3.845 \(\int \frac{\left (a+b x^2\right )^2}{(e x)^{7/2} \sqrt{c+d x^2}} \, dx\)

Optimal. Leaf size=387 \[ \frac{2 \sqrt{e x} \sqrt{c+d x^2} \left (-3 a^2 d^2+10 a b c d+5 b^2 c^2\right )}{5 c^2 \sqrt{d} e^4 \left (\sqrt{c}+\sqrt{d} x\right )}+\frac{\left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (-3 a^2 d^2+10 a b c d+5 b^2 c^2\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{5 c^{7/4} d^{3/4} e^{7/2} \sqrt{c+d x^2}}-\frac{2 \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (-3 a^2 d^2+10 a b c d+5 b^2 c^2\right ) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{5 c^{7/4} d^{3/4} e^{7/2} \sqrt{c+d x^2}}-\frac{2 a^2 \sqrt{c+d x^2}}{5 c e (e x)^{5/2}}-\frac{2 a \sqrt{c+d x^2} (10 b c-3 a d)}{5 c^2 e^3 \sqrt{e x}} \]

[Out]

(-2*a^2*Sqrt[c + d*x^2])/(5*c*e*(e*x)^(5/2)) - (2*a*(10*b*c - 3*a*d)*Sqrt[c + d*
x^2])/(5*c^2*e^3*Sqrt[e*x]) + (2*(5*b^2*c^2 + 10*a*b*c*d - 3*a^2*d^2)*Sqrt[e*x]*
Sqrt[c + d*x^2])/(5*c^2*Sqrt[d]*e^4*(Sqrt[c] + Sqrt[d]*x)) - (2*(5*b^2*c^2 + 10*
a*b*c*d - 3*a^2*d^2)*(Sqrt[c] + Sqrt[d]*x)*Sqrt[(c + d*x^2)/(Sqrt[c] + Sqrt[d]*x
)^2]*EllipticE[2*ArcTan[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], 1/2])/(5*c^(7/4)
*d^(3/4)*e^(7/2)*Sqrt[c + d*x^2]) + ((5*b^2*c^2 + 10*a*b*c*d - 3*a^2*d^2)*(Sqrt[
c] + Sqrt[d]*x)*Sqrt[(c + d*x^2)/(Sqrt[c] + Sqrt[d]*x)^2]*EllipticF[2*ArcTan[(d^
(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], 1/2])/(5*c^(7/4)*d^(3/4)*e^(7/2)*Sqrt[c + d
*x^2])

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Rubi [A]  time = 0.837622, antiderivative size = 387, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214 \[ \frac{2 \sqrt{e x} \sqrt{c+d x^2} \left (-3 a^2 d^2+10 a b c d+5 b^2 c^2\right )}{5 c^2 \sqrt{d} e^4 \left (\sqrt{c}+\sqrt{d} x\right )}+\frac{\left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (-3 a^2 d^2+10 a b c d+5 b^2 c^2\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{5 c^{7/4} d^{3/4} e^{7/2} \sqrt{c+d x^2}}-\frac{2 \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (-3 a^2 d^2+10 a b c d+5 b^2 c^2\right ) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{5 c^{7/4} d^{3/4} e^{7/2} \sqrt{c+d x^2}}-\frac{2 a^2 \sqrt{c+d x^2}}{5 c e (e x)^{5/2}}-\frac{2 a \sqrt{c+d x^2} (10 b c-3 a d)}{5 c^2 e^3 \sqrt{e x}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^2)^2/((e*x)^(7/2)*Sqrt[c + d*x^2]),x]

[Out]

(-2*a^2*Sqrt[c + d*x^2])/(5*c*e*(e*x)^(5/2)) - (2*a*(10*b*c - 3*a*d)*Sqrt[c + d*
x^2])/(5*c^2*e^3*Sqrt[e*x]) + (2*(5*b^2*c^2 + 10*a*b*c*d - 3*a^2*d^2)*Sqrt[e*x]*
Sqrt[c + d*x^2])/(5*c^2*Sqrt[d]*e^4*(Sqrt[c] + Sqrt[d]*x)) - (2*(5*b^2*c^2 + 10*
a*b*c*d - 3*a^2*d^2)*(Sqrt[c] + Sqrt[d]*x)*Sqrt[(c + d*x^2)/(Sqrt[c] + Sqrt[d]*x
)^2]*EllipticE[2*ArcTan[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], 1/2])/(5*c^(7/4)
*d^(3/4)*e^(7/2)*Sqrt[c + d*x^2]) + ((5*b^2*c^2 + 10*a*b*c*d - 3*a^2*d^2)*(Sqrt[
c] + Sqrt[d]*x)*Sqrt[(c + d*x^2)/(Sqrt[c] + Sqrt[d]*x)^2]*EllipticF[2*ArcTan[(d^
(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], 1/2])/(5*c^(7/4)*d^(3/4)*e^(7/2)*Sqrt[c + d
*x^2])

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Rubi in Sympy [A]  time = 84.3162, size = 360, normalized size = 0.93 \[ - \frac{2 a^{2} \sqrt{c + d x^{2}}}{5 c e \left (e x\right )^{\frac{5}{2}}} + \frac{2 a \sqrt{c + d x^{2}} \left (3 a d - 10 b c\right )}{5 c^{2} e^{3} \sqrt{e x}} + \frac{2 \sqrt{e x} \sqrt{c + d x^{2}} \left (- a d \left (3 a d - 10 b c\right ) + 5 b^{2} c^{2}\right )}{5 c^{2} \sqrt{d} e^{4} \left (\sqrt{c} + \sqrt{d} x\right )} - \frac{2 \sqrt{\frac{c + d x^{2}}{\left (\sqrt{c} + \sqrt{d} x\right )^{2}}} \left (\sqrt{c} + \sqrt{d} x\right ) \left (- a d \left (3 a d - 10 b c\right ) + 5 b^{2} c^{2}\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}} \right )}\middle | \frac{1}{2}\right )}{5 c^{\frac{7}{4}} d^{\frac{3}{4}} e^{\frac{7}{2}} \sqrt{c + d x^{2}}} + \frac{\sqrt{\frac{c + d x^{2}}{\left (\sqrt{c} + \sqrt{d} x\right )^{2}}} \left (\sqrt{c} + \sqrt{d} x\right ) \left (- a d \left (3 a d - 10 b c\right ) + 5 b^{2} c^{2}\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}} \right )}\middle | \frac{1}{2}\right )}{5 c^{\frac{7}{4}} d^{\frac{3}{4}} e^{\frac{7}{2}} \sqrt{c + d x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a)**2/(e*x)**(7/2)/(d*x**2+c)**(1/2),x)

[Out]

-2*a**2*sqrt(c + d*x**2)/(5*c*e*(e*x)**(5/2)) + 2*a*sqrt(c + d*x**2)*(3*a*d - 10
*b*c)/(5*c**2*e**3*sqrt(e*x)) + 2*sqrt(e*x)*sqrt(c + d*x**2)*(-a*d*(3*a*d - 10*b
*c) + 5*b**2*c**2)/(5*c**2*sqrt(d)*e**4*(sqrt(c) + sqrt(d)*x)) - 2*sqrt((c + d*x
**2)/(sqrt(c) + sqrt(d)*x)**2)*(sqrt(c) + sqrt(d)*x)*(-a*d*(3*a*d - 10*b*c) + 5*
b**2*c**2)*elliptic_e(2*atan(d**(1/4)*sqrt(e*x)/(c**(1/4)*sqrt(e))), 1/2)/(5*c**
(7/4)*d**(3/4)*e**(7/2)*sqrt(c + d*x**2)) + sqrt((c + d*x**2)/(sqrt(c) + sqrt(d)
*x)**2)*(sqrt(c) + sqrt(d)*x)*(-a*d*(3*a*d - 10*b*c) + 5*b**2*c**2)*elliptic_f(2
*atan(d**(1/4)*sqrt(e*x)/(c**(1/4)*sqrt(e))), 1/2)/(5*c**(7/4)*d**(3/4)*e**(7/2)
*sqrt(c + d*x**2))

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Mathematica [C]  time = 1.43669, size = 217, normalized size = 0.56 \[ \frac{x^{7/2} \left (-\frac{2 a \sqrt{c+d x^2} \left (a \left (c-3 d x^2\right )+10 b c x^2\right )}{c^2 x^{5/2}}-\frac{2 x \left (-3 a^2 d^2+10 a b c d+5 b^2 c^2\right ) \left (-\sqrt{x} \left (\frac{c}{x^2}+d\right )+\frac{i c \sqrt{\frac{c}{d x^2}+1} \left (E\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{c}}{\sqrt{d}}}}{\sqrt{x}}\right )\right |-1\right )-F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{c}}{\sqrt{d}}}}{\sqrt{x}}\right )\right |-1\right )\right )}{\left (\frac{i \sqrt{c}}{\sqrt{d}}\right )^{3/2}}\right )}{c^2 d \sqrt{c+d x^2}}\right )}{5 (e x)^{7/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^2)^2/((e*x)^(7/2)*Sqrt[c + d*x^2]),x]

[Out]

(x^(7/2)*((-2*a*Sqrt[c + d*x^2]*(10*b*c*x^2 + a*(c - 3*d*x^2)))/(c^2*x^(5/2)) -
(2*(5*b^2*c^2 + 10*a*b*c*d - 3*a^2*d^2)*x*(-((d + c/x^2)*Sqrt[x]) + (I*c*Sqrt[1
+ c/(d*x^2)]*(EllipticE[I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[d]]/Sqrt[x]], -1] - Elli
pticF[I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[d]]/Sqrt[x]], -1]))/((I*Sqrt[c])/Sqrt[d])^
(3/2)))/(c^2*d*Sqrt[c + d*x^2])))/(5*(e*x)^(7/2))

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Maple [A]  time = 0.03, size = 626, normalized size = 1.6 \[ -{\frac{1}{5\,d{x}^{2}{e}^{3}{c}^{2}} \left ( 6\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticE} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ){x}^{2}{a}^{2}c{d}^{2}-20\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticE} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ){x}^{2}ab{c}^{2}d-10\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticE} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ){x}^{2}{b}^{2}{c}^{3}-3\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticF} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ){x}^{2}{a}^{2}c{d}^{2}+10\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticF} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ){x}^{2}ab{c}^{2}d+5\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticF} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ){x}^{2}{b}^{2}{c}^{3}-6\,{x}^{4}{a}^{2}{d}^{3}+20\,{x}^{4}abc{d}^{2}-4\,{x}^{2}{a}^{2}c{d}^{2}+20\,{x}^{2}ab{c}^{2}d+2\,{a}^{2}{c}^{2}d \right ){\frac{1}{\sqrt{d{x}^{2}+c}}}{\frac{1}{\sqrt{ex}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^2/(e*x)^(7/2)/(d*x^2+c)^(1/2),x)

[Out]

-1/5/x^2*(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^2*a^2*c*d^2-20*((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^2*a*b*c^2*d-1
0*((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^2*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^2*a^2*c*d^2+10*((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^2*a*b*c^2*d+5*((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^2*b^2*c^3-6*x^4*a^2*d^3+20*x^4*a*b*c*
d^2-4*x^2*a^2*c*d^2+20*x^2*a*b*c^2*d+2*a^2*c^2*d)/(d*x^2+c)^(1/2)/d/e^3/(e*x)^(1
/2)/c^2

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{2} + a\right )}^{2}}{\sqrt{d x^{2} + c} \left (e x\right )^{\frac{7}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x^2 + a)^2/(sqrt(d*x^2 + c)*(e*x)^(7/2)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}{\sqrt{d x^{2} + c} \sqrt{e x} e^{3} x^{3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((b^2*x^4 + 2*a*b*x^2 + a^2)/(sqrt(d*x^2 + c)*sqrt(e*x)*e^3*x^3), 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/(e*x)**(7/2)/(d*x**2+c)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{2} + a\right )}^{2}}{\sqrt{d x^{2} + c} \left (e x\right )^{\frac{7}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x^2 + a)^2/(sqrt(d*x^2 + c)*(e*x)^(7/2)), x)