3.4.87 \(\int \sqrt {x} (A+B x) (a+c x^2) \, dx\) [387]

Optimal. Leaf size=45 \[ \frac {2}{3} a A x^{3/2}+\frac {2}{5} a B x^{5/2}+\frac {2}{7} A c x^{7/2}+\frac {2}{9} B c x^{9/2} \]

[Out]

2/3*a*A*x^(3/2)+2/5*a*B*x^(5/2)+2/7*A*c*x^(7/2)+2/9*B*c*x^(9/2)

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Rubi [A]
time = 0.01, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {780} \begin {gather*} \frac {2}{3} a A x^{3/2}+\frac {2}{5} a B x^{5/2}+\frac {2}{7} A c x^{7/2}+\frac {2}{9} B c x^{9/2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[x]*(A + B*x)*(a + c*x^2),x]

[Out]

(2*a*A*x^(3/2))/3 + (2*a*B*x^(5/2))/5 + (2*A*c*x^(7/2))/7 + (2*B*c*x^(9/2))/9

Rule 780

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

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Mathematica [A]
time = 0.02, size = 35, normalized size = 0.78 \begin {gather*} \frac {2}{315} x^{3/2} \left (21 a (5 A+3 B x)+5 c x^2 (9 A+7 B x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*x^(3/2)*(21*a*(5*A + 3*B*x) + 5*c*x^2*(9*A + 7*B*x)))/315

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Maple [A]
time = 0.19, size = 30, normalized size = 0.67

method	result	size
gosper	\(\frac {2 x^{\frac {3}{2}} \left (35 B c \,x^{3}+45 A c \,x^{2}+63 B a x +105 A a \right )}{315}\)	\(30\)
derivativedivides	\(\frac {2 a A \,x^{\frac {3}{2}}}{3}+\frac {2 a B \,x^{\frac {5}{2}}}{5}+\frac {2 A c \,x^{\frac {7}{2}}}{7}+\frac {2 B c \,x^{\frac {9}{2}}}{9}\)	\(30\)
default	\(\frac {2 a A \,x^{\frac {3}{2}}}{3}+\frac {2 a B \,x^{\frac {5}{2}}}{5}+\frac {2 A c \,x^{\frac {7}{2}}}{7}+\frac {2 B c \,x^{\frac {9}{2}}}{9}\)	\(30\)
trager	\(\frac {2 x^{\frac {3}{2}} \left (35 B c \,x^{3}+45 A c \,x^{2}+63 B a x +105 A a \right )}{315}\)	\(30\)
risch	\(\frac {2 x^{\frac {3}{2}} \left (35 B c \,x^{3}+45 A c \,x^{2}+63 B a x +105 A a \right )}{315}\)	\(30\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(c*x^2+a)*x^(1/2),x,method=_RETURNVERBOSE)

[Out]

2/3*a*A*x^(3/2)+2/5*a*B*x^(5/2)+2/7*A*c*x^(7/2)+2/9*B*c*x^(9/2)

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Maxima [A]
time = 0.26, size = 29, normalized size = 0.64 \begin {gather*} \frac {2}{9} \, B c x^{\frac {9}{2}} + \frac {2}{7} \, A c x^{\frac {7}{2}} + \frac {2}{5} \, B a x^{\frac {5}{2}} + \frac {2}{3} \, A a x^{\frac {3}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/9*B*c*x^(9/2) + 2/7*A*c*x^(7/2) + 2/5*B*a*x^(5/2) + 2/3*A*a*x^(3/2)

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Fricas [A]
time = 2.59, size = 32, normalized size = 0.71 \begin {gather*} \frac {2}{315} \, {\left (35 \, B c x^{4} + 45 \, A c x^{3} + 63 \, B a x^{2} + 105 \, A a x\right )} \sqrt {x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/315*(35*B*c*x^4 + 45*A*c*x^3 + 63*B*a*x^2 + 105*A*a*x)*sqrt(x)

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Sympy [A]
time = 1.00, size = 46, normalized size = 1.02 \begin {gather*} \frac {2 A a x^{\frac {3}{2}}}{3} + \frac {2 A c x^{\frac {7}{2}}}{7} + \frac {2 B a x^{\frac {5}{2}}}{5} + \frac {2 B c x^{\frac {9}{2}}}{9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*A*a*x**(3/2)/3 + 2*A*c*x**(7/2)/7 + 2*B*a*x**(5/2)/5 + 2*B*c*x**(9/2)/9

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Giac [A]
time = 0.57, size = 29, normalized size = 0.64 \begin {gather*} \frac {2}{9} \, B c x^{\frac {9}{2}} + \frac {2}{7} \, A c x^{\frac {7}{2}} + \frac {2}{5} \, B a x^{\frac {5}{2}} + \frac {2}{3} \, A a x^{\frac {3}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/9*B*c*x^(9/2) + 2/7*A*c*x^(7/2) + 2/5*B*a*x^(5/2) + 2/3*A*a*x^(3/2)

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Mupad [B]
time = 0.04, size = 29, normalized size = 0.64 \begin {gather*} \frac {2\,A\,a\,x^{3/2}}{3}+\frac {2\,B\,a\,x^{5/2}}{5}+\frac {2\,A\,c\,x^{7/2}}{7}+\frac {2\,B\,c\,x^{9/2}}{9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(1/2)*(a + c*x^2)*(A + B*x),x)

[Out]

(2*A*a*x^(3/2))/3 + (2*B*a*x^(5/2))/5 + (2*A*c*x^(7/2))/7 + (2*B*c*x^(9/2))/9

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