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

Optimal. Leaf size=421 \[ -\frac{2 a^2 \left (c+d x^2\right )^{3/2}}{5 c e (e x)^{5/2}}+\frac{2 (e x)^{3/2} \sqrt{c+d x^2} \left (a d (a d+10 b c)+b^2 c^2\right )}{5 c^2 e^5}+\frac{4 \sqrt{e x} \sqrt{c+d x^2} \left (a d (a d+10 b c)+b^2 c^2\right )}{5 c \sqrt{d} e^4 \left (\sqrt{c}+\sqrt{d} x\right )}+\frac{2 \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (a d (a d+10 b c)+b^2 c^2\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{5 c^{3/4} d^{3/4} e^{7/2} \sqrt{c+d x^2}}-\frac{4 \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (a d (a d+10 b c)+b^2 c^2\right ) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{5 c^{3/4} d^{3/4} e^{7/2} \sqrt{c+d x^2}}-\frac{2 a \left (c+d x^2\right )^{3/2} (a d+10 b c)}{5 c^2 e^3 \sqrt{e x}} \]

[Out]

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

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Rubi [A]  time = 0.963847, antiderivative size = 421, 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 \[ -\frac{2 a^2 \left (c+d x^2\right )^{3/2}}{5 c e (e x)^{5/2}}+\frac{2 (e x)^{3/2} \sqrt{c+d x^2} \left (a d (a d+10 b c)+b^2 c^2\right )}{5 c^2 e^5}+\frac{4 \sqrt{e x} \sqrt{c+d x^2} \left (a d (a d+10 b c)+b^2 c^2\right )}{5 c \sqrt{d} e^4 \left (\sqrt{c}+\sqrt{d} x\right )}+\frac{2 \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (a d (a d+10 b c)+b^2 c^2\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{5 c^{3/4} d^{3/4} e^{7/2} \sqrt{c+d x^2}}-\frac{4 \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (a d (a d+10 b c)+b^2 c^2\right ) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{5 c^{3/4} d^{3/4} e^{7/2} \sqrt{c+d x^2}}-\frac{2 a \left (c+d x^2\right )^{3/2} (a d+10 b c)}{5 c^2 e^3 \sqrt{e x}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 94.1369, size = 394, normalized size = 0.94 \[ - \frac{2 a^{2} \left (c + d x^{2}\right )^{\frac{3}{2}}}{5 c e \left (e x\right )^{\frac{5}{2}}} - \frac{2 a \left (c + d x^{2}\right )^{\frac{3}{2}} \left (a d + 10 b c\right )}{5 c^{2} e^{3} \sqrt{e x}} + \frac{4 \sqrt{e x} \sqrt{c + d x^{2}} \left (a d \left (a d + 10 b c\right ) + b^{2} c^{2}\right )}{5 c \sqrt{d} e^{4} \left (\sqrt{c} + \sqrt{d} x\right )} + \frac{2 \left (e x\right )^{\frac{3}{2}} \sqrt{c + d x^{2}} \left (a d \left (a d + 10 b c\right ) + b^{2} c^{2}\right )}{5 c^{2} e^{5}} - \frac{4 \sqrt{\frac{c + d x^{2}}{\left (\sqrt{c} + \sqrt{d} x\right )^{2}}} \left (\sqrt{c} + \sqrt{d} x\right ) \left (a d \left (a d + 10 b c\right ) + b^{2} c^{2}\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}} \right )}\middle | \frac{1}{2}\right )}{5 c^{\frac{3}{4}} d^{\frac{3}{4}} e^{\frac{7}{2}} \sqrt{c + d x^{2}}} + \frac{2 \sqrt{\frac{c + d x^{2}}{\left (\sqrt{c} + \sqrt{d} x\right )^{2}}} \left (\sqrt{c} + \sqrt{d} x\right ) \left (a d \left (a d + 10 b c\right ) + b^{2} c^{2}\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}} \right )}\middle | \frac{1}{2}\right )}{5 c^{\frac{3}{4}} d^{\frac{3}{4}} e^{\frac{7}{2}} \sqrt{c + d x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 1.40011, size = 226, normalized size = 0.54 \[ \frac{x^{7/2} \left (\frac{2 \sqrt{c+d x^2} \left (-a^2 \left (c+2 d x^2\right )-10 a b c x^2+b^2 c x^4\right )}{c x^{5/2}}-\frac{4 x \left (a^2 d^2+10 a b c d+b^2 c^2\right ) \left (-\sqrt{x} \left (\frac{c}{x^2}+d\right )+\frac{i c \sqrt{\frac{c}{d x^2}+1} \left (E\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{c}}{\sqrt{d}}}}{\sqrt{x}}\right )\right |-1\right )-F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{c}}{\sqrt{d}}}}{\sqrt{x}}\right )\right |-1\right )\right )}{\left (\frac{i \sqrt{c}}{\sqrt{d}}\right )^{3/2}}\right )}{c d \sqrt{c+d x^2}}\right )}{5 (e x)^{7/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.074, size = 648, normalized size = 1.5 \[{\frac{2}{5\,d{x}^{2}{e}^{3}c} \left ( 2\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticE} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ){x}^{2}{a}^{2}c{d}^{2}+20\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticE} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ){x}^{2}ab{c}^{2}d+2\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticE} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ){x}^{2}{b}^{2}{c}^{3}-\sqrt{{1 \left ( dx+\sqrt{-cd} \right ){\frac{1}{\sqrt{-cd}}}}}\sqrt{2}\sqrt{{1 \left ( -dx+\sqrt{-cd} \right ){\frac{1}{\sqrt{-cd}}}}}\sqrt{-{dx{\frac{1}{\sqrt{-cd}}}}}{\it EllipticF} \left ( \sqrt{{1 \left ( dx+\sqrt{-cd} \right ){\frac{1}{\sqrt{-cd}}}}},{\frac{\sqrt{2}}{2}} \right ){x}^{2}{a}^{2}c{d}^{2}-10\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticF} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ){x}^{2}ab{c}^{2}d-\sqrt{{1 \left ( dx+\sqrt{-cd} \right ){\frac{1}{\sqrt{-cd}}}}}\sqrt{2}\sqrt{{1 \left ( -dx+\sqrt{-cd} \right ){\frac{1}{\sqrt{-cd}}}}}\sqrt{-{dx{\frac{1}{\sqrt{-cd}}}}}{\it EllipticF} \left ( \sqrt{{1 \left ( dx+\sqrt{-cd} \right ){\frac{1}{\sqrt{-cd}}}}},{\frac{\sqrt{2}}{2}} \right ){x}^{2}{b}^{2}{c}^{3}+{x}^{6}{b}^{2}c{d}^{2}-2\,{x}^{4}{a}^{2}{d}^{3}-10\,{x}^{4}abc{d}^{2}+{x}^{4}{b}^{2}{c}^{2}d-3\,{x}^{2}{a}^{2}c{d}^{2}-10\,{x}^{2}ab{c}^{2}d-{a}^{2}{c}^{2}d \right ){\frac{1}{\sqrt{d{x}^{2}+c}}}{\frac{1}{\sqrt{ex}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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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. \[{\rm integral}\left (\frac{{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )} \sqrt{d x^{2} + c}}{\sqrt{e x} e^{3} x^{3}}, x\right ) \]

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*(d*x**2+c)**(1/2)/(e*x)**(7/2),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)