3.829 \(\int \frac{\left (a+b x^2\right )^2 \sqrt{c+d x^2}}{x^{11/2}} \, dx\)

Optimal. Leaf size=386 \[ -\frac{2 a^2 \left (c+d x^2\right )^{3/2}}{9 c x^{9/2}}-\frac{2 \sqrt{c+d x^2} \left (a d (6 b c-a d)+15 b^2 c^2\right )}{15 c^2 \sqrt{x}}+\frac{4 \sqrt{d} \sqrt{x} \sqrt{c+d x^2} \left (a d (6 b c-a d)+15 b^2 c^2\right )}{15 c^2 \left (\sqrt{c}+\sqrt{d} x\right )}+\frac{2 \sqrt [4]{d} \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (a d (6 b c-a d)+15 b^2 c^2\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )|\frac{1}{2}\right )}{15 c^{7/4} \sqrt{c+d x^2}}-\frac{4 \sqrt [4]{d} \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (a d (6 b c-a d)+15 b^2 c^2\right ) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )|\frac{1}{2}\right )}{15 c^{7/4} \sqrt{c+d x^2}}-\frac{2 a \left (c+d x^2\right )^{3/2} (6 b c-a d)}{15 c^2 x^{5/2}} \]

[Out]

(-2*(15*b^2*c^2 + a*d*(6*b*c - a*d))*Sqrt[c + d*x^2])/(15*c^2*Sqrt[x]) + (4*Sqrt
[d]*(15*b^2*c^2 + a*d*(6*b*c - a*d))*Sqrt[x]*Sqrt[c + d*x^2])/(15*c^2*(Sqrt[c] +
Sqrt[d]*x)) - (2*a^2*(c + d*x^2)^(3/2))/(9*c*x^(9/2)) - (2*a*(6*b*c - a*d)*(c +
d*x^2)^(3/2))/(15*c^2*x^(5/2)) - (4*d^(1/4)*(15*b^2*c^2 + a*d*(6*b*c - a*d))*(S
qrt[c] + Sqrt[d]*x)*Sqrt[(c + d*x^2)/(Sqrt[c] + Sqrt[d]*x)^2]*EllipticE[2*ArcTan
[(d^(1/4)*Sqrt[x])/c^(1/4)], 1/2])/(15*c^(7/4)*Sqrt[c + d*x^2]) + (2*d^(1/4)*(15
*b^2*c^2 + a*d*(6*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])/(15*c^(7/4)*S
qrt[c + d*x^2])

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Rubi [A]  time = 0.768105, antiderivative size = 383, normalized size of antiderivative = 0.99, number of steps used = 7, number of rules used = 7, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.269 \[ -\frac{2 a^2 \left (c+d x^2\right )^{3/2}}{9 c x^{9/2}}-\frac{2 \sqrt{c+d x^2} \left (\frac{a d (6 b c-a d)}{c^2}+15 b^2\right )}{15 \sqrt{x}}+\frac{4 \sqrt{d} \sqrt{x} \sqrt{c+d x^2} \left (a d (6 b c-a d)+15 b^2 c^2\right )}{15 c^2 \left (\sqrt{c}+\sqrt{d} x\right )}+\frac{2 \sqrt [4]{d} \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (a d (6 b c-a d)+15 b^2 c^2\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )|\frac{1}{2}\right )}{15 c^{7/4} \sqrt{c+d x^2}}-\frac{4 \sqrt [4]{d} \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (a d (6 b c-a d)+15 b^2 c^2\right ) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )|\frac{1}{2}\right )}{15 c^{7/4} \sqrt{c+d x^2}}-\frac{2 a \left (c+d x^2\right )^{3/2} (6 b c-a d)}{15 c^2 x^{5/2}} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x^2)^2*Sqrt[c + d*x^2])/x^(11/2),x]

[Out]

(-2*(15*b^2 + (a*d*(6*b*c - a*d))/c^2)*Sqrt[c + d*x^2])/(15*Sqrt[x]) + (4*Sqrt[d
]*(15*b^2*c^2 + a*d*(6*b*c - a*d))*Sqrt[x]*Sqrt[c + d*x^2])/(15*c^2*(Sqrt[c] + S
qrt[d]*x)) - (2*a^2*(c + d*x^2)^(3/2))/(9*c*x^(9/2)) - (2*a*(6*b*c - a*d)*(c + d
*x^2)^(3/2))/(15*c^2*x^(5/2)) - (4*d^(1/4)*(15*b^2*c^2 + a*d*(6*b*c - a*d))*(Sqr
t[c] + Sqrt[d]*x)*Sqrt[(c + d*x^2)/(Sqrt[c] + Sqrt[d]*x)^2]*EllipticE[2*ArcTan[(
d^(1/4)*Sqrt[x])/c^(1/4)], 1/2])/(15*c^(7/4)*Sqrt[c + d*x^2]) + (2*d^(1/4)*(15*b
^2*c^2 + a*d*(6*b*c - a*d))*(Sqrt[c] + Sqrt[d]*x)*Sqrt[(c + d*x^2)/(Sqrt[c] + Sq
rt[d]*x)^2]*EllipticF[2*ArcTan[(d^(1/4)*Sqrt[x])/c^(1/4)], 1/2])/(15*c^(7/4)*Sqr
t[c + d*x^2])

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Rubi in Sympy [A]  time = 64.9998, size = 360, normalized size = 0.93 \[ - \frac{2 a^{2} \left (c + d x^{2}\right )^{\frac{3}{2}}}{9 c x^{\frac{9}{2}}} + \frac{2 a \left (c + d x^{2}\right )^{\frac{3}{2}} \left (a d - 6 b c\right )}{15 c^{2} x^{\frac{5}{2}}} + \frac{4 \sqrt{d} \sqrt{x} \sqrt{c + d x^{2}} \left (- a d \left (a d - 6 b c\right ) + 15 b^{2} c^{2}\right )}{15 c^{2} \left (\sqrt{c} + \sqrt{d} x\right )} - \frac{2 \sqrt{c + d x^{2}} \left (- a d \left (a d - 6 b c\right ) + 15 b^{2} c^{2}\right )}{15 c^{2} \sqrt{x}} - \frac{4 \sqrt [4]{d} \sqrt{\frac{c + d x^{2}}{\left (\sqrt{c} + \sqrt{d} x\right )^{2}}} \left (\sqrt{c} + \sqrt{d} x\right ) \left (- a d \left (a d - 6 b c\right ) + 15 b^{2} c^{2}\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}} \right )}\middle | \frac{1}{2}\right )}{15 c^{\frac{7}{4}} \sqrt{c + d x^{2}}} + \frac{2 \sqrt [4]{d} \sqrt{\frac{c + d x^{2}}{\left (\sqrt{c} + \sqrt{d} x\right )^{2}}} \left (\sqrt{c} + \sqrt{d} x\right ) \left (- a d \left (a d - 6 b c\right ) + 15 b^{2} c^{2}\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}} \right )}\middle | \frac{1}{2}\right )}{15 c^{\frac{7}{4}} \sqrt{c + d x^{2}}} \]

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**(11/2),x)

[Out]

-2*a**2*(c + d*x**2)**(3/2)/(9*c*x**(9/2)) + 2*a*(c + d*x**2)**(3/2)*(a*d - 6*b*
c)/(15*c**2*x**(5/2)) + 4*sqrt(d)*sqrt(x)*sqrt(c + d*x**2)*(-a*d*(a*d - 6*b*c) +
15*b**2*c**2)/(15*c**2*(sqrt(c) + sqrt(d)*x)) - 2*sqrt(c + d*x**2)*(-a*d*(a*d -
6*b*c) + 15*b**2*c**2)/(15*c**2*sqrt(x)) - 4*d**(1/4)*sqrt((c + d*x**2)/(sqrt(c
) + sqrt(d)*x)**2)*(sqrt(c) + sqrt(d)*x)*(-a*d*(a*d - 6*b*c) + 15*b**2*c**2)*ell
iptic_e(2*atan(d**(1/4)*sqrt(x)/c**(1/4)), 1/2)/(15*c**(7/4)*sqrt(c + d*x**2)) +
2*d**(1/4)*sqrt((c + d*x**2)/(sqrt(c) + sqrt(d)*x)**2)*(sqrt(c) + sqrt(d)*x)*(-
a*d*(a*d - 6*b*c) + 15*b**2*c**2)*elliptic_f(2*atan(d**(1/4)*sqrt(x)/c**(1/4)),
1/2)/(15*c**(7/4)*sqrt(c + d*x**2))

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Mathematica [C]  time = 0.845727, size = 283, normalized size = 0.73 \[ \frac{2 \left (\left (-c-d x^2\right ) \left (3 x^4 \left (-2 a^2 d^2+12 a b c d+15 b^2 c^2\right )+5 a^2 c^2+2 a c x^2 (a d+9 b c)\right )+\frac{6 x^4 \left (-a^2 d^2+6 a b c d+15 b^2 c^2\right ) \left (\sqrt{\frac{i \sqrt{c}}{\sqrt{d}}} \left (c+d x^2\right )+\sqrt{c} \sqrt{d} x^{3/2} \sqrt{\frac{c}{d x^2}+1} F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{c}}{\sqrt{d}}}}{\sqrt{x}}\right )\right |-1\right )-\sqrt{c} \sqrt{d} x^{3/2} \sqrt{\frac{c}{d x^2}+1} E\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{c}}{\sqrt{d}}}}{\sqrt{x}}\right )\right |-1\right )\right )}{\sqrt{\frac{i \sqrt{c}}{\sqrt{d}}}}\right )}{45 c^2 x^{9/2} \sqrt{c+d x^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x^2)^2*Sqrt[c + d*x^2])/x^(11/2),x]

[Out]

(2*((-c - d*x^2)*(5*a^2*c^2 + 2*a*c*(9*b*c + a*d)*x^2 + 3*(15*b^2*c^2 + 12*a*b*c
*d - 2*a^2*d^2)*x^4) + (6*(15*b^2*c^2 + 6*a*b*c*d - a^2*d^2)*x^4*(Sqrt[(I*Sqrt[c
])/Sqrt[d]]*(c + d*x^2) - Sqrt[c]*Sqrt[d]*Sqrt[1 + c/(d*x^2)]*x^(3/2)*EllipticE[
I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[d]]/Sqrt[x]], -1] + Sqrt[c]*Sqrt[d]*Sqrt[1 + c/(
d*x^2)]*x^(3/2)*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[d]]/Sqrt[x]], -1]))/Sq
rt[(I*Sqrt[c])/Sqrt[d]]))/(45*c^2*x^(9/2)*Sqrt[c + d*x^2])

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Maple [A]  time = 0.084, size = 659, normalized size = 1.7 \[ -{\frac{2}{45\,{c}^{2}} \left ( 6\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticE} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ){x}^{4}{a}^{2}c{d}^{2}-36\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticE} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ){x}^{4}ab{c}^{2}d-90\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticE} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ){x}^{4}{b}^{2}{c}^{3}-3\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticF} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ){x}^{4}{a}^{2}c{d}^{2}+18\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticF} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ){x}^{4}ab{c}^{2}d+45\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticF} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ){x}^{4}{b}^{2}{c}^{3}-6\,{x}^{6}{a}^{2}{d}^{3}+36\,{x}^{6}abc{d}^{2}+45\,{x}^{6}{b}^{2}{c}^{2}d-4\,{x}^{4}{a}^{2}c{d}^{2}+54\,{x}^{4}ab{c}^{2}d+45\,{x}^{4}{b}^{2}{c}^{3}+7\,{x}^{2}{a}^{2}{c}^{2}d+18\,{x}^{2}ab{c}^{3}+5\,{a}^{2}{c}^{3} \right ){\frac{1}{\sqrt{d{x}^{2}+c}}}{x}^{-{\frac{9}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^2*(d*x^2+c)^(1/2)/x^(11/2),x)

[Out]

-2/45*(6*((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)*EllipticE(((d*x+(-c*d)^(1/2))/(-c*d)
^(1/2))^(1/2),1/2*2^(1/2))*x^4*a^2*c*d^2-36*((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)*E
llipticE(((d*x+(-c*d)^(1/2))/(-c*d)^(1/2))^(1/2),1/2*2^(1/2))*x^4*a*b*c^2*d-90*(
(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)*EllipticE(((d*x+(-c*d)^(1/2))/(-c*d)^(1/2))^(1
/2),1/2*2^(1/2))*x^4*b^2*c^3-3*((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))*x^4*a^2*c*d^2+18*((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))*x^4*a*b*c^2*d+45*((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))*x^4*b^2*c^3-6*x^6*a^2*d^3+36*x^6*a*b*c*d^
2+45*x^6*b^2*c^2*d-4*x^4*a^2*c*d^2+54*x^4*a*b*c^2*d+45*x^4*b^2*c^3+7*x^2*a^2*c^2
*d+18*x^2*a*b*c^3+5*a^2*c^3)/(d*x^2+c)^(1/2)/x^(9/2)/c^2

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{2} + a\right )}^{2} \sqrt{d x^{2} + c}}{x^{\frac{11}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^2*sqrt(d*x^2 + c)/x^(11/2),x, algorithm="maxima")

[Out]

integrate((b*x^2 + a)^2*sqrt(d*x^2 + c)/x^(11/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )} \sqrt{d x^{2} + c}}{x^{\frac{11}{2}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^2*sqrt(d*x^2 + c)/x^(11/2),x, algorithm="fricas")

[Out]

integral((b^2*x^4 + 2*a*b*x^2 + a^2)*sqrt(d*x^2 + c)/x^(11/2), 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**(11/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{2} + a\right )}^{2} \sqrt{d x^{2} + c}}{x^{\frac{11}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^2*sqrt(d*x^2 + c)/x^(11/2),x, algorithm="giac")

[Out]

integrate((b*x^2 + a)^2*sqrt(d*x^2 + c)/x^(11/2), x)