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

Optimal. Leaf size=438 \[ \frac{\sqrt [4]{d} \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (7 a^2 d^2-18 a b c d+15 b^2 c^2\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{15 c^{11/4} e^{11/2} \sqrt{c+d x^2}}-\frac{2 \sqrt [4]{d} \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (7 a^2 d^2-18 a b c d+15 b^2 c^2\right ) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{15 c^{11/4} e^{11/2} \sqrt{c+d x^2}}+\frac{2 \sqrt{d} \sqrt{e x} \sqrt{c+d x^2} \left (7 a^2 d^2-18 a b c d+15 b^2 c^2\right )}{15 c^3 e^6 \left (\sqrt{c}+\sqrt{d} x\right )}-\frac{2 \sqrt{c+d x^2} \left (7 a^2 d^2-18 a b c d+15 b^2 c^2\right )}{15 c^3 e^5 \sqrt{e x}}-\frac{2 a^2 \sqrt{c+d x^2}}{9 c e (e x)^{9/2}}-\frac{2 a \sqrt{c+d x^2} (18 b c-7 a d)}{45 c^2 e^3 (e x)^{5/2}} \]

[Out]

(-2*a^2*Sqrt[c + d*x^2])/(9*c*e*(e*x)^(9/2)) - (2*a*(18*b*c - 7*a*d)*Sqrt[c + d*
x^2])/(45*c^2*e^3*(e*x)^(5/2)) - (2*(15*b^2*c^2 - 18*a*b*c*d + 7*a^2*d^2)*Sqrt[c
+ d*x^2])/(15*c^3*e^5*Sqrt[e*x]) + (2*Sqrt[d]*(15*b^2*c^2 - 18*a*b*c*d + 7*a^2*
d^2)*Sqrt[e*x]*Sqrt[c + d*x^2])/(15*c^3*e^6*(Sqrt[c] + Sqrt[d]*x)) - (2*d^(1/4)*
(15*b^2*c^2 - 18*a*b*c*d + 7*a^2*d^2)*(Sqrt[c] + Sqrt[d]*x)*Sqrt[(c + d*x^2)/(Sq
rt[c] + Sqrt[d]*x)^2]*EllipticE[2*ArcTan[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])],
1/2])/(15*c^(11/4)*e^(11/2)*Sqrt[c + d*x^2]) + (d^(1/4)*(15*b^2*c^2 - 18*a*b*c*
d + 7*a^2*d^2)*(Sqrt[c] + Sqrt[d]*x)*Sqrt[(c + d*x^2)/(Sqrt[c] + Sqrt[d]*x)^2]*E
llipticF[2*ArcTan[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], 1/2])/(15*c^(11/4)*e^(
11/2)*Sqrt[c + d*x^2])

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Rubi [A]  time = 0.980644, antiderivative size = 438, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{\sqrt [4]{d} \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (7 a^2 d^2-18 a b c d+15 b^2 c^2\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{15 c^{11/4} e^{11/2} \sqrt{c+d x^2}}-\frac{2 \sqrt [4]{d} \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (7 a^2 d^2-18 a b c d+15 b^2 c^2\right ) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{15 c^{11/4} e^{11/2} \sqrt{c+d x^2}}+\frac{2 \sqrt{d} \sqrt{e x} \sqrt{c+d x^2} \left (7 a^2 d^2-18 a b c d+15 b^2 c^2\right )}{15 c^3 e^6 \left (\sqrt{c}+\sqrt{d} x\right )}-\frac{2 \sqrt{c+d x^2} \left (7 a^2 d^2-18 a b c d+15 b^2 c^2\right )}{15 c^3 e^5 \sqrt{e x}}-\frac{2 a^2 \sqrt{c+d x^2}}{9 c e (e x)^{9/2}}-\frac{2 a \sqrt{c+d x^2} (18 b c-7 a d)}{45 c^2 e^3 (e x)^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*a^2*Sqrt[c + d*x^2])/(9*c*e*(e*x)^(9/2)) - (2*a*(18*b*c - 7*a*d)*Sqrt[c + d*
x^2])/(45*c^2*e^3*(e*x)^(5/2)) - (2*(15*b^2*c^2 - 18*a*b*c*d + 7*a^2*d^2)*Sqrt[c
+ d*x^2])/(15*c^3*e^5*Sqrt[e*x]) + (2*Sqrt[d]*(15*b^2*c^2 - 18*a*b*c*d + 7*a^2*
d^2)*Sqrt[e*x]*Sqrt[c + d*x^2])/(15*c^3*e^6*(Sqrt[c] + Sqrt[d]*x)) - (2*d^(1/4)*
(15*b^2*c^2 - 18*a*b*c*d + 7*a^2*d^2)*(Sqrt[c] + Sqrt[d]*x)*Sqrt[(c + d*x^2)/(Sq
rt[c] + Sqrt[d]*x)^2]*EllipticE[2*ArcTan[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])],
1/2])/(15*c^(11/4)*e^(11/2)*Sqrt[c + d*x^2]) + (d^(1/4)*(15*b^2*c^2 - 18*a*b*c*
d + 7*a^2*d^2)*(Sqrt[c] + Sqrt[d]*x)*Sqrt[(c + d*x^2)/(Sqrt[c] + Sqrt[d]*x)^2]*E
llipticF[2*ArcTan[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], 1/2])/(15*c^(11/4)*e^(
11/2)*Sqrt[c + d*x^2])

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Rubi in Sympy [A]  time = 98.4659, size = 410, normalized size = 0.94 \[ - \frac{2 a^{2} \sqrt{c + d x^{2}}}{9 c e \left (e x\right )^{\frac{9}{2}}} + \frac{2 a \sqrt{c + d x^{2}} \left (7 a d - 18 b c\right )}{45 c^{2} e^{3} \left (e x\right )^{\frac{5}{2}}} + \frac{2 \sqrt{d} \sqrt{e x} \sqrt{c + d x^{2}} \left (a d \left (7 a d - 18 b c\right ) + 15 b^{2} c^{2}\right )}{15 c^{3} e^{6} \left (\sqrt{c} + \sqrt{d} x\right )} - \frac{2 \sqrt{c + d x^{2}} \left (a d \left (7 a d - 18 b c\right ) + 15 b^{2} c^{2}\right )}{15 c^{3} e^{5} \sqrt{e x}} - \frac{2 \sqrt [4]{d} \sqrt{\frac{c + d x^{2}}{\left (\sqrt{c} + \sqrt{d} x\right )^{2}}} \left (\sqrt{c} + \sqrt{d} x\right ) \left (a d \left (7 a d - 18 b c\right ) + 15 b^{2} c^{2}\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}} \right )}\middle | \frac{1}{2}\right )}{15 c^{\frac{11}{4}} e^{\frac{11}{2}} \sqrt{c + d x^{2}}} + \frac{\sqrt [4]{d} \sqrt{\frac{c + d x^{2}}{\left (\sqrt{c} + \sqrt{d} x\right )^{2}}} \left (\sqrt{c} + \sqrt{d} x\right ) \left (a d \left (7 a d - 18 b c\right ) + 15 b^{2} c^{2}\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}} \right )}\middle | \frac{1}{2}\right )}{15 c^{\frac{11}{4}} e^{\frac{11}{2}} \sqrt{c + d x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*a**2*sqrt(c + d*x**2)/(9*c*e*(e*x)**(9/2)) + 2*a*sqrt(c + d*x**2)*(7*a*d - 18
*b*c)/(45*c**2*e**3*(e*x)**(5/2)) + 2*sqrt(d)*sqrt(e*x)*sqrt(c + d*x**2)*(a*d*(7
*a*d - 18*b*c) + 15*b**2*c**2)/(15*c**3*e**6*(sqrt(c) + sqrt(d)*x)) - 2*sqrt(c +
d*x**2)*(a*d*(7*a*d - 18*b*c) + 15*b**2*c**2)/(15*c**3*e**5*sqrt(e*x)) - 2*d**(
1/4)*sqrt((c + d*x**2)/(sqrt(c) + sqrt(d)*x)**2)*(sqrt(c) + sqrt(d)*x)*(a*d*(7*a
*d - 18*b*c) + 15*b**2*c**2)*elliptic_e(2*atan(d**(1/4)*sqrt(e*x)/(c**(1/4)*sqrt
(e))), 1/2)/(15*c**(11/4)*e**(11/2)*sqrt(c + d*x**2)) + d**(1/4)*sqrt((c + d*x**
2)/(sqrt(c) + sqrt(d)*x)**2)*(sqrt(c) + sqrt(d)*x)*(a*d*(7*a*d - 18*b*c) + 15*b*
*2*c**2)*elliptic_f(2*atan(d**(1/4)*sqrt(e*x)/(c**(1/4)*sqrt(e))), 1/2)/(15*c**(
11/4)*e**(11/2)*sqrt(c + d*x**2))

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Mathematica [C]  time = 0.694552, size = 288, normalized size = 0.66 \[ \frac{\sqrt{e x} \left (-6 \sqrt{c} \sqrt{d} x^5 \sqrt{\frac{d x^2}{c}+1} \left (7 a^2 d^2-18 a b c d+15 b^2 c^2\right ) F\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{d} x}{\sqrt{c}}}\right )\right |-1\right )+6 \sqrt{c} \sqrt{d} x^5 \sqrt{\frac{d x^2}{c}+1} \left (7 a^2 d^2-18 a b c d+15 b^2 c^2\right ) E\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{d} x}{\sqrt{c}}}\right )\right |-1\right )-2 \sqrt{\frac{i \sqrt{d} x}{\sqrt{c}}} \left (c+d x^2\right ) \left (a^2 \left (5 c^2-7 c d x^2+21 d^2 x^4\right )+18 a b c x^2 \left (c-3 d x^2\right )+45 b^2 c^2 x^4\right )\right )}{45 c^3 e^6 x^5 \sqrt{\frac{i \sqrt{d} x}{\sqrt{c}}} \sqrt{c+d x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[e*x]*(-2*Sqrt[(I*Sqrt[d]*x)/Sqrt[c]]*(c + d*x^2)*(45*b^2*c^2*x^4 + 18*a*b*
c*x^2*(c - 3*d*x^2) + a^2*(5*c^2 - 7*c*d*x^2 + 21*d^2*x^4)) + 6*Sqrt[c]*Sqrt[d]*
(15*b^2*c^2 - 18*a*b*c*d + 7*a^2*d^2)*x^5*Sqrt[1 + (d*x^2)/c]*EllipticE[I*ArcSin
h[Sqrt[(I*Sqrt[d]*x)/Sqrt[c]]], -1] - 6*Sqrt[c]*Sqrt[d]*(15*b^2*c^2 - 18*a*b*c*d
+ 7*a^2*d^2)*x^5*Sqrt[1 + (d*x^2)/c]*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[d]*x)/Sqr
t[c]]], -1]))/(45*c^3*e^6*x^5*Sqrt[(I*Sqrt[d]*x)/Sqrt[c]]*Sqrt[c + d*x^2])

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Maple [A]  time = 0.055, size = 667, normalized size = 1.5 \[{\frac{1}{45\,{x}^{4}{e}^{5}{c}^{3}} \left ( 42\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticE} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ){x}^{4}{a}^{2}c{d}^{2}-108\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticE} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ){x}^{4}ab{c}^{2}d+90\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticE} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ){x}^{4}{b}^{2}{c}^{3}-21\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticF} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ){x}^{4}{a}^{2}c{d}^{2}+54\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticF} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ){x}^{4}ab{c}^{2}d-45\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticF} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ){x}^{4}{b}^{2}{c}^{3}-42\,{x}^{6}{a}^{2}{d}^{3}+108\,{x}^{6}abc{d}^{2}-90\,{x}^{6}{b}^{2}{c}^{2}d-28\,{x}^{4}{a}^{2}c{d}^{2}+72\,{x}^{4}ab{c}^{2}d-90\,{x}^{4}{b}^{2}{c}^{3}+4\,{x}^{2}{a}^{2}{c}^{2}d-36\,{x}^{2}ab{c}^{3}-10\,{a}^{2}{c}^{3} \right ){\frac{1}{\sqrt{d{x}^{2}+c}}}{\frac{1}{\sqrt{ex}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/45/x^4*(42*((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^2-108*((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*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^3-21*((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)*Elliptic
F(((d*x+(-c*d)^(1/2))/(-c*d)^(1/2))^(1/2),1/2*2^(1/2))*x^4*a^2*c*d^2+54*((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^3-42*x^6*a^2*d^3+108*x^6*
a*b*c*d^2-90*x^6*b^2*c^2*d-28*x^4*a^2*c*d^2+72*x^4*a*b*c^2*d-90*x^4*b^2*c^3+4*x^
2*a^2*c^2*d-36*x^2*a*b*c^3-10*a^2*c^3)/(d*x^2+c)^(1/2)/e^5/(e*x)^(1/2)/c^3

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}{\sqrt{d x^{2} + c} \sqrt{e x} e^{5} x^{5}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^2/(sqrt(d*x^2 + c)*(e*x)^(11/2)),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^5*x^5), 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)**(11/2)/(d*x**2+c)**(1/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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