3.4.19 \(\int x^{7/2} (a+b x) (A+B x) \, dx\) [319]

Optimal. Leaf size=39 \[ \frac {2}{9} a A x^{9/2}+\frac {2}{11} (A b+a B) x^{11/2}+\frac {2}{13} b B x^{13/2} \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {77} \begin {gather*} \frac {2}{11} x^{11/2} (a B+A b)+\frac {2}{9} a A x^{9/2}+\frac {2}{13} b B x^{13/2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^(7/2)*(a + b*x)*(A + B*x),x]

[Out]

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

Rule 77

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])

Rubi steps

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

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Mathematica [A]
time = 0.02, size = 31, normalized size = 0.79 \begin {gather*} \frac {2 x^{9/2} \left (143 a A+117 A b x+117 a B x+99 b B x^2\right )}{1287} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.05, size = 28, normalized size = 0.72

method	result	size
gosper	\(\frac {2 x^{\frac {9}{2}} \left (99 b B \,x^{2}+117 A b x +117 B a x +143 A a \right )}{1287}\)	\(28\)
derivativedivides	\(\frac {2 a A \,x^{\frac {9}{2}}}{9}+\frac {2 \left (A b +B a \right ) x^{\frac {11}{2}}}{11}+\frac {2 b B \,x^{\frac {13}{2}}}{13}\)	\(28\)
default	\(\frac {2 a A \,x^{\frac {9}{2}}}{9}+\frac {2 \left (A b +B a \right ) x^{\frac {11}{2}}}{11}+\frac {2 b B \,x^{\frac {13}{2}}}{13}\)	\(28\)
trager	\(\frac {2 x^{\frac {9}{2}} \left (99 b B \,x^{2}+117 A b x +117 B a x +143 A a \right )}{1287}\)	\(28\)
risch	\(\frac {2 x^{\frac {9}{2}} \left (99 b B \,x^{2}+117 A b x +117 B a x +143 A a \right )}{1287}\)	\(28\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(7/2)*(b*x+a)*(B*x+A),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.27, size = 27, normalized size = 0.69 \begin {gather*} \frac {2}{13} \, B b x^{\frac {13}{2}} + \frac {2}{9} \, A a x^{\frac {9}{2}} + \frac {2}{11} \, {\left (B a + A b\right )} x^{\frac {11}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 1.14, size = 32, normalized size = 0.82 \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),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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Sympy [A]
time = 0.55, size = 46, normalized size = 1.18 \begin {gather*} \frac {2 A a x^{\frac {9}{2}}}{9} + \frac {2 A b x^{\frac {11}{2}}}{11} + \frac {2 B a x^{\frac {11}{2}}}{11} + \frac {2 B b x^{\frac {13}{2}}}{13} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 0.76, size = 29, normalized size = 0.74 \begin {gather*} \frac {2}{13} \, B b x^{\frac {13}{2}} + \frac {2}{11} \, B a x^{\frac {11}{2}} + \frac {2}{11} \, A b x^{\frac {11}{2}} + \frac {2}{9} \, A a x^{\frac {9}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 0.06, size = 27, normalized size = 0.69 \begin {gather*} \frac {2\,x^{9/2}\,\left (143\,A\,a+117\,A\,b\,x+117\,B\,a\,x+99\,B\,b\,x^2\right )}{1287} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(7/2)*(A + B*x)*(a + b*x),x)

[Out]

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

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