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

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

[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))

________________________________________________________________________________________

Rubi [A]  time = 0.0823243, antiderivative size = 297, normalized size of antiderivative = 1., 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 = {1112, 270} \[ \frac{2 b^5 (d x)^{27/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{27 d^{11} \left (a+b x^2\right )}+\frac{10 a b^4 (d x)^{23/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{23 d^9 \left (a+b x^2\right )}+\frac{20 a^2 b^3 (d x)^{19/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{19 d^7 \left (a+b x^2\right )}+\frac{4 a^3 b^2 (d x)^{15/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{3 d^5 \left (a+b x^2\right )}+\frac{10 a^4 b (d x)^{11/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{11 d^3 \left (a+b x^2\right )}+\frac{2 a^5 (d x)^{7/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{7 d \left (a+b x^2\right )} \]

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 1112

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]

Rule 270

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 (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*}

Mathematica [A]  time = 0.042513, size = 88, normalized size = 0.3 \[ \frac{2 x (d x)^{5/2} \sqrt{\left (a+b x^2\right )^2} \left (478170 a^2 b^3 x^6+605682 a^3 b^2 x^4+412965 a^4 b x^2+129789 a^5+197505 a b^4 x^8+33649 b^5 x^{10}\right )}{908523 \left (a+b x^2\right )} \]

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))

________________________________________________________________________________________

Maple [A]  time = 0.174, size = 83, normalized size = 0.3 \begin{align*}{\frac{2\,x \left ( 33649\,{b}^{5}{x}^{10}+197505\,a{b}^{4}{x}^{8}+478170\,{a}^{2}{b}^{3}{x}^{6}+605682\,{b}^{2}{a}^{3}{x}^{4}+412965\,{a}^{4}b{x}^{2}+129789\,{a}^{5} \right ) }{908523\, \left ( b{x}^{2}+a \right ) ^{5}} \left ( dx \right ) ^{{\frac{5}{2}}} \left ( \left ( b{x}^{2}+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \end{align*}

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)

[Out]

2/908523*x*(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)
*(d*x)^(5/2)*((b*x^2+a)^2)^(5/2)/(b*x^2+a)^5

________________________________________________________________________________________

Maxima [A]  time = 1.01273, size = 198, normalized size = 0.67 \begin{align*} \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{align*}

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)

________________________________________________________________________________________

Fricas [A]  time = 1.47708, size = 215, normalized size = 0.72 \begin{align*} \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{align*}

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)

________________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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]

Timed out

________________________________________________________________________________________

Giac [A]  time = 1.24999, size = 207, normalized size = 0.7 \begin{align*} \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{align*}

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)