3.29 \(\int \left (a+b x^2\right ) \sqrt{c+d x^2} \left (e+f x^2\right )^{3/2} \, dx\)

Optimal. Leaf size=543 \[ \frac{x \sqrt{c+d x^2} \sqrt{e+f x^2} \left (14 a d f (3 d e-c f)+b \left (8 c^2 f^2-15 c d e f+3 d^2 e^2\right )\right )}{105 d^2 f}+\frac{e^{3/2} \sqrt{c+d x^2} \left (7 a d f (9 d e-c f)-b \left (-4 c^2 f^2+9 c d e f+3 d^2 e^2\right )\right ) F\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{105 d^2 f^{3/2} \sqrt{e+f x^2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac{x \sqrt{c+d x^2} \left (7 a d f \left (-2 c^2 f^2+7 c d e f+3 d^2 e^2\right )-b \left (-8 c^3 f^3+19 c^2 d e f^2-9 c d^2 e^2 f+6 d^3 e^3\right )\right )}{105 d^3 f \sqrt{e+f x^2}}-\frac{\sqrt{e} \sqrt{c+d x^2} \left (7 a d f \left (-2 c^2 f^2+7 c d e f+3 d^2 e^2\right )-b \left (-8 c^3 f^3+19 c^2 d e f^2-9 c d^2 e^2 f+6 d^3 e^3\right )\right ) E\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{105 d^3 f^{3/2} \sqrt{e+f x^2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac{x \left (c+d x^2\right )^{3/2} \sqrt{e+f x^2} (7 a d f-4 b c f+3 b d e)}{35 d^2}+\frac{b x \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^{3/2}}{7 d} \]

[Out]

((7*a*d*f*(3*d^2*e^2 + 7*c*d*e*f - 2*c^2*f^2) - b*(6*d^3*e^3 - 9*c*d^2*e^2*f + 1
9*c^2*d*e*f^2 - 8*c^3*f^3))*x*Sqrt[c + d*x^2])/(105*d^3*f*Sqrt[e + f*x^2]) + ((1
4*a*d*f*(3*d*e - c*f) + b*(3*d^2*e^2 - 15*c*d*e*f + 8*c^2*f^2))*x*Sqrt[c + d*x^2
]*Sqrt[e + f*x^2])/(105*d^2*f) + ((3*b*d*e - 4*b*c*f + 7*a*d*f)*x*(c + d*x^2)^(3
/2)*Sqrt[e + f*x^2])/(35*d^2) + (b*x*(c + d*x^2)^(3/2)*(e + f*x^2)^(3/2))/(7*d)
- (Sqrt[e]*(7*a*d*f*(3*d^2*e^2 + 7*c*d*e*f - 2*c^2*f^2) - b*(6*d^3*e^3 - 9*c*d^2
*e^2*f + 19*c^2*d*e*f^2 - 8*c^3*f^3))*Sqrt[c + d*x^2]*EllipticE[ArcTan[(Sqrt[f]*
x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(105*d^3*f^(3/2)*Sqrt[(e*(c + d*x^2))/(c*(e + f*x
^2))]*Sqrt[e + f*x^2]) + (e^(3/2)*(7*a*d*f*(9*d*e - c*f) - b*(3*d^2*e^2 + 9*c*d*
e*f - 4*c^2*f^2))*Sqrt[c + d*x^2]*EllipticF[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*
e)/(c*f)])/(105*d^2*f^(3/2)*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2
])

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Rubi [A]  time = 1.71365, antiderivative size = 543, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{x \sqrt{c+d x^2} \sqrt{e+f x^2} \left (14 a d f (3 d e-c f)+b \left (8 c^2 f^2-15 c d e f+3 d^2 e^2\right )\right )}{105 d^2 f}+\frac{e^{3/2} \sqrt{c+d x^2} \left (7 a d f (9 d e-c f)-b \left (-4 c^2 f^2+9 c d e f+3 d^2 e^2\right )\right ) F\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{105 d^2 f^{3/2} \sqrt{e+f x^2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac{x \sqrt{c+d x^2} \left (7 a d f \left (-2 c^2 f^2+7 c d e f+3 d^2 e^2\right )-b \left (-8 c^3 f^3+19 c^2 d e f^2-9 c d^2 e^2 f+6 d^3 e^3\right )\right )}{105 d^3 f \sqrt{e+f x^2}}-\frac{\sqrt{e} \sqrt{c+d x^2} \left (7 a d f \left (-2 c^2 f^2+7 c d e f+3 d^2 e^2\right )-b \left (-8 c^3 f^3+19 c^2 d e f^2-9 c d^2 e^2 f+6 d^3 e^3\right )\right ) E\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{105 d^3 f^{3/2} \sqrt{e+f x^2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac{x \left (c+d x^2\right )^{3/2} \sqrt{e+f x^2} (7 a d f-4 b c f+3 b d e)}{35 d^2}+\frac{b x \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^{3/2}}{7 d} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^2)*Sqrt[c + d*x^2]*(e + f*x^2)^(3/2),x]

[Out]

((7*a*d*f*(3*d^2*e^2 + 7*c*d*e*f - 2*c^2*f^2) - b*(6*d^3*e^3 - 9*c*d^2*e^2*f + 1
9*c^2*d*e*f^2 - 8*c^3*f^3))*x*Sqrt[c + d*x^2])/(105*d^3*f*Sqrt[e + f*x^2]) + ((1
4*a*d*f*(3*d*e - c*f) + b*(3*d^2*e^2 - 15*c*d*e*f + 8*c^2*f^2))*x*Sqrt[c + d*x^2
]*Sqrt[e + f*x^2])/(105*d^2*f) + ((3*b*d*e - 4*b*c*f + 7*a*d*f)*x*(c + d*x^2)^(3
/2)*Sqrt[e + f*x^2])/(35*d^2) + (b*x*(c + d*x^2)^(3/2)*(e + f*x^2)^(3/2))/(7*d)
- (Sqrt[e]*(7*a*d*f*(3*d^2*e^2 + 7*c*d*e*f - 2*c^2*f^2) - b*(6*d^3*e^3 - 9*c*d^2
*e^2*f + 19*c^2*d*e*f^2 - 8*c^3*f^3))*Sqrt[c + d*x^2]*EllipticE[ArcTan[(Sqrt[f]*
x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(105*d^3*f^(3/2)*Sqrt[(e*(c + d*x^2))/(c*(e + f*x
^2))]*Sqrt[e + f*x^2]) + (e^(3/2)*(7*a*d*f*(9*d*e - c*f) - b*(3*d^2*e^2 + 9*c*d*
e*f - 4*c^2*f^2))*Sqrt[c + d*x^2]*EllipticF[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*
e)/(c*f)])/(105*d^2*f^(3/2)*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2
])

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a)*(d*x**2+c)**(1/2)*(f*x**2+e)**(3/2),x)

[Out]

Timed out

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Mathematica [C]  time = 2.1212, size = 372, normalized size = 0.69 \[ \frac{i e \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} (c f-d e) \left (b \left (4 c^2 f^2-6 c d e f+6 d^2 e^2\right )-7 a d f (c f+3 d e)\right ) F\left (i \sinh ^{-1}\left (\sqrt{\frac{d}{c}} x\right )|\frac{c f}{d e}\right )+f x \left (-\sqrt{\frac{d}{c}}\right ) \left (c+d x^2\right ) \left (e+f x^2\right ) \left (-7 a d f \left (c f+6 d e+3 d f x^2\right )+4 b c^2 f^2-3 b c d f \left (3 e+f x^2\right )-3 b d^2 \left (e^2+8 e f x^2+5 f^2 x^4\right )\right )-i e \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} \left (7 a d f \left (-2 c^2 f^2+7 c d e f+3 d^2 e^2\right )+b \left (8 c^3 f^3-19 c^2 d e f^2+9 c d^2 e^2 f-6 d^3 e^3\right )\right ) E\left (i \sinh ^{-1}\left (\sqrt{\frac{d}{c}} x\right )|\frac{c f}{d e}\right )}{105 c^2 f^2 \left (\frac{d}{c}\right )^{5/2} \sqrt{c+d x^2} \sqrt{e+f x^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^2)*Sqrt[c + d*x^2]*(e + f*x^2)^(3/2),x]

[Out]

(-(Sqrt[d/c]*f*x*(c + d*x^2)*(e + f*x^2)*(4*b*c^2*f^2 - 3*b*c*d*f*(3*e + f*x^2)
- 7*a*d*f*(6*d*e + c*f + 3*d*f*x^2) - 3*b*d^2*(e^2 + 8*e*f*x^2 + 5*f^2*x^4))) -
I*e*(7*a*d*f*(3*d^2*e^2 + 7*c*d*e*f - 2*c^2*f^2) + b*(-6*d^3*e^3 + 9*c*d^2*e^2*f
- 19*c^2*d*e*f^2 + 8*c^3*f^3))*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*Elliptic
E[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)] + I*e*(-(d*e) + c*f)*(-7*a*d*f*(3*d*e + c
*f) + b*(6*d^2*e^2 - 6*c*d*e*f + 4*c^2*f^2))*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2
)/e]*EllipticF[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)])/(105*c^2*(d/c)^(5/2)*f^2*Sq
rt[c + d*x^2]*Sqrt[e + f*x^2])

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Maple [B]  time = 0.028, size = 1332, normalized size = 2.5 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)*(d*x^2+c)^(1/2)*(f*x^2+e)^(3/2),x)

[Out]

1/105*(d*x^2+c)^(1/2)*(f*x^2+e)^(1/2)*(9*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)
*EllipticE(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*b*c*d^2*e^3*f+6*((d*x^2+c)/c)^(1/2)*(
(f*x^2+e)/e)^(1/2)*EllipticF(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*b*d^3*e^4-6*((d*x^2
+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)*EllipticE(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*b*d^3
*e^4+18*(-d/c)^(1/2)*x^7*b*c*d^2*f^4+39*(-d/c)^(1/2)*x^7*b*d^3*e*f^3+28*(-d/c)^(
1/2)*x^5*a*c*d^2*f^4+63*(-d/c)^(1/2)*x^5*a*d^3*e*f^3-(-d/c)^(1/2)*x^5*b*c^2*d*f^
4+27*(-d/c)^(1/2)*x^5*b*d^3*e^2*f^2+7*(-d/c)^(1/2)*x^3*a*c^2*d*f^4+42*(-d/c)^(1/
2)*x^3*a*d^3*e^2*f^2+3*(-d/c)^(1/2)*x^3*b*d^3*e^3*f-4*(-d/c)^(1/2)*x*b*c^3*e*f^3
-4*(-d/c)^(1/2)*x^3*b*c^3*f^4+15*(-d/c)^(1/2)*x^9*b*d^3*f^4+7*(-d/c)^(1/2)*x*a*c
^2*d*e*f^3+42*(-d/c)^(1/2)*x*a*c*d^2*e^2*f^2-21*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e
)^(1/2)*EllipticF(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*a*d^3*e^3*f-4*((d*x^2+c)/c)^(1
/2)*((f*x^2+e)/e)^(1/2)*EllipticF(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*b*c^3*e*f^3+9*
(-d/c)^(1/2)*x*b*c^2*d*e^2*f^2+3*(-d/c)^(1/2)*x*b*c*d^2*e^3*f+21*((d*x^2+c)/c)^(
1/2)*((f*x^2+e)/e)^(1/2)*EllipticE(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*a*d^3*e^3*f+8
*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)*EllipticE(x*(-d/c)^(1/2),(c*f/d/e)^(1/2
))*b*c^3*e*f^3+21*(-d/c)^(1/2)*x^7*a*d^3*f^4+7*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)
^(1/2)*EllipticF(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*a*c^2*d*e*f^3-19*((d*x^2+c)/c)^
(1/2)*((f*x^2+e)/e)^(1/2)*EllipticE(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*b*c^2*d*e^2*
f^2-14*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)*EllipticE(x*(-d/c)^(1/2),(c*f/d/e
)^(1/2))*a*c^2*d*e*f^3+49*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)*EllipticE(x*(-
d/c)^(1/2),(c*f/d/e)^(1/2))*a*c*d^2*e^2*f^2+14*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)
^(1/2)*EllipticF(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*a*c*d^2*e^2*f^2+10*((d*x^2+c)/c
)^(1/2)*((f*x^2+e)/e)^(1/2)*EllipticF(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*b*c^2*d*e^
2*f^2-12*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)*EllipticF(x*(-d/c)^(1/2),(c*f/d
/e)^(1/2))*b*c*d^2*e^3*f+51*(-d/c)^(1/2)*x^5*b*c*d^2*e*f^3+70*(-d/c)^(1/2)*x^3*a
*c*d^2*e*f^3+8*(-d/c)^(1/2)*x^3*b*c^2*d*e*f^3+36*(-d/c)^(1/2)*x^3*b*c*d^2*e^2*f^
2)/f^2/(d*f*x^4+c*f*x^2+d*e*x^2+c*e)/d^2/(-d/c)^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)*sqrt(d*x^2 + c)*(f*x^2 + e)^(3/2),x, algorithm="maxima")

[Out]

integrate((b*x^2 + a)*sqrt(d*x^2 + c)*(f*x^2 + e)^(3/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)*sqrt(d*x^2 + c)*(f*x^2 + e)^(3/2),x, algorithm="fricas")

[Out]

integral((b*f*x^4 + (b*e + a*f)*x^2 + a*e)*sqrt(d*x^2 + c)*sqrt(f*x^2 + e), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \left (a + b x^{2}\right ) \sqrt{c + d x^{2}} \left (e + f x^{2}\right )^{\frac{3}{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**2+a)*(d*x**2+c)**(1/2)*(f*x**2+e)**(3/2),x)

[Out]

Integral((a + b*x**2)*sqrt(c + d*x**2)*(e + f*x**2)**(3/2), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (b x^{2} + a\right )} \sqrt{d x^{2} + c}{\left (f x^{2} + e\right )}^{\frac{3}{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)*sqrt(d*x^2 + c)*(f*x^2 + e)^(3/2),x, algorithm="giac")

[Out]

integrate((b*x^2 + a)*sqrt(d*x^2 + c)*(f*x^2 + e)^(3/2), x)