3.2.64 \(\int \frac {(A+B x^2) (b x^2+c x^4)}{x^{3/2}} \, dx\) [164]

Optimal. Leaf size=39 \[ \frac {2}{3} A b x^{3/2}+\frac {2}{7} (b B+A c) x^{7/2}+\frac {2}{11} B c x^{11/2} \]

[Out]

2/3*A*b*x^(3/2)+2/7*(A*c+B*b)*x^(7/2)+2/11*B*c*x^(11/2)

________________________________________________________________________________________

Rubi [A]
time = 0.01, antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {1598, 459} \begin {gather*} \frac {2}{7} x^{7/2} (A c+b B)+\frac {2}{3} A b x^{3/2}+\frac {2}{11} B c x^{11/2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((A + B*x^2)*(b*x^2 + c*x^4))/x^(3/2),x]

[Out]

(2*A*b*x^(3/2))/3 + (2*(b*B + A*c)*x^(7/2))/7 + (2*B*c*x^(11/2))/11

Rule 459

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Int[ExpandI
ntegrand[(e*x)^m*(a + b*x^n)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[b*c - a*d, 0] &
& IGtQ[p, 0] && IGtQ[q, 0]

Rule 1598

Int[(u_.)*(x_)^(m_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(m + n*p)*(a + b*x^(q -
p))^n, x] /; FreeQ[{a, b, m, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rubi steps

\begin {align*} \int \frac {\left (A+B x^2\right ) \left (b x^2+c x^4\right )}{x^{3/2}} \, dx &=\int \sqrt {x} \left (A+B x^2\right ) \left (b+c x^2\right ) \, dx\\ &=\int \left (A b \sqrt {x}+(b B+A c) x^{5/2}+B c x^{9/2}\right ) \, dx\\ &=\frac {2}{3} A b x^{3/2}+\frac {2}{7} (b B+A c) x^{7/2}+\frac {2}{11} B c x^{11/2}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.02, size = 35, normalized size = 0.90 \begin {gather*} \frac {2}{231} x^{3/2} \left (77 A b+33 b B x^2+33 A c x^2+21 B c x^4\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x^2)*(b*x^2 + c*x^4))/x^(3/2),x]

[Out]

(2*x^(3/2)*(77*A*b + 33*b*B*x^2 + 33*A*c*x^2 + 21*B*c*x^4))/231

________________________________________________________________________________________

Maple [A]
time = 0.13, size = 28, normalized size = 0.72

method	result	size
derivativedivides	\(\frac {2 A b \,x^{\frac {3}{2}}}{3}+\frac {2 \left (A c +B b \right ) x^{\frac {7}{2}}}{7}+\frac {2 B c \,x^{\frac {11}{2}}}{11}\)	\(28\)
default	\(\frac {2 A b \,x^{\frac {3}{2}}}{3}+\frac {2 \left (A c +B b \right ) x^{\frac {7}{2}}}{7}+\frac {2 B c \,x^{\frac {11}{2}}}{11}\)	\(28\)
gosper	\(\frac {2 x^{\frac {3}{2}} \left (21 B c \,x^{4}+33 A c \,x^{2}+33 b B \,x^{2}+77 A b \right )}{231}\)	\(32\)
trager	\(\frac {2 x^{\frac {3}{2}} \left (21 B c \,x^{4}+33 A c \,x^{2}+33 b B \,x^{2}+77 A b \right )}{231}\)	\(32\)
risch	\(\frac {2 x^{\frac {3}{2}} \left (21 B c \,x^{4}+33 A c \,x^{2}+33 b B \,x^{2}+77 A b \right )}{231}\)	\(32\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x^2+A)*(c*x^4+b*x^2)/x^(3/2),x,method=_RETURNVERBOSE)

[Out]

2/3*A*b*x^(3/2)+2/7*(A*c+B*b)*x^(7/2)+2/11*B*c*x^(11/2)

________________________________________________________________________________________

Maxima [A]
time = 0.27, size = 27, normalized size = 0.69 \begin {gather*} \frac {2}{11} \, B c x^{\frac {11}{2}} + \frac {2}{7} \, {\left (B b + A c\right )} x^{\frac {7}{2}} + \frac {2}{3} \, A b x^{\frac {3}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x^2+A)*(c*x^4+b*x^2)/x^(3/2),x, algorithm="maxima")

[Out]

2/11*B*c*x^(11/2) + 2/7*(B*b + A*c)*x^(7/2) + 2/3*A*b*x^(3/2)

________________________________________________________________________________________

Fricas [A]
time = 1.87, size = 30, normalized size = 0.77 \begin {gather*} \frac {2}{231} \, {\left (21 \, B c x^{5} + 33 \, {\left (B b + A c\right )} x^{3} + 77 \, A b x\right )} \sqrt {x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x^2+A)*(c*x^4+b*x^2)/x^(3/2),x, algorithm="fricas")

[Out]

2/231*(21*B*c*x^5 + 33*(B*b + A*c)*x^3 + 77*A*b*x)*sqrt(x)

________________________________________________________________________________________

Sympy [A]
time = 0.32, size = 46, normalized size = 1.18 \begin {gather*} \frac {2 A b x^{\frac {3}{2}}}{3} + \frac {2 A c x^{\frac {7}{2}}}{7} + \frac {2 B b x^{\frac {7}{2}}}{7} + \frac {2 B c x^{\frac {11}{2}}}{11} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x**2+A)*(c*x**4+b*x**2)/x**(3/2),x)

[Out]

2*A*b*x**(3/2)/3 + 2*A*c*x**(7/2)/7 + 2*B*b*x**(7/2)/7 + 2*B*c*x**(11/2)/11

________________________________________________________________________________________

Giac [A]
time = 0.46, size = 29, normalized size = 0.74 \begin {gather*} \frac {2}{11} \, B c x^{\frac {11}{2}} + \frac {2}{7} \, B b x^{\frac {7}{2}} + \frac {2}{7} \, A c x^{\frac {7}{2}} + \frac {2}{3} \, A b x^{\frac {3}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x^2+A)*(c*x^4+b*x^2)/x^(3/2),x, algorithm="giac")

[Out]

2/11*B*c*x^(11/2) + 2/7*B*b*x^(7/2) + 2/7*A*c*x^(7/2) + 2/3*A*b*x^(3/2)

________________________________________________________________________________________

Mupad [B]
time = 0.04, size = 31, normalized size = 0.79 \begin {gather*} \frac {2\,x^{3/2}\,\left (77\,A\,b+33\,A\,c\,x^2+33\,B\,b\,x^2+21\,B\,c\,x^4\right )}{231} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x^2)*(b*x^2 + c*x^4))/x^(3/2),x)

[Out]

(2*x^(3/2)*(77*A*b + 33*A*c*x^2 + 33*B*b*x^2 + 21*B*c*x^4))/231

________________________________________________________________________________________