3.5.91 \(\int \frac {a^2+2 a b x^2+b^2 x^4}{(d x)^{3/2}} \, dx\)

Optimal. Leaf size=49 \[ -\frac {2 a^2}{d \sqrt {d x}}+\frac {4 a b (d x)^{3/2}}{3 d^3}+\frac {2 b^2 (d x)^{7/2}}{7 d^5} \]

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Rubi [A]  time = 0.01, antiderivative size = 49, 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 \sqrt {d x}}+\frac {4 a b (d x)^{3/2}}{3 d^3}+\frac {2 b^2 (d x)^{7/2}}{7 d^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*a^2)/(d*Sqrt[d*x]) + (4*a*b*(d*x)^(3/2))/(3*d^3) + (2*b^2*(d*x)^(7/2))/(7*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 \frac {a^2+2 a b x^2+b^2 x^4}{(d x)^{3/2}} \, dx &=\int \left (\frac {a^2}{(d x)^{3/2}}+\frac {2 a b \sqrt {d x}}{d^2}+\frac {b^2 (d x)^{5/2}}{d^4}\right ) \, dx\\ &=-\frac {2 a^2}{d \sqrt {d x}}+\frac {4 a b (d x)^{3/2}}{3 d^3}+\frac {2 b^2 (d x)^{7/2}}{7 d^5}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 33, normalized size = 0.67 \begin {gather*} \frac {2 x \left (-21 a^2+14 a b x^2+3 b^2 x^4\right )}{21 (d x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*x*(-21*a^2 + 14*a*b*x^2 + 3*b^2*x^4))/(21*(d*x)^(3/2))

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IntegrateAlgebraic [A]  time = 0.04, size = 44, normalized size = 0.90 \begin {gather*} \frac {2 \left (-21 a^2 d^4+14 a b d^4 x^2+3 b^2 d^4 x^4\right )}{21 d^5 \sqrt {d x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(-21*a^2*d^4 + 14*a*b*d^4*x^2 + 3*b^2*d^4*x^4))/(21*d^5*Sqrt[d*x])

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fricas [A]  time = 1.92, size = 34, normalized size = 0.69 \begin {gather*} \frac {2 \, {\left (3 \, b^{2} x^{4} + 14 \, a b x^{2} - 21 \, a^{2}\right )} \sqrt {d x}}{21 \, d^{2} x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/21*(3*b^2*x^4 + 14*a*b*x^2 - 21*a^2)*sqrt(d*x)/(d^2*x)

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giac [A]  time = 0.22, size = 51, normalized size = 1.04 \begin {gather*} -\frac {2 \, {\left (\frac {21 \, a^{2}}{\sqrt {d x}} - \frac {3 \, \sqrt {d x} b^{2} d^{27} x^{3} + 14 \, \sqrt {d x} a b d^{27} x}{d^{28}}\right )}}{21 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/21*(21*a^2/sqrt(d*x) - (3*sqrt(d*x)*b^2*d^27*x^3 + 14*sqrt(d*x)*a*b*d^27*x)/d^28)/d

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maple [A]  time = 0.01, size = 30, normalized size = 0.61 \begin {gather*} -\frac {2 \left (-3 b^{2} x^{4}-14 a b \,x^{2}+21 a^{2}\right ) x}{21 \left (d x \right )^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/21*(-3*b^2*x^4-14*a*b*x^2+21*a^2)*x/(d*x)^(3/2)

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maxima [A]  time = 1.36, size = 44, normalized size = 0.90 \begin {gather*} -\frac {2 \, {\left (\frac {21 \, a^{2}}{\sqrt {d x}} - \frac {3 \, \left (d x\right )^{\frac {7}{2}} b^{2} + 14 \, \left (d x\right )^{\frac {3}{2}} a b d^{2}}{d^{4}}\right )}}{21 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/21*(21*a^2/sqrt(d*x) - (3*(d*x)^(7/2)*b^2 + 14*(d*x)^(3/2)*a*b*d^2)/d^4)/d

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mupad [B]  time = 0.05, size = 31, normalized size = 0.63 \begin {gather*} \frac {-42\,a^2+28\,a\,b\,x^2+6\,b^2\,x^4}{21\,d\,\sqrt {d\,x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(6*b^2*x^4 - 42*a^2 + 28*a*b*x^2)/(21*d*(d*x)^(1/2))

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sympy [A]  time = 0.66, size = 48, normalized size = 0.98 \begin {gather*} - \frac {2 a^{2}}{d^{\frac {3}{2}} \sqrt {x}} + \frac {4 a b x^{\frac {3}{2}}}{3 d^{\frac {3}{2}}} + \frac {2 b^{2} x^{\frac {7}{2}}}{7 d^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*a**2/(d**(3/2)*sqrt(x)) + 4*a*b*x**(3/2)/(3*d**(3/2)) + 2*b**2*x**(7/2)/(7*d**(3/2))

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