3.7.66 \(\int (d x)^{3/2} (a^2+2 a b x^2+b^2 x^4) \, dx\) [666]

Optimal. Leaf size=51 \[ \frac {2 a^2 (d x)^{5/2}}{5 d}+\frac {4 a b (d x)^{9/2}}{9 d^3}+\frac {2 b^2 (d x)^{13/2}}{13 d^5} \]

[Out]

2/5*a^2*(d*x)^(5/2)/d+4/9*a*b*(d*x)^(9/2)/d^3+2/13*b^2*(d*x)^(13/2)/d^5

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Rubi [A]
time = 0.01, antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.038, Rules used = {14} \begin {gather*} \frac {2 a^2 (d x)^{5/2}}{5 d}+\frac {4 a b (d x)^{9/2}}{9 d^3}+\frac {2 b^2 (d x)^{13/2}}{13 d^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d*x)^(3/2)*(a^2 + 2*a*b*x^2 + b^2*x^4),x]

[Out]

(2*a^2*(d*x)^(5/2))/(5*d) + (4*a*b*(d*x)^(9/2))/(9*d^3) + (2*b^2*(d*x)^(13/2))/(13*d^5)

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rubi steps

\begin {align*} \int (d x)^{3/2} \left (a^2+2 a b x^2+b^2 x^4\right ) \, dx &=\int \left (a^2 (d x)^{3/2}+\frac {2 a b (d x)^{7/2}}{d^2}+\frac {b^2 (d x)^{11/2}}{d^4}\right ) \, dx\\ &=\frac {2 a^2 (d x)^{5/2}}{5 d}+\frac {4 a b (d x)^{9/2}}{9 d^3}+\frac {2 b^2 (d x)^{13/2}}{13 d^5}\\ \end {align*}

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Mathematica [A]
time = 0.02, size = 33, normalized size = 0.65 \begin {gather*} \frac {2}{585} x (d x)^{3/2} \left (117 a^2+130 a b x^2+45 b^2 x^4\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d*x)^(3/2)*(a^2 + 2*a*b*x^2 + b^2*x^4),x]

[Out]

(2*x*(d*x)^(3/2)*(117*a^2 + 130*a*b*x^2 + 45*b^2*x^4))/585

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Maple [A]
time = 0.02, size = 42, normalized size = 0.82

method	result	size
gosper	\(\frac {2 x \left (45 b^{2} x^{4}+130 a b \,x^{2}+117 a^{2}\right ) \left (d x \right )^{\frac {3}{2}}}{585}\)	\(30\)
trager	\(\frac {2 d \,x^{2} \left (45 b^{2} x^{4}+130 a b \,x^{2}+117 a^{2}\right ) \sqrt {d x}}{585}\)	\(33\)
risch	\(\frac {2 d^{2} x^{3} \left (45 b^{2} x^{4}+130 a b \,x^{2}+117 a^{2}\right )}{585 \sqrt {d x}}\)	\(35\)
derivativedivides	\(\frac {\frac {2 b^{2} \left (d x \right )^{\frac {13}{2}}}{13}+\frac {4 a b \,d^{2} \left (d x \right )^{\frac {9}{2}}}{9}+\frac {2 a^{2} d^{4} \left (d x \right )^{\frac {5}{2}}}{5}}{d^{5}}\)	\(42\)
default	\(\frac {\frac {2 b^{2} \left (d x \right )^{\frac {13}{2}}}{13}+\frac {4 a b \,d^{2} \left (d x \right )^{\frac {9}{2}}}{9}+\frac {2 a^{2} d^{4} \left (d x \right )^{\frac {5}{2}}}{5}}{d^{5}}\)	\(42\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x)^(3/2)*(b^2*x^4+2*a*b*x^2+a^2),x,method=_RETURNVERBOSE)

[Out]

2/d^5*(1/13*b^2*(d*x)^(13/2)+2/9*a*b*d^2*(d*x)^(9/2)+1/5*a^2*d^4*(d*x)^(5/2))

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Maxima [A]
time = 0.28, size = 41, normalized size = 0.80 \begin {gather*} \frac {2 \, {\left (45 \, \left (d x\right )^{\frac {13}{2}} b^{2} + 130 \, \left (d x\right )^{\frac {9}{2}} a b d^{2} + 117 \, \left (d x\right )^{\frac {5}{2}} a^{2} d^{4}\right )}}{585 \, d^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^(3/2)*(b^2*x^4+2*a*b*x^2+a^2),x, algorithm="maxima")

[Out]

2/585*(45*(d*x)^(13/2)*b^2 + 130*(d*x)^(9/2)*a*b*d^2 + 117*(d*x)^(5/2)*a^2*d^4)/d^5

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Fricas [A]
time = 0.34, size = 34, normalized size = 0.67 \begin {gather*} \frac {2}{585} \, {\left (45 \, b^{2} d x^{6} + 130 \, a b d x^{4} + 117 \, a^{2} d x^{2}\right )} \sqrt {d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^(3/2)*(b^2*x^4+2*a*b*x^2+a^2),x, algorithm="fricas")

[Out]

2/585*(45*b^2*d*x^6 + 130*a*b*d*x^4 + 117*a^2*d*x^2)*sqrt(d*x)

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Sympy [A]
time = 0.23, size = 48, normalized size = 0.94 \begin {gather*} \frac {2 a^{2} x \left (d x\right )^{\frac {3}{2}}}{5} + \frac {4 a b x^{3} \left (d x\right )^{\frac {3}{2}}}{9} + \frac {2 b^{2} x^{5} \left (d x\right )^{\frac {3}{2}}}{13} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)**(3/2)*(b**2*x**4+2*a*b*x**2+a**2),x)

[Out]

2*a**2*x*(d*x)**(3/2)/5 + 4*a*b*x**3*(d*x)**(3/2)/9 + 2*b**2*x**5*(d*x)**(3/2)/13

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Giac [A]
time = 3.69, size = 42, normalized size = 0.82 \begin {gather*} \frac {2}{585} \, {\left (45 \, \sqrt {d x} b^{2} x^{6} + 130 \, \sqrt {d x} a b x^{4} + 117 \, \sqrt {d x} a^{2} x^{2}\right )} d \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^(3/2)*(b^2*x^4+2*a*b*x^2+a^2),x, algorithm="giac")

[Out]

2/585*(45*sqrt(d*x)*b^2*x^6 + 130*sqrt(d*x)*a*b*x^4 + 117*sqrt(d*x)*a^2*x^2)*d

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Mupad [B]
time = 4.22, size = 41, normalized size = 0.80 \begin {gather*} \frac {90\,b^2\,{\left (d\,x\right )}^{13/2}+234\,a^2\,d^4\,{\left (d\,x\right )}^{5/2}+260\,a\,b\,d^2\,{\left (d\,x\right )}^{9/2}}{585\,d^5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x)^(3/2)*(a^2 + b^2*x^4 + 2*a*b*x^2),x)

[Out]

(90*b^2*(d*x)^(13/2) + 234*a^2*d^4*(d*x)^(5/2) + 260*a*b*d^2*(d*x)^(9/2))/(585*d^5)

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