3.7.68 \(\int \frac {a^2+2 a b x^2+b^2 x^4}{\sqrt {d x}} \, dx\) [668]

Optimal. Leaf size=49 \[ \frac {2 a^2 \sqrt {d x}}{d}+\frac {4 a b (d x)^{5/2}}{5 d^3}+\frac {2 b^2 (d x)^{9/2}}{9 d^5} \]

[Out]

4/5*a*b*(d*x)^(5/2)/d^3+2/9*b^2*(d*x)^(9/2)/d^5+2*a^2*(d*x)^(1/2)/d

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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 \sqrt {d x}}{d}+\frac {4 a b (d x)^{5/2}}{5 d^3}+\frac {2 b^2 (d x)^{9/2}}{9 d^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Mathematica [A]
time = 0.02, size = 33, normalized size = 0.67 \begin {gather*} \frac {2 x \left (45 a^2+18 a b x^2+5 b^2 x^4\right )}{45 \sqrt {d x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*x*(45*a^2 + 18*a*b*x^2 + 5*b^2*x^4))/(45*Sqrt[d*x])

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Maple [A]
time = 0.02, size = 41, normalized size = 0.84

method	result	size
gosper	\(\frac {2 \left (5 b^{2} x^{4}+18 a b \,x^{2}+45 a^{2}\right ) x}{45 \sqrt {d x}}\)	\(30\)
risch	\(\frac {2 \left (5 b^{2} x^{4}+18 a b \,x^{2}+45 a^{2}\right ) x}{45 \sqrt {d x}}\)	\(30\)
trager	\(\frac {\left (\frac {2}{9} b^{2} x^{4}+\frac {4}{5} a b \,x^{2}+2 a^{2}\right ) \sqrt {d x}}{d}\)	\(31\)
derivativedivides	\(\frac {\frac {2 b^{2} \left (d x \right )^{\frac {9}{2}}}{9}+\frac {4 a b \,d^{2} \left (d x \right )^{\frac {5}{2}}}{5}+2 a^{2} d^{4} \sqrt {d x}}{d^{5}}\)	\(41\)
default	\(\frac {\frac {2 b^{2} \left (d x \right )^{\frac {9}{2}}}{9}+\frac {4 a b \,d^{2} \left (d x \right )^{\frac {5}{2}}}{5}+2 a^{2} d^{4} \sqrt {d x}}{d^{5}}\)	\(41\)

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)^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.34, size = 41, normalized size = 0.84 \begin {gather*} \frac {2 \, {\left (45 \, \sqrt {d x} a^{2} + \frac {5 \, \left (d x\right )^{\frac {9}{2}} b^{2}}{d^{4}} + \frac {18 \, \left (d x\right )^{\frac {5}{2}} a b}{d^{2}}\right )}}{45 \, 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)^(1/2),x, algorithm="maxima")

[Out]

2/45*(45*sqrt(d*x)*a^2 + 5*(d*x)^(9/2)*b^2/d^4 + 18*(d*x)^(5/2)*a*b/d^2)/d

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Fricas [A]
time = 0.35, size = 31, normalized size = 0.63 \begin {gather*} \frac {2 \, {\left (5 \, b^{2} x^{4} + 18 \, a b x^{2} + 45 \, a^{2}\right )} \sqrt {d x}}{45 \, 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)^(1/2),x, algorithm="fricas")

[Out]

2/45*(5*b^2*x^4 + 18*a*b*x^2 + 45*a^2)*sqrt(d*x)/d

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Sympy [A]
time = 0.22, size = 46, normalized size = 0.94 \begin {gather*} \frac {2 a^{2} x}{\sqrt {d x}} + \frac {4 a b x^{3}}{5 \sqrt {d x}} + \frac {2 b^{2} x^{5}}{9 \sqrt {d 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)**(1/2),x)

[Out]

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

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Giac [A]
time = 4.52, size = 41, normalized size = 0.84 \begin {gather*} \frac {2 \, {\left (5 \, \sqrt {d x} b^{2} x^{4} + 18 \, \sqrt {d x} a b x^{2} + 45 \, \sqrt {d x} a^{2}\right )}}{45 \, 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)^(1/2),x, algorithm="giac")

[Out]

2/45*(5*sqrt(d*x)*b^2*x^4 + 18*sqrt(d*x)*a*b*x^2 + 45*sqrt(d*x)*a^2)/d

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Mupad [B]
time = 0.05, size = 41, normalized size = 0.84 \begin {gather*} \frac {10\,b^2\,{\left (d\,x\right )}^{9/2}+90\,a^2\,d^4\,\sqrt {d\,x}+36\,a\,b\,d^2\,{\left (d\,x\right )}^{5/2}}{45\,d^5} \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)^(1/2),x)

[Out]

(10*b^2*(d*x)^(9/2) + 90*a^2*d^4*(d*x)^(1/2) + 36*a*b*d^2*(d*x)^(5/2))/(45*d^5)

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