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

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

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Rubi [A]  time = 0.08, 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 = {1112, 270} \begin {gather*} \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 )} \end {gather*}

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 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]

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]

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*}

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Mathematica [A]  time = 0.03, size = 88, normalized size = 0.30 \begin {gather*} \frac {2 x (d x)^{3/2} \sqrt {\left (a+b x^2\right )^2} \left (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}\right )}{348075 \left (a+b x^2\right )} \end {gather*}

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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IntegrateAlgebraic [A]  time = 119.00, size = 141, normalized size = 0.47 \begin {gather*} \frac {2 \left (a d^2+b d^2 x^2\right ) \left (69615 a^5 d^{10} (d x)^{5/2}+193375 a^4 b d^8 (d x)^{9/2}+267750 a^3 b^2 d^6 (d x)^{13/2}+204750 a^2 b^3 d^4 (d x)^{17/2}+82875 a b^4 d^2 (d x)^{21/2}+13923 b^5 (d x)^{25/2}\right )}{348075 d^{13} \sqrt {\frac {\left (a d^2+b d^2 x^2\right )^2}{d^4}}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(a*d^2 + b*d^2*x^2)*(69615*a^5*d^10*(d*x)^(5/2) + 193375*a^4*b*d^8*(d*x)^(9/2) + 267750*a^3*b^2*d^6*(d*x)^(
13/2) + 204750*a^2*b^3*d^4*(d*x)^(17/2) + 82875*a*b^4*d^2*(d*x)^(21/2) + 13923*b^5*(d*x)^(25/2)))/(348075*d^13
*Sqrt[(a*d^2 + b*d^2*x^2)^2/d^4])

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fricas [A]  time = 1.10, size = 70, normalized size = 0.24 \begin {gather*} \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 {gather*}

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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giac [A]  time = 0.20, size = 138, normalized size = 0.46 \begin {gather*} \frac {2}{348075} \, {\left (13923 \, \sqrt {d x} b^{5} x^{12} \mathrm {sgn}\left (b x^{2} + a\right ) + 82875 \, \sqrt {d x} a b^{4} x^{10} \mathrm {sgn}\left (b x^{2} + a\right ) + 204750 \, \sqrt {d x} a^{2} b^{3} x^{8} \mathrm {sgn}\left (b x^{2} + a\right ) + 267750 \, \sqrt {d x} a^{3} b^{2} x^{6} \mathrm {sgn}\left (b x^{2} + a\right ) + 193375 \, \sqrt {d x} a^{4} b x^{4} \mathrm {sgn}\left (b x^{2} + a\right ) + 69615 \, \sqrt {d x} a^{5} x^{2} \mathrm {sgn}\left (b x^{2} + a\right )\right )} d \end {gather*}

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]

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

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maple [A]  time = 0.01, size = 83, normalized size = 0.28 \begin {gather*} \frac {2 \left (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}\right ) \left (d x \right )^{\frac {3}{2}} \left (\left (b \,x^{2}+a \right )^{2}\right )^{\frac {5}{2}} x}{348075 \left (b \,x^{2}+a \right )^{5}} \end {gather*}

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 = 1.42, size = 147, normalized size = 0.49 \begin {gather*} \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 {gather*}

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int {\left (d\,x\right )}^{3/2}\,{\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)^(3/2)*(a^2 + b^2*x^4 + 2*a*b*x^2)^(5/2),x)

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (d x\right )^{\frac {3}{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)**(3/2)*(b**2*x**4+2*a*b*x**2+a**2)**(5/2),x)

[Out]

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

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