3.743 \(\int (d x)^{3/2} (a^2+2 a b x^2+b^2 x^4)^{5/2} \, dx\)

Optimal. Leaf size=297 \[ \frac{2 b^5 (d x)^{25/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{25 d^{11} \left (a+b x^2\right )}+\frac{10 a b^4 (d x)^{21/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{21 d^9 \left (a+b x^2\right )}+\frac{20 a^2 b^3 (d x)^{17/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{17 d^7 \left (a+b x^2\right )}+\frac{20 a^3 b^2 (d x)^{13/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{13 d^5 \left (a+b x^2\right )}+\frac{10 a^4 b (d x)^{9/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{9 d^3 \left (a+b x^2\right )}+\frac{2 a^5 (d x)^{5/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{5 d \left (a+b x^2\right )} \]

[Out]

(2*a^5*(d*x)^(5/2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(5*d*(a + b*x^2)) + (10*a^4*b*(d*x)^(9/2)*Sqrt[a^2 + 2*a*b
*x^2 + b^2*x^4])/(9*d^3*(a + b*x^2)) + (20*a^3*b^2*(d*x)^(13/2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(13*d^5*(a +
b*x^2)) + (20*a^2*b^3*(d*x)^(17/2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(17*d^7*(a + b*x^2)) + (10*a*b^4*(d*x)^(21
/2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(21*d^9*(a + b*x^2)) + (2*b^5*(d*x)^(25/2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4
])/(25*d^11*(a + b*x^2))

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Rubi [A]  time = 0.0772811, 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} \[ \frac{2 b^5 (d x)^{25/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{25 d^{11} \left (a+b x^2\right )}+\frac{10 a b^4 (d x)^{21/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{21 d^9 \left (a+b x^2\right )}+\frac{20 a^2 b^3 (d x)^{17/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{17 d^7 \left (a+b x^2\right )}+\frac{20 a^3 b^2 (d x)^{13/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{13 d^5 \left (a+b x^2\right )}+\frac{10 a^4 b (d x)^{9/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{9 d^3 \left (a+b x^2\right )}+\frac{2 a^5 (d x)^{5/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{5 d \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]

Int[(d*x)^(3/2)*(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2),x]

[Out]

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

Mathematica [A]  time = 0.0338699, size = 88, normalized size = 0.3 \[ \frac{2 x (d x)^{3/2} \sqrt{\left (a+b x^2\right )^2} \left (204750 a^2 b^3 x^6+267750 a^3 b^2 x^4+193375 a^4 b x^2+69615 a^5+82875 a b^4 x^8+13923 b^5 x^{10}\right )}{348075 \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[(d*x)^(3/2)*(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2),x]

[Out]

(2*x*(d*x)^(3/2)*Sqrt[(a + b*x^2)^2]*(69615*a^5 + 193375*a^4*b*x^2 + 267750*a^3*b^2*x^4 + 204750*a^2*b^3*x^6 +
82875*a*b^4*x^8 + 13923*b^5*x^10))/(348075*(a + b*x^2))

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Maple [A]  time = 0.174, size = 83, normalized size = 0.3 \begin{align*}{\frac{2\,x \left ( 13923\,{b}^{5}{x}^{10}+82875\,a{b}^{4}{x}^{8}+204750\,{a}^{2}{b}^{3}{x}^{6}+267750\,{b}^{2}{a}^{3}{x}^{4}+193375\,{a}^{4}b{x}^{2}+69615\,{a}^{5} \right ) }{348075\, \left ( b{x}^{2}+a \right ) ^{5}} \left ( dx \right ) ^{{\frac{3}{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)^(3/2)*(b^2*x^4+2*a*b*x^2+a^2)^(5/2),x)

[Out]

2/348075*x*(13923*b^5*x^10+82875*a*b^4*x^8+204750*a^2*b^3*x^6+267750*a^3*b^2*x^4+193375*a^4*b*x^2+69615*a^5)*(
d*x)^(3/2)*((b*x^2+a)^2)^(5/2)/(b*x^2+a)^5

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Maxima [A]  time = 1.01767, size = 198, normalized size = 0.67 \begin{align*} \frac{2}{525} \,{\left (21 \, b^{5} d^{\frac{3}{2}} x^{3} + 25 \, a b^{4} d^{\frac{3}{2}} x\right )} x^{\frac{19}{2}} + \frac{8}{357} \,{\left (17 \, a b^{4} d^{\frac{3}{2}} x^{3} + 21 \, a^{2} b^{3} d^{\frac{3}{2}} x\right )} x^{\frac{15}{2}} + \frac{12}{221} \,{\left (13 \, a^{2} b^{3} d^{\frac{3}{2}} x^{3} + 17 \, a^{3} b^{2} d^{\frac{3}{2}} x\right )} x^{\frac{11}{2}} + \frac{8}{117} \,{\left (9 \, a^{3} b^{2} d^{\frac{3}{2}} x^{3} + 13 \, a^{4} b d^{\frac{3}{2}} x\right )} x^{\frac{7}{2}} + \frac{2}{45} \,{\left (5 \, a^{4} b d^{\frac{3}{2}} x^{3} + 9 \, a^{5} d^{\frac{3}{2}} x\right )} x^{\frac{3}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^(3/2)*(b^2*x^4+2*a*b*x^2+a^2)^(5/2),x, algorithm="maxima")

[Out]

2/525*(21*b^5*d^(3/2)*x^3 + 25*a*b^4*d^(3/2)*x)*x^(19/2) + 8/357*(17*a*b^4*d^(3/2)*x^3 + 21*a^2*b^3*d^(3/2)*x)
*x^(15/2) + 12/221*(13*a^2*b^3*d^(3/2)*x^3 + 17*a^3*b^2*d^(3/2)*x)*x^(11/2) + 8/117*(9*a^3*b^2*d^(3/2)*x^3 + 1
3*a^4*b*d^(3/2)*x)*x^(7/2) + 2/45*(5*a^4*b*d^(3/2)*x^3 + 9*a^5*d^(3/2)*x)*x^(3/2)

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Fricas [A]  time = 1.49071, size = 196, normalized size = 0.66 \begin{align*} \frac{2}{348075} \,{\left (13923 \, b^{5} d x^{12} + 82875 \, a b^{4} d x^{10} + 204750 \, a^{2} b^{3} d x^{8} + 267750 \, a^{3} b^{2} d x^{6} + 193375 \, a^{4} b d x^{4} + 69615 \, a^{5} d x^{2}\right )} \sqrt{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^(3/2)*(b^2*x^4+2*a*b*x^2+a^2)^(5/2),x, algorithm="fricas")

[Out]

2/348075*(13923*b^5*d*x^12 + 82875*a*b^4*d*x^10 + 204750*a^2*b^3*d*x^8 + 267750*a^3*b^2*d*x^6 + 193375*a^4*b*d
*x^4 + 69615*a^5*d*x^2)*sqrt(d*x)

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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)**(3/2)*(b**2*x**4+2*a*b*x**2+a**2)**(5/2),x)

[Out]

Timed out

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Giac [A]  time = 1.27631, size = 190, normalized size = 0.64 \begin{align*} \frac{2}{25} \, \sqrt{d x} b^{5} d x^{12} \mathrm{sgn}\left (b x^{2} + a\right ) + \frac{10}{21} \, \sqrt{d x} a b^{4} d x^{10} \mathrm{sgn}\left (b x^{2} + a\right ) + \frac{20}{17} \, \sqrt{d x} a^{2} b^{3} d x^{8} \mathrm{sgn}\left (b x^{2} + a\right ) + \frac{20}{13} \, \sqrt{d x} a^{3} b^{2} d x^{6} \mathrm{sgn}\left (b x^{2} + a\right ) + \frac{10}{9} \, \sqrt{d x} a^{4} b d x^{4} \mathrm{sgn}\left (b x^{2} + a\right ) + \frac{2}{5} \, \sqrt{d x} a^{5} d x^{2} \mathrm{sgn}\left (b x^{2} + a\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^(3/2)*(b^2*x^4+2*a*b*x^2+a^2)^(5/2),x, algorithm="giac")

[Out]

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