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

Optimal. Leaf size=77 \[ \frac {2}{7} a^2 A x^{7/2}+\frac {2}{9} a^2 B x^{9/2}+\frac {4}{11} a A c x^{11/2}+\frac {4}{13} a B c x^{13/2}+\frac {2}{15} A c^2 x^{15/2}+\frac {2}{17} B c^2 x^{17/2} \]

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Rubi [A]  time = 0.03, antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {766} \begin {gather*} \frac {2}{7} a^2 A x^{7/2}+\frac {2}{9} a^2 B x^{9/2}+\frac {4}{11} a A c x^{11/2}+\frac {4}{13} a B c x^{13/2}+\frac {2}{15} A c^2 x^{15/2}+\frac {2}{17} B c^2 x^{17/2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*a^2*A*x^(7/2))/7 + (2*a^2*B*x^(9/2))/9 + (4*a*A*c*x^(11/2))/11 + (4*a*B*c*x^(13/2))/13 + (2*A*c^2*x^(15/2))
/15 + (2*B*c^2*x^(17/2))/17

Rule 766

Int[((e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(e*x
)^m*(f + g*x)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, e, f, g, m}, x] && IGtQ[p, 0]

Rubi steps

\begin {align*} \int x^{5/2} (A+B x) \left (a+c x^2\right )^2 \, dx &=\int \left (a^2 A x^{5/2}+a^2 B x^{7/2}+2 a A c x^{9/2}+2 a B c x^{11/2}+A c^2 x^{13/2}+B c^2 x^{15/2}\right ) \, dx\\ &=\frac {2}{7} a^2 A x^{7/2}+\frac {2}{9} a^2 B x^{9/2}+\frac {4}{11} a A c x^{11/2}+\frac {4}{13} a B c x^{13/2}+\frac {2}{15} A c^2 x^{15/2}+\frac {2}{17} B c^2 x^{17/2}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 60, normalized size = 0.78 \begin {gather*} \frac {2}{63} a^2 x^{7/2} (9 A+7 B x)+\frac {4}{143} a c x^{11/2} (13 A+11 B x)+\frac {2}{255} c^2 x^{15/2} (17 A+15 B x) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*a^2*x^(7/2)*(9*A + 7*B*x))/63 + (4*a*c*x^(11/2)*(13*A + 11*B*x))/143 + (2*c^2*x^(15/2)*(17*A + 15*B*x))/255

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IntegrateAlgebraic [A]  time = 0.03, size = 69, normalized size = 0.90 \begin {gather*} \frac {2 \left (109395 a^2 A x^{7/2}+85085 a^2 B x^{9/2}+139230 a A c x^{11/2}+117810 a B c x^{13/2}+51051 A c^2 x^{15/2}+45045 B c^2 x^{17/2}\right )}{765765} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(109395*a^2*A*x^(7/2) + 85085*a^2*B*x^(9/2) + 139230*a*A*c*x^(11/2) + 117810*a*B*c*x^(13/2) + 51051*A*c^2*x
^(15/2) + 45045*B*c^2*x^(17/2)))/765765

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fricas [A]  time = 0.41, size = 58, normalized size = 0.75 \begin {gather*} \frac {2}{765765} \, {\left (45045 \, B c^{2} x^{8} + 51051 \, A c^{2} x^{7} + 117810 \, B a c x^{6} + 139230 \, A a c x^{5} + 85085 \, B a^{2} x^{4} + 109395 \, A a^{2} x^{3}\right )} \sqrt {x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/765765*(45045*B*c^2*x^8 + 51051*A*c^2*x^7 + 117810*B*a*c*x^6 + 139230*A*a*c*x^5 + 85085*B*a^2*x^4 + 109395*A
*a^2*x^3)*sqrt(x)

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giac [A]  time = 0.15, size = 53, normalized size = 0.69 \begin {gather*} \frac {2}{17} \, B c^{2} x^{\frac {17}{2}} + \frac {2}{15} \, A c^{2} x^{\frac {15}{2}} + \frac {4}{13} \, B a c x^{\frac {13}{2}} + \frac {4}{11} \, A a c x^{\frac {11}{2}} + \frac {2}{9} \, B a^{2} x^{\frac {9}{2}} + \frac {2}{7} \, A a^{2} x^{\frac {7}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/17*B*c^2*x^(17/2) + 2/15*A*c^2*x^(15/2) + 4/13*B*a*c*x^(13/2) + 4/11*A*a*c*x^(11/2) + 2/9*B*a^2*x^(9/2) + 2/
7*A*a^2*x^(7/2)

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maple [A]  time = 0.05, size = 54, normalized size = 0.70 \begin {gather*} \frac {2 \left (45045 B \,c^{2} x^{5}+51051 A \,c^{2} x^{4}+117810 B a c \,x^{3}+139230 A a c \,x^{2}+85085 B \,a^{2} x +109395 A \,a^{2}\right ) x^{\frac {7}{2}}}{765765} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/765765*x^(7/2)*(45045*B*c^2*x^5+51051*A*c^2*x^4+117810*B*a*c*x^3+139230*A*a*c*x^2+85085*B*a^2*x+109395*A*a^2
)

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maxima [A]  time = 0.53, size = 53, normalized size = 0.69 \begin {gather*} \frac {2}{17} \, B c^{2} x^{\frac {17}{2}} + \frac {2}{15} \, A c^{2} x^{\frac {15}{2}} + \frac {4}{13} \, B a c x^{\frac {13}{2}} + \frac {4}{11} \, A a c x^{\frac {11}{2}} + \frac {2}{9} \, B a^{2} x^{\frac {9}{2}} + \frac {2}{7} \, A a^{2} x^{\frac {7}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/17*B*c^2*x^(17/2) + 2/15*A*c^2*x^(15/2) + 4/13*B*a*c*x^(13/2) + 4/11*A*a*c*x^(11/2) + 2/9*B*a^2*x^(9/2) + 2/
7*A*a^2*x^(7/2)

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mupad [B]  time = 0.03, size = 53, normalized size = 0.69 \begin {gather*} \frac {2\,A\,a^2\,x^{7/2}}{7}+\frac {2\,B\,a^2\,x^{9/2}}{9}+\frac {2\,A\,c^2\,x^{15/2}}{15}+\frac {2\,B\,c^2\,x^{17/2}}{17}+\frac {4\,A\,a\,c\,x^{11/2}}{11}+\frac {4\,B\,a\,c\,x^{13/2}}{13} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*A*a^2*x^(7/2))/7 + (2*B*a^2*x^(9/2))/9 + (2*A*c^2*x^(15/2))/15 + (2*B*c^2*x^(17/2))/17 + (4*A*a*c*x^(11/2))
/11 + (4*B*a*c*x^(13/2))/13

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sympy [A]  time = 8.50, size = 80, normalized size = 1.04 \begin {gather*} \frac {2 A a^{2} x^{\frac {7}{2}}}{7} + \frac {4 A a c x^{\frac {11}{2}}}{11} + \frac {2 A c^{2} x^{\frac {15}{2}}}{15} + \frac {2 B a^{2} x^{\frac {9}{2}}}{9} + \frac {4 B a c x^{\frac {13}{2}}}{13} + \frac {2 B c^{2} x^{\frac {17}{2}}}{17} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(5/2)*(B*x+A)*(c*x**2+a)**2,x)

[Out]

2*A*a**2*x**(7/2)/7 + 4*A*a*c*x**(11/2)/11 + 2*A*c**2*x**(15/2)/15 + 2*B*a**2*x**(9/2)/9 + 4*B*a*c*x**(13/2)/1
3 + 2*B*c**2*x**(17/2)/17

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