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

Optimal. Leaf size=295 \[ \frac {2 b^5 (d x)^{19/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{19 d^{11} \left (a+b x^2\right )}+\frac {2 a b^4 (d x)^{15/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{3 d^9 \left (a+b x^2\right )}+\frac {20 a^2 b^3 (d x)^{11/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{11 d^7 \left (a+b x^2\right )}-\frac {2 a^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}{d \sqrt {d x} \left (a+b x^2\right )}+\frac {10 a^4 b (d x)^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{3 d^3 \left (a+b x^2\right )}+\frac {20 a^3 b^2 (d x)^{7/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{7 d^5 \left (a+b x^2\right )} \]

[Out]

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

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Rubi [A]  time = 0.08, antiderivative size = 295, 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} \[ \frac {2 b^5 (d x)^{19/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{19 d^{11} \left (a+b x^2\right )}+\frac {2 a b^4 (d x)^{15/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{3 d^9 \left (a+b x^2\right )}+\frac {20 a^2 b^3 (d x)^{11/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{11 d^7 \left (a+b x^2\right )}+\frac {20 a^3 b^2 (d x)^{7/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{7 d^5 \left (a+b x^2\right )}+\frac {10 a^4 b (d x)^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{3 d^3 \left (a+b x^2\right )}-\frac {2 a^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}{d \sqrt {d x} \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*a^5*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(d*Sqrt[d*x]*(a + b*x^2)) + (10*a^4*b*(d*x)^(3/2)*Sqrt[a^2 + 2*a*b*x^
2 + b^2*x^4])/(3*d^3*(a + b*x^2)) + (20*a^3*b^2*(d*x)^(7/2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(7*d^5*(a + b*x^2
)) + (20*a^2*b^3*(d*x)^(11/2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(11*d^7*(a + b*x^2)) + (2*a*b^4*(d*x)^(15/2)*Sq
rt[a^2 + 2*a*b*x^2 + b^2*x^4])/(3*d^9*(a + b*x^2)) + (2*b^5*(d*x)^(19/2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(19*
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 \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}{(d x)^{3/2}} \, dx &=\frac {\sqrt {a^2+2 a b x^2+b^2 x^4} \int \frac {\left (a b+b^2 x^2\right )^5}{(d x)^{3/2}} \, 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 (\frac {a^5 b^5}{(d x)^{3/2}}+\frac {5 a^4 b^6 \sqrt {d x}}{d^2}+\frac {10 a^3 b^7 (d x)^{5/2}}{d^4}+\frac {10 a^2 b^8 (d x)^{9/2}}{d^6}+\frac {5 a b^9 (d x)^{13/2}}{d^8}+\frac {b^{10} (d x)^{17/2}}{d^{10}}\right ) \, dx}{b^4 \left (a b+b^2 x^2\right )}\\ &=-\frac {2 a^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}{d \sqrt {d x} \left (a+b x^2\right )}+\frac {10 a^4 b (d x)^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{3 d^3 \left (a+b x^2\right )}+\frac {20 a^3 b^2 (d x)^{7/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{7 d^5 \left (a+b x^2\right )}+\frac {20 a^2 b^3 (d x)^{11/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{11 d^7 \left (a+b x^2\right )}+\frac {2 a b^4 (d x)^{15/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{3 d^9 \left (a+b x^2\right )}+\frac {2 b^5 (d x)^{19/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{19 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 \[ \frac {2 x \sqrt {\left (a+b x^2\right )^2} \left (-4389 a^5+7315 a^4 b x^2+6270 a^3 b^2 x^4+3990 a^2 b^3 x^6+1463 a b^4 x^8+231 b^5 x^{10}\right )}{4389 (d x)^{3/2} \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*x*Sqrt[(a + b*x^2)^2]*(-4389*a^5 + 7315*a^4*b*x^2 + 6270*a^3*b^2*x^4 + 3990*a^2*b^3*x^6 + 1463*a*b^4*x^8 +
231*b^5*x^10))/(4389*(d*x)^(3/2)*(a + b*x^2))

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fricas [A]  time = 0.82, size = 67, normalized size = 0.23 \[ \frac {2 \, {\left (231 \, b^{5} x^{10} + 1463 \, a b^{4} x^{8} + 3990 \, a^{2} b^{3} x^{6} + 6270 \, a^{3} b^{2} x^{4} + 7315 \, a^{4} b x^{2} - 4389 \, a^{5}\right )} \sqrt {d x}}{4389 \, d^{2} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/4389*(231*b^5*x^10 + 1463*a*b^4*x^8 + 3990*a^2*b^3*x^6 + 6270*a^3*b^2*x^4 + 7315*a^4*b*x^2 - 4389*a^5)*sqrt(
d*x)/(d^2*x)

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giac [A]  time = 0.19, size = 156, normalized size = 0.53 \[ -\frac {2 \, {\left (\frac {4389 \, a^{5} \mathrm {sgn}\left (b x^{2} + a\right )}{\sqrt {d x}} - \frac {231 \, \sqrt {d x} b^{5} d^{189} x^{9} \mathrm {sgn}\left (b x^{2} + a\right ) + 1463 \, \sqrt {d x} a b^{4} d^{189} x^{7} \mathrm {sgn}\left (b x^{2} + a\right ) + 3990 \, \sqrt {d x} a^{2} b^{3} d^{189} x^{5} \mathrm {sgn}\left (b x^{2} + a\right ) + 6270 \, \sqrt {d x} a^{3} b^{2} d^{189} x^{3} \mathrm {sgn}\left (b x^{2} + a\right ) + 7315 \, \sqrt {d x} a^{4} b d^{189} x \mathrm {sgn}\left (b x^{2} + a\right )}{d^{190}}\right )}}{4389 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/4389*(4389*a^5*sgn(b*x^2 + a)/sqrt(d*x) - (231*sqrt(d*x)*b^5*d^189*x^9*sgn(b*x^2 + a) + 1463*sqrt(d*x)*a*b^
4*d^189*x^7*sgn(b*x^2 + a) + 3990*sqrt(d*x)*a^2*b^3*d^189*x^5*sgn(b*x^2 + a) + 6270*sqrt(d*x)*a^3*b^2*d^189*x^
3*sgn(b*x^2 + a) + 7315*sqrt(d*x)*a^4*b*d^189*x*sgn(b*x^2 + a))/d^190)/d

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maple [A]  time = 0.00, size = 83, normalized size = 0.28 \[ -\frac {2 \left (-231 b^{5} x^{10}-1463 a \,b^{4} x^{8}-3990 a^{2} b^{3} x^{6}-6270 a^{3} b^{2} x^{4}-7315 a^{4} b \,x^{2}+4389 a^{5}\right ) \left (\left (b \,x^{2}+a \right )^{2}\right )^{\frac {5}{2}} x}{4389 \left (b \,x^{2}+a \right )^{5} \left (d x \right )^{\frac {3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/4389*x*(-231*b^5*x^10-1463*a*b^4*x^8-3990*a^2*b^3*x^6-6270*a^3*b^2*x^4-7315*a^4*b*x^2+4389*a^5)*((b*x^2+a)^
2)^(5/2)/(b*x^2+a)^5/(d*x)^(3/2)

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maxima [A]  time = 1.50, size = 151, normalized size = 0.51 \[ \frac {2 \, {\left (77 \, {\left (15 \, b^{5} \sqrt {d} x^{3} + 19 \, a b^{4} \sqrt {d} x\right )} x^{\frac {13}{2}} + 532 \, {\left (11 \, a b^{4} \sqrt {d} x^{3} + 15 \, a^{2} b^{3} \sqrt {d} x\right )} x^{\frac {9}{2}} + 1710 \, {\left (7 \, a^{2} b^{3} \sqrt {d} x^{3} + 11 \, a^{3} b^{2} \sqrt {d} x\right )} x^{\frac {5}{2}} + 4180 \, {\left (3 \, a^{3} b^{2} \sqrt {d} x^{3} + 7 \, a^{4} b \sqrt {d} x\right )} \sqrt {x} + \frac {7315 \, {\left (a^{4} b \sqrt {d} x^{3} - 3 \, a^{5} \sqrt {d} x\right )}}{x^{\frac {3}{2}}}\right )}}{21945 \, d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/21945*(77*(15*b^5*sqrt(d)*x^3 + 19*a*b^4*sqrt(d)*x)*x^(13/2) + 532*(11*a*b^4*sqrt(d)*x^3 + 15*a^2*b^3*sqrt(d
)*x)*x^(9/2) + 1710*(7*a^2*b^3*sqrt(d)*x^3 + 11*a^3*b^2*sqrt(d)*x)*x^(5/2) + 4180*(3*a^3*b^2*sqrt(d)*x^3 + 7*a
^4*b*sqrt(d)*x)*sqrt(x) + 7315*(a^4*b*sqrt(d)*x^3 - 3*a^5*sqrt(d)*x)/x^(3/2))/d^2

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mupad [B]  time = 4.54, size = 116, normalized size = 0.39 \[ \frac {2\,\sqrt {a^2+2\,a\,b\,x^2+b^2\,x^4}\,\left (3803\,a^4+3512\,a^3\,b\,x^2+2758\,a^2\,b^2\,x^4+1232\,a\,b^3\,x^6+231\,b^4\,x^8\right )}{4389\,d\,\sqrt {d\,x}}-\frac {16384\,a^5\,\sqrt {a^2+2\,a\,b\,x^2+b^2\,x^4}}{4389\,d\,\sqrt {d\,x}\,\left (b\,x^2+a\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*(a^2 + b^2*x^4 + 2*a*b*x^2)^(1/2)*(3803*a^4 + 231*b^4*x^8 + 3512*a^3*b*x^2 + 1232*a*b^3*x^6 + 2758*a^2*b^2*
x^4))/(4389*d*(d*x)^(1/2)) - (16384*a^5*(a^2 + b^2*x^4 + 2*a*b*x^2)^(1/2))/(4389*d*(d*x)^(1/2)*(a + b*x^2))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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