3.828 \(\int \frac{\left (a+b x^2\right )^2 \sqrt{c+d x^2}}{x^{9/2}} \, dx\)

Optimal. Leaf size=213 \[ -\frac{2 a^2 \left (c+d x^2\right )^{3/2}}{7 c x^{7/2}}+\frac{2 \sqrt{x} \sqrt{c+d x^2} \left (a d (14 b c-a d)+7 b^2 c^2\right )}{21 c^2}+\frac{2 \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (a d (14 b c-a d)+7 b^2 c^2\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )|\frac{1}{2}\right )}{21 c^{5/4} \sqrt [4]{d} \sqrt{c+d x^2}}-\frac{2 a \left (c+d x^2\right )^{3/2} (14 b c-a d)}{21 c^2 x^{3/2}} \]

[Out]

(2*(7*b^2*c^2 + a*d*(14*b*c - a*d))*Sqrt[x]*Sqrt[c + d*x^2])/(21*c^2) - (2*a^2*(
c + d*x^2)^(3/2))/(7*c*x^(7/2)) - (2*a*(14*b*c - a*d)*(c + d*x^2)^(3/2))/(21*c^2
*x^(3/2)) + (2*(7*b^2*c^2 + a*d*(14*b*c - a*d))*(Sqrt[c] + Sqrt[d]*x)*Sqrt[(c +
d*x^2)/(Sqrt[c] + Sqrt[d]*x)^2]*EllipticF[2*ArcTan[(d^(1/4)*Sqrt[x])/c^(1/4)], 1
/2])/(21*c^(5/4)*d^(1/4)*Sqrt[c + d*x^2])

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Rubi [A]  time = 0.432012, antiderivative size = 210, normalized size of antiderivative = 0.99, number of steps used = 5, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192 \[ -\frac{2 a^2 \left (c+d x^2\right )^{3/2}}{7 c x^{7/2}}+\frac{2}{21} \sqrt{x} \sqrt{c+d x^2} \left (\frac{a d (14 b c-a d)}{c^2}+7 b^2\right )+\frac{2 \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (a d (14 b c-a d)+7 b^2 c^2\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )|\frac{1}{2}\right )}{21 c^{5/4} \sqrt [4]{d} \sqrt{c+d x^2}}-\frac{2 a \left (c+d x^2\right )^{3/2} (14 b c-a d)}{21 c^2 x^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x^2)^2*Sqrt[c + d*x^2])/x^(9/2),x]

[Out]

(2*(7*b^2 + (a*d*(14*b*c - a*d))/c^2)*Sqrt[x]*Sqrt[c + d*x^2])/21 - (2*a^2*(c +
d*x^2)^(3/2))/(7*c*x^(7/2)) - (2*a*(14*b*c - a*d)*(c + d*x^2)^(3/2))/(21*c^2*x^(
3/2)) + (2*(7*b^2*c^2 + a*d*(14*b*c - a*d))*(Sqrt[c] + Sqrt[d]*x)*Sqrt[(c + d*x^
2)/(Sqrt[c] + Sqrt[d]*x)^2]*EllipticF[2*ArcTan[(d^(1/4)*Sqrt[x])/c^(1/4)], 1/2])
/(21*c^(5/4)*d^(1/4)*Sqrt[c + d*x^2])

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Rubi in Sympy [A]  time = 37.7216, size = 199, normalized size = 0.93 \[ - \frac{2 a^{2} \left (c + d x^{2}\right )^{\frac{3}{2}}}{7 c x^{\frac{7}{2}}} + \frac{2 a \left (c + d x^{2}\right )^{\frac{3}{2}} \left (a d - 14 b c\right )}{21 c^{2} x^{\frac{3}{2}}} + \frac{2 \sqrt{x} \sqrt{c + d x^{2}} \left (- a d \left (a d - 14 b c\right ) + 7 b^{2} c^{2}\right )}{21 c^{2}} + \frac{2 \sqrt{\frac{c + d x^{2}}{\left (\sqrt{c} + \sqrt{d} x\right )^{2}}} \left (\sqrt{c} + \sqrt{d} x\right ) \left (- a d \left (a d - 14 b c\right ) + 7 b^{2} c^{2}\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}} \right )}\middle | \frac{1}{2}\right )}{21 c^{\frac{5}{4}} \sqrt [4]{d} \sqrt{c + d x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a)**2*(d*x**2+c)**(1/2)/x**(9/2),x)

[Out]

-2*a**2*(c + d*x**2)**(3/2)/(7*c*x**(7/2)) + 2*a*(c + d*x**2)**(3/2)*(a*d - 14*b
*c)/(21*c**2*x**(3/2)) + 2*sqrt(x)*sqrt(c + d*x**2)*(-a*d*(a*d - 14*b*c) + 7*b**
2*c**2)/(21*c**2) + 2*sqrt((c + d*x**2)/(sqrt(c) + sqrt(d)*x)**2)*(sqrt(c) + sqr
t(d)*x)*(-a*d*(a*d - 14*b*c) + 7*b**2*c**2)*elliptic_f(2*atan(d**(1/4)*sqrt(x)/c
**(1/4)), 1/2)/(21*c**(5/4)*d**(1/4)*sqrt(c + d*x**2))

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Mathematica [C]  time = 0.32464, size = 160, normalized size = 0.75 \[ \frac{2 \left (\left (c+d x^2\right ) \left (-a^2 \left (3 c+2 d x^2\right )-14 a b c x^2+7 b^2 c x^4\right )+\frac{2 i x^{9/2} \sqrt{\frac{c}{d x^2}+1} \left (-a^2 d^2+14 a b c d+7 b^2 c^2\right ) F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{c}}{\sqrt{d}}}}{\sqrt{x}}\right )\right |-1\right )}{\sqrt{\frac{i \sqrt{c}}{\sqrt{d}}}}\right )}{21 c x^{7/2} \sqrt{c+d x^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x^2)^2*Sqrt[c + d*x^2])/x^(9/2),x]

[Out]

(2*((c + d*x^2)*(-14*a*b*c*x^2 + 7*b^2*c*x^4 - a^2*(3*c + 2*d*x^2)) + ((2*I)*(7*
b^2*c^2 + 14*a*b*c*d - a^2*d^2)*Sqrt[1 + c/(d*x^2)]*x^(9/2)*EllipticF[I*ArcSinh[
Sqrt[(I*Sqrt[c])/Sqrt[d]]/Sqrt[x]], -1])/Sqrt[(I*Sqrt[c])/Sqrt[d]]))/(21*c*x^(7/
2)*Sqrt[c + d*x^2])

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Maple [A]  time = 0.079, size = 385, normalized size = 1.8 \[ -{\frac{2}{21\,cd} \left ( \sqrt{{1 \left ( dx+\sqrt{-cd} \right ){\frac{1}{\sqrt{-cd}}}}}\sqrt{2}\sqrt{{1 \left ( -dx+\sqrt{-cd} \right ){\frac{1}{\sqrt{-cd}}}}}\sqrt{-{dx{\frac{1}{\sqrt{-cd}}}}}{\it EllipticF} \left ( \sqrt{{1 \left ( dx+\sqrt{-cd} \right ){\frac{1}{\sqrt{-cd}}}}},{\frac{\sqrt{2}}{2}} \right ) \sqrt{-cd}{x}^{3}{a}^{2}{d}^{2}-14\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticF} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ) \sqrt{-cd}{x}^{3}abcd-7\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticF} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ) \sqrt{-cd}{x}^{3}{b}^{2}{c}^{2}-7\,{x}^{6}{b}^{2}c{d}^{2}+2\,{x}^{4}{a}^{2}{d}^{3}+14\,{x}^{4}abc{d}^{2}-7\,{x}^{4}{b}^{2}{c}^{2}d+5\,{x}^{2}{a}^{2}c{d}^{2}+14\,{x}^{2}ab{c}^{2}d+3\,{a}^{2}{c}^{2}d \right ){\frac{1}{\sqrt{d{x}^{2}+c}}}{x}^{-{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^2*(d*x^2+c)^(1/2)/x^(9/2),x)

[Out]

-2/21/(d*x^2+c)^(1/2)*(((d*x+(-c*d)^(1/2))/(-c*d)^(1/2))^(1/2)*2^(1/2)*((-d*x+(-
c*d)^(1/2))/(-c*d)^(1/2))^(1/2)*(-x/(-c*d)^(1/2)*d)^(1/2)*EllipticF(((d*x+(-c*d)
^(1/2))/(-c*d)^(1/2))^(1/2),1/2*2^(1/2))*(-c*d)^(1/2)*x^3*a^2*d^2-14*((d*x+(-c*d
)^(1/2))/(-c*d)^(1/2))^(1/2)*2^(1/2)*((-d*x+(-c*d)^(1/2))/(-c*d)^(1/2))^(1/2)*(-
x/(-c*d)^(1/2)*d)^(1/2)*EllipticF(((d*x+(-c*d)^(1/2))/(-c*d)^(1/2))^(1/2),1/2*2^
(1/2))*(-c*d)^(1/2)*x^3*a*b*c*d-7*((d*x+(-c*d)^(1/2))/(-c*d)^(1/2))^(1/2)*2^(1/2
)*((-d*x+(-c*d)^(1/2))/(-c*d)^(1/2))^(1/2)*(-x/(-c*d)^(1/2)*d)^(1/2)*EllipticF((
(d*x+(-c*d)^(1/2))/(-c*d)^(1/2))^(1/2),1/2*2^(1/2))*(-c*d)^(1/2)*x^3*b^2*c^2-7*x
^6*b^2*c*d^2+2*x^4*a^2*d^3+14*x^4*a*b*c*d^2-7*x^4*b^2*c^2*d+5*x^2*a^2*c*d^2+14*x
^2*a*b*c^2*d+3*a^2*c^2*d)/x^(7/2)/c/d

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{2} + a\right )}^{2} \sqrt{d x^{2} + c}}{x^{\frac{9}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^2*sqrt(d*x^2 + c)/x^(9/2),x, algorithm="maxima")

[Out]

integrate((b*x^2 + a)^2*sqrt(d*x^2 + c)/x^(9/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )} \sqrt{d x^{2} + c}}{x^{\frac{9}{2}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^2*sqrt(d*x^2 + c)/x^(9/2),x, algorithm="fricas")

[Out]

integral((b^2*x^4 + 2*a*b*x^2 + a^2)*sqrt(d*x^2 + c)/x^(9/2), x)

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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*x**2+a)**2*(d*x**2+c)**(1/2)/x**(9/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{2} + a\right )}^{2} \sqrt{d x^{2} + c}}{x^{\frac{9}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^2*sqrt(d*x^2 + c)/x^(9/2),x, algorithm="giac")

[Out]

integrate((b*x^2 + a)^2*sqrt(d*x^2 + c)/x^(9/2), x)