*x*(126*b^4*e^5*x^9+560*a*b^3*e^5*x^8+700*b^4*d*e^4*x^8+945*a^2*b^2*e^5*x^7+3150*a*b^3*d*e^4*x^7+1575*b^
4*d^2*e^3*x^7+720*a^3*b*e^5*x^6+5400*a^2*b^2*d*e^4*x^6+7200*a*b^3*d^2*e^3*x^6+1800*b^4*d^3*e^2*x^6+210*a^4*e^5
*x^5+4200*a^3*b*d*e^4*x^5+12600*a^2*b^2*d^2*e^3*x^5+8400*a*b^3*d^3*e^2*x^5+1050*b^4*d^4*e*x^5+1260*a^4*d*e^4*x
^4+10080*a^3*b*d^2*e^3*x^4+15120*a^2*b^2*d^3*e^2*x^4+5040*a*b^3*d^4*e*x^4+252*b^4*d^5*x^4+3150*a^4*d^2*e^3*x^3
+12600*a^3*b*d^3*e^2*x^3+9450*a^2*b^2*d^4*e*x^3+1260*a*b^3*d^5*x^3+4200*a^4*d^3*e^2*x^2+8400*a^3*b*d^4*e*x^2+2
520*a^2*b^2*d^5*x^2+3150*a^4*d^4*e*x+2520*a^3*b*d^5*x+1260*a^4*d^5)*((b*x+a)^2)^(3/2)/(b*x+a)^3

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 1282 vs. \(2 (193) = 386\).
time = 0.29, size = 1282, normalized size = 5.05 \begin {gather*} \frac {1}{4} \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a d^{5} x + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a^{2} d^{5}}{4 \, b} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} x^{5} e^{5}}{10 \, b} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a x^{4} e^{5}}{6 \, b^{2}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} {\left (5 \, b d e^{4} + a e^{5}\right )} x^{4}}{9 \, b^{2}} + \frac {5 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a^{2} x^{3} e^{5}}{24 \, b^{3}} - \frac {13 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} {\left (5 \, b d e^{4} + a e^{5}\right )} a x^{3}}{72 \, b^{3}} + \frac {5 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} {\left (2 \, b d^{2} e^{3} + a d e^{4}\right )} x^{3}}{8 \, b^{2}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a^{6} x e^{5}}{4 \, b^{5}} - \frac {13 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a^{3} x^{2} e^{5}}{56 \, b^{4}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} {\left (5 \, b d e^{4} + a e^{5}\right )} a^{5} x}{4 \, b^{5}} + \frac {5 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} {\left (2 \, b d^{2} e^{3} + a d e^{4}\right )} a^{4} x}{4 \, b^{4}} - \frac {5 \, {\left (b d^{3} e^{2} + a d^{2} e^{3}\right )} {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a^{3} x}{2 \, b^{3}} + \frac {5 \, {\left (b d^{4} e + 2 \, a d^{3} e^{2}\right )} {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a^{2} x}{4 \, b^{2}} - \frac {{\left (b d^{5} + 5 \, a d^{4} e\right )} {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a x}{4 \, b} + \frac {37 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} {\left (5 \, b d e^{4} + a e^{5}\right )} a^{2} x^{2}}{168 \, b^{4}} - \frac {55 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} {\left (2 \, b d^{2} e^{3} + a d e^{4}\right )} a x^{2}}{56 \, b^{3}} + \frac {10 \, {\left (b d^{3} e^{2} + a d^{2} e^{3}\right )} {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} x^{2}}{7 \, b^{2}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a^{7} e^{5}}{4 \, b^{6}} + \frac {41 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a^{4} x e^{5}}{168 \, b^{5}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} {\left (5 \, b d e^{4} + a e^{5}\right )} a^{6}}{4 \, b^{6}} + \frac {5 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} {\left (2 \, b d^{2} e^{3} + a d e^{4}\right )} a^{5}}{4 \, b^{5}} - \frac {5 \, {\left (b d^{3} e^{2} + a d^{2} e^{3}\right )} {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a^{4}}{2 \, b^{4}} + \frac {5 \, {\left (b d^{4} e + 2 \, a d^{3} e^{2}\right )} {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a^{3}}{4 \, b^{3}} - \frac {{\left (b d^{5} + 5 \, a d^{4} e\right )} {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a^{2}}{4 \, b^{2}} - \frac {121 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} {\left (5 \, b d e^{4} + a e^{5}\right )} a^{3} x}{504 \, b^{5}} + \frac {65 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} {\left (2 \, b d^{2} e^{3} + a d e^{4}\right )} a^{2} x}{56 \, b^{4}} - \frac {15 \, {\left (b d^{3} e^{2} + a d^{2} e^{3}\right )} {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a x}{7 \, b^{3}} + \frac {5 \, {\left (b d^{4} e + 2 \, a d^{3} e^{2}\right )} {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} x}{6 \, b^{2}} - \frac {209 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a^{5} e^{5}}{840 \, b^{6}} + \frac {125 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} {\left (5 \, b d e^{4} + a e^{5}\right )} a^{4}}{504 \, b^{6}} - \frac {69 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} {\left (2 \, b d^{2} e^{3} + a d e^{4}\right )} a^{3}}{56 \, b^{5}} + \frac {17 \, {\left (b d^{3} e^{2} + a d^{2} e^{3}\right )} {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a^{2}}{7 \, b^{4}} - \frac {7 \, {\left (b d^{4} e + 2 \, a d^{3} e^{2}\right )} {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a}{6 \, b^{3}} + \frac {{\left (b d^{5} + 5 \, a d^{4} e\right )} {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}}}{5 \, b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(e*x+d)^5*(b^2*x^2+2*a*b*x+a^2)^(3/2),x, algorithm="maxima")

[Out]

1/4*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*a*d^5*x + 1/4*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*a^2*d^5/b + 1/10*(b^2*x^2 +
2*a*b*x + a^2)^(5/2)*x^5*e^5/b - 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a*x^4*e^5/b^2 + 1/9*(b^2*x^2 + 2*a*b*x +
a^2)^(5/2)*(5*b*d*e^4 + a*e^5)*x^4/b^2 + 5/24*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^2*x^3*e^5/b^3 - 13/72*(b^2*x^2
+ 2*a*b*x + a^2)^(5/2)*(5*b*d*e^4 + a*e^5)*a*x^3/b^3 + 5/8*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*(2*b*d^2*e^3 + a*d
*e^4)*x^3/b^2 + 1/4*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*a^6*x*e^5/b^5 - 13/56*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^3*
x^2*e^5/b^4 - 1/4*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*(5*b*d*e^4 + a*e^5)*a^5*x/b^5 + 5/4*(b^2*x^2 + 2*a*b*x + a^2
)^(3/2)*(2*b*d^2*e^3 + a*d*e^4)*a^4*x/b^4 - 5/2*(b*d^3*e^2 + a*d^2*e^3)*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*a^3*x/
b^3 + 5/4*(b*d^4*e + 2*a*d^3*e^2)*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*a^2*x/b^2 - 1/4*(b*d^5 + 5*a*d^4*e)*(b^2*x^2
+ 2*a*b*x + a^2)^(3/2)*a*x/b + 37/168*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*(5*b*d*e^4 + a*e^5)*a^2*x^2/b^4 - 55/56
*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*(2*b*d^2*e^3 + a*d*e^4)*a*x^2/b^3 + 10/7*(b*d^3*e^2 + a*d^2*e^3)*(b^2*x^2 + 2
*a*b*x + a^2)^(5/2)*x^2/b^2 + 1/4*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*a^7*e^5/b^6 + 41/168*(b^2*x^2 + 2*a*b*x + a^
2)^(5/2)*a^4*x*e^5/b^5 - 1/4*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*(5*b*d*e^4 + a*e^5)*a^6/b^6 + 5/4*(b^2*x^2 + 2*a*
b*x + a^2)^(3/2)*(2*b*d^2*e^3 + a*d*e^4)*a^5/b^5 - 5/2*(b*d^3*e^2 + a*d^2*e^3)*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)
*a^4/b^4 + 5/4*(b*d^4*e + 2*a*d^3*e^2)*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*a^3/b^3 - 1/4*(b*d^5 + 5*a*d^4*e)*(b^2*
x^2 + 2*a*b*x + a^2)^(3/2)*a^2/b^2 - 121/504*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*(5*b*d*e^4 + a*e^5)*a^3*x/b^5 + 6
5/56*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*(2*b*d^2*e^3 + a*d*e^4)*a^2*x/b^4 - 15/7*(b*d^3*e^2 + a*d^2*e^3)*(b^2*x^2
+ 2*a*b*x + a^2)^(5/2)*a*x/b^3 + 5/6*(b*d^4*e + 2*a*d^3*e^2)*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*x/b^2 - 209/840*
(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^5*e^5/b^6 + 125/504*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*(5*b*d*e^4 + a*e^5)*a^4/
b^6 - 69/56*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*(2*b*d^2*e^3 + a*d*e^4)*a^3/b^5 + 17/7*(b*d^3*e^2 + a*d^2*e^3)*(b^
2*x^2 + 2*a*b*x + a^2)^(5/2)*a^2/b^4 - 7/6*(b*d^4*e + 2*a*d^3*e^2)*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a/b^3 + 1/5
*(b*d^5 + 5*a*d^4*e)*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)/b^2

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Fricas [A]
time = 2.21, size = 357, normalized size = 1.41 \begin {gather*} \frac {1}{5} \, b^{4} d^{5} x^{5} + a b^{3} d^{5} x^{4} + 2 \, a^{2} b^{2} d^{5} x^{3} + 2 \, a^{3} b d^{5} x^{2} + a^{4} d^{5} x + \frac {1}{1260} \, {\left (126 \, b^{4} x^{10} + 560 \, a b^{3} x^{9} + 945 \, a^{2} b^{2} x^{8} + 720 \, a^{3} b x^{7} + 210 \, a^{4} x^{6}\right )} e^{5} + \frac {1}{126} \, {\left (70 \, b^{4} d x^{9} + 315 \, a b^{3} d x^{8} + 540 \, a^{2} b^{2} d x^{7} + 420 \, a^{3} b d x^{6} + 126 \, a^{4} d x^{5}\right )} e^{4} + \frac {1}{28} \, {\left (35 \, b^{4} d^{2} x^{8} + 160 \, a b^{3} d^{2} x^{7} + 280 \, a^{2} b^{2} d^{2} x^{6} + 224 \, a^{3} b d^{2} x^{5} + 70 \, a^{4} d^{2} x^{4}\right )} e^{3} + \frac {2}{21} \, {\left (15 \, b^{4} d^{3} x^{7} + 70 \, a b^{3} d^{3} x^{6} + 126 \, a^{2} b^{2} d^{3} x^{5} + 105 \, a^{3} b d^{3} x^{4} + 35 \, a^{4} d^{3} x^{3}\right )} e^{2} + \frac {1}{6} \, {\left (5 \, b^{4} d^{4} x^{6} + 24 \, a b^{3} d^{4} x^{5} + 45 \, a^{2} b^{2} d^{4} x^{4} + 40 \, a^{3} b d^{4} x^{3} + 15 \, a^{4} d^{4} x^{2}\right )} e \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(e*x+d)^5*(b^2*x^2+2*a*b*x+a^2)^(3/2),x, algorithm="fricas")

[Out]

1/5*b^4*d^5*x^5 + a*b^3*d^5*x^4 + 2*a^2*b^2*d^5*x^3 + 2*a^3*b*d^5*x^2 + a^4*d^5*x + 1/1260*(126*b^4*x^10 + 560
*a*b^3*x^9 + 945*a^2*b^2*x^8 + 720*a^3*b*x^7 + 210*a^4*x^6)*e^5 + 1/126*(70*b^4*d*x^9 + 315*a*b^3*d*x^8 + 540*
a^2*b^2*d*x^7 + 420*a^3*b*d*x^6 + 126*a^4*d*x^5)*e^4 + 1/28*(35*b^4*d^2*x^8 + 160*a*b^3*d^2*x^7 + 280*a^2*b^2*
d^2*x^6 + 224*a^3*b*d^2*x^5 + 70*a^4*d^2*x^4)*e^3 + 2/21*(15*b^4*d^3*x^7 + 70*a*b^3*d^3*x^6 + 126*a^2*b^2*d^3*
x^5 + 105*a^3*b*d^3*x^4 + 35*a^4*d^3*x^3)*e^2 + 1/6*(5*b^4*d^4*x^6 + 24*a*b^3*d^4*x^5 + 45*a^2*b^2*d^4*x^4 + 4
0*a^3*b*d^4*x^3 + 15*a^4*d^4*x^2)*e

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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 )^{5} \left (\left (a + b x\right )^{2}\right )^{\frac {3}{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(e*x+d)**5*(b**2*x**2+2*a*b*x+a**2)**(3/2),x)

[Out]

Integral((a + b*x)*(d + e*x)**5*((a + b*x)**2)**(3/2), x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 561 vs. \(2 (193) = 386\).
time = 0.94, size = 561, normalized size = 2.21 \begin {gather*} \frac {1}{10} \, b^{4} x^{10} e^{5} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{9} \, b^{4} d x^{9} e^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{4} \, b^{4} d^{2} x^{8} e^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {10}{7} \, b^{4} d^{3} x^{7} e^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{6} \, b^{4} d^{4} x^{6} e \mathrm {sgn}\left (b x + a\right ) + \frac {1}{5} \, b^{4} d^{5} x^{5} \mathrm {sgn}\left (b x + a\right ) + \frac {4}{9} \, a b^{3} x^{9} e^{5} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{2} \, a b^{3} d x^{8} e^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {40}{7} \, a b^{3} d^{2} x^{7} e^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {20}{3} \, a b^{3} d^{3} x^{6} e^{2} \mathrm {sgn}\left (b x + a\right ) + 4 \, a b^{3} d^{4} x^{5} e \mathrm {sgn}\left (b x + a\right ) + a b^{3} d^{5} x^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {3}{4} \, a^{2} b^{2} x^{8} e^{5} \mathrm {sgn}\left (b x + a\right ) + \frac {30}{7} \, a^{2} b^{2} d x^{7} e^{4} \mathrm {sgn}\left (b x + a\right ) + 10 \, a^{2} b^{2} d^{2} x^{6} e^{3} \mathrm {sgn}\left (b x + a\right ) + 12 \, a^{2} b^{2} d^{3} x^{5} e^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {15}{2} \, a^{2} b^{2} d^{4} x^{4} e \mathrm {sgn}\left (b x + a\right ) + 2 \, a^{2} b^{2} d^{5} x^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {4}{7} \, a^{3} b x^{7} e^{5} \mathrm {sgn}\left (b x + a\right ) + \frac {10}{3} \, a^{3} b d x^{6} e^{4} \mathrm {sgn}\left (b x + a\right ) + 8 \, a^{3} b d^{2} x^{5} e^{3} \mathrm {sgn}\left (b x + a\right ) + 10 \, a^{3} b d^{3} x^{4} e^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {20}{3} \, a^{3} b d^{4} x^{3} e \mathrm {sgn}\left (b x + a\right ) + 2 \, a^{3} b d^{5} x^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{6} \, a^{4} x^{6} e^{5} \mathrm {sgn}\left (b x + a\right ) + a^{4} d x^{5} e^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{2} \, a^{4} d^{2} x^{4} e^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {10}{3} \, a^{4} d^{3} x^{3} e^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{2} \, a^{4} d^{4} x^{2} e \mathrm {sgn}\left (b x + a\right ) + a^{4} d^{5} x \mathrm {sgn}\left (b x + a\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(e*x+d)^5*(b^2*x^2+2*a*b*x+a^2)^(3/2),x, algorithm="giac")

[Out]

1/10*b^4*x^10*e^5*sgn(b*x + a) + 5/9*b^4*d*x^9*e^4*sgn(b*x + a) + 5/4*b^4*d^2*x^8*e^3*sgn(b*x + a) + 10/7*b^4*
d^3*x^7*e^2*sgn(b*x + a) + 5/6*b^4*d^4*x^6*e*sgn(b*x + a) + 1/5*b^4*d^5*x^5*sgn(b*x + a) + 4/9*a*b^3*x^9*e^5*s
gn(b*x + a) + 5/2*a*b^3*d*x^8*e^4*sgn(b*x + a) + 40/7*a*b^3*d^2*x^7*e^3*sgn(b*x + a) + 20/3*a*b^3*d^3*x^6*e^2*
sgn(b*x + a) + 4*a*b^3*d^4*x^5*e*sgn(b*x + a) + a*b^3*d^5*x^4*sgn(b*x + a) + 3/4*a^2*b^2*x^8*e^5*sgn(b*x + a)
+ 30/7*a^2*b^2*d*x^7*e^4*sgn(b*x + a) + 10*a^2*b^2*d^2*x^6*e^3*sgn(b*x + a) + 12*a^2*b^2*d^3*x^5*e^2*sgn(b*x +
a) + 15/2*a^2*b^2*d^4*x^4*e*sgn(b*x + a) + 2*a^2*b^2*d^5*x^3*sgn(b*x + a) + 4/7*a^3*b*x^7*e^5*sgn(b*x + a) +
10/3*a^3*b*d*x^6*e^4*sgn(b*x + a) + 8*a^3*b*d^2*x^5*e^3*sgn(b*x + a) + 10*a^3*b*d^3*x^4*e^2*sgn(b*x + a) + 20/
3*a^3*b*d^4*x^3*e*sgn(b*x + a) + 2*a^3*b*d^5*x^2*sgn(b*x + a) + 1/6*a^4*x^6*e^5*sgn(b*x + a) + a^4*d*x^5*e^4*s
gn(b*x + a) + 5/2*a^4*d^2*x^4*e^3*sgn(b*x + a) + 10/3*a^4*d^3*x^3*e^2*sgn(b*x + a) + 5/2*a^4*d^4*x^2*e*sgn(b*x
+ a) + a^4*d^5*x*sgn(b*x + a)

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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 )}^5\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)*(d + e*x)^5*(a^2 + b^2*x^2 + 2*a*b*x)^(3/2),x)

[Out]

int((a + b*x)*(d + e*x)^5*(a^2 + b^2*x^2 + 2*a*b*x)^(3/2), x)

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