3.18.29 \(\int (a+b x) (d+e x)^5 \sqrt {a^2+2 a b x+b^2 x^2} \, dx\)

Optimal. Leaf size=146 \[ \frac {b^2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^8}{8 e^3 (a+b x)}-\frac {2 b \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^7 (b d-a e)}{7 e^3 (a+b x)}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^6 (b d-a e)^2}{6 e^3 (a+b x)} \]

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Rubi [A]  time = 0.15, 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)^8}{8 e^3 (a+b x)}-\frac {2 b \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^7 (b d-a e)}{7 e^3 (a+b x)}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^6 (b d-a e)^2}{6 e^3 (a+b x)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((b*d - a*e)^2*(d + e*x)^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(6*e^3*(a + b*x)) - (2*b*(b*d - a*e)*(d + e*x)^7*Sqr
t[a^2 + 2*a*b*x + b^2*x^2])/(7*e^3*(a + b*x)) + (b^2*(d + e*x)^8*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(8*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)^5 \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)^5 \, 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)^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)^2 (d+e x)^5}{e^2}-\frac {2 b (b d-a e) (d+e x)^6}{e^2}+\frac {b^2 (d+e x)^7}{e^2}\right ) \, dx}{a b+b^2 x}\\ &=\frac {(b d-a e)^2 (d+e x)^6 \sqrt {a^2+2 a b x+b^2 x^2}}{6 e^3 (a+b x)}-\frac {2 b (b d-a e) (d+e x)^7 \sqrt {a^2+2 a b x+b^2 x^2}}{7 e^3 (a+b x)}+\frac {b^2 (d+e x)^8 \sqrt {a^2+2 a b x+b^2 x^2}}{8 e^3 (a+b x)}\\ \end {align*}

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Mathematica [A]  time = 0.06, size = 196, normalized size = 1.34 \begin {gather*} \frac {x \sqrt {(a+b x)^2} \left (28 a^2 \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 )+8 a 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 )+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 )\right )}{168 (a+b x)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(x*Sqrt[(a + b*x)^2]*(28*a^2*(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) +
8*a*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) + 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)))/(168*(a + b*x))

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IntegrateAlgebraic [F]  time = 2.24, size = 0, normalized size = 0.00 \begin {gather*} \int (a+b x) (d+e x)^5 \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)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2],x]

[Out]

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

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fricas [A]  time = 0.41, size = 197, normalized size = 1.35 \begin {gather*} \frac {1}{8} \, b^{2} e^{5} x^{8} + a^{2} d^{5} x + \frac {1}{7} \, {\left (5 \, b^{2} d e^{4} + 2 \, a b e^{5}\right )} x^{7} + \frac {1}{6} \, {\left (10 \, b^{2} d^{2} e^{3} + 10 \, a b d e^{4} + a^{2} e^{5}\right )} x^{6} + {\left (2 \, b^{2} d^{3} e^{2} + 4 \, a b d^{2} e^{3} + a^{2} d e^{4}\right )} x^{5} + \frac {5}{4} \, {\left (b^{2} d^{4} e + 4 \, a b d^{3} e^{2} + 2 \, a^{2} d^{2} e^{3}\right )} x^{4} + \frac {1}{3} \, {\left (b^{2} d^{5} + 10 \, a b d^{4} e + 10 \, a^{2} d^{3} e^{2}\right )} x^{3} + \frac {1}{2} \, {\left (2 \, a b d^{5} + 5 \, a^{2} d^{4} e\right )} x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(e*x+d)^5*((b*x+a)^2)^(1/2),x, algorithm="fricas")

[Out]

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

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giac [B]  time = 0.17, size = 311, normalized size = 2.13 \begin {gather*} \frac {1}{8} \, b^{2} x^{8} e^{5} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{7} \, b^{2} d x^{7} e^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{3} \, b^{2} d^{2} x^{6} e^{3} \mathrm {sgn}\left (b x + a\right ) + 2 \, b^{2} d^{3} x^{5} e^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{4} \, b^{2} d^{4} x^{4} e \mathrm {sgn}\left (b x + a\right ) + \frac {1}{3} \, b^{2} d^{5} x^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {2}{7} \, a b x^{7} e^{5} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{3} \, a b d x^{6} e^{4} \mathrm {sgn}\left (b x + a\right ) + 4 \, a b d^{2} x^{5} e^{3} \mathrm {sgn}\left (b x + a\right ) + 5 \, a b d^{3} x^{4} e^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {10}{3} \, a b d^{4} x^{3} e \mathrm {sgn}\left (b x + a\right ) + a b d^{5} x^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{6} \, a^{2} x^{6} e^{5} \mathrm {sgn}\left (b x + a\right ) + a^{2} d x^{5} e^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{2} \, a^{2} d^{2} x^{4} e^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {10}{3} \, a^{2} d^{3} x^{3} e^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{2} \, a^{2} d^{4} x^{2} e \mathrm {sgn}\left (b x + a\right ) + a^{2} 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*x+a)^2)^(1/2),x, algorithm="giac")

[Out]

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

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maple [B]  time = 0.05, size = 230, normalized size = 1.58 \begin {gather*} \frac {\left (21 b^{2} e^{5} x^{7}+48 x^{6} a b \,e^{5}+120 x^{6} b^{2} d \,e^{4}+28 x^{5} a^{2} e^{5}+280 x^{5} a b d \,e^{4}+280 x^{5} b^{2} d^{2} e^{3}+168 a^{2} d \,e^{4} x^{4}+672 a b \,d^{2} e^{3} x^{4}+336 b^{2} d^{3} e^{2} x^{4}+420 x^{3} a^{2} d^{2} e^{3}+840 x^{3} a b \,d^{3} e^{2}+210 x^{3} b^{2} d^{4} e +560 x^{2} a^{2} d^{3} e^{2}+560 x^{2} a b \,d^{4} e +56 x^{2} b^{2} d^{5}+420 x \,a^{2} d^{4} e +168 x a b \,d^{5}+168 a^{2} d^{5}\right ) \sqrt {\left (b x +a \right )^{2}}\, x}{168 b x +168 a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)*(e*x+d)^5*((b*x+a)^2)^(1/2),x)

[Out]

1/168*x*(21*b^2*e^5*x^7+48*a*b*e^5*x^6+120*b^2*d*e^4*x^6+28*a^2*e^5*x^5+280*a*b*d*e^4*x^5+280*b^2*d^2*e^3*x^5+
168*a^2*d*e^4*x^4+672*a*b*d^2*e^3*x^4+336*b^2*d^3*e^2*x^4+420*a^2*d^2*e^3*x^3+840*a*b*d^3*e^2*x^3+210*b^2*d^4*
e*x^3+560*a^2*d^3*e^2*x^2+560*a*b*d^4*e*x^2+56*b^2*d^5*x^2+420*a^2*d^4*e*x+168*a*b*d^5*x+168*a^2*d^5)*((b*x+a)
^2)^(1/2)/(b*x+a)

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maxima [B]  time = 0.58, size = 1323, normalized size = 9.06

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(e*x+d)^5*((b*x+a)^2)^(1/2),x, algorithm="maxima")

[Out]

1/8*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*e^5*x^5/b - 13/56*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*a*e^5*x^4/b^2 + 9/28*(b^
2*x^2 + 2*a*b*x + a^2)^(3/2)*a^2*e^5*x^3/b^3 + 1/2*sqrt(b^2*x^2 + 2*a*b*x + a^2)*a*d^5*x + 1/2*sqrt(b^2*x^2 +
2*a*b*x + a^2)*a^6*e^5*x/b^5 - 11/28*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*a^3*e^5*x^2/b^4 + 1/2*sqrt(b^2*x^2 + 2*a*
b*x + a^2)*a^2*d^5/b + 1/2*sqrt(b^2*x^2 + 2*a*b*x + a^2)*a^7*e^5/b^6 + 25/56*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*a
^4*e^5*x/b^5 - 27/56*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*a^5*e^5/b^6 + 1/7*(5*b*d*e^4 + a*e^5)*(b^2*x^2 + 2*a*b*x
+ a^2)^(3/2)*x^4/b^2 - 11/42*(5*b*d*e^4 + a*e^5)*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*a*x^3/b^3 + 5/6*(2*b*d^2*e^3
+ a*d*e^4)*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*x^3/b^2 - 1/2*(5*b*d*e^4 + a*e^5)*sqrt(b^2*x^2 + 2*a*b*x + a^2)*a^5
*x/b^5 + 5/2*(2*b*d^2*e^3 + a*d*e^4)*sqrt(b^2*x^2 + 2*a*b*x + a^2)*a^4*x/b^4 - 5*(b*d^3*e^2 + a*d^2*e^3)*sqrt(
b^2*x^2 + 2*a*b*x + a^2)*a^3*x/b^3 + 5/2*(b*d^4*e + 2*a*d^3*e^2)*sqrt(b^2*x^2 + 2*a*b*x + a^2)*a^2*x/b^2 - 1/2
*(b*d^5 + 5*a*d^4*e)*sqrt(b^2*x^2 + 2*a*b*x + a^2)*a*x/b + 5/14*(5*b*d*e^4 + a*e^5)*(b^2*x^2 + 2*a*b*x + a^2)^
(3/2)*a^2*x^2/b^4 - 3/2*(2*b*d^2*e^3 + a*d*e^4)*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*a*x^2/b^3 + 2*(b*d^3*e^2 + a*d
^2*e^3)*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*x^2/b^2 - 1/2*(5*b*d*e^4 + a*e^5)*sqrt(b^2*x^2 + 2*a*b*x + a^2)*a^6/b^
6 + 5/2*(2*b*d^2*e^3 + a*d*e^4)*sqrt(b^2*x^2 + 2*a*b*x + a^2)*a^5/b^5 - 5*(b*d^3*e^2 + a*d^2*e^3)*sqrt(b^2*x^2
+ 2*a*b*x + a^2)*a^4/b^4 + 5/2*(b*d^4*e + 2*a*d^3*e^2)*sqrt(b^2*x^2 + 2*a*b*x + a^2)*a^3/b^3 - 1/2*(b*d^5 + 5
*a*d^4*e)*sqrt(b^2*x^2 + 2*a*b*x + a^2)*a^2/b^2 - 3/7*(5*b*d*e^4 + a*e^5)*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*a^3*
x/b^5 + 2*(2*b*d^2*e^3 + a*d*e^4)*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*a^2*x/b^4 - 7/2*(b*d^3*e^2 + a*d^2*e^3)*(b^2
*x^2 + 2*a*b*x + a^2)^(3/2)*a*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)*x/b^2 + 10/2
1*(5*b*d*e^4 + a*e^5)*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*a^4/b^6 - 7/3*(2*b*d^2*e^3 + a*d*e^4)*(b^2*x^2 + 2*a*b*x
+ a^2)^(3/2)*a^3/b^5 + 9/2*(b*d^3*e^2 + a*d^2*e^3)*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*a^2/b^4 - 25/12*(b*d^4*e +
2*a*d^3*e^2)*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*a/b^3 + 1/3*(b*d^5 + 5*a*d^4*e)*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)/
b^2

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mupad [B]  time = 4.77, size = 1541, normalized size = 10.55

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a + b*x)^2)^(1/2)*(a + b*x)*(d + e*x)^5,x)

[Out]

a*d^5*(x/2 + a/(2*b))*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2) + (d^5*(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^5*x^5*(a^2 + b^2*x^2 + 2*a*b*x)^(3/2))/(8*b) + (a*e^5*x^4*(a^2
+ b^2*x^2 + 2*a*b*x)^(3/2))/(7*b^2) + (5*d*e^4*x^4*(a^2 + b^2*x^2 + 2*a*b*x)^(3/2))/(7*b) - (13*a*e^5*(a^2 + b
^2*x^2 + 2*a*b*x)^(1/2)*(6*b^4*x^4*(a^2 + b^2*x^2 + 2*a*b*x) - a^6 + 20*a^4*b^2*x^2 + 19*a^5*b*x - 11*a*b^3*x^
3*(a^2 + b^2*x^2 + 2*a*b*x) + 15*a^2*b^2*x^2*(a^2 + b^2*x^2 + 2*a*b*x) - 18*a^3*b*x*(a^2 + b^2*x^2 + 2*a*b*x))
)/(336*b^6) + (2*d^3*e^2*x^2*(a^2 + b^2*x^2 + 2*a*b*x)^(3/2))/b + (5*d^2*e^3*x^3*(a^2 + b^2*x^2 + 2*a*b*x)^(3/
2))/(3*b) + (5*d^4*e*x*(a^2 + b^2*x^2 + 2*a*b*x)^(3/2))/(4*b) - (a^3*e^5*(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)))/(35*b^
6) - (41*a^2*e^5*(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)))/(560*b^6) - (5*a*
d^4*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))/(96*b^4) - (5*a^3*d^3*
e^2*(x/2 + a/(2*b))*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/(2*b^2) - (3*a*d^2*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)))/(4*
b^4) - (29*a^2*d*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)))/(56*b^5) - (7*a*d^3*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))/(6*b^3) - (5*a^3*d*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) + (5*a*d^3*e^2*x*(a^2 +
b^2*x^2 + 2*a*b*x)^(3/2))/(2*b^2) + (5*a*d*e^4*x^3*(a^2 + b^2*x^2 + 2*a*b*x)^(3/2))/(6*b^2) - (5*a^2*d^4*e*(x/
2 + a/(2*b))*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/(4*b) - (19*a^2*d^2*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))/(12*b^4) - (11*a^2*d^3*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))/(16*b^5) - (a^3*d^2*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))/(6*b^6) - (11*a*d*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)))/(42*b^5) + (2*a*d^2*e^3*x^2*(a^2 + b^2*x^2 + 2*a*b*x)^(3/2))/b^2

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sympy [B]  time = 0.16, size = 218, normalized size = 1.49 \begin {gather*} a^{2} d^{5} x + \frac {b^{2} e^{5} x^{8}}{8} + x^{7} \left (\frac {2 a b e^{5}}{7} + \frac {5 b^{2} d e^{4}}{7}\right ) + x^{6} \left (\frac {a^{2} e^{5}}{6} + \frac {5 a b d e^{4}}{3} + \frac {5 b^{2} d^{2} e^{3}}{3}\right ) + x^{5} \left (a^{2} d e^{4} + 4 a b d^{2} e^{3} + 2 b^{2} d^{3} e^{2}\right ) + x^{4} \left (\frac {5 a^{2} d^{2} e^{3}}{2} + 5 a b d^{3} e^{2} + \frac {5 b^{2} d^{4} e}{4}\right ) + x^{3} \left (\frac {10 a^{2} d^{3} e^{2}}{3} + \frac {10 a b d^{4} e}{3} + \frac {b^{2} d^{5}}{3}\right ) + x^{2} \left (\frac {5 a^{2} d^{4} e}{2} + a b d^{5}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(e*x+d)**5*((b*x+a)**2)**(1/2),x)

[Out]

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

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