3.18.30 \(\int (a+b x) (d+e x)^4 \sqrt {a^2+2 a b x+b^2 x^2} \, dx\)

Optimal. Leaf size=146 \[ \frac {b^2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^7}{7 e^3 (a+b x)}-\frac {b \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^6 (b d-a e)}{3 e^3 (a+b x)}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^5 (b d-a e)^2}{5 e^3 (a+b x)} \]

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Rubi [A]  time = 0.13, antiderivative size = 146, 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^2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^7}{7 e^3 (a+b x)}-\frac {b \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^6 (b d-a e)}{3 e^3 (a+b x)}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^5 (b d-a e)^2}{5 e^3 (a+b x)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Mathematica [A]  time = 0.05, size = 163, normalized size = 1.12 \begin {gather*} \frac {x \sqrt {(a+b x)^2} \left (21 a^2 \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 )+7 a 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 )+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 )\right )}{105 (a+b x)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(x*Sqrt[(a + b*x)^2]*(21*a^2*(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) + 7*a*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) + b^2*x^2*(35*d^4 + 105*d^3*e*x + 126*d^2*e^2*x^2 + 7
0*d*e^3*x^3 + 15*e^4*x^4)))/(105*(a + b*x))

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IntegrateAlgebraic [F]  time = 1.62, size = 0, normalized size = 0.00 \begin {gather*} \int (a+b x) (d+e x)^4 \sqrt {a^2+2 a b x+b^2 x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(a + b*x)*(d + e*x)^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2],x]

[Out]

Defer[IntegrateAlgebraic][(a + b*x)*(d + e*x)^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2], x]

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fricas [A]  time = 0.39, size = 156, normalized size = 1.07 \begin {gather*} \frac {1}{7} \, b^{2} e^{4} x^{7} + a^{2} d^{4} x + \frac {1}{3} \, {\left (2 \, b^{2} d e^{3} + a b e^{4}\right )} x^{6} + \frac {1}{5} \, {\left (6 \, b^{2} d^{2} e^{2} + 8 \, a b d e^{3} + a^{2} e^{4}\right )} x^{5} + {\left (b^{2} d^{3} e + 3 \, a b d^{2} e^{2} + a^{2} d e^{3}\right )} x^{4} + \frac {1}{3} \, {\left (b^{2} d^{4} + 8 \, a b d^{3} e + 6 \, a^{2} d^{2} e^{2}\right )} x^{3} + {\left (a b d^{4} + 2 \, a^{2} d^{3} e\right )} x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(e*x+d)^4*((b*x+a)^2)^(1/2),x, algorithm="fricas")

[Out]

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

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giac [B]  time = 0.18, size = 254, normalized size = 1.74 \begin {gather*} \frac {1}{7} \, b^{2} x^{7} e^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {2}{3} \, b^{2} d x^{6} e^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {6}{5} \, b^{2} d^{2} x^{5} e^{2} \mathrm {sgn}\left (b x + a\right ) + b^{2} d^{3} x^{4} e \mathrm {sgn}\left (b x + a\right ) + \frac {1}{3} \, b^{2} d^{4} x^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{3} \, a b x^{6} e^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {8}{5} \, a b d x^{5} e^{3} \mathrm {sgn}\left (b x + a\right ) + 3 \, a b d^{2} x^{4} e^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {8}{3} \, a b d^{3} x^{3} e \mathrm {sgn}\left (b x + a\right ) + a b d^{4} x^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{5} \, a^{2} x^{5} e^{4} \mathrm {sgn}\left (b x + a\right ) + a^{2} d x^{4} e^{3} \mathrm {sgn}\left (b x + a\right ) + 2 \, a^{2} d^{2} x^{3} e^{2} \mathrm {sgn}\left (b x + a\right ) + 2 \, a^{2} d^{3} x^{2} e \mathrm {sgn}\left (b x + a\right ) + a^{2} d^{4} 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)^4*((b*x+a)^2)^(1/2),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.05, size = 189, normalized size = 1.29 \begin {gather*} \frac {\left (15 b^{2} e^{4} x^{6}+35 x^{5} a b \,e^{4}+70 x^{5} b^{2} d \,e^{3}+21 x^{4} a^{2} e^{4}+168 x^{4} a b d \,e^{3}+126 x^{4} b^{2} d^{2} e^{2}+105 a^{2} d \,e^{3} x^{3}+315 a b \,d^{2} e^{2} x^{3}+105 b^{2} d^{3} e \,x^{3}+210 x^{2} a^{2} d^{2} e^{2}+280 x^{2} a b \,d^{3} e +35 x^{2} b^{2} d^{4}+210 a^{2} d^{3} e x +105 a b \,d^{4} x +105 a^{2} d^{4}\right ) \sqrt {\left (b x +a \right )^{2}}\, x}{105 b x +105 a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)*(e*x+d)^4*((b*x+a)^2)^(1/2),x)

[Out]

1/105*x*(15*b^2*e^4*x^6+35*a*b*e^4*x^5+70*b^2*d*e^3*x^5+21*a^2*e^4*x^4+168*a*b*d*e^3*x^4+126*b^2*d^2*e^2*x^4+1
05*a^2*d*e^3*x^3+315*a*b*d^2*e^2*x^3+105*b^2*d^3*e*x^3+210*a^2*d^2*e^2*x^2+280*a*b*d^3*e*x^2+35*b^2*d^4*x^2+21
0*a^2*d^3*e*x+105*a*b*d^4*x+105*a^2*d^4)*((b*x+a)^2)^(1/2)/(b*x+a)

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maxima [B]  time = 0.70, size = 996, normalized size = 6.82

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(e*x+d)^4*((b*x+a)^2)^(1/2),x, algorithm="maxima")

[Out]

1/7*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*e^4*x^4/b - 11/42*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*a*e^4*x^3/b^2 + 1/2*sqrt
(b^2*x^2 + 2*a*b*x + a^2)*a*d^4*x - 1/2*sqrt(b^2*x^2 + 2*a*b*x + a^2)*a^5*e^4*x/b^4 + 5/14*(b^2*x^2 + 2*a*b*x
+ a^2)^(3/2)*a^2*e^4*x^2/b^3 + 1/2*sqrt(b^2*x^2 + 2*a*b*x + a^2)*a^2*d^4/b - 1/2*sqrt(b^2*x^2 + 2*a*b*x + a^2)
*a^6*e^4/b^5 - 3/7*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*a^3*e^4*x/b^4 + 10/21*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*a^4*e
^4/b^5 + 1/6*(4*b*d*e^3 + a*e^4)*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*x^3/b^2 + 1/2*(4*b*d*e^3 + a*e^4)*sqrt(b^2*x^
2 + 2*a*b*x + a^2)*a^4*x/b^4 - (3*b*d^2*e^2 + 2*a*d*e^3)*sqrt(b^2*x^2 + 2*a*b*x + a^2)*a^3*x/b^3 + (2*b*d^3*e
+ 3*a*d^2*e^2)*sqrt(b^2*x^2 + 2*a*b*x + a^2)*a^2*x/b^2 - 1/2*(b*d^4 + 4*a*d^3*e)*sqrt(b^2*x^2 + 2*a*b*x + a^2)
*a*x/b - 3/10*(4*b*d*e^3 + a*e^4)*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*a*x^2/b^3 + 2/5*(3*b*d^2*e^2 + 2*a*d*e^3)*(b
^2*x^2 + 2*a*b*x + a^2)^(3/2)*x^2/b^2 + 1/2*(4*b*d*e^3 + a*e^4)*sqrt(b^2*x^2 + 2*a*b*x + a^2)*a^5/b^5 - (3*b*d
^2*e^2 + 2*a*d*e^3)*sqrt(b^2*x^2 + 2*a*b*x + a^2)*a^4/b^4 + (2*b*d^3*e + 3*a*d^2*e^2)*sqrt(b^2*x^2 + 2*a*b*x +
a^2)*a^3/b^3 - 1/2*(b*d^4 + 4*a*d^3*e)*sqrt(b^2*x^2 + 2*a*b*x + a^2)*a^2/b^2 + 2/5*(4*b*d*e^3 + a*e^4)*(b^2*x
^2 + 2*a*b*x + a^2)^(3/2)*a^2*x/b^4 - 7/10*(3*b*d^2*e^2 + 2*a*d*e^3)*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*a*x/b^3 +
1/2*(2*b*d^3*e + 3*a*d^2*e^2)*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*x/b^2 - 7/15*(4*b*d*e^3 + a*e^4)*(b^2*x^2 + 2*a
*b*x + a^2)^(3/2)*a^3/b^5 + 9/10*(3*b*d^2*e^2 + 2*a*d*e^3)*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*a^2/b^4 - 5/6*(2*b*
d^3*e + 3*a*d^2*e^2)*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*a/b^3 + 1/3*(b*d^4 + 4*a*d^3*e)*(b^2*x^2 + 2*a*b*x + a^2)
^(3/2)/b^2

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mupad [B]  time = 3.59, size = 1095, normalized size = 7.50 \begin {gather*} a\,d^4\,\left (\frac {x}{2}+\frac {a}{2\,b}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}+\frac {d^4\,\left (8\,b^2\,\left (a^2+b^2\,x^2\right )-12\,a^2\,b^2+4\,a\,b^3\,x\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{24\,b^3}+\frac {e^4\,x^4\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2}}{7\,b}-\frac {a^3\,e^4\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}\,\left (a^3-5\,a\,b^2\,x^2+3\,b\,x\,\left (a^2+2\,a\,b\,x+b^2\,x^2\right )-4\,a^2\,b\,x\right )}{24\,b^5}+\frac {a\,e^4\,x^3\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2}}{6\,b^2}+\frac {2\,d\,e^3\,x^3\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2}}{3\,b}-\frac {11\,a\,e^4\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}\,\left (a^5+5\,b^3\,x^3\,\left (a^2+2\,a\,b\,x+b^2\,x^2\right )-14\,a^3\,b^2\,x^2-13\,a^4\,b\,x-9\,a\,b^2\,x^2\,\left (a^2+2\,a\,b\,x+b^2\,x^2\right )+12\,a^2\,b\,x\,\left (a^2+2\,a\,b\,x+b^2\,x^2\right )\right )}{210\,b^5}+\frac {6\,d^2\,e^2\,x^2\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2}}{5\,b}+\frac {d^3\,e\,x\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2}}{b}-\frac {29\,a^2\,e^4\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}\,\left (4\,b^2\,x^2\,\left (a^2+2\,a\,b\,x+b^2\,x^2\right )-a^4+9\,a^2\,b^2\,x^2+8\,a^3\,b\,x-7\,a\,b\,x\,\left (a^2+2\,a\,b\,x+b^2\,x^2\right )\right )}{280\,b^5}-\frac {a\,d^3\,e\,\left (8\,b^2\,\left (a^2+b^2\,x^2\right )-12\,a^2\,b^2+4\,a\,b^3\,x\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{24\,b^4}-\frac {3\,a^3\,d^2\,e^2\,\left (\frac {x}{2}+\frac {a}{2\,b}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{2\,b^2}-\frac {7\,a\,d^2\,e^2\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}\,\left (a^3-5\,a\,b^2\,x^2+3\,b\,x\,\left (a^2+2\,a\,b\,x+b^2\,x^2\right )-4\,a^2\,b\,x\right )}{10\,b^3}-\frac {19\,a^2\,d\,e^3\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}\,\left (a^3-5\,a\,b^2\,x^2+3\,b\,x\,\left (a^2+2\,a\,b\,x+b^2\,x^2\right )-4\,a^2\,b\,x\right )}{30\,b^4}-\frac {a^3\,d\,e^3\,\left (8\,b^2\,\left (a^2+b^2\,x^2\right )-12\,a^2\,b^2+4\,a\,b^3\,x\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{15\,b^6}+\frac {3\,a\,d^2\,e^2\,x\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2}}{2\,b^2}+\frac {4\,a\,d\,e^3\,x^2\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2}}{5\,b^2}-\frac {a^2\,d^3\,e\,\left (\frac {x}{2}+\frac {a}{2\,b}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{b}-\frac {3\,a\,d\,e^3\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}\,\left (4\,b^2\,x^2\,\left (a^2+2\,a\,b\,x+b^2\,x^2\right )-a^4+9\,a^2\,b^2\,x^2+8\,a^3\,b\,x-7\,a\,b\,x\,\left (a^2+2\,a\,b\,x+b^2\,x^2\right )\right )}{10\,b^4}-\frac {33\,a^2\,d^2\,e^2\,\left (8\,b^2\,\left (a^2+b^2\,x^2\right )-12\,a^2\,b^2+4\,a\,b^3\,x\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{80\,b^5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a + b*x)^2)^(1/2)*(a + b*x)*(d + e*x)^4,x)

[Out]

a*d^4*(x/2 + a/(2*b))*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2) + (d^4*(8*b^2*(a^2 + b^2*x^2) - 12*a^2*b^2 + 4*a*b^3*x)*
(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/(24*b^3) + (e^4*x^4*(a^2 + b^2*x^2 + 2*a*b*x)^(3/2))/(7*b) - (a^3*e^4*(a^2 +
b^2*x^2 + 2*a*b*x)^(1/2)*(a^3 - 5*a*b^2*x^2 + 3*b*x*(a^2 + b^2*x^2 + 2*a*b*x) - 4*a^2*b*x))/(24*b^5) + (a*e^4*
x^3*(a^2 + b^2*x^2 + 2*a*b*x)^(3/2))/(6*b^2) + (2*d*e^3*x^3*(a^2 + b^2*x^2 + 2*a*b*x)^(3/2))/(3*b) - (11*a*e^4
*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2)*(a^5 + 5*b^3*x^3*(a^2 + b^2*x^2 + 2*a*b*x) - 14*a^3*b^2*x^2 - 13*a^4*b*x - 9*
a*b^2*x^2*(a^2 + b^2*x^2 + 2*a*b*x) + 12*a^2*b*x*(a^2 + b^2*x^2 + 2*a*b*x)))/(210*b^5) + (6*d^2*e^2*x^2*(a^2 +
b^2*x^2 + 2*a*b*x)^(3/2))/(5*b) + (d^3*e*x*(a^2 + b^2*x^2 + 2*a*b*x)^(3/2))/b - (29*a^2*e^4*(a^2 + b^2*x^2 +
2*a*b*x)^(1/2)*(4*b^2*x^2*(a^2 + b^2*x^2 + 2*a*b*x) - a^4 + 9*a^2*b^2*x^2 + 8*a^3*b*x - 7*a*b*x*(a^2 + b^2*x^2
+ 2*a*b*x)))/(280*b^5) - (a*d^3*e*(8*b^2*(a^2 + b^2*x^2) - 12*a^2*b^2 + 4*a*b^3*x)*(a^2 + b^2*x^2 + 2*a*b*x)^
(1/2))/(24*b^4) - (3*a^3*d^2*e^2*(x/2 + a/(2*b))*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/(2*b^2) - (7*a*d^2*e^2*(a^2
+ b^2*x^2 + 2*a*b*x)^(1/2)*(a^3 - 5*a*b^2*x^2 + 3*b*x*(a^2 + b^2*x^2 + 2*a*b*x) - 4*a^2*b*x))/(10*b^3) - (19*a
^2*d*e^3*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2)*(a^3 - 5*a*b^2*x^2 + 3*b*x*(a^2 + b^2*x^2 + 2*a*b*x) - 4*a^2*b*x))/(3
0*b^4) - (a^3*d*e^3*(8*b^2*(a^2 + b^2*x^2) - 12*a^2*b^2 + 4*a*b^3*x)*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/(15*b^6)
+ (3*a*d^2*e^2*x*(a^2 + b^2*x^2 + 2*a*b*x)^(3/2))/(2*b^2) + (4*a*d*e^3*x^2*(a^2 + b^2*x^2 + 2*a*b*x)^(3/2))/(
5*b^2) - (a^2*d^3*e*(x/2 + a/(2*b))*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/b - (3*a*d*e^3*(a^2 + b^2*x^2 + 2*a*b*x)^
(1/2)*(4*b^2*x^2*(a^2 + b^2*x^2 + 2*a*b*x) - a^4 + 9*a^2*b^2*x^2 + 8*a^3*b*x - 7*a*b*x*(a^2 + b^2*x^2 + 2*a*b*
x)))/(10*b^4) - (33*a^2*d^2*e^2*(8*b^2*(a^2 + b^2*x^2) - 12*a^2*b^2 + 4*a*b^3*x)*(a^2 + b^2*x^2 + 2*a*b*x)^(1/
2))/(80*b^5)

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sympy [A]  time = 0.16, size = 168, normalized size = 1.15 \begin {gather*} a^{2} d^{4} x + \frac {b^{2} e^{4} x^{7}}{7} + x^{6} \left (\frac {a b e^{4}}{3} + \frac {2 b^{2} d e^{3}}{3}\right ) + x^{5} \left (\frac {a^{2} e^{4}}{5} + \frac {8 a b d e^{3}}{5} + \frac {6 b^{2} d^{2} e^{2}}{5}\right ) + x^{4} \left (a^{2} d e^{3} + 3 a b d^{2} e^{2} + b^{2} d^{3} e\right ) + x^{3} \left (2 a^{2} d^{2} e^{2} + \frac {8 a b d^{3} e}{3} + \frac {b^{2} d^{4}}{3}\right ) + x^{2} \left (2 a^{2} d^{3} e + a b d^{4}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(e*x+d)**4*((b*x+a)**2)**(1/2),x)

[Out]

a**2*d**4*x + b**2*e**4*x**7/7 + x**6*(a*b*e**4/3 + 2*b**2*d*e**3/3) + x**5*(a**2*e**4/5 + 8*a*b*d*e**3/5 + 6*
b**2*d**2*e**2/5) + x**4*(a**2*d*e**3 + 3*a*b*d**2*e**2 + b**2*d**3*e) + x**3*(2*a**2*d**2*e**2 + 8*a*b*d**3*e
/3 + b**2*d**4/3) + x**2*(2*a**2*d**3*e + a*b*d**4)

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