3.18.72 \(\int (a+b x) (a^2+2 a b x+b^2 x^2)^{5/2} \, dx\)

Optimal. Leaf size=27 \[ \frac {\left (a^2+2 a b x+b^2 x^2\right )^{7/2}}{7 b} \]

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Rubi [A]  time = 0.01, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.038, Rules used = {629} \begin {gather*} \frac {\left (a^2+2 a b x+b^2 x^2\right )^{7/2}}{7 b} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(a^2 + 2*a*b*x + b^2*x^2)^(7/2)/(7*b)

Rule 629

Int[((d_) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d*(a + b*x + c*x^2)^(p +
1))/(b*(p + 1)), x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[2*c*d - b*e, 0] && NeQ[p, -1]

Rubi steps

\begin {align*} \int (a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx &=\frac {\left (a^2+2 a b x+b^2 x^2\right )^{7/2}}{7 b}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 25, normalized size = 0.93 \begin {gather*} \frac {(a+b x)^6 \sqrt {(a+b x)^2}}{7 b} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((a + b*x)^6*Sqrt[(a + b*x)^2])/(7*b)

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IntegrateAlgebraic [A]  time = 0.03, size = 18, normalized size = 0.67 \begin {gather*} \frac {\left ((a+b x)^2\right )^{7/2}}{7 b} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((a + b*x)^2)^(7/2)/(7*b)

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fricas [B]  time = 0.40, size = 64, normalized size = 2.37 \begin {gather*} \frac {1}{7} \, b^{6} x^{7} + a b^{5} x^{6} + 3 \, a^{2} b^{4} x^{5} + 5 \, a^{3} b^{3} x^{4} + 5 \, a^{4} b^{2} x^{3} + 3 \, a^{5} b x^{2} + a^{6} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*b^6*x^7 + a*b^5*x^6 + 3*a^2*b^4*x^5 + 5*a^3*b^3*x^4 + 5*a^4*b^2*x^3 + 3*a^5*b*x^2 + a^6*x

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giac [B]  time = 0.19, size = 120, normalized size = 4.44 \begin {gather*} \frac {1}{7} \, b^{6} x^{7} \mathrm {sgn}\left (b x + a\right ) + a b^{5} x^{6} \mathrm {sgn}\left (b x + a\right ) + 3 \, a^{2} b^{4} x^{5} \mathrm {sgn}\left (b x + a\right ) + 5 \, a^{3} b^{3} x^{4} \mathrm {sgn}\left (b x + a\right ) + 5 \, a^{4} b^{2} x^{3} \mathrm {sgn}\left (b x + a\right ) + 3 \, a^{5} b x^{2} \mathrm {sgn}\left (b x + a\right ) + a^{6} x \mathrm {sgn}\left (b x + a\right ) + \frac {a^{7} \mathrm {sgn}\left (b x + a\right )}{7 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.04, size = 82, normalized size = 3.04 \begin {gather*} \frac {\left (b^{6} x^{6}+7 a \,b^{5} x^{5}+21 a^{2} b^{4} x^{4}+35 a^{3} b^{3} x^{3}+35 a^{4} b^{2} x^{2}+21 a^{5} b x +7 a^{6}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}} x}{7 \left (b x +a \right )^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*x*(b^6*x^6+7*a*b^5*x^5+21*a^2*b^4*x^4+35*a^3*b^3*x^3+35*a^4*b^2*x^2+21*a^5*b*x+7*a^6)*((b*x+a)^2)^(5/2)/(b
*x+a)^5

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maxima [A]  time = 0.68, size = 23, normalized size = 0.85 \begin {gather*} \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}}}{7 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)/b

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mupad [B]  time = 2.24, size = 14, normalized size = 0.52 \begin {gather*} \frac {{\left ({\left (a+b\,x\right )}^2\right )}^{7/2}}{7\,b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((a + b*x)^2)^(7/2)/(7*b)

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sympy [A]  time = 5.52, size = 226, normalized size = 8.37 \begin {gather*} \begin {cases} \frac {a^{6} \sqrt {a^{2} + 2 a b x + b^{2} x^{2}}}{7 b} + \frac {6 a^{5} x \sqrt {a^{2} + 2 a b x + b^{2} x^{2}}}{7} + \frac {15 a^{4} b x^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2}}}{7} + \frac {20 a^{3} b^{2} x^{3} \sqrt {a^{2} + 2 a b x + b^{2} x^{2}}}{7} + \frac {15 a^{2} b^{3} x^{4} \sqrt {a^{2} + 2 a b x + b^{2} x^{2}}}{7} + \frac {6 a b^{4} x^{5} \sqrt {a^{2} + 2 a b x + b^{2} x^{2}}}{7} + \frac {b^{5} x^{6} \sqrt {a^{2} + 2 a b x + b^{2} x^{2}}}{7} & \text {for}\: b \neq 0 \\a x \left (a^{2}\right )^{\frac {5}{2}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a**6*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(7*b) + 6*a**5*x*sqrt(a**2 + 2*a*b*x + b**2*x**2)/7 + 15*a**4
*b*x**2*sqrt(a**2 + 2*a*b*x + b**2*x**2)/7 + 20*a**3*b**2*x**3*sqrt(a**2 + 2*a*b*x + b**2*x**2)/7 + 15*a**2*b*
*3*x**4*sqrt(a**2 + 2*a*b*x + b**2*x**2)/7 + 6*a*b**4*x**5*sqrt(a**2 + 2*a*b*x + b**2*x**2)/7 + b**5*x**6*sqrt
(a**2 + 2*a*b*x + b**2*x**2)/7, Ne(b, 0)), (a*x*(a**2)**(5/2), True))

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