3.960 \(\int \frac{x^4 \left (a+b x^2\right )^{5/2}}{\sqrt{c+d x^2}} \, dx\)

Optimal. Leaf size=553 \[ \frac{x^3 \sqrt{a+b x^2} \sqrt{c+d x^2} \left (75 a^2 d^2-115 a b c d+48 b^2 c^2\right )}{315 d^3}-\frac{x \sqrt{a+b x^2} \sqrt{c+d x^2} \left (-5 a^3 d^3+105 a^2 b c d^2-156 a b^2 c^2 d+64 b^3 c^3\right )}{315 b d^4}+\frac{c^{3/2} \sqrt{a+b x^2} \left (-5 a^3 d^3+105 a^2 b c d^2-156 a b^2 c^2 d+64 b^3 c^3\right ) F\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{315 b d^{9/2} \sqrt{c+d x^2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac{x \sqrt{a+b x^2} \left (-10 a^4 d^4-25 a^3 b c d^3+243 a^2 b^2 c^2 d^2-328 a b^3 c^3 d+128 b^4 c^4\right )}{315 b^2 d^4 \sqrt{c+d x^2}}-\frac{\sqrt{c} \sqrt{a+b x^2} \left (-10 a^4 d^4-25 a^3 b c d^3+243 a^2 b^2 c^2 d^2-328 a b^3 c^3 d+128 b^4 c^4\right ) E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{315 b^2 d^{9/2} \sqrt{c+d x^2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac{4 b x^5 \sqrt{a+b x^2} \sqrt{c+d x^2} (2 b c-3 a d)}{63 d^2}+\frac{b x^5 \left (a+b x^2\right )^{3/2} \sqrt{c+d x^2}}{9 d} \]

[Out]

((128*b^4*c^4 - 328*a*b^3*c^3*d + 243*a^2*b^2*c^2*d^2 - 25*a^3*b*c*d^3 - 10*a^4*
d^4)*x*Sqrt[a + b*x^2])/(315*b^2*d^4*Sqrt[c + d*x^2]) - ((64*b^3*c^3 - 156*a*b^2
*c^2*d + 105*a^2*b*c*d^2 - 5*a^3*d^3)*x*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(315*b*
d^4) + ((48*b^2*c^2 - 115*a*b*c*d + 75*a^2*d^2)*x^3*Sqrt[a + b*x^2]*Sqrt[c + d*x
^2])/(315*d^3) - (4*b*(2*b*c - 3*a*d)*x^5*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(63*d
^2) + (b*x^5*(a + b*x^2)^(3/2)*Sqrt[c + d*x^2])/(9*d) - (Sqrt[c]*(128*b^4*c^4 -
328*a*b^3*c^3*d + 243*a^2*b^2*c^2*d^2 - 25*a^3*b*c*d^3 - 10*a^4*d^4)*Sqrt[a + b*
x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(315*b^2*d^(9/2)*S
qrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) + (c^(3/2)*(64*b^3*c^3 - 1
56*a*b^2*c^2*d + 105*a^2*b*c*d^2 - 5*a^3*d^3)*Sqrt[a + b*x^2]*EllipticF[ArcTan[(
Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(315*b*d^(9/2)*Sqrt[(c*(a + b*x^2))/(a*(c
+ d*x^2))]*Sqrt[c + d*x^2])

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Rubi [A]  time = 1.85435, antiderivative size = 553, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.269 \[ \frac{x^3 \sqrt{a+b x^2} \sqrt{c+d x^2} \left (75 a^2 d^2-115 a b c d+48 b^2 c^2\right )}{315 d^3}-\frac{x \sqrt{a+b x^2} \sqrt{c+d x^2} \left (-5 a^3 d^3+105 a^2 b c d^2-156 a b^2 c^2 d+64 b^3 c^3\right )}{315 b d^4}+\frac{c^{3/2} \sqrt{a+b x^2} \left (-5 a^3 d^3+105 a^2 b c d^2-156 a b^2 c^2 d+64 b^3 c^3\right ) F\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{315 b d^{9/2} \sqrt{c+d x^2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac{x \sqrt{a+b x^2} \left (-10 a^4 d^4-25 a^3 b c d^3+243 a^2 b^2 c^2 d^2-328 a b^3 c^3 d+128 b^4 c^4\right )}{315 b^2 d^4 \sqrt{c+d x^2}}-\frac{\sqrt{c} \sqrt{a+b x^2} \left (-10 a^4 d^4-25 a^3 b c d^3+243 a^2 b^2 c^2 d^2-328 a b^3 c^3 d+128 b^4 c^4\right ) E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{315 b^2 d^{9/2} \sqrt{c+d x^2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac{4 b x^5 \sqrt{a+b x^2} \sqrt{c+d x^2} (2 b c-3 a d)}{63 d^2}+\frac{b x^5 \left (a+b x^2\right )^{3/2} \sqrt{c+d x^2}}{9 d} \]

Antiderivative was successfully verified.

[In]  Int[(x^4*(a + b*x^2)^(5/2))/Sqrt[c + d*x^2],x]

[Out]

((128*b^4*c^4 - 328*a*b^3*c^3*d + 243*a^2*b^2*c^2*d^2 - 25*a^3*b*c*d^3 - 10*a^4*
d^4)*x*Sqrt[a + b*x^2])/(315*b^2*d^4*Sqrt[c + d*x^2]) - ((64*b^3*c^3 - 156*a*b^2
*c^2*d + 105*a^2*b*c*d^2 - 5*a^3*d^3)*x*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(315*b*
d^4) + ((48*b^2*c^2 - 115*a*b*c*d + 75*a^2*d^2)*x^3*Sqrt[a + b*x^2]*Sqrt[c + d*x
^2])/(315*d^3) - (4*b*(2*b*c - 3*a*d)*x^5*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(63*d
^2) + (b*x^5*(a + b*x^2)^(3/2)*Sqrt[c + d*x^2])/(9*d) - (Sqrt[c]*(128*b^4*c^4 -
328*a*b^3*c^3*d + 243*a^2*b^2*c^2*d^2 - 25*a^3*b*c*d^3 - 10*a^4*d^4)*Sqrt[a + b*
x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(315*b^2*d^(9/2)*S
qrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) + (c^(3/2)*(64*b^3*c^3 - 1
56*a*b^2*c^2*d + 105*a^2*b*c*d^2 - 5*a^3*d^3)*Sqrt[a + b*x^2]*EllipticF[ArcTan[(
Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(315*b*d^(9/2)*Sqrt[(c*(a + b*x^2))/(a*(c
+ d*x^2))]*Sqrt[c + d*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(x**4*(b*x**2+a)**(5/2)/(d*x**2+c)**(1/2),x)

[Out]

Timed out

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Mathematica [C]  time = 3.05645, size = 379, normalized size = 0.69 \[ \frac{d x \sqrt{\frac{b}{a}} \left (a+b x^2\right ) \left (c+d x^2\right ) \left (5 a^3 d^3+15 a^2 b d^2 \left (5 d x^2-7 c\right )+a b^2 d \left (156 c^2-115 c d x^2+95 d^2 x^4\right )+b^3 \left (-64 c^3+48 c^2 d x^2-40 c d^2 x^4+35 d^3 x^6\right )\right )-i c \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} \left (5 a^4 d^4+130 a^3 b c d^3-399 a^2 b^2 c^2 d^2+392 a b^3 c^3 d-128 b^4 c^4\right ) F\left (i \sinh ^{-1}\left (\sqrt{\frac{b}{a}} x\right )|\frac{a d}{b c}\right )+i c \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} \left (10 a^4 d^4+25 a^3 b c d^3-243 a^2 b^2 c^2 d^2+328 a b^3 c^3 d-128 b^4 c^4\right ) E\left (i \sinh ^{-1}\left (\sqrt{\frac{b}{a}} x\right )|\frac{a d}{b c}\right )}{315 b d^5 \sqrt{\frac{b}{a}} \sqrt{a+b x^2} \sqrt{c+d x^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^4*(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)*(5*a^3*d^3 + 15*a^2*b*d^2*(-7*c + 5*d*x^2
) + a*b^2*d*(156*c^2 - 115*c*d*x^2 + 95*d^2*x^4) + b^3*(-64*c^3 + 48*c^2*d*x^2 -
40*c*d^2*x^4 + 35*d^3*x^6)) + I*c*(-128*b^4*c^4 + 328*a*b^3*c^3*d - 243*a^2*b^2
*c^2*d^2 + 25*a^3*b*c*d^3 + 10*a^4*d^4)*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*
EllipticE[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] - I*c*(-128*b^4*c^4 + 392*a*b^3*c
^3*d - 399*a^2*b^2*c^2*d^2 + 130*a^3*b*c*d^3 + 5*a^4*d^4)*Sqrt[1 + (b*x^2)/a]*Sq
rt[1 + (d*x^2)/c]*EllipticF[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)])/(315*b*Sqrt[b/
a]*d^5*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])

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Maple [A]  time = 0.037, size = 1047, normalized size = 1.9 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4*(b*x^2+a)^(5/2)/(d*x^2+c)^(1/2),x)

[Out]

1/315*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)*(-25*(-b/a)^(1/2)*x^7*a*b^3*c*d^4+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))*b^
4*c^5-64*(-b/a)^(1/2)*x^3*b^4*c^4*d+5*(-b/a)^(1/2)*x*a^4*c*d^4+130*(-b/a)^(1/2)*
x^9*a*b^3*d^5-5*(-b/a)^(1/2)*x^9*b^4*c*d^4+170*(-b/a)^(1/2)*x^7*a^2*b^2*d^5+8*(-
b/a)^(1/2)*x^7*b^4*c^2*d^3+80*(-b/a)^(1/2)*x^5*a^3*b*d^5-16*(-b/a)^(1/2)*x^5*b^4
*c^3*d^2-50*(-b/a)^(1/2)*x^5*a^2*b^2*c*d^4+49*(-b/a)^(1/2)*x^5*a*b^3*c^2*d^3-25*
(-b/a)^(1/2)*x^3*a^3*b*c*d^4-64*(-b/a)^(1/2)*x^3*a^2*b^2*c^2*d^3+140*(-b/a)^(1/2
)*x^3*a*b^3*c^3*d^2-105*(-b/a)^(1/2)*x*a^3*b*c^2*d^3+156*(-b/a)^(1/2)*x*a^2*b^2*
c^3*d^2-64*(-b/a)^(1/2)*x*a*b^3*c^4*d+5*((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^4*c*d^4-10*((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^4*c*d^4-25*((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*b*c^
2*d^3+243*((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^2*c^3*d^2-128*((b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/2)*Elliptic
F(x*(-b/a)^(1/2),(a*d/b/c)^(1/2))*b^4*c^5-328*((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^3*c^4*d+130*((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*b*c^2*d^3-
399*((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^2*c^3*d^2+392*((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^3*c^4*d+35*(-b/a)^(1/2)*x^11*b^4*d^5+5*(-b/a)^(1
/2)*x^3*a^4*d^5)/b/d^5/(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. \[ \int \frac{{\left (b x^{2} + a\right )}^{\frac{5}{2}} x^{4}}{\sqrt{d x^{2} + c}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^(5/2)*x^4/sqrt(d*x^2 + c),x, algorithm="maxima")

[Out]

integrate((b*x^2 + a)^(5/2)*x^4/sqrt(d*x^2 + c), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (b^{2} x^{8} + 2 \, a b x^{6} + a^{2} x^{4}\right )} \sqrt{b x^{2} + a}}{\sqrt{d x^{2} + c}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^(5/2)*x^4/sqrt(d*x^2 + c),x, algorithm="fricas")

[Out]

integral((b^2*x^8 + 2*a*b*x^6 + a^2*x^4)*sqrt(b*x^2 + a)/sqrt(d*x^2 + c), 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(x**4*(b*x**2+a)**(5/2)/(d*x**2+c)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{2} + a\right )}^{\frac{5}{2}} x^{4}}{\sqrt{d x^{2} + c}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^(5/2)*x^4/sqrt(d*x^2 + c),x, algorithm="giac")

[Out]

integrate((b*x^2 + a)^(5/2)*x^4/sqrt(d*x^2 + c), x)