3.5.87 \(\int (d x)^{5/2} (a^2+2 a b x^2+b^2 x^4) \, dx\)

Optimal. Leaf size=51 \[ \frac {2 a^2 (d x)^{7/2}}{7 d}+\frac {4 a b (d x)^{11/2}}{11 d^3}+\frac {2 b^2 (d x)^{15/2}}{15 d^5} \]

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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)^{7/2}}{7 d}+\frac {4 a b (d x)^{11/2}}{11 d^3}+\frac {2 b^2 (d x)^{15/2}}{15 d^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d*x)^(5/2)*(a^2 + 2*a*b*x^2 + b^2*x^4),x]

[Out]

(2*a^2*(d*x)^(7/2))/(7*d) + (4*a*b*(d*x)^(11/2))/(11*d^3) + (2*b^2*(d*x)^(15/2))/(15*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)^{5/2} \left (a^2+2 a b x^2+b^2 x^4\right ) \, dx &=\int \left (a^2 (d x)^{5/2}+\frac {2 a b (d x)^{9/2}}{d^2}+\frac {b^2 (d x)^{13/2}}{d^4}\right ) \, dx\\ &=\frac {2 a^2 (d x)^{7/2}}{7 d}+\frac {4 a b (d x)^{11/2}}{11 d^3}+\frac {2 b^2 (d x)^{15/2}}{15 d^5}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 33, normalized size = 0.65 \begin {gather*} \frac {2 x (d x)^{5/2} \left (165 a^2+210 a b x^2+77 b^2 x^4\right )}{1155} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d*x)^(5/2)*(a^2 + 2*a*b*x^2 + b^2*x^4),x]

[Out]

(2*x*(d*x)^(5/2)*(165*a^2 + 210*a*b*x^2 + 77*b^2*x^4))/1155

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IntegrateAlgebraic [A]  time = 0.03, size = 44, normalized size = 0.86 \begin {gather*} \frac {2 (d x)^{7/2} \left (165 a^2 d^4+210 a b d^4 x^2+77 b^2 d^4 x^4\right )}{1155 d^5} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(d*x)^(5/2)*(a^2 + 2*a*b*x^2 + b^2*x^4),x]

[Out]

(2*(d*x)^(7/2)*(165*a^2*d^4 + 210*a*b*d^4*x^2 + 77*b^2*d^4*x^4))/(1155*d^5)

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fricas [A]  time = 1.01, size = 40, normalized size = 0.78 \begin {gather*} \frac {2}{1155} \, {\left (77 \, b^{2} d^{2} x^{7} + 210 \, a b d^{2} x^{5} + 165 \, a^{2} d^{2} x^{3}\right )} \sqrt {d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^(5/2)*(b^2*x^4+2*a*b*x^2+a^2),x, algorithm="fricas")

[Out]

2/1155*(77*b^2*d^2*x^7 + 210*a*b*d^2*x^5 + 165*a^2*d^2*x^3)*sqrt(d*x)

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giac [A]  time = 0.17, size = 48, normalized size = 0.94 \begin {gather*} \frac {2}{15} \, \sqrt {d x} b^{2} d^{2} x^{7} + \frac {4}{11} \, \sqrt {d x} a b d^{2} x^{5} + \frac {2}{7} \, \sqrt {d x} a^{2} d^{2} x^{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^(5/2)*(b^2*x^4+2*a*b*x^2+a^2),x, algorithm="giac")

[Out]

2/15*sqrt(d*x)*b^2*d^2*x^7 + 4/11*sqrt(d*x)*a*b*d^2*x^5 + 2/7*sqrt(d*x)*a^2*d^2*x^3

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maple [A]  time = 0.01, size = 30, normalized size = 0.59 \begin {gather*} \frac {2 \left (77 b^{2} x^{4}+210 a b \,x^{2}+165 a^{2}\right ) \left (d x \right )^{\frac {5}{2}} x}{1155} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x)^(5/2)*(b^2*x^4+2*a*b*x^2+a^2),x)

[Out]

2/1155*x*(77*b^2*x^4+210*a*b*x^2+165*a^2)*(d*x)^(5/2)

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maxima [A]  time = 1.30, size = 41, normalized size = 0.80 \begin {gather*} \frac {2 \, {\left (77 \, \left (d x\right )^{\frac {15}{2}} b^{2} + 210 \, \left (d x\right )^{\frac {11}{2}} a b d^{2} + 165 \, \left (d x\right )^{\frac {7}{2}} a^{2} d^{4}\right )}}{1155 \, d^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^(5/2)*(b^2*x^4+2*a*b*x^2+a^2),x, algorithm="maxima")

[Out]

2/1155*(77*(d*x)^(15/2)*b^2 + 210*(d*x)^(11/2)*a*b*d^2 + 165*(d*x)^(7/2)*a^2*d^4)/d^5

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mupad [B]  time = 0.07, size = 40, normalized size = 0.78 \begin {gather*} \frac {\frac {2\,b^2\,{\left (d\,x\right )}^{15/2}}{15}+\frac {2\,a^2\,d^4\,{\left (d\,x\right )}^{7/2}}{7}+\frac {4\,a\,b\,d^2\,{\left (d\,x\right )}^{11/2}}{11}}{d^5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x)^(5/2)*(a^2 + b^2*x^4 + 2*a*b*x^2),x)

[Out]

((2*b^2*(d*x)^(15/2))/15 + (2*a^2*d^4*(d*x)^(7/2))/7 + (4*a*b*d^2*(d*x)^(11/2))/11)/d^5

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sympy [A]  time = 2.67, size = 49, normalized size = 0.96 \begin {gather*} \frac {2 a^{2} d^{\frac {5}{2}} x^{\frac {7}{2}}}{7} + \frac {4 a b d^{\frac {5}{2}} x^{\frac {11}{2}}}{11} + \frac {2 b^{2} d^{\frac {5}{2}} x^{\frac {15}{2}}}{15} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)**(5/2)*(b**2*x**4+2*a*b*x**2+a**2),x)

[Out]

2*a**2*d**(5/2)*x**(7/2)/7 + 4*a*b*d**(5/2)*x**(11/2)/11 + 2*b**2*d**(5/2)*x**(15/2)/15

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