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

Optimal. Leaf size=195 \[ \frac{2 b^3 (d x)^{19/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{19 d^7 \left (a+b x^2\right )}+\frac{2 a b^2 (d x)^{15/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{5 d^5 \left (a+b x^2\right )}+\frac{6 a^2 b (d x)^{11/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{11 d^3 \left (a+b x^2\right )}+\frac{2 a^3 (d x)^{7/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{7 d \left (a+b x^2\right )} \]

[Out]

(2*a^3*(d*x)^(7/2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(7*d*(a + b*x^2)) + (6*a^2*b*(d*x)^(11/2)*Sqrt[a^2 + 2*a*b
*x^2 + b^2*x^4])/(11*d^3*(a + b*x^2)) + (2*a*b^2*(d*x)^(15/2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(5*d^5*(a + b*x
^2)) + (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))

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Rubi [A]  time = 0.0594125, antiderivative size = 195, 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} \[ \frac{2 b^3 (d x)^{19/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{19 d^7 \left (a+b x^2\right )}+\frac{2 a b^2 (d x)^{15/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{5 d^5 \left (a+b x^2\right )}+\frac{6 a^2 b (d x)^{11/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{11 d^3 \left (a+b x^2\right )}+\frac{2 a^3 (d x)^{7/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{7 d \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*a^3*(d*x)^(7/2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(7*d*(a + b*x^2)) + (6*a^2*b*(d*x)^(11/2)*Sqrt[a^2 + 2*a*b
*x^2 + b^2*x^4])/(11*d^3*(a + b*x^2)) + (2*a*b^2*(d*x)^(15/2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(5*d^5*(a + b*x
^2)) + (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))

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 )^{3/2} \, 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 )^3 \, dx}{b^2 \left (a b+b^2 x^2\right )}\\ &=\frac{\sqrt{a^2+2 a b x^2+b^2 x^4} \int \left (a^3 b^3 (d x)^{5/2}+\frac{3 a^2 b^4 (d x)^{9/2}}{d^2}+\frac{3 a b^5 (d x)^{13/2}}{d^4}+\frac{b^6 (d x)^{17/2}}{d^6}\right ) \, dx}{b^2 \left (a b+b^2 x^2\right )}\\ &=\frac{2 a^3 (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{6 a^2 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 b^2 (d x)^{15/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{5 d^5 \left (a+b x^2\right )}+\frac{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 )}\\ \end{align*}

Mathematica [A]  time = 0.0308913, size = 66, normalized size = 0.34 \[ \frac{2 x (d x)^{5/2} \sqrt{\left (a+b x^2\right )^2} \left (1995 a^2 b x^2+1045 a^3+1463 a b^2 x^4+385 b^3 x^6\right )}{7315 \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*x*(d*x)^(5/2)*Sqrt[(a + b*x^2)^2]*(1045*a^3 + 1995*a^2*b*x^2 + 1463*a*b^2*x^4 + 385*b^3*x^6))/(7315*(a + b*
x^2))

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Maple [A]  time = 0.167, size = 61, normalized size = 0.3 \begin{align*}{\frac{2\,x \left ( 385\,{b}^{3}{x}^{6}+1463\,a{x}^{4}{b}^{2}+1995\,{a}^{2}b{x}^{2}+1045\,{a}^{3} \right ) }{7315\, \left ( b{x}^{2}+a \right ) ^{3}} \left ( dx \right ) ^{{\frac{5}{2}}} \left ( \left ( b{x}^{2}+a \right ) ^{2} \right ) ^{{\frac{3}{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)^(3/2),x)

[Out]

2/7315*x*(385*b^3*x^6+1463*a*b^2*x^4+1995*a^2*b*x^2+1045*a^3)*(d*x)^(5/2)*((b*x^2+a)^2)^(3/2)/(b*x^2+a)^3

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Maxima [A]  time = 1.0142, size = 112, normalized size = 0.57 \begin{align*} \frac{2}{285} \,{\left (15 \, b^{3} d^{\frac{5}{2}} x^{3} + 19 \, a b^{2} d^{\frac{5}{2}} x\right )} x^{\frac{13}{2}} + \frac{4}{165} \,{\left (11 \, a b^{2} d^{\frac{5}{2}} x^{3} + 15 \, a^{2} b d^{\frac{5}{2}} x\right )} x^{\frac{9}{2}} + \frac{2}{77} \,{\left (7 \, a^{2} b d^{\frac{5}{2}} x^{3} + 11 \, a^{3} 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)^(3/2),x, algorithm="maxima")

[Out]

2/285*(15*b^3*d^(5/2)*x^3 + 19*a*b^2*d^(5/2)*x)*x^(13/2) + 4/165*(11*a*b^2*d^(5/2)*x^3 + 15*a^2*b*d^(5/2)*x)*x
^(9/2) + 2/77*(7*a^2*b*d^(5/2)*x^3 + 11*a^3*d^(5/2)*x)*x^(5/2)

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Fricas [A]  time = 1.24644, size = 131, normalized size = 0.67 \begin{align*} \frac{2}{7315} \,{\left (385 \, b^{3} d^{2} x^{9} + 1463 \, a b^{2} d^{2} x^{7} + 1995 \, a^{2} b d^{2} x^{5} + 1045 \, a^{3} 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)^(3/2),x, algorithm="fricas")

[Out]

2/7315*(385*b^3*d^2*x^9 + 1463*a*b^2*d^2*x^7 + 1995*a^2*b*d^2*x^5 + 1045*a^3*d^2*x^3)*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)**(5/2)*(b**2*x**4+2*a*b*x**2+a**2)**(3/2),x)

[Out]

Timed out

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Giac [A]  time = 1.27354, size = 134, normalized size = 0.69 \begin{align*} \frac{2}{19} \, \sqrt{d x} b^{3} d^{2} x^{9} \mathrm{sgn}\left (b x^{2} + a\right ) + \frac{2}{5} \, \sqrt{d x} a b^{2} d^{2} x^{7} \mathrm{sgn}\left (b x^{2} + a\right ) + \frac{6}{11} \, \sqrt{d x} a^{2} b d^{2} x^{5} \mathrm{sgn}\left (b x^{2} + a\right ) + \frac{2}{7} \, \sqrt{d x} a^{3} 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)^(3/2),x, algorithm="giac")

[Out]

2/19*sqrt(d*x)*b^3*d^2*x^9*sgn(b*x^2 + a) + 2/5*sqrt(d*x)*a*b^2*d^2*x^7*sgn(b*x^2 + a) + 6/11*sqrt(d*x)*a^2*b*
d^2*x^5*sgn(b*x^2 + a) + 2/7*sqrt(d*x)*a^3*d^2*x^3*sgn(b*x^2 + a)