3.7.82 \(\int \frac {(a^2+2 a b x^2+b^2 x^4)^3}{\sqrt {d x}} \, dx\) [682]

Optimal. Leaf size=129 \[ \frac {2 a^6 \sqrt {d x}}{d}+\frac {12 a^5 b (d x)^{5/2}}{5 d^3}+\frac {10 a^4 b^2 (d x)^{9/2}}{3 d^5}+\frac {40 a^3 b^3 (d x)^{13/2}}{13 d^7}+\frac {30 a^2 b^4 (d x)^{17/2}}{17 d^9}+\frac {4 a b^5 (d x)^{21/2}}{7 d^{11}}+\frac {2 b^6 (d x)^{25/2}}{25 d^{13}} \]

[Out]

12/5*a^5*b*(d*x)^(5/2)/d^3+10/3*a^4*b^2*(d*x)^(9/2)/d^5+40/13*a^3*b^3*(d*x)^(13/2)/d^7+30/17*a^2*b^4*(d*x)^(17
/2)/d^9+4/7*a*b^5*(d*x)^(21/2)/d^11+2/25*b^6*(d*x)^(25/2)/d^13+2*a^6*(d*x)^(1/2)/d

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Rubi [A]
time = 0.04, antiderivative size = 129, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {28, 276} \begin {gather*} \frac {2 a^6 \sqrt {d x}}{d}+\frac {12 a^5 b (d x)^{5/2}}{5 d^3}+\frac {10 a^4 b^2 (d x)^{9/2}}{3 d^5}+\frac {40 a^3 b^3 (d x)^{13/2}}{13 d^7}+\frac {30 a^2 b^4 (d x)^{17/2}}{17 d^9}+\frac {4 a b^5 (d x)^{21/2}}{7 d^{11}}+\frac {2 b^6 (d x)^{25/2}}{25 d^{13}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*a^6*Sqrt[d*x])/d + (12*a^5*b*(d*x)^(5/2))/(5*d^3) + (10*a^4*b^2*(d*x)^(9/2))/(3*d^5) + (40*a^3*b^3*(d*x)^(1
3/2))/(13*d^7) + (30*a^2*b^4*(d*x)^(17/2))/(17*d^9) + (4*a*b^5*(d*x)^(21/2))/(7*d^11) + (2*b^6*(d*x)^(25/2))/(
25*d^13)

Rule 28

Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Dist[1/c^p, Int[u*(b/2 + c*x^n)^(2*
p), x], x] /; FreeQ[{a, b, c, n}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 276

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rubi steps

\begin {align*} \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^3}{\sqrt {d x}} \, dx &=\frac {\int \frac {\left (a b+b^2 x^2\right )^6}{\sqrt {d x}} \, dx}{b^6}\\ &=\frac {\int \left (\frac {a^6 b^6}{\sqrt {d x}}+\frac {6 a^5 b^7 (d x)^{3/2}}{d^2}+\frac {15 a^4 b^8 (d x)^{7/2}}{d^4}+\frac {20 a^3 b^9 (d x)^{11/2}}{d^6}+\frac {15 a^2 b^{10} (d x)^{15/2}}{d^8}+\frac {6 a b^{11} (d x)^{19/2}}{d^{10}}+\frac {b^{12} (d x)^{23/2}}{d^{12}}\right ) \, dx}{b^6}\\ &=\frac {2 a^6 \sqrt {d x}}{d}+\frac {12 a^5 b (d x)^{5/2}}{5 d^3}+\frac {10 a^4 b^2 (d x)^{9/2}}{3 d^5}+\frac {40 a^3 b^3 (d x)^{13/2}}{13 d^7}+\frac {30 a^2 b^4 (d x)^{17/2}}{17 d^9}+\frac {4 a b^5 (d x)^{21/2}}{7 d^{11}}+\frac {2 b^6 (d x)^{25/2}}{25 d^{13}}\\ \end {align*}

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Mathematica [A]
time = 0.02, size = 77, normalized size = 0.60 \begin {gather*} \frac {2 x \left (116025 a^6+139230 a^5 b x^2+193375 a^4 b^2 x^4+178500 a^3 b^3 x^6+102375 a^2 b^4 x^8+33150 a b^5 x^{10}+4641 b^6 x^{12}\right )}{116025 \sqrt {d x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*x*(116025*a^6 + 139230*a^5*b*x^2 + 193375*a^4*b^2*x^4 + 178500*a^3*b^3*x^6 + 102375*a^2*b^4*x^8 + 33150*a*b
^5*x^10 + 4641*b^6*x^12))/(116025*Sqrt[d*x])

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Maple [A]
time = 0.04, size = 105, normalized size = 0.81

method	result	size
gosper	\(\frac {2 \left (4641 b^{6} x^{12}+33150 a \,b^{5} x^{10}+102375 a^{2} b^{4} x^{8}+178500 a^{3} b^{3} x^{6}+193375 a^{4} b^{2} x^{4}+139230 a^{5} b \,x^{2}+116025 a^{6}\right ) x}{116025 \sqrt {d x}}\)	\(74\)
risch	\(\frac {2 \left (4641 b^{6} x^{12}+33150 a \,b^{5} x^{10}+102375 a^{2} b^{4} x^{8}+178500 a^{3} b^{3} x^{6}+193375 a^{4} b^{2} x^{4}+139230 a^{5} b \,x^{2}+116025 a^{6}\right ) x}{116025 \sqrt {d x}}\)	\(74\)
trager	\(\frac {\left (\frac {2}{25} b^{6} x^{12}+\frac {4}{7} a \,b^{5} x^{10}+\frac {30}{17} a^{2} b^{4} x^{8}+\frac {40}{13} a^{3} b^{3} x^{6}+\frac {10}{3} a^{4} b^{2} x^{4}+\frac {12}{5} a^{5} b \,x^{2}+2 a^{6}\right ) \sqrt {d x}}{d}\)	\(75\)
derivativedivides	\(\frac {\frac {2 b^{6} \left (d x \right )^{\frac {25}{2}}}{25}+\frac {4 a \,b^{5} d^{2} \left (d x \right )^{\frac {21}{2}}}{7}+\frac {30 a^{2} d^{4} b^{4} \left (d x \right )^{\frac {17}{2}}}{17}+\frac {40 a^{3} d^{6} b^{3} \left (d x \right )^{\frac {13}{2}}}{13}+\frac {10 a^{4} d^{8} b^{2} \left (d x \right )^{\frac {9}{2}}}{3}+\frac {12 a^{5} d^{10} b \left (d x \right )^{\frac {5}{2}}}{5}+2 a^{6} d^{12} \sqrt {d x}}{d^{13}}\)	\(105\)
default	\(\frac {\frac {2 b^{6} \left (d x \right )^{\frac {25}{2}}}{25}+\frac {4 a \,b^{5} d^{2} \left (d x \right )^{\frac {21}{2}}}{7}+\frac {30 a^{2} d^{4} b^{4} \left (d x \right )^{\frac {17}{2}}}{17}+\frac {40 a^{3} d^{6} b^{3} \left (d x \right )^{\frac {13}{2}}}{13}+\frac {10 a^{4} d^{8} b^{2} \left (d x \right )^{\frac {9}{2}}}{3}+\frac {12 a^{5} d^{10} b \left (d x \right )^{\frac {5}{2}}}{5}+2 a^{6} d^{12} \sqrt {d x}}{d^{13}}\)	\(105\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b^2*x^4+2*a*b*x^2+a^2)^3/(d*x)^(1/2),x,method=_RETURNVERBOSE)

[Out]

2/d^13*(1/25*b^6*(d*x)^(25/2)+2/7*a*b^5*d^2*(d*x)^(21/2)+15/17*a^2*d^4*b^4*(d*x)^(17/2)+20/13*a^3*d^6*b^3*(d*x
)^(13/2)+5/3*a^4*d^8*b^2*(d*x)^(9/2)+6/5*a^5*d^10*b*(d*x)^(5/2)+a^6*d^12*(d*x)^(1/2))

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Maxima [A]
time = 0.29, size = 155, normalized size = 1.20 \begin {gather*} \frac {2 \, {\left (116025 \, \sqrt {d x} a^{6} + \frac {4641 \, \left (d x\right )^{\frac {25}{2}} b^{6}}{d^{12}} + \frac {33150 \, \left (d x\right )^{\frac {21}{2}} a b^{5}}{d^{10}} + \frac {81900 \, \left (d x\right )^{\frac {17}{2}} a^{2} b^{4}}{d^{8}} + \frac {71400 \, \left (d x\right )^{\frac {13}{2}} a^{3} b^{3}}{d^{6}} + 7735 \, {\left (\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 )} a^{4} + 175 \, {\left (\frac {117 \, \left (d x\right )^{\frac {17}{2}} b^{4}}{d^{8}} + \frac {612 \, \left (d x\right )^{\frac {13}{2}} a b^{3}}{d^{6}} + \frac {884 \, \left (d x\right )^{\frac {9}{2}} a^{2} b^{2}}{d^{4}}\right )} a^{2}\right )}}{116025 \, 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)^3/(d*x)^(1/2),x, algorithm="maxima")

[Out]

2/116025*(116025*sqrt(d*x)*a^6 + 4641*(d*x)^(25/2)*b^6/d^12 + 33150*(d*x)^(21/2)*a*b^5/d^10 + 81900*(d*x)^(17/
2)*a^2*b^4/d^8 + 71400*(d*x)^(13/2)*a^3*b^3/d^6 + 7735*(5*(d*x)^(9/2)*b^2/d^4 + 18*(d*x)^(5/2)*a*b/d^2)*a^4 +
175*(117*(d*x)^(17/2)*b^4/d^8 + 612*(d*x)^(13/2)*a*b^3/d^6 + 884*(d*x)^(9/2)*a^2*b^2/d^4)*a^2)/d

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Fricas [A]
time = 0.34, size = 75, normalized size = 0.58 \begin {gather*} \frac {2 \, {\left (4641 \, b^{6} x^{12} + 33150 \, a b^{5} x^{10} + 102375 \, a^{2} b^{4} x^{8} + 178500 \, a^{3} b^{3} x^{6} + 193375 \, a^{4} b^{2} x^{4} + 139230 \, a^{5} b x^{2} + 116025 \, a^{6}\right )} \sqrt {d x}}{116025 \, 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)^3/(d*x)^(1/2),x, algorithm="fricas")

[Out]

2/116025*(4641*b^6*x^12 + 33150*a*b^5*x^10 + 102375*a^2*b^4*x^8 + 178500*a^3*b^3*x^6 + 193375*a^4*b^2*x^4 + 13
9230*a^5*b*x^2 + 116025*a^6)*sqrt(d*x)/d

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Sympy [A]
time = 0.52, size = 128, normalized size = 0.99 \begin {gather*} \frac {2 a^{6} x}{\sqrt {d x}} + \frac {12 a^{5} b x^{3}}{5 \sqrt {d x}} + \frac {10 a^{4} b^{2} x^{5}}{3 \sqrt {d x}} + \frac {40 a^{3} b^{3} x^{7}}{13 \sqrt {d x}} + \frac {30 a^{2} b^{4} x^{9}}{17 \sqrt {d x}} + \frac {4 a b^{5} x^{11}}{7 \sqrt {d x}} + \frac {2 b^{6} x^{13}}{25 \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)**3/(d*x)**(1/2),x)

[Out]

2*a**6*x/sqrt(d*x) + 12*a**5*b*x**3/(5*sqrt(d*x)) + 10*a**4*b**2*x**5/(3*sqrt(d*x)) + 40*a**3*b**3*x**7/(13*sq
rt(d*x)) + 30*a**2*b**4*x**9/(17*sqrt(d*x)) + 4*a*b**5*x**11/(7*sqrt(d*x)) + 2*b**6*x**13/(25*sqrt(d*x))

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Giac [A]
time = 3.88, size = 105, normalized size = 0.81 \begin {gather*} \frac {2 \, {\left (4641 \, \sqrt {d x} b^{6} x^{12} + 33150 \, \sqrt {d x} a b^{5} x^{10} + 102375 \, \sqrt {d x} a^{2} b^{4} x^{8} + 178500 \, \sqrt {d x} a^{3} b^{3} x^{6} + 193375 \, \sqrt {d x} a^{4} b^{2} x^{4} + 139230 \, \sqrt {d x} a^{5} b x^{2} + 116025 \, \sqrt {d x} a^{6}\right )}}{116025 \, 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)^3/(d*x)^(1/2),x, algorithm="giac")

[Out]

2/116025*(4641*sqrt(d*x)*b^6*x^12 + 33150*sqrt(d*x)*a*b^5*x^10 + 102375*sqrt(d*x)*a^2*b^4*x^8 + 178500*sqrt(d*
x)*a^3*b^3*x^6 + 193375*sqrt(d*x)*a^4*b^2*x^4 + 139230*sqrt(d*x)*a^5*b*x^2 + 116025*sqrt(d*x)*a^6)/d

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Mupad [B]
time = 0.04, size = 103, normalized size = 0.80 \begin {gather*} \frac {2\,a^6\,\sqrt {d\,x}}{d}+\frac {2\,b^6\,{\left (d\,x\right )}^{25/2}}{25\,d^{13}}+\frac {10\,a^4\,b^2\,{\left (d\,x\right )}^{9/2}}{3\,d^5}+\frac {40\,a^3\,b^3\,{\left (d\,x\right )}^{13/2}}{13\,d^7}+\frac {30\,a^2\,b^4\,{\left (d\,x\right )}^{17/2}}{17\,d^9}+\frac {12\,a^5\,b\,{\left (d\,x\right )}^{5/2}}{5\,d^3}+\frac {4\,a\,b^5\,{\left (d\,x\right )}^{21/2}}{7\,d^{11}} \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)^3/(d*x)^(1/2),x)

[Out]

(2*a^6*(d*x)^(1/2))/d + (2*b^6*(d*x)^(25/2))/(25*d^13) + (10*a^4*b^2*(d*x)^(9/2))/(3*d^5) + (40*a^3*b^3*(d*x)^
(13/2))/(13*d^7) + (30*a^2*b^4*(d*x)^(17/2))/(17*d^9) + (12*a^5*b*(d*x)^(5/2))/(5*d^3) + (4*a*b^5*(d*x)^(21/2)
)/(7*d^11)

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