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

Optimal. Leaf size=69 \[ \frac{(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2} (A b-a B)}{6 b^2}+\frac{B \left (a^2+2 a b x+b^2 x^2\right )^{7/2}}{7 b^2} \]

[Out]

((A*b - a*B)*(a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/(6*b^2) + (B*(a^2 + 2*a*b*x + b^2*x^2)^(7/2))/(7*b^2)

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Rubi [A]  time = 0.0213433, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {640, 609} \[ \frac{(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2} (A b-a B)}{6 b^2}+\frac{B \left (a^2+2 a b x+b^2 x^2\right )^{7/2}}{7 b^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((A*b - a*B)*(a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/(6*b^2) + (B*(a^2 + 2*a*b*x + b^2*x^2)^(7/2))/(7*b^2)

Rule 640

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

Rule 609

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((b + 2*c*x)*(a + b*x + c*x^2)^p)/(2*c*(2*p + 1
)), x] /; FreeQ[{a, b, c, p}, x] && EqQ[b^2 - 4*a*c, 0] && NeQ[p, -2^(-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{B \left (a^2+2 a b x+b^2 x^2\right )^{7/2}}{7 b^2}+\frac{\left (2 A b^2-2 a b B\right ) \int \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx}{2 b^2}\\ &=\frac{(A b-a B) (a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{6 b^2}+\frac{B \left (a^2+2 a b x+b^2 x^2\right )^{7/2}}{7 b^2}\\ \end{align*}

Mathematica [A]  time = 0.0444619, size = 121, normalized size = 1.75 \[ \frac{x \sqrt{(a+b x)^2} \left (35 a^3 b^2 x^2 (4 A+3 B x)+21 a^2 b^3 x^3 (5 A+4 B x)+35 a^4 b x (3 A+2 B x)+21 a^5 (2 A+B x)+7 a b^4 x^4 (6 A+5 B x)+b^5 x^5 (7 A+6 B x)\right )}{42 (a+b x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x*Sqrt[(a + b*x)^2]*(21*a^5*(2*A + B*x) + 35*a^4*b*x*(3*A + 2*B*x) + 35*a^3*b^2*x^2*(4*A + 3*B*x) + 21*a^2*b^
3*x^3*(5*A + 4*B*x) + 7*a*b^4*x^4*(6*A + 5*B*x) + b^5*x^5*(7*A + 6*B*x)))/(42*(a + b*x))

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Maple [B]  time = 0.004, size = 138, normalized size = 2. \begin{align*}{\frac{x \left ( 6\,B{b}^{5}{x}^{6}+7\,{x}^{5}A{b}^{5}+35\,{x}^{5}Ba{b}^{4}+42\,Aa{b}^{4}{x}^{4}+84\,B{a}^{2}{b}^{3}{x}^{4}+105\,{x}^{3}A{a}^{2}{b}^{3}+105\,{x}^{3}B{a}^{3}{b}^{2}+140\,{x}^{2}A{a}^{3}{b}^{2}+70\,{x}^{2}B{a}^{4}b+105\,xA{a}^{4}b+21\,xB{a}^{5}+42\,A{a}^{5} \right ) }{42\, \left ( bx+a \right ) ^{5}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \end{align*}

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]

1/42*x*(6*B*b^5*x^6+7*A*b^5*x^5+35*B*a*b^4*x^5+42*A*a*b^4*x^4+84*B*a^2*b^3*x^4+105*A*a^2*b^3*x^3+105*B*a^3*b^2
*x^3+140*A*a^3*b^2*x^2+70*B*a^4*b*x^2+105*A*a^4*b*x+21*B*a^5*x+42*A*a^5)*((b*x+a)^2)^(5/2)/(b*x+a)^5

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

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]

Exception raised: ValueError

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Fricas [A]  time = 1.28833, size = 247, normalized size = 3.58 \begin{align*} \frac{1}{7} \, B b^{5} x^{7} + A a^{5} x + \frac{1}{6} \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{6} +{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{5} + \frac{5}{2} \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{4} + \frac{5}{3} \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{3} + \frac{1}{2} \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x^{2} \end{align*}

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (A + B x\right ) \left (\left (a + b x\right )^{2}\right )^{\frac{5}{2}}\, dx \end{align*}

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]

Integral((A + B*x)*((a + b*x)**2)**(5/2), x)

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Giac [B]  time = 1.18094, size = 293, normalized size = 4.25 \begin{align*} \frac{1}{7} \, B b^{5} x^{7} \mathrm{sgn}\left (b x + a\right ) + \frac{5}{6} \, B a b^{4} x^{6} \mathrm{sgn}\left (b x + a\right ) + \frac{1}{6} \, A b^{5} x^{6} \mathrm{sgn}\left (b x + a\right ) + 2 \, B a^{2} b^{3} x^{5} \mathrm{sgn}\left (b x + a\right ) + A a b^{4} x^{5} \mathrm{sgn}\left (b x + a\right ) + \frac{5}{2} \, B a^{3} b^{2} x^{4} \mathrm{sgn}\left (b x + a\right ) + \frac{5}{2} \, A a^{2} b^{3} x^{4} \mathrm{sgn}\left (b x + a\right ) + \frac{5}{3} \, B a^{4} b x^{3} \mathrm{sgn}\left (b x + a\right ) + \frac{10}{3} \, A a^{3} b^{2} x^{3} \mathrm{sgn}\left (b x + a\right ) + \frac{1}{2} \, B a^{5} x^{2} \mathrm{sgn}\left (b x + a\right ) + \frac{5}{2} \, A a^{4} b x^{2} \mathrm{sgn}\left (b x + a\right ) + A a^{5} x \mathrm{sgn}\left (b x + a\right ) - \frac{{\left (B a^{7} - 7 \, A a^{6} b\right )} \mathrm{sgn}\left (b x + a\right )}{42 \, b^{2}} \end{align*}

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*b^5*x^7*sgn(b*x + a) + 5/6*B*a*b^4*x^6*sgn(b*x + a) + 1/6*A*b^5*x^6*sgn(b*x + a) + 2*B*a^2*b^3*x^5*sgn(b
*x + a) + A*a*b^4*x^5*sgn(b*x + a) + 5/2*B*a^3*b^2*x^4*sgn(b*x + a) + 5/2*A*a^2*b^3*x^4*sgn(b*x + a) + 5/3*B*a
^4*b*x^3*sgn(b*x + a) + 10/3*A*a^3*b^2*x^3*sgn(b*x + a) + 1/2*B*a^5*x^2*sgn(b*x + a) + 5/2*A*a^4*b*x^2*sgn(b*x
+ a) + A*a^5*x*sgn(b*x + a) - 1/42*(B*a^7 - 7*A*a^6*b)*sgn(b*x + a)/b^2