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

Optimal. Leaf size=320 \[ \frac {20 a^2 b^2 x^{13/2} \sqrt {a^2+2 a b x+b^2 x^2} (a B+A b)}{13 (a+b x)}+\frac {2 b^4 x^{17/2} \sqrt {a^2+2 a b x+b^2 x^2} (5 a B+A b)}{17 (a+b x)}+\frac {2 a b^3 x^{15/2} \sqrt {a^2+2 a b x+b^2 x^2} (2 a B+A b)}{3 (a+b x)}+\frac {2 b^5 B x^{19/2} \sqrt {a^2+2 a b x+b^2 x^2}}{19 (a+b x)}+\frac {2 a^5 A x^{7/2} \sqrt {a^2+2 a b x+b^2 x^2}}{7 (a+b x)}+\frac {2 a^4 x^{9/2} \sqrt {a^2+2 a b x+b^2 x^2} (a B+5 A b)}{9 (a+b x)}+\frac {10 a^3 b x^{11/2} \sqrt {a^2+2 a b x+b^2 x^2} (a B+2 A b)}{11 (a+b x)} \]

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Rubi [A]  time = 0.12, antiderivative size = 320, 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 b^4 x^{17/2} \sqrt {a^2+2 a b x+b^2 x^2} (5 a B+A b)}{17 (a+b x)}+\frac {2 a b^3 x^{15/2} \sqrt {a^2+2 a b x+b^2 x^2} (2 a B+A b)}{3 (a+b x)}+\frac {20 a^2 b^2 x^{13/2} \sqrt {a^2+2 a b x+b^2 x^2} (a B+A b)}{13 (a+b x)}+\frac {10 a^3 b x^{11/2} \sqrt {a^2+2 a b x+b^2 x^2} (a B+2 A b)}{11 (a+b x)}+\frac {2 a^4 x^{9/2} \sqrt {a^2+2 a b x+b^2 x^2} (a B+5 A b)}{9 (a+b x)}+\frac {2 a^5 A x^{7/2} \sqrt {a^2+2 a b x+b^2 x^2}}{7 (a+b x)}+\frac {2 b^5 B x^{19/2} \sqrt {a^2+2 a b x+b^2 x^2}}{19 (a+b x)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*a^5*A*x^(7/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*(a + b*x)) + (2*a^4*(5*A*b + a*B)*x^(9/2)*Sqrt[a^2 + 2*a*b*
x + b^2*x^2])/(9*(a + b*x)) + (10*a^3*b*(2*A*b + a*B)*x^(11/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(11*(a + b*x)) +
(20*a^2*b^2*(A*b + a*B)*x^(13/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(13*(a + b*x)) + (2*a*b^3*(A*b + 2*a*B)*x^(15
/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*(a + b*x)) + (2*b^4*(A*b + 5*a*B)*x^(17/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]
)/(17*(a + b*x)) + (2*b^5*B*x^(19/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(19*(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^{5/2} (A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int x^{5/2} \left (a b+b^2 x\right )^5 (A+B x) \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (a^5 A b^5 x^{5/2}+a^4 b^5 (5 A b+a B) x^{7/2}+5 a^3 b^6 (2 A b+a B) x^{9/2}+10 a^2 b^7 (A b+a B) x^{11/2}+5 a b^8 (A b+2 a B) x^{13/2}+b^9 (A b+5 a B) x^{15/2}+b^{10} B x^{17/2}\right ) \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=\frac {2 a^5 A x^{7/2} \sqrt {a^2+2 a b x+b^2 x^2}}{7 (a+b x)}+\frac {2 a^4 (5 A b+a B) x^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}}{9 (a+b x)}+\frac {10 a^3 b (2 A b+a B) x^{11/2} \sqrt {a^2+2 a b x+b^2 x^2}}{11 (a+b x)}+\frac {20 a^2 b^2 (A b+a B) x^{13/2} \sqrt {a^2+2 a b x+b^2 x^2}}{13 (a+b x)}+\frac {2 a b^3 (A b+2 a B) x^{15/2} \sqrt {a^2+2 a b x+b^2 x^2}}{3 (a+b x)}+\frac {2 b^4 (A b+5 a B) x^{17/2} \sqrt {a^2+2 a b x+b^2 x^2}}{17 (a+b x)}+\frac {2 b^5 B x^{19/2} \sqrt {a^2+2 a b x+b^2 x^2}}{19 (a+b x)}\\ \end {align*}

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Mathematica [A]  time = 0.08, size = 110, normalized size = 0.34 \begin {gather*} \frac {2 \sqrt {(a+b x)^2} \left (\frac {x^{7/2} \left (21879 a^5+85085 a^4 b x+139230 a^3 b^2 x^2+117810 a^2 b^3 x^3+51051 a b^4 x^4+9009 b^5 x^5\right ) (19 A b-7 a B)}{153153}+B x^{7/2} (a+b x)^6\right )}{19 b (a+b x)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[(a + b*x)^2]*(B*x^(7/2)*(a + b*x)^6 + ((19*A*b - 7*a*B)*x^(7/2)*(21879*a^5 + 85085*a^4*b*x + 139230*a^
3*b^2*x^2 + 117810*a^2*b^3*x^3 + 51051*a*b^4*x^4 + 9009*b^5*x^5))/153153))/(19*b*(a + b*x))

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IntegrateAlgebraic [A]  time = 16.13, size = 171, normalized size = 0.53 \begin {gather*} \frac {2 \sqrt {(a+b x)^2} \left (415701 a^5 A x^{7/2}+323323 a^5 B x^{9/2}+1616615 a^4 A b x^{9/2}+1322685 a^4 b B x^{11/2}+2645370 a^3 A b^2 x^{11/2}+2238390 a^3 b^2 B x^{13/2}+2238390 a^2 A b^3 x^{13/2}+1939938 a^2 b^3 B x^{15/2}+969969 a A b^4 x^{15/2}+855855 a b^4 B x^{17/2}+171171 A b^5 x^{17/2}+153153 b^5 B x^{19/2}\right )}{2909907 (a+b x)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[(a + b*x)^2]*(415701*a^5*A*x^(7/2) + 1616615*a^4*A*b*x^(9/2) + 323323*a^5*B*x^(9/2) + 2645370*a^3*A*b^
2*x^(11/2) + 1322685*a^4*b*B*x^(11/2) + 2238390*a^2*A*b^3*x^(13/2) + 2238390*a^3*b^2*B*x^(13/2) + 969969*a*A*b
^4*x^(15/2) + 1939938*a^2*b^3*B*x^(15/2) + 171171*A*b^5*x^(17/2) + 855855*a*b^4*B*x^(17/2) + 153153*b^5*B*x^(1
9/2)))/(2909907*(a + b*x))

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fricas [A]  time = 0.43, size = 124, normalized size = 0.39 \begin {gather*} \frac {2}{2909907} \, {\left (153153 \, B b^{5} x^{9} + 415701 \, A a^{5} x^{3} + 171171 \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{8} + 969969 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{7} + 2238390 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{6} + 1322685 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{5} + 323323 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x^{4}\right )} \sqrt {x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/2909907*(153153*B*b^5*x^9 + 415701*A*a^5*x^3 + 171171*(5*B*a*b^4 + A*b^5)*x^8 + 969969*(2*B*a^2*b^3 + A*a*b^
4)*x^7 + 2238390*(B*a^3*b^2 + A*a^2*b^3)*x^6 + 1322685*(B*a^4*b + 2*A*a^3*b^2)*x^5 + 323323*(B*a^5 + 5*A*a^4*b
)*x^4)*sqrt(x)

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giac [A]  time = 0.19, size = 197, normalized size = 0.62 \begin {gather*} \frac {2}{19} \, B b^{5} x^{\frac {19}{2}} \mathrm {sgn}\left (b x + a\right ) + \frac {10}{17} \, B a b^{4} x^{\frac {17}{2}} \mathrm {sgn}\left (b x + a\right ) + \frac {2}{17} \, A b^{5} x^{\frac {17}{2}} \mathrm {sgn}\left (b x + a\right ) + \frac {4}{3} \, B a^{2} b^{3} x^{\frac {15}{2}} \mathrm {sgn}\left (b x + a\right ) + \frac {2}{3} \, A a b^{4} x^{\frac {15}{2}} \mathrm {sgn}\left (b x + a\right ) + \frac {20}{13} \, B a^{3} b^{2} x^{\frac {13}{2}} \mathrm {sgn}\left (b x + a\right ) + \frac {20}{13} \, A a^{2} b^{3} x^{\frac {13}{2}} \mathrm {sgn}\left (b x + a\right ) + \frac {10}{11} \, B a^{4} b x^{\frac {11}{2}} \mathrm {sgn}\left (b x + a\right ) + \frac {20}{11} \, A a^{3} b^{2} x^{\frac {11}{2}} \mathrm {sgn}\left (b x + a\right ) + \frac {2}{9} \, B a^{5} x^{\frac {9}{2}} \mathrm {sgn}\left (b x + a\right ) + \frac {10}{9} \, A a^{4} b x^{\frac {9}{2}} \mathrm {sgn}\left (b x + a\right ) + \frac {2}{7} \, A a^{5} x^{\frac {7}{2}} \mathrm {sgn}\left (b x + a\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/19*B*b^5*x^(19/2)*sgn(b*x + a) + 10/17*B*a*b^4*x^(17/2)*sgn(b*x + a) + 2/17*A*b^5*x^(17/2)*sgn(b*x + a) + 4/
3*B*a^2*b^3*x^(15/2)*sgn(b*x + a) + 2/3*A*a*b^4*x^(15/2)*sgn(b*x + a) + 20/13*B*a^3*b^2*x^(13/2)*sgn(b*x + a)
+ 20/13*A*a^2*b^3*x^(13/2)*sgn(b*x + a) + 10/11*B*a^4*b*x^(11/2)*sgn(b*x + a) + 20/11*A*a^3*b^2*x^(11/2)*sgn(b
*x + a) + 2/9*B*a^5*x^(9/2)*sgn(b*x + a) + 10/9*A*a^4*b*x^(9/2)*sgn(b*x + a) + 2/7*A*a^5*x^(7/2)*sgn(b*x + a)

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maple [A]  time = 0.05, size = 140, normalized size = 0.44 \begin {gather*} \frac {2 \left (153153 B \,b^{5} x^{6}+171171 A \,b^{5} x^{5}+855855 B a \,b^{4} x^{5}+969969 A a \,b^{4} x^{4}+1939938 B \,a^{2} b^{3} x^{4}+2238390 A \,a^{2} b^{3} x^{3}+2238390 B \,a^{3} b^{2} x^{3}+2645370 A \,a^{3} b^{2} x^{2}+1322685 B \,a^{4} b \,x^{2}+1616615 A \,a^{4} b x +323323 B \,a^{5} x +415701 A \,a^{5}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}} x^{\frac {7}{2}}}{2909907 \left (b x +a \right )^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/2909907*x^(7/2)*(153153*B*b^5*x^6+171171*A*b^5*x^5+855855*B*a*b^4*x^5+969969*A*a*b^4*x^4+1939938*B*a^2*b^3*x
^4+2238390*A*a^2*b^3*x^3+2238390*B*a^3*b^2*x^3+2645370*A*a^3*b^2*x^2+1322685*B*a^4*b*x^2+1616615*A*a^4*b*x+323
323*B*a^5*x+415701*A*a^5)*((b*x+a)^2)^(5/2)/(b*x+a)^5

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maxima [A]  time = 0.73, size = 241, normalized size = 0.75 \begin {gather*} \frac {2}{765765} \, {\left (3003 \, {\left (15 \, b^{5} x^{2} + 17 \, a b^{4} x\right )} x^{\frac {13}{2}} + 15708 \, {\left (13 \, a b^{4} x^{2} + 15 \, a^{2} b^{3} x\right )} x^{\frac {11}{2}} + 32130 \, {\left (11 \, a^{2} b^{3} x^{2} + 13 \, a^{3} b^{2} x\right )} x^{\frac {9}{2}} + 30940 \, {\left (9 \, a^{3} b^{2} x^{2} + 11 \, a^{4} b x\right )} x^{\frac {7}{2}} + 12155 \, {\left (7 \, a^{4} b x^{2} + 9 \, a^{5} x\right )} x^{\frac {5}{2}}\right )} A + \frac {2}{2078505} \, {\left (6435 \, {\left (17 \, b^{5} x^{2} + 19 \, a b^{4} x\right )} x^{\frac {15}{2}} + 32604 \, {\left (15 \, a b^{4} x^{2} + 17 \, a^{2} b^{3} x\right )} x^{\frac {13}{2}} + 63954 \, {\left (13 \, a^{2} b^{3} x^{2} + 15 \, a^{3} b^{2} x\right )} x^{\frac {11}{2}} + 58140 \, {\left (11 \, a^{3} b^{2} x^{2} + 13 \, a^{4} b x\right )} x^{\frac {9}{2}} + 20995 \, {\left (9 \, a^{4} b x^{2} + 11 \, a^{5} x\right )} x^{\frac {7}{2}}\right )} B \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/765765*(3003*(15*b^5*x^2 + 17*a*b^4*x)*x^(13/2) + 15708*(13*a*b^4*x^2 + 15*a^2*b^3*x)*x^(11/2) + 32130*(11*a
^2*b^3*x^2 + 13*a^3*b^2*x)*x^(9/2) + 30940*(9*a^3*b^2*x^2 + 11*a^4*b*x)*x^(7/2) + 12155*(7*a^4*b*x^2 + 9*a^5*x
)*x^(5/2))*A + 2/2078505*(6435*(17*b^5*x^2 + 19*a*b^4*x)*x^(15/2) + 32604*(15*a*b^4*x^2 + 17*a^2*b^3*x)*x^(13/
2) + 63954*(13*a^2*b^3*x^2 + 15*a^3*b^2*x)*x^(11/2) + 58140*(11*a^3*b^2*x^2 + 13*a^4*b*x)*x^(9/2) + 20995*(9*a
^4*b*x^2 + 11*a^5*x)*x^(7/2))*B

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int x^{5/2}\,\left (A+B\,x\right )\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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