3.741 \(\int \frac{\left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}}{(d x)^{7/2}} \, dx\)

Optimal. Leaf size=191 \[ \frac{2 a b^2 (d x)^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{d^5 \left (a+b x^2\right )}-\frac{6 a^2 b \sqrt{a^2+2 a b x^2+b^2 x^4}}{d^3 \sqrt{d x} \left (a+b x^2\right )}+\frac{2 b^3 (d x)^{7/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{7 d^7 \left (a+b x^2\right )}-\frac{2 a^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}{5 d (d x)^{5/2} \left (a+b x^2\right )} \]

[Out]

(-2*a^3*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(5*d*(d*x)^(5/2)*(a + b*x^2)) - (6*a^2*
b*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(d^3*Sqrt[d*x]*(a + b*x^2)) + (2*a*b^2*(d*x)^
(3/2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(d^5*(a + b*x^2)) + (2*b^3*(d*x)^(7/2)*Sq
rt[a^2 + 2*a*b*x^2 + b^2*x^4])/(7*d^7*(a + b*x^2))

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Rubi [A]  time = 0.157869, antiderivative size = 191, 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 \[ \frac{2 a b^2 (d x)^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{d^5 \left (a+b x^2\right )}-\frac{6 a^2 b \sqrt{a^2+2 a b x^2+b^2 x^4}}{d^3 \sqrt{d x} \left (a+b x^2\right )}+\frac{2 b^3 (d x)^{7/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{7 d^7 \left (a+b x^2\right )}-\frac{2 a^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}{5 d (d x)^{5/2} \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]  Int[(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2)/(d*x)^(7/2),x]

[Out]

(-2*a^3*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(5*d*(d*x)^(5/2)*(a + b*x^2)) - (6*a^2*
b*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(d^3*Sqrt[d*x]*(a + b*x^2)) + (2*a*b^2*(d*x)^
(3/2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(d^5*(a + b*x^2)) + (2*b^3*(d*x)^(7/2)*Sq
rt[a^2 + 2*a*b*x^2 + b^2*x^4])/(7*d^7*(a + b*x^2))

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Rubi in Sympy [A]  time = 18.3959, size = 156, normalized size = 0.82 \[ \frac{256 a^{3} \sqrt{a^{2} + 2 a b x^{2} + b^{2} x^{4}}}{35 d \left (d x\right )^{\frac{5}{2}} \left (a + b x^{2}\right )} - \frac{64 a^{2} \sqrt{a^{2} + 2 a b x^{2} + b^{2} x^{4}}}{7 d \left (d x\right )^{\frac{5}{2}}} + \frac{8 a \left (a + b x^{2}\right ) \sqrt{a^{2} + 2 a b x^{2} + b^{2} x^{4}}}{7 d \left (d x\right )^{\frac{5}{2}}} + \frac{2 \left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{\frac{3}{2}}}{7 d \left (d x\right )^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b**2*x**4+2*a*b*x**2+a**2)**(3/2)/(d*x)**(7/2),x)

[Out]

256*a**3*sqrt(a**2 + 2*a*b*x**2 + b**2*x**4)/(35*d*(d*x)**(5/2)*(a + b*x**2)) -
64*a**2*sqrt(a**2 + 2*a*b*x**2 + b**2*x**4)/(7*d*(d*x)**(5/2)) + 8*a*(a + b*x**2
)*sqrt(a**2 + 2*a*b*x**2 + b**2*x**4)/(7*d*(d*x)**(5/2)) + 2*(a**2 + 2*a*b*x**2
+ b**2*x**4)**(3/2)/(7*d*(d*x)**(5/2))

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Mathematica [A]  time = 0.0405466, size = 66, normalized size = 0.35 \[ \frac{2 x \sqrt{\left (a+b x^2\right )^2} \left (-7 a^3-105 a^2 b x^2+35 a b^2 x^4+5 b^3 x^6\right )}{35 (d x)^{7/2} \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]  Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2)/(d*x)^(7/2),x]

[Out]

(2*x*Sqrt[(a + b*x^2)^2]*(-7*a^3 - 105*a^2*b*x^2 + 35*a*b^2*x^4 + 5*b^3*x^6))/(3
5*(d*x)^(7/2)*(a + b*x^2))

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Maple [A]  time = 0.009, size = 61, normalized size = 0.3 \[ -{\frac{2\, \left ( -5\,{b}^{3}{x}^{6}-35\,a{x}^{4}{b}^{2}+105\,{a}^{2}b{x}^{2}+7\,{a}^{3} \right ) x}{35\, \left ( b{x}^{2}+a \right ) ^{3}} \left ( \left ( b{x}^{2}+a \right ) ^{2} \right ) ^{{\frac{3}{2}}} \left ( dx \right ) ^{-{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b^2*x^4+2*a*b*x^2+a^2)^(3/2)/(d*x)^(7/2),x)

[Out]

-2/35*x*(-5*b^3*x^6-35*a*b^2*x^4+105*a^2*b*x^2+7*a^3)*((b*x^2+a)^2)^(3/2)/(b*x^2
+a)^3/(d*x)^(7/2)

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Maxima [A]  time = 0.72334, size = 116, normalized size = 0.61 \[ \frac{2 \,{\left (5 \,{\left (3 \, b^{3} \sqrt{d} x^{3} + 7 \, a b^{2} \sqrt{d} x\right )} \sqrt{x} + \frac{70 \,{\left (a b^{2} \sqrt{d} x^{3} - 3 \, a^{2} b \sqrt{d} x\right )}}{x^{\frac{3}{2}}} - \frac{21 \,{\left (5 \, a^{2} b \sqrt{d} x^{3} + a^{3} \sqrt{d} x\right )}}{x^{\frac{7}{2}}}\right )}}{105 \, d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^4 + 2*a*b*x^2 + a^2)^(3/2)/(d*x)^(7/2),x, algorithm="maxima")

[Out]

2/105*(5*(3*b^3*sqrt(d)*x^3 + 7*a*b^2*sqrt(d)*x)*sqrt(x) + 70*(a*b^2*sqrt(d)*x^3
- 3*a^2*b*sqrt(d)*x)/x^(3/2) - 21*(5*a^2*b*sqrt(d)*x^3 + a^3*sqrt(d)*x)/x^(7/2)
)/d^4

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Fricas [A]  time = 0.270475, size = 61, normalized size = 0.32 \[ \frac{2 \,{\left (5 \, b^{3} x^{6} + 35 \, a b^{2} x^{4} - 105 \, a^{2} b x^{2} - 7 \, a^{3}\right )}}{35 \, \sqrt{d x} d^{3} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^4 + 2*a*b*x^2 + a^2)^(3/2)/(d*x)^(7/2),x, algorithm="fricas")

[Out]

2/35*(5*b^3*x^6 + 35*a*b^2*x^4 - 105*a^2*b*x^2 - 7*a^3)/(sqrt(d*x)*d^3*x^2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b**2*x**4+2*a*b*x**2+a**2)**(3/2)/(d*x)**(7/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.269321, size = 144, normalized size = 0.75 \[ -\frac{2 \,{\left (\frac{7 \,{\left (15 \, a^{2} b d^{3} x^{2}{\rm sign}\left (b x^{2} + a\right ) + a^{3} d^{3}{\rm sign}\left (b x^{2} + a\right )\right )}}{\sqrt{d x} d^{2} x^{2}} - \frac{5 \,{\left (\sqrt{d x} b^{3} d^{21} x^{3}{\rm sign}\left (b x^{2} + a\right ) + 7 \, \sqrt{d x} a b^{2} d^{21} x{\rm sign}\left (b x^{2} + a\right )\right )}}{d^{21}}\right )}}{35 \, d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^4 + 2*a*b*x^2 + a^2)^(3/2)/(d*x)^(7/2),x, algorithm="giac")

[Out]

-2/35*(7*(15*a^2*b*d^3*x^2*sign(b*x^2 + a) + a^3*d^3*sign(b*x^2 + a))/(sqrt(d*x)
*d^2*x^2) - 5*(sqrt(d*x)*b^3*d^21*x^3*sign(b*x^2 + a) + 7*sqrt(d*x)*a*b^2*d^21*x
*sign(b*x^2 + a))/d^21)/d^4