3.8.42 \(\int (d x)^{5/2} (a^2+2 a b x^2+b^2 x^4)^{5/2} \, dx\) [742]

Optimal. Leaf size=297 \[ \frac {2 a^5 (d x)^{7/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{7 d \left (a+b x^2\right )}+\frac {10 a^4 b (d x)^{11/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{11 d^3 \left (a+b x^2\right )}+\frac {4 a^3 b^2 (d x)^{15/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{3 d^5 \left (a+b x^2\right )}+\frac {20 a^2 b^3 (d x)^{19/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{19 d^7 \left (a+b x^2\right )}+\frac {10 a b^4 (d x)^{23/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{23 d^9 \left (a+b x^2\right )}+\frac {2 b^5 (d x)^{27/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{27 d^{11} \left (a+b x^2\right )} \]

[Out]

2/7*a^5*(d*x)^(7/2)*((b*x^2+a)^2)^(1/2)/d/(b*x^2+a)+10/11*a^4*b*(d*x)^(11/2)*((b*x^2+a)^2)^(1/2)/d^3/(b*x^2+a)
+4/3*a^3*b^2*(d*x)^(15/2)*((b*x^2+a)^2)^(1/2)/d^5/(b*x^2+a)+20/19*a^2*b^3*(d*x)^(19/2)*((b*x^2+a)^2)^(1/2)/d^7
/(b*x^2+a)+10/23*a*b^4*(d*x)^(23/2)*((b*x^2+a)^2)^(1/2)/d^9/(b*x^2+a)+2/27*b^5*(d*x)^(27/2)*((b*x^2+a)^2)^(1/2
)/d^11/(b*x^2+a)

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Rubi [A]
time = 0.06, antiderivative size = 297, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {1126, 276} \begin {gather*} \frac {2 b^5 (d x)^{27/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{27 d^{11} \left (a+b x^2\right )}+\frac {10 a b^4 (d x)^{23/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{23 d^9 \left (a+b x^2\right )}+\frac {20 a^2 b^3 (d x)^{19/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{19 d^7 \left (a+b x^2\right )}+\frac {2 a^5 (d x)^{7/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{7 d \left (a+b x^2\right )}+\frac {10 a^4 b (d x)^{11/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{11 d^3 \left (a+b x^2\right )}+\frac {4 a^3 b^2 (d x)^{15/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{3 d^5 \left (a+b x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*a^5*(d*x)^(7/2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(7*d*(a + b*x^2)) + (10*a^4*b*(d*x)^(11/2)*Sqrt[a^2 + 2*a*
b*x^2 + b^2*x^4])/(11*d^3*(a + b*x^2)) + (4*a^3*b^2*(d*x)^(15/2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(3*d^5*(a +
b*x^2)) + (20*a^2*b^3*(d*x)^(19/2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(19*d^7*(a + b*x^2)) + (10*a*b^4*(d*x)^(23
/2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(23*d^9*(a + b*x^2)) + (2*b^5*(d*x)^(27/2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4
])/(27*d^11*(a + b*x^2))

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]

Rule 1126

Int[((d_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Dist[(a + b*x^2 + c*x^4)^FracPa
rt[p]/(c^IntPart[p]*(b/2 + c*x^2)^(2*FracPart[p])), Int[(d*x)^m*(b/2 + c*x^2)^(2*p), x], x] /; FreeQ[{a, b, c,
d, m, p}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p - 1/2]

Rubi steps

\begin {align*} \int (d x)^{5/2} \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2} \, dx &=\frac {\sqrt {a^2+2 a b x^2+b^2 x^4} \int (d x)^{5/2} \left (a b+b^2 x^2\right )^5 \, dx}{b^4 \left (a b+b^2 x^2\right )}\\ &=\frac {\sqrt {a^2+2 a b x^2+b^2 x^4} \int \left (a^5 b^5 (d x)^{5/2}+\frac {5 a^4 b^6 (d x)^{9/2}}{d^2}+\frac {10 a^3 b^7 (d x)^{13/2}}{d^4}+\frac {10 a^2 b^8 (d x)^{17/2}}{d^6}+\frac {5 a b^9 (d x)^{21/2}}{d^8}+\frac {b^{10} (d x)^{25/2}}{d^{10}}\right ) \, dx}{b^4 \left (a b+b^2 x^2\right )}\\ &=\frac {2 a^5 (d x)^{7/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{7 d \left (a+b x^2\right )}+\frac {10 a^4 b (d x)^{11/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{11 d^3 \left (a+b x^2\right )}+\frac {4 a^3 b^2 (d x)^{15/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{3 d^5 \left (a+b x^2\right )}+\frac {20 a^2 b^3 (d x)^{19/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{19 d^7 \left (a+b x^2\right )}+\frac {10 a b^4 (d x)^{23/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{23 d^9 \left (a+b x^2\right )}+\frac {2 b^5 (d x)^{27/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{27 d^{11} \left (a+b x^2\right )}\\ \end {align*}

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Mathematica [A]
time = 0.04, size = 88, normalized size = 0.30 \begin {gather*} \frac {2 x (d x)^{5/2} \sqrt {\left (a+b x^2\right )^2} \left (129789 a^5+412965 a^4 b x^2+605682 a^3 b^2 x^4+478170 a^2 b^3 x^6+197505 a b^4 x^8+33649 b^5 x^{10}\right )}{908523 \left (a+b x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*x*(d*x)^(5/2)*Sqrt[(a + b*x^2)^2]*(129789*a^5 + 412965*a^4*b*x^2 + 605682*a^3*b^2*x^4 + 478170*a^2*b^3*x^6
+ 197505*a*b^4*x^8 + 33649*b^5*x^10))/(908523*(a + b*x^2))

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Maple [A]
time = 0.04, size = 85, normalized size = 0.29

method	result	size
gosper	\(\frac {2 x \left (33649 b^{5} x^{10}+197505 b^{4} a \,x^{8}+478170 a^{2} b^{3} x^{6}+605682 b^{2} a^{3} x^{4}+412965 b \,a^{4} x^{2}+129789 a^{5}\right ) \left (d x \right )^{\frac {5}{2}} \left (\left (b \,x^{2}+a \right )^{2}\right )^{\frac {5}{2}}}{908523 \left (b \,x^{2}+a \right )^{5}}\)	\(83\)
default	\(\frac {2 \left (\left (b \,x^{2}+a \right )^{2}\right )^{\frac {5}{2}} \left (d x \right )^{\frac {7}{2}} \left (33649 b^{5} x^{10}+197505 b^{4} a \,x^{8}+478170 a^{2} b^{3} x^{6}+605682 b^{2} a^{3} x^{4}+412965 b \,a^{4} x^{2}+129789 a^{5}\right )}{908523 d \left (b \,x^{2}+a \right )^{5}}\)	\(85\)
risch	\(\frac {2 d^{3} \sqrt {\left (b \,x^{2}+a \right )^{2}}\, x^{4} \left (33649 b^{5} x^{10}+197505 b^{4} a \,x^{8}+478170 a^{2} b^{3} x^{6}+605682 b^{2} a^{3} x^{4}+412965 b \,a^{4} x^{2}+129789 a^{5}\right )}{908523 \left (b \,x^{2}+a \right ) \sqrt {d x}}\)	\(88\)

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)^(5/2),x,method=_RETURNVERBOSE)

[Out]

2/908523*((b*x^2+a)^2)^(5/2)*(d*x)^(7/2)/d*(33649*b^5*x^10+197505*a*b^4*x^8+478170*a^2*b^3*x^6+605682*a^3*b^2*
x^4+412965*a^4*b*x^2+129789*a^5)/(b*x^2+a)^5

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Maxima [A]
time = 0.28, size = 147, normalized size = 0.49 \begin {gather*} \frac {2}{621} \, {\left (23 \, b^{5} d^{\frac {5}{2}} x^{3} + 27 \, a b^{4} d^{\frac {5}{2}} x\right )} x^{\frac {21}{2}} + \frac {8}{437} \, {\left (19 \, a b^{4} d^{\frac {5}{2}} x^{3} + 23 \, a^{2} b^{3} d^{\frac {5}{2}} x\right )} x^{\frac {17}{2}} + \frac {4}{95} \, {\left (15 \, a^{2} b^{3} d^{\frac {5}{2}} x^{3} + 19 \, a^{3} b^{2} d^{\frac {5}{2}} x\right )} x^{\frac {13}{2}} + \frac {8}{165} \, {\left (11 \, a^{3} b^{2} d^{\frac {5}{2}} x^{3} + 15 \, a^{4} b d^{\frac {5}{2}} x\right )} x^{\frac {9}{2}} + \frac {2}{77} \, {\left (7 \, a^{4} b d^{\frac {5}{2}} x^{3} + 11 \, a^{5} d^{\frac {5}{2}} x\right )} x^{\frac {5}{2}} \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)^(5/2),x, algorithm="maxima")

[Out]

2/621*(23*b^5*d^(5/2)*x^3 + 27*a*b^4*d^(5/2)*x)*x^(21/2) + 8/437*(19*a*b^4*d^(5/2)*x^3 + 23*a^2*b^3*d^(5/2)*x)
*x^(17/2) + 4/95*(15*a^2*b^3*d^(5/2)*x^3 + 19*a^3*b^2*d^(5/2)*x)*x^(13/2) + 8/165*(11*a^3*b^2*d^(5/2)*x^3 + 15
*a^4*b*d^(5/2)*x)*x^(9/2) + 2/77*(7*a^4*b*d^(5/2)*x^3 + 11*a^5*d^(5/2)*x)*x^(5/2)

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Fricas [A]
time = 0.35, size = 82, normalized size = 0.28 \begin {gather*} \frac {2}{908523} \, {\left (33649 \, b^{5} d^{2} x^{13} + 197505 \, a b^{4} d^{2} x^{11} + 478170 \, a^{2} b^{3} d^{2} x^{9} + 605682 \, a^{3} b^{2} d^{2} x^{7} + 412965 \, a^{4} b d^{2} x^{5} + 129789 \, a^{5} 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)^(5/2),x, algorithm="fricas")

[Out]

2/908523*(33649*b^5*d^2*x^13 + 197505*a*b^4*d^2*x^11 + 478170*a^2*b^3*d^2*x^9 + 605682*a^3*b^2*d^2*x^7 + 41296
5*a^4*b*d^2*x^5 + 129789*a^5*d^2*x^3)*sqrt(d*x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (d x\right )^{\frac {5}{2}} \left (\left (a + b x^{2}\right )^{2}\right )^{\frac {5}{2}}\, dx \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)**(5/2),x)

[Out]

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

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Giac [A]
time = 4.01, size = 153, normalized size = 0.52 \begin {gather*} \frac {2}{27} \, \sqrt {d x} b^{5} d^{2} x^{13} \mathrm {sgn}\left (b x^{2} + a\right ) + \frac {10}{23} \, \sqrt {d x} a b^{4} d^{2} x^{11} \mathrm {sgn}\left (b x^{2} + a\right ) + \frac {20}{19} \, \sqrt {d x} a^{2} b^{3} d^{2} x^{9} \mathrm {sgn}\left (b x^{2} + a\right ) + \frac {4}{3} \, \sqrt {d x} a^{3} b^{2} d^{2} x^{7} \mathrm {sgn}\left (b x^{2} + a\right ) + \frac {10}{11} \, \sqrt {d x} a^{4} b d^{2} x^{5} \mathrm {sgn}\left (b x^{2} + a\right ) + \frac {2}{7} \, \sqrt {d x} a^{5} d^{2} x^{3} \mathrm {sgn}\left (b x^{2} + a\right ) \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)^(5/2),x, algorithm="giac")

[Out]

2/27*sqrt(d*x)*b^5*d^2*x^13*sgn(b*x^2 + a) + 10/23*sqrt(d*x)*a*b^4*d^2*x^11*sgn(b*x^2 + a) + 20/19*sqrt(d*x)*a
^2*b^3*d^2*x^9*sgn(b*x^2 + a) + 4/3*sqrt(d*x)*a^3*b^2*d^2*x^7*sgn(b*x^2 + a) + 10/11*sqrt(d*x)*a^4*b*d^2*x^5*s
gn(b*x^2 + a) + 2/7*sqrt(d*x)*a^5*d^2*x^3*sgn(b*x^2 + a)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int {\left (d\,x\right )}^{5/2}\,{\left (a^2+2\,a\,b\,x^2+b^2\,x^4\right )}^{5/2} \,d x \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)^(5/2),x)

[Out]

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

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