3.8.9 \(\int x^{7/2} (A+B x) \sqrt {a^2+2 a b x+b^2 x^2} \, dx\)

Optimal. Leaf size=120 \[ \frac {2 x^{11/2} \sqrt {a^2+2 a b x+b^2 x^2} (a B+A b)}{11 (a+b x)}+\frac {2 a A x^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}}{9 (a+b x)}+\frac {2 b B x^{13/2} \sqrt {a^2+2 a b x+b^2 x^2}}{13 (a+b x)} \]

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Rubi [A]  time = 0.05, antiderivative size = 120, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {770, 76} \begin {gather*} \frac {2 x^{11/2} \sqrt {a^2+2 a b x+b^2 x^2} (a B+A b)}{11 (a+b x)}+\frac {2 a A x^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}}{9 (a+b x)}+\frac {2 b B x^{13/2} \sqrt {a^2+2 a b x+b^2 x^2}}{13 (a+b x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^(7/2)*(A + B*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2],x]

[Out]

(2*a*A*x^(9/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(9*(a + b*x)) + (2*(A*b + a*B)*x^(11/2)*Sqrt[a^2 + 2*a*b*x + b^2
*x^2])/(11*(a + b*x)) + (2*b*B*x^(13/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(13*(a + b*x))

Rule 76

Int[((d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_))*((e_) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*
x)*(d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && IGtQ[p, 0] && (NeQ[n, -1] || EqQ[p, 1]) && N
eQ[b*e + a*f, 0] && ( !IntegerQ[n] || LtQ[9*p + 5*n, 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && Rational
Q[a, b, d, e, f])) && (NeQ[n + p + 3, 0] || EqQ[p, 1])

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 x^{7/2} (A+B x) \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 x^{7/2} \left (a b+b^2 x\right ) (A+B x) \, dx}{a b+b^2 x}\\ &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (a A b x^{7/2}+b (A b+a B) x^{9/2}+b^2 B x^{11/2}\right ) \, dx}{a b+b^2 x}\\ &=\frac {2 a A x^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}}{9 (a+b x)}+\frac {2 (A b+a B) x^{11/2} \sqrt {a^2+2 a b x+b^2 x^2}}{11 (a+b x)}+\frac {2 b B x^{13/2} \sqrt {a^2+2 a b x+b^2 x^2}}{13 (a+b x)}\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 51, normalized size = 0.42 \begin {gather*} \frac {2 x^{9/2} \sqrt {(a+b x)^2} (13 a (11 A+9 B x)+9 b x (13 A+11 B x))}{1287 (a+b x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^(7/2)*(A + B*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2],x]

[Out]

(2*x^(9/2)*Sqrt[(a + b*x)^2]*(13*a*(11*A + 9*B*x) + 9*b*x*(13*A + 11*B*x)))/(1287*(a + b*x))

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IntegrateAlgebraic [A]  time = 9.97, size = 59, normalized size = 0.49 \begin {gather*} \frac {2 \sqrt {(a+b x)^2} \left (143 a A x^{9/2}+117 a B x^{11/2}+117 A b x^{11/2}+99 b B x^{13/2}\right )}{1287 (a+b x)} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[x^(7/2)*(A + B*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2],x]

[Out]

(2*Sqrt[(a + b*x)^2]*(143*a*A*x^(9/2) + 117*A*b*x^(11/2) + 117*a*B*x^(11/2) + 99*b*B*x^(13/2)))/(1287*(a + b*x
))

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fricas [A]  time = 0.42, size = 32, normalized size = 0.27 \begin {gather*} \frac {2}{1287} \, {\left (99 \, B b x^{6} + 143 \, A a x^{4} + 117 \, {\left (B a + A b\right )} x^{5}\right )} \sqrt {x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(7/2)*(B*x+A)*((b*x+a)^2)^(1/2),x, algorithm="fricas")

[Out]

2/1287*(99*B*b*x^6 + 143*A*a*x^4 + 117*(B*a + A*b)*x^5)*sqrt(x)

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giac [A]  time = 0.18, size = 53, normalized size = 0.44 \begin {gather*} \frac {2}{13} \, B b x^{\frac {13}{2}} \mathrm {sgn}\left (b x + a\right ) + \frac {2}{11} \, B a x^{\frac {11}{2}} \mathrm {sgn}\left (b x + a\right ) + \frac {2}{11} \, A b x^{\frac {11}{2}} \mathrm {sgn}\left (b x + a\right ) + \frac {2}{9} \, A a x^{\frac {9}{2}} \mathrm {sgn}\left (b x + a\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(7/2)*(B*x+A)*((b*x+a)^2)^(1/2),x, algorithm="giac")

[Out]

2/13*B*b*x^(13/2)*sgn(b*x + a) + 2/11*B*a*x^(11/2)*sgn(b*x + a) + 2/11*A*b*x^(11/2)*sgn(b*x + a) + 2/9*A*a*x^(
9/2)*sgn(b*x + a)

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maple [A]  time = 0.05, size = 44, normalized size = 0.37 \begin {gather*} \frac {2 \left (99 B b \,x^{2}+117 A b x +117 B a x +143 A a \right ) \sqrt {\left (b x +a \right )^{2}}\, x^{\frac {9}{2}}}{1287 \left (b x +a \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(7/2)*(B*x+A)*((b*x+a)^2)^(1/2),x)

[Out]

2/1287*x^(9/2)*(99*B*b*x^2+117*A*b*x+117*B*a*x+143*A*a)*((b*x+a)^2)^(1/2)/(b*x+a)

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maxima [A]  time = 0.54, size = 35, normalized size = 0.29 \begin {gather*} \frac {2}{143} \, {\left (11 \, b x^{2} + 13 \, a x\right )} B x^{\frac {9}{2}} + \frac {2}{99} \, {\left (9 \, b x^{2} + 11 \, a x\right )} A x^{\frac {7}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(7/2)*(B*x+A)*((b*x+a)^2)^(1/2),x, algorithm="maxima")

[Out]

2/143*(11*b*x^2 + 13*a*x)*B*x^(9/2) + 2/99*(9*b*x^2 + 11*a*x)*A*x^(7/2)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int x^{7/2}\,\sqrt {{\left (a+b\,x\right )}^2}\,\left (A+B\,x\right ) \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(7/2)*((a + b*x)^2)^(1/2)*(A + B*x),x)

[Out]

int(x^(7/2)*((a + b*x)^2)^(1/2)*(A + B*x), x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(7/2)*(B*x+A)*((b*x+a)**2)**(1/2),x)

[Out]

Timed out

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