3.2.65 \(\int \frac {(A+B x^2) (b x^2+c x^4)}{x^{5/2}} \, dx\) [165]

Optimal. Leaf size=37 \[ 2 A b \sqrt {x}+\frac {2}{5} (b B+A c) x^{5/2}+\frac {2}{9} B c x^{9/2} \]

[Out]

2/5*(A*c+B*b)*x^(5/2)+2/9*B*c*x^(9/2)+2*A*b*x^(1/2)

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Rubi [A]
time = 0.02, antiderivative size = 37, 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}{5} x^{5/2} (A c+b B)+2 A b \sqrt {x}+\frac {2}{9} B c x^{9/2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((A + B*x^2)*(b*x^2 + c*x^4))/x^(5/2),x]

[Out]

2*A*b*Sqrt[x] + (2*(b*B + A*c)*x^(5/2))/5 + (2*B*c*x^(9/2))/9

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

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Mathematica [A]
time = 0.02, size = 35, normalized size = 0.95 \begin {gather*} \frac {2}{45} \sqrt {x} \left (45 A b+9 b B x^2+9 A c x^2+5 B c x^4\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x^2)*(b*x^2 + c*x^4))/x^(5/2),x]

[Out]

(2*Sqrt[x]*(45*A*b + 9*b*B*x^2 + 9*A*c*x^2 + 5*B*c*x^4))/45

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Maple [A]
time = 0.13, size = 28, normalized size = 0.76

method	result	size
derivativedivides	\(\frac {2 \left (A c +B b \right ) x^{\frac {5}{2}}}{5}+\frac {2 B c \,x^{\frac {9}{2}}}{9}+2 A b \sqrt {x}\)	\(28\)
default	\(\frac {2 \left (A c +B b \right ) x^{\frac {5}{2}}}{5}+\frac {2 B c \,x^{\frac {9}{2}}}{9}+2 A b \sqrt {x}\)	\(28\)
trager	\(\left (\frac {2}{9} B c \,x^{4}+\frac {2}{5} A c \,x^{2}+\frac {2}{5} b B \,x^{2}+2 A b \right ) \sqrt {x}\)	\(31\)
gosper	\(\frac {2 \sqrt {x}\, \left (5 B c \,x^{4}+9 A c \,x^{2}+9 b B \,x^{2}+45 A b \right )}{45}\)	\(32\)
risch	\(\frac {2 \sqrt {x}\, \left (5 B c \,x^{4}+9 A c \,x^{2}+9 b B \,x^{2}+45 A b \right )}{45}\)	\(32\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x^2+A)*(c*x^4+b*x^2)/x^(5/2),x,method=_RETURNVERBOSE)

[Out]

2/5*(A*c+B*b)*x^(5/2)+2/9*B*c*x^(9/2)+2*A*b*x^(1/2)

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Maxima [A]
time = 0.27, size = 27, normalized size = 0.73 \begin {gather*} \frac {2}{9} \, B c x^{\frac {9}{2}} + \frac {2}{5} \, {\left (B b + A c\right )} x^{\frac {5}{2}} + 2 \, A b \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^(5/2),x, algorithm="maxima")

[Out]

2/9*B*c*x^(9/2) + 2/5*(B*b + A*c)*x^(5/2) + 2*A*b*sqrt(x)

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Fricas [A]
time = 1.63, size = 29, normalized size = 0.78 \begin {gather*} \frac {2}{45} \, {\left (5 \, B c x^{4} + 9 \, {\left (B b + A c\right )} x^{2} + 45 \, A b\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^(5/2),x, algorithm="fricas")

[Out]

2/45*(5*B*c*x^4 + 9*(B*b + A*c)*x^2 + 45*A*b)*sqrt(x)

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Sympy [A]
time = 0.37, size = 44, normalized size = 1.19 \begin {gather*} 2 A b \sqrt {x} + \frac {2 A c x^{\frac {5}{2}}}{5} + \frac {2 B b 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**2+A)*(c*x**4+b*x**2)/x**(5/2),x)

[Out]

2*A*b*sqrt(x) + 2*A*c*x**(5/2)/5 + 2*B*b*x**(5/2)/5 + 2*B*c*x**(9/2)/9

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Giac [A]
time = 0.57, size = 29, normalized size = 0.78 \begin {gather*} \frac {2}{9} \, B c x^{\frac {9}{2}} + \frac {2}{5} \, B b x^{\frac {5}{2}} + \frac {2}{5} \, A c x^{\frac {5}{2}} + 2 \, A b \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^(5/2),x, algorithm="giac")

[Out]

2/9*B*c*x^(9/2) + 2/5*B*b*x^(5/2) + 2/5*A*c*x^(5/2) + 2*A*b*sqrt(x)

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Mupad [B]
time = 0.04, size = 31, normalized size = 0.84 \begin {gather*} \frac {2\,\sqrt {x}\,\left (45\,A\,b+9\,A\,c\,x^2+9\,B\,b\,x^2+5\,B\,c\,x^4\right )}{45} \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^(5/2),x)

[Out]

(2*x^(1/2)*(45*A*b + 9*A*c*x^2 + 9*B*b*x^2 + 5*B*c*x^4))/45

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