3.961 \(\int \frac{x^2 \left (a+b x^2\right )^{5/2}}{\sqrt{c+d x^2}} \, dx\)

Optimal. Leaf size=436 \[ \frac{x \sqrt{a+b x^2} \sqrt{c+d x^2} \left (45 a^2 d^2-61 a b c d+24 b^2 c^2\right )}{105 d^3}-\frac{c^{3/2} \sqrt{a+b x^2} \left (45 a^2 d^2-61 a b c d+24 b^2 c^2\right ) F\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{105 d^{7/2} \sqrt{c+d x^2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac{x \sqrt{a+b x^2} \left (-15 a^3 d^3+103 a^2 b c d^2-128 a b^2 c^2 d+48 b^3 c^3\right )}{105 b d^3 \sqrt{c+d x^2}}+\frac{\sqrt{c} \sqrt{a+b x^2} \left (-15 a^3 d^3+103 a^2 b c d^2-128 a b^2 c^2 d+48 b^3 c^3\right ) E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{105 b d^{7/2} \sqrt{c+d x^2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac{2 b x^3 \sqrt{a+b x^2} \sqrt{c+d x^2} (3 b c-5 a d)}{35 d^2}+\frac{b x^3 \left (a+b x^2\right )^{3/2} \sqrt{c+d x^2}}{7 d} \]

[Out]

-((48*b^3*c^3 - 128*a*b^2*c^2*d + 103*a^2*b*c*d^2 - 15*a^3*d^3)*x*Sqrt[a + b*x^2
])/(105*b*d^3*Sqrt[c + d*x^2]) + ((24*b^2*c^2 - 61*a*b*c*d + 45*a^2*d^2)*x*Sqrt[
a + b*x^2]*Sqrt[c + d*x^2])/(105*d^3) - (2*b*(3*b*c - 5*a*d)*x^3*Sqrt[a + b*x^2]
*Sqrt[c + d*x^2])/(35*d^2) + (b*x^3*(a + b*x^2)^(3/2)*Sqrt[c + d*x^2])/(7*d) + (
Sqrt[c]*(48*b^3*c^3 - 128*a*b^2*c^2*d + 103*a^2*b*c*d^2 - 15*a^3*d^3)*Sqrt[a + b
*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(105*b*d^(7/2)*Sq
rt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) - (c^(3/2)*(24*b^2*c^2 - 61
*a*b*c*d + 45*a^2*d^2)*Sqrt[a + b*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1
- (b*c)/(a*d)])/(105*d^(7/2)*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^
2])

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Rubi [A]  time = 1.27749, antiderivative size = 436, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.269 \[ \frac{x \sqrt{a+b x^2} \sqrt{c+d x^2} \left (45 a^2 d^2-61 a b c d+24 b^2 c^2\right )}{105 d^3}-\frac{c^{3/2} \sqrt{a+b x^2} \left (45 a^2 d^2-61 a b c d+24 b^2 c^2\right ) F\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{105 d^{7/2} \sqrt{c+d x^2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac{x \sqrt{a+b x^2} \left (-15 a^3 d^3+103 a^2 b c d^2-128 a b^2 c^2 d+48 b^3 c^3\right )}{105 b d^3 \sqrt{c+d x^2}}+\frac{\sqrt{c} \sqrt{a+b x^2} \left (-15 a^3 d^3+103 a^2 b c d^2-128 a b^2 c^2 d+48 b^3 c^3\right ) E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{105 b d^{7/2} \sqrt{c+d x^2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac{2 b x^3 \sqrt{a+b x^2} \sqrt{c+d x^2} (3 b c-5 a d)}{35 d^2}+\frac{b x^3 \left (a+b x^2\right )^{3/2} \sqrt{c+d x^2}}{7 d} \]

Antiderivative was successfully verified.

[In]  Int[(x^2*(a + b*x^2)^(5/2))/Sqrt[c + d*x^2],x]

[Out]

-((48*b^3*c^3 - 128*a*b^2*c^2*d + 103*a^2*b*c*d^2 - 15*a^3*d^3)*x*Sqrt[a + b*x^2
])/(105*b*d^3*Sqrt[c + d*x^2]) + ((24*b^2*c^2 - 61*a*b*c*d + 45*a^2*d^2)*x*Sqrt[
a + b*x^2]*Sqrt[c + d*x^2])/(105*d^3) - (2*b*(3*b*c - 5*a*d)*x^3*Sqrt[a + b*x^2]
*Sqrt[c + d*x^2])/(35*d^2) + (b*x^3*(a + b*x^2)^(3/2)*Sqrt[c + d*x^2])/(7*d) + (
Sqrt[c]*(48*b^3*c^3 - 128*a*b^2*c^2*d + 103*a^2*b*c*d^2 - 15*a^3*d^3)*Sqrt[a + b
*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(105*b*d^(7/2)*Sq
rt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) - (c^(3/2)*(24*b^2*c^2 - 61
*a*b*c*d + 45*a^2*d^2)*Sqrt[a + b*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1
- (b*c)/(a*d)])/(105*d^(7/2)*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^
2])

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Rubi in Sympy [A]  time = 149.728, size = 410, normalized size = 0.94 \[ \frac{b x^{3} \left (a + b x^{2}\right )^{\frac{3}{2}} \sqrt{c + d x^{2}}}{7 d} + \frac{2 b x^{3} \sqrt{a + b x^{2}} \sqrt{c + d x^{2}} \left (5 a d - 3 b c\right )}{35 d^{2}} - \frac{c^{\frac{3}{2}} \sqrt{a + b x^{2}} \left (45 a^{2} d^{2} - 61 a b c d + 24 b^{2} c^{2}\right ) F\left (\operatorname{atan}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}\middle | 1 - \frac{b c}{a d}\right )}{105 d^{\frac{7}{2}} \sqrt{\frac{c \left (a + b x^{2}\right )}{a \left (c + d x^{2}\right )}} \sqrt{c + d x^{2}}} + \frac{x \sqrt{a + b x^{2}} \sqrt{c + d x^{2}} \left (45 a^{2} d^{2} - 61 a b c d + 24 b^{2} c^{2}\right )}{105 d^{3}} - \frac{\sqrt{c} \sqrt{a + b x^{2}} \left (15 a^{3} d^{3} - 103 a^{2} b c d^{2} + 128 a b^{2} c^{2} d - 48 b^{3} c^{3}\right ) E\left (\operatorname{atan}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}\middle | 1 - \frac{b c}{a d}\right )}{105 b d^{\frac{7}{2}} \sqrt{\frac{c \left (a + b x^{2}\right )}{a \left (c + d x^{2}\right )}} \sqrt{c + d x^{2}}} + \frac{x \sqrt{a + b x^{2}} \left (15 a^{3} d^{3} - 103 a^{2} b c d^{2} + 128 a b^{2} c^{2} d - 48 b^{3} c^{3}\right )}{105 b d^{3} \sqrt{c + d x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**2*(b*x**2+a)**(5/2)/(d*x**2+c)**(1/2),x)

[Out]

b*x**3*(a + b*x**2)**(3/2)*sqrt(c + d*x**2)/(7*d) + 2*b*x**3*sqrt(a + b*x**2)*sq
rt(c + d*x**2)*(5*a*d - 3*b*c)/(35*d**2) - c**(3/2)*sqrt(a + b*x**2)*(45*a**2*d*
*2 - 61*a*b*c*d + 24*b**2*c**2)*elliptic_f(atan(sqrt(d)*x/sqrt(c)), 1 - b*c/(a*d
))/(105*d**(7/2)*sqrt(c*(a + b*x**2)/(a*(c + d*x**2)))*sqrt(c + d*x**2)) + x*sqr
t(a + b*x**2)*sqrt(c + d*x**2)*(45*a**2*d**2 - 61*a*b*c*d + 24*b**2*c**2)/(105*d
**3) - sqrt(c)*sqrt(a + b*x**2)*(15*a**3*d**3 - 103*a**2*b*c*d**2 + 128*a*b**2*c
**2*d - 48*b**3*c**3)*elliptic_e(atan(sqrt(d)*x/sqrt(c)), 1 - b*c/(a*d))/(105*b*
d**(7/2)*sqrt(c*(a + b*x**2)/(a*(c + d*x**2)))*sqrt(c + d*x**2)) + x*sqrt(a + b*
x**2)*(15*a**3*d**3 - 103*a**2*b*c*d**2 + 128*a*b**2*c**2*d - 48*b**3*c**3)/(105
*b*d**3*sqrt(c + d*x**2))

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Mathematica [C]  time = 1.10101, size = 306, normalized size = 0.7 \[ \frac{d x \sqrt{\frac{b}{a}} \left (a+b x^2\right ) \left (c+d x^2\right ) \left (45 a^2 d^2+a b d \left (45 d x^2-61 c\right )+3 b^2 \left (8 c^2-6 c d x^2+5 d^2 x^4\right )\right )+4 i c \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} \left (15 a^3 d^3-41 a^2 b c d^2+38 a b^2 c^2 d-12 b^3 c^3\right ) F\left (i \sinh ^{-1}\left (\sqrt{\frac{b}{a}} x\right )|\frac{a d}{b c}\right )-i c \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} \left (15 a^3 d^3-103 a^2 b c d^2+128 a b^2 c^2 d-48 b^3 c^3\right ) E\left (i \sinh ^{-1}\left (\sqrt{\frac{b}{a}} x\right )|\frac{a d}{b c}\right )}{105 d^4 \sqrt{\frac{b}{a}} \sqrt{a+b x^2} \sqrt{c+d x^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^2*(a + b*x^2)^(5/2))/Sqrt[c + d*x^2],x]

[Out]

(Sqrt[b/a]*d*x*(a + b*x^2)*(c + d*x^2)*(45*a^2*d^2 + a*b*d*(-61*c + 45*d*x^2) +
3*b^2*(8*c^2 - 6*c*d*x^2 + 5*d^2*x^4)) - I*c*(-48*b^3*c^3 + 128*a*b^2*c^2*d - 10
3*a^2*b*c*d^2 + 15*a^3*d^3)*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticE[I*
ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] + (4*I)*c*(-12*b^3*c^3 + 38*a*b^2*c^2*d - 41*
a^2*b*c*d^2 + 15*a^3*d^3)*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticF[I*Ar
cSinh[Sqrt[b/a]*x], (a*d)/(b*c)])/(105*Sqrt[b/a]*d^4*Sqrt[a + b*x^2]*Sqrt[c + d*
x^2])

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Maple [A]  time = 0.03, size = 782, normalized size = 1.8 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2*(b*x^2+a)^(5/2)/(d*x^2+c)^(1/2),x)

[Out]

-1/105*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)*(-15*(-b/a)^(1/2)*x^9*b^3*d^4-60*(-b/a)^(
1/2)*x^7*a*b^2*d^4+3*(-b/a)^(1/2)*x^7*b^3*c*d^3-90*(-b/a)^(1/2)*x^5*a^2*b*d^4+19
*(-b/a)^(1/2)*x^5*a*b^2*c*d^3-6*(-b/a)^(1/2)*x^5*b^3*c^2*d^2-45*(-b/a)^(1/2)*x^3
*a^3*d^4-29*(-b/a)^(1/2)*x^3*a^2*b*c*d^3+55*(-b/a)^(1/2)*x^3*a*b^2*c^2*d^2-24*(-
b/a)^(1/2)*x^3*b^3*c^3*d+60*((b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/2)*EllipticF(x*
(-b/a)^(1/2),(a*d/b/c)^(1/2))*a^3*c*d^3-164*((b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1
/2)*EllipticF(x*(-b/a)^(1/2),(a*d/b/c)^(1/2))*a^2*b*c^2*d^2+152*((b*x^2+a)/a)^(1
/2)*((d*x^2+c)/c)^(1/2)*EllipticF(x*(-b/a)^(1/2),(a*d/b/c)^(1/2))*a*b^2*c^3*d-48
*((b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/2)*EllipticF(x*(-b/a)^(1/2),(a*d/b/c)^(1/2
))*b^3*c^4-15*((b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/2)*EllipticE(x*(-b/a)^(1/2),(
a*d/b/c)^(1/2))*a^3*c*d^3+103*((b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/2)*EllipticE(
x*(-b/a)^(1/2),(a*d/b/c)^(1/2))*a^2*b*c^2*d^2-128*((b*x^2+a)/a)^(1/2)*((d*x^2+c)
/c)^(1/2)*EllipticE(x*(-b/a)^(1/2),(a*d/b/c)^(1/2))*a*b^2*c^3*d+48*((b*x^2+a)/a)
^(1/2)*((d*x^2+c)/c)^(1/2)*EllipticE(x*(-b/a)^(1/2),(a*d/b/c)^(1/2))*b^3*c^4-45*
(-b/a)^(1/2)*x*a^3*c*d^3+61*(-b/a)^(1/2)*x*a^2*b*c^2*d^2-24*(-b/a)^(1/2)*x*a*b^2
*c^3*d)/d^4/(b*d*x^4+a*d*x^2+b*c*x^2+a*c)/(-b/a)^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^(5/2)*x^2/sqrt(d*x^2 + c),x, algorithm="maxima")

[Out]

integrate((b*x^2 + a)^(5/2)*x^2/sqrt(d*x^2 + c), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (b^{2} x^{6} + 2 \, a b x^{4} + a^{2} x^{2}\right )} \sqrt{b x^{2} + a}}{\sqrt{d x^{2} + c}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^(5/2)*x^2/sqrt(d*x^2 + c),x, algorithm="fricas")

[Out]

integral((b^2*x^6 + 2*a*b*x^4 + a^2*x^2)*sqrt(b*x^2 + a)/sqrt(d*x^2 + c), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2} \left (a + b x^{2}\right )^{\frac{5}{2}}}{\sqrt{c + d x^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(x**2*(a + b*x**2)**(5/2)/sqrt(c + d*x**2), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{2} + a\right )}^{\frac{5}{2}} x^{2}}{\sqrt{d x^{2} + c}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^(5/2)*x^2/sqrt(d*x^2 + c),x, algorithm="giac")

[Out]

integrate((b*x^2 + a)^(5/2)*x^2/sqrt(d*x^2 + c), x)