3.2.42 \(\int x^{3/2} (A+B x) (b x+c x^2) \, dx\) [142]

Optimal. Leaf size=39 \[ \frac {2}{7} A b x^{7/2}+\frac {2}{9} (b B+A c) x^{9/2}+\frac {2}{11} B c x^{11/2} \]

[Out]

2/7*A*b*x^(7/2)+2/9*(A*c+B*b)*x^(9/2)+2/11*B*c*x^(11/2)

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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 = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {779} \begin {gather*} \frac {2}{9} x^{9/2} (A c+b B)+\frac {2}{7} A b x^{7/2}+\frac {2}{11} B c x^{11/2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^(3/2)*(A + B*x)*(b*x + c*x^2),x]

[Out]

(2*A*b*x^(7/2))/7 + (2*(b*B + A*c)*x^(9/2))/9 + (2*B*c*x^(11/2))/11

Rule 779

Int[((e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[Expand
Integrand[(e*x)^m*(f + g*x)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, e, f, g, m}, x] && IntegerQ[p] && (
GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

\begin {align*} \int x^{3/2} (A+B x) \left (b x+c x^2\right ) \, dx &=\int \left (A b x^{5/2}+(b B+A c) x^{7/2}+B c x^{9/2}\right ) \, dx\\ &=\frac {2}{7} A b x^{7/2}+\frac {2}{9} (b B+A c) x^{9/2}+\frac {2}{11} B c x^{11/2}\\ \end {align*}

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Mathematica [A]
time = 0.02, size = 31, normalized size = 0.79 \begin {gather*} \frac {2}{693} x^{7/2} \left (99 A b+77 b B x+77 A c x+63 B c x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^(3/2)*(A + B*x)*(b*x + c*x^2),x]

[Out]

(2*x^(7/2)*(99*A*b + 77*b*B*x + 77*A*c*x + 63*B*c*x^2))/693

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Maple [A]
time = 0.20, size = 28, normalized size = 0.72

method	result	size
gosper	\(\frac {2 x^{\frac {7}{2}} \left (63 B c \,x^{2}+77 A c x +77 b B x +99 A b \right )}{693}\)	\(28\)
derivativedivides	\(\frac {2 A b \,x^{\frac {7}{2}}}{7}+\frac {2 \left (A c +B b \right ) x^{\frac {9}{2}}}{9}+\frac {2 B c \,x^{\frac {11}{2}}}{11}\)	\(28\)
default	\(\frac {2 A b \,x^{\frac {7}{2}}}{7}+\frac {2 \left (A c +B b \right ) x^{\frac {9}{2}}}{9}+\frac {2 B c \,x^{\frac {11}{2}}}{11}\)	\(28\)
trager	\(\frac {2 x^{\frac {7}{2}} \left (63 B c \,x^{2}+77 A c x +77 b B x +99 A b \right )}{693}\)	\(28\)
risch	\(\frac {2 x^{\frac {7}{2}} \left (63 B c \,x^{2}+77 A c x +77 b B x +99 A b \right )}{693}\)	\(28\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(3/2)*(B*x+A)*(c*x^2+b*x),x,method=_RETURNVERBOSE)

[Out]

2/7*A*b*x^(7/2)+2/9*(A*c+B*b)*x^(9/2)+2/11*B*c*x^(11/2)

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Maxima [A]
time = 0.26, size = 27, normalized size = 0.69 \begin {gather*} \frac {2}{11} \, B c x^{\frac {11}{2}} + \frac {2}{7} \, A b x^{\frac {7}{2}} + \frac {2}{9} \, {\left (B b + A c\right )} x^{\frac {9}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(3/2)*(B*x+A)*(c*x^2+b*x),x, algorithm="maxima")

[Out]

2/11*B*c*x^(11/2) + 2/7*A*b*x^(7/2) + 2/9*(B*b + A*c)*x^(9/2)

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Fricas [A]
time = 3.59, size = 32, normalized size = 0.82 \begin {gather*} \frac {2}{693} \, {\left (63 \, B c x^{5} + 99 \, A b x^{3} + 77 \, {\left (B b + A c\right )} x^{4}\right )} \sqrt {x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(3/2)*(B*x+A)*(c*x^2+b*x),x, algorithm="fricas")

[Out]

2/693*(63*B*c*x^5 + 99*A*b*x^3 + 77*(B*b + A*c)*x^4)*sqrt(x)

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Sympy [A]
time = 0.22, size = 46, normalized size = 1.18 \begin {gather*} \frac {2 A b x^{\frac {7}{2}}}{7} + \frac {2 A c x^{\frac {9}{2}}}{9} + \frac {2 B b x^{\frac {9}{2}}}{9} + \frac {2 B c x^{\frac {11}{2}}}{11} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(3/2)*(B*x+A)*(c*x**2+b*x),x)

[Out]

2*A*b*x**(7/2)/7 + 2*A*c*x**(9/2)/9 + 2*B*b*x**(9/2)/9 + 2*B*c*x**(11/2)/11

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Giac [A]
time = 1.02, size = 29, normalized size = 0.74 \begin {gather*} \frac {2}{11} \, B c x^{\frac {11}{2}} + \frac {2}{9} \, B b x^{\frac {9}{2}} + \frac {2}{9} \, A c x^{\frac {9}{2}} + \frac {2}{7} \, A b x^{\frac {7}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(3/2)*(B*x+A)*(c*x^2+b*x),x, algorithm="giac")

[Out]

2/11*B*c*x^(11/2) + 2/9*B*b*x^(9/2) + 2/9*A*c*x^(9/2) + 2/7*A*b*x^(7/2)

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Mupad [B]
time = 1.02, size = 27, normalized size = 0.69 \begin {gather*} \frac {2\,x^{7/2}\,\left (99\,A\,b+77\,A\,c\,x+77\,B\,b\,x+63\,B\,c\,x^2\right )}{693} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(3/2)*(b*x + c*x^2)*(A + B*x),x)

[Out]

(2*x^(7/2)*(99*A*b + 77*A*c*x + 77*B*b*x + 63*B*c*x^2))/693

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