3.16.70 \(\int (d+e x)^4 (a^2+2 a b x+b^2 x^2)^{5/2} \, dx\) [1570]

Optimal. Leaf size=219 \[ \frac {(b d-a e)^4 (a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{6 b^5}+\frac {4 e (b d-a e)^3 (a+b x)^6 \sqrt {a^2+2 a b x+b^2 x^2}}{7 b^5}+\frac {3 e^2 (b d-a e)^2 (a+b x)^7 \sqrt {a^2+2 a b x+b^2 x^2}}{4 b^5}+\frac {4 e^3 (b d-a e) (a+b x)^8 \sqrt {a^2+2 a b x+b^2 x^2}}{9 b^5}+\frac {e^4 (a+b x)^9 \sqrt {a^2+2 a b x+b^2 x^2}}{10 b^5} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.06, antiderivative size = 219, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.036, Rules used = {659} \begin {gather*} \frac {4 e^3 \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^8 (b d-a e)}{9 b^5}+\frac {3 e^2 \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^7 (b d-a e)^2}{4 b^5}+\frac {4 e \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^6 (b d-a e)^3}{7 b^5}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^5 (b d-a e)^4}{6 b^5}+\frac {e^4 \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^9}{10 b^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^4*(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]

[Out]

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

Rule 659

Int[((d_.) + (e_.)*(x_))^(m_)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[(a + b*x + c*x^2)^Fra
cPart[p]/(c^IntPart[p]*(b/2 + c*x)^(2*FracPart[p])), Int[ExpandLinearProduct[(b/2 + c*x)^(2*p), (d + e*x)^m, b
/2, c, x], x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && EqQ[b^2 - 4*a*c, 0] &&  !IntegerQ[p] && NeQ[2*c*d - b*
e, 0] && IGtQ[m, 0] && EqQ[m - 2*p + 1, 0]

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]
time = 0.06, size = 319, normalized size = 1.46 \begin {gather*} \frac {x \sqrt {(a+b x)^2} \left (252 a^5 \left (5 d^4+10 d^3 e x+10 d^2 e^2 x^2+5 d e^3 x^3+e^4 x^4\right )+210 a^4 b x \left (15 d^4+40 d^3 e x+45 d^2 e^2 x^2+24 d e^3 x^3+5 e^4 x^4\right )+120 a^3 b^2 x^2 \left (35 d^4+105 d^3 e x+126 d^2 e^2 x^2+70 d e^3 x^3+15 e^4 x^4\right )+45 a^2 b^3 x^3 \left (70 d^4+224 d^3 e x+280 d^2 e^2 x^2+160 d e^3 x^3+35 e^4 x^4\right )+10 a b^4 x^4 \left (126 d^4+420 d^3 e x+540 d^2 e^2 x^2+315 d e^3 x^3+70 e^4 x^4\right )+b^5 x^5 \left (210 d^4+720 d^3 e x+945 d^2 e^2 x^2+560 d e^3 x^3+126 e^4 x^4\right )\right )}{1260 (a+b x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^4*(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]

[Out]

(x*Sqrt[(a + b*x)^2]*(252*a^5*(5*d^4 + 10*d^3*e*x + 10*d^2*e^2*x^2 + 5*d*e^3*x^3 + e^4*x^4) + 210*a^4*b*x*(15*
d^4 + 40*d^3*e*x + 45*d^2*e^2*x^2 + 24*d*e^3*x^3 + 5*e^4*x^4) + 120*a^3*b^2*x^2*(35*d^4 + 105*d^3*e*x + 126*d^
2*e^2*x^2 + 70*d*e^3*x^3 + 15*e^4*x^4) + 45*a^2*b^3*x^3*(70*d^4 + 224*d^3*e*x + 280*d^2*e^2*x^2 + 160*d*e^3*x^
3 + 35*e^4*x^4) + 10*a*b^4*x^4*(126*d^4 + 420*d^3*e*x + 540*d^2*e^2*x^2 + 315*d*e^3*x^3 + 70*e^4*x^4) + b^5*x^
5*(210*d^4 + 720*d^3*e*x + 945*d^2*e^2*x^2 + 560*d*e^3*x^3 + 126*e^4*x^4)))/(1260*(a + b*x))

________________________________________________________________________________________

Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(413\) vs. \(2(154)=308\).
time = 0.62, size = 414, normalized size = 1.89

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^4*(b^2*x^2+2*a*b*x+a^2)^(5/2),x,method=_RETURNVERBOSE)

[Out]

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

________________________________________________________________________________________

Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 576 vs. \(2 (156) = 312\).
time = 0.31, size = 576, normalized size = 2.63 \begin {gather*} \frac {1}{6} \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} d^{4} x - \frac {2 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a d^{3} x e}{3 \, b} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a d^{4}}{6 \, b} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a^{2} d^{2} x e^{2}}{b^{2}} - \frac {2 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a^{2} d^{3} e}{3 \, b^{2}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} x^{3} e^{4}}{10 \, b^{2}} - \frac {2 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a^{3} d x e^{3}}{3 \, b^{3}} + \frac {4 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} d x^{2} e^{3}}{9 \, b^{2}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a^{3} d^{2} e^{2}}{b^{3}} + \frac {3 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} d^{2} x e^{2}}{4 \, b^{2}} + \frac {4 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} d^{3} e}{7 \, b^{2}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a^{4} x e^{4}}{6 \, b^{4}} - \frac {13 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} a x^{2} e^{4}}{90 \, b^{3}} - \frac {2 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a^{4} d e^{3}}{3 \, b^{4}} - \frac {11 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} a d x e^{3}}{18 \, b^{3}} - \frac {27 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} a d^{2} e^{2}}{28 \, b^{3}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a^{5} e^{4}}{6 \, b^{5}} + \frac {29 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} a^{2} x e^{4}}{180 \, b^{4}} + \frac {83 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} a^{2} d e^{3}}{126 \, b^{4}} - \frac {209 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} a^{3} e^{4}}{1260 \, b^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^4*(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="maxima")

[Out]

1/6*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*d^4*x - 2/3*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a*d^3*x*e/b + 1/6*(b^2*x^2 + 2
*a*b*x + a^2)^(5/2)*a*d^4/b + (b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^2*d^2*x*e^2/b^2 - 2/3*(b^2*x^2 + 2*a*b*x + a^2
)^(5/2)*a^2*d^3*e/b^2 + 1/10*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*x^3*e^4/b^2 - 2/3*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)
*a^3*d*x*e^3/b^3 + 4/9*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*d*x^2*e^3/b^2 + (b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^3*d^2
*e^2/b^3 + 3/4*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*d^2*x*e^2/b^2 + 4/7*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*d^3*e/b^2 +
1/6*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^4*x*e^4/b^4 - 13/90*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a*x^2*e^4/b^3 - 2/3
*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^4*d*e^3/b^4 - 11/18*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a*d*x*e^3/b^3 - 27/28*(
b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a*d^2*e^2/b^3 + 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^5*e^4/b^5 + 29/180*(b^2*x
^2 + 2*a*b*x + a^2)^(7/2)*a^2*x*e^4/b^4 + 83/126*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a^2*d*e^3/b^4 - 209/1260*(b^2
*x^2 + 2*a*b*x + a^2)^(7/2)*a^3*e^4/b^5

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 357 vs. \(2 (156) = 312\).
time = 3.17, size = 357, normalized size = 1.63 \begin {gather*} \frac {1}{6} \, b^{5} d^{4} x^{6} + a b^{4} d^{4} x^{5} + \frac {5}{2} \, a^{2} b^{3} d^{4} x^{4} + \frac {10}{3} \, a^{3} b^{2} d^{4} x^{3} + \frac {5}{2} \, a^{4} b d^{4} x^{2} + a^{5} d^{4} x + \frac {1}{1260} \, {\left (126 \, b^{5} x^{10} + 700 \, a b^{4} x^{9} + 1575 \, a^{2} b^{3} x^{8} + 1800 \, a^{3} b^{2} x^{7} + 1050 \, a^{4} b x^{6} + 252 \, a^{5} x^{5}\right )} e^{4} + \frac {1}{126} \, {\left (56 \, b^{5} d x^{9} + 315 \, a b^{4} d x^{8} + 720 \, a^{2} b^{3} d x^{7} + 840 \, a^{3} b^{2} d x^{6} + 504 \, a^{4} b d x^{5} + 126 \, a^{5} d x^{4}\right )} e^{3} + \frac {1}{28} \, {\left (21 \, b^{5} d^{2} x^{8} + 120 \, a b^{4} d^{2} x^{7} + 280 \, a^{2} b^{3} d^{2} x^{6} + 336 \, a^{3} b^{2} d^{2} x^{5} + 210 \, a^{4} b d^{2} x^{4} + 56 \, a^{5} d^{2} x^{3}\right )} e^{2} + \frac {2}{21} \, {\left (6 \, b^{5} d^{3} x^{7} + 35 \, a b^{4} d^{3} x^{6} + 84 \, a^{2} b^{3} d^{3} x^{5} + 105 \, a^{3} b^{2} d^{3} x^{4} + 70 \, a^{4} b d^{3} x^{3} + 21 \, a^{5} d^{3} x^{2}\right )} e \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^4*(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="fricas")

[Out]

1/6*b^5*d^4*x^6 + a*b^4*d^4*x^5 + 5/2*a^2*b^3*d^4*x^4 + 10/3*a^3*b^2*d^4*x^3 + 5/2*a^4*b*d^4*x^2 + a^5*d^4*x +
1/1260*(126*b^5*x^10 + 700*a*b^4*x^9 + 1575*a^2*b^3*x^8 + 1800*a^3*b^2*x^7 + 1050*a^4*b*x^6 + 252*a^5*x^5)*e^
4 + 1/126*(56*b^5*d*x^9 + 315*a*b^4*d*x^8 + 720*a^2*b^3*d*x^7 + 840*a^3*b^2*d*x^6 + 504*a^4*b*d*x^5 + 126*a^5*
d*x^4)*e^3 + 1/28*(21*b^5*d^2*x^8 + 120*a*b^4*d^2*x^7 + 280*a^2*b^3*d^2*x^6 + 336*a^3*b^2*d^2*x^5 + 210*a^4*b*
d^2*x^4 + 56*a^5*d^2*x^3)*e^2 + 2/21*(6*b^5*d^3*x^7 + 35*a*b^4*d^3*x^6 + 84*a^2*b^3*d^3*x^5 + 105*a^3*b^2*d^3*
x^4 + 70*a^4*b*d^3*x^3 + 21*a^5*d^3*x^2)*e

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (d + e x\right )^{4} \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((e*x+d)**4*(b**2*x**2+2*a*b*x+a**2)**(5/2),x)

[Out]

Integral((d + e*x)**4*((a + b*x)**2)**(5/2), x)

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 564 vs. \(2 (156) = 312\).
time = 2.06, size = 564, normalized size = 2.58 \begin {gather*} \frac {1}{10} \, b^{5} x^{10} e^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {4}{9} \, b^{5} d x^{9} e^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {3}{4} \, b^{5} d^{2} x^{8} e^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {4}{7} \, b^{5} d^{3} x^{7} e \mathrm {sgn}\left (b x + a\right ) + \frac {1}{6} \, b^{5} d^{4} x^{6} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{9} \, a b^{4} x^{9} e^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{2} \, a b^{4} d x^{8} e^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {30}{7} \, a b^{4} d^{2} x^{7} e^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {10}{3} \, a b^{4} d^{3} x^{6} e \mathrm {sgn}\left (b x + a\right ) + a b^{4} d^{4} x^{5} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{4} \, a^{2} b^{3} x^{8} e^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {40}{7} \, a^{2} b^{3} d x^{7} e^{3} \mathrm {sgn}\left (b x + a\right ) + 10 \, a^{2} b^{3} d^{2} x^{6} e^{2} \mathrm {sgn}\left (b x + a\right ) + 8 \, a^{2} b^{3} d^{3} x^{5} e \mathrm {sgn}\left (b x + a\right ) + \frac {5}{2} \, a^{2} b^{3} d^{4} x^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {10}{7} \, a^{3} b^{2} x^{7} e^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {20}{3} \, a^{3} b^{2} d x^{6} e^{3} \mathrm {sgn}\left (b x + a\right ) + 12 \, a^{3} b^{2} d^{2} x^{5} e^{2} \mathrm {sgn}\left (b x + a\right ) + 10 \, a^{3} b^{2} d^{3} x^{4} e \mathrm {sgn}\left (b x + a\right ) + \frac {10}{3} \, a^{3} b^{2} d^{4} x^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{6} \, a^{4} b x^{6} e^{4} \mathrm {sgn}\left (b x + a\right ) + 4 \, a^{4} b d x^{5} e^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {15}{2} \, a^{4} b d^{2} x^{4} e^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {20}{3} \, a^{4} b d^{3} x^{3} e \mathrm {sgn}\left (b x + a\right ) + \frac {5}{2} \, a^{4} b d^{4} x^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{5} \, a^{5} x^{5} e^{4} \mathrm {sgn}\left (b x + a\right ) + a^{5} d x^{4} e^{3} \mathrm {sgn}\left (b x + a\right ) + 2 \, a^{5} d^{2} x^{3} e^{2} \mathrm {sgn}\left (b x + a\right ) + 2 \, a^{5} d^{3} x^{2} e \mathrm {sgn}\left (b x + a\right ) + a^{5} d^{4} x \mathrm {sgn}\left (b x + a\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^4*(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="giac")

[Out]

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

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int {\left (d+e\,x\right )}^4\,{\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((d + e*x)^4*(a^2 + b^2*x^2 + 2*a*b*x)^(5/2),x)

[Out]

int((d + e*x)^4*(a^2 + b^2*x^2 + 2*a*b*x)^(5/2), x)

________________________________________________________________________________________