3.1089 \(\int (d x)^{3/2} \sqrt{a+b x^2+c x^4} \, dx\)

Optimal. Leaf size=147 \[ \frac{2 (d x)^{5/2} \sqrt{a+b x^2+c x^4} F_1\left (\frac{5}{4};-\frac{1}{2},-\frac{1}{2};\frac{9}{4};-\frac{2 c x^2}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}}\right )}{5 d \sqrt{\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^2}{\sqrt{b^2-4 a c}+b}+1}} \]

[Out]

(2*(d*x)^(5/2)*Sqrt[a + b*x^2 + c*x^4]*AppellF1[5/4, -1/2, -1/2, 9/4, (-2*c*x^2)
/(b - Sqrt[b^2 - 4*a*c]), (-2*c*x^2)/(b + Sqrt[b^2 - 4*a*c])])/(5*d*Sqrt[1 + (2*
c*x^2)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[1 + (2*c*x^2)/(b + Sqrt[b^2 - 4*a*c])])

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Rubi [A]  time = 0.552309, antiderivative size = 147, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ \frac{2 (d x)^{5/2} \sqrt{a+b x^2+c x^4} F_1\left (\frac{5}{4};-\frac{1}{2},-\frac{1}{2};\frac{9}{4};-\frac{2 c x^2}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}}\right )}{5 d \sqrt{\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^2}{\sqrt{b^2-4 a c}+b}+1}} \]

Antiderivative was successfully verified.

[In]  Int[(d*x)^(3/2)*Sqrt[a + b*x^2 + c*x^4],x]

[Out]

(2*(d*x)^(5/2)*Sqrt[a + b*x^2 + c*x^4]*AppellF1[5/4, -1/2, -1/2, 9/4, (-2*c*x^2)
/(b - Sqrt[b^2 - 4*a*c]), (-2*c*x^2)/(b + Sqrt[b^2 - 4*a*c])])/(5*d*Sqrt[1 + (2*
c*x^2)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[1 + (2*c*x^2)/(b + Sqrt[b^2 - 4*a*c])])

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Rubi in Sympy [A]  time = 31.355, size = 131, normalized size = 0.89 \[ \frac{2 \left (d x\right )^{\frac{5}{2}} \sqrt{a + b x^{2} + c x^{4}} \operatorname{appellf_{1}}{\left (\frac{5}{4},- \frac{1}{2},- \frac{1}{2},\frac{9}{4},- \frac{2 c x^{2}}{b - \sqrt{- 4 a c + b^{2}}},- \frac{2 c x^{2}}{b + \sqrt{- 4 a c + b^{2}}} \right )}}{5 d \sqrt{\frac{2 c x^{2}}{b - \sqrt{- 4 a c + b^{2}}} + 1} \sqrt{\frac{2 c x^{2}}{b + \sqrt{- 4 a c + b^{2}}} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x)**(3/2)*(c*x**4+b*x**2+a)**(1/2),x)

[Out]

2*(d*x)**(5/2)*sqrt(a + b*x**2 + c*x**4)*appellf1(5/4, -1/2, -1/2, 9/4, -2*c*x**
2/(b - sqrt(-4*a*c + b**2)), -2*c*x**2/(b + sqrt(-4*a*c + b**2)))/(5*d*sqrt(2*c*
x**2/(b - sqrt(-4*a*c + b**2)) + 1)*sqrt(2*c*x**2/(b + sqrt(-4*a*c + b**2)) + 1)
)

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Mathematica [B]  time = 5.33786, size = 1048, normalized size = 7.13 \[ \frac{d \sqrt{d x} \left (-\frac{25 b \left (2 c x^2+b-\sqrt{b^2-4 a c}\right ) \left (2 c x^2+b+\sqrt{b^2-4 a c}\right ) F_1\left (\frac{1}{4};\frac{1}{2},\frac{1}{2};\frac{5}{4};-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}},\frac{2 c x^2}{\sqrt{b^2-4 a c}-b}\right ) a^2}{5 a F_1\left (\frac{1}{4};\frac{1}{2},\frac{1}{2};\frac{5}{4};-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}},\frac{2 c x^2}{\sqrt{b^2-4 a c}-b}\right )-x^2 \left (\left (b+\sqrt{b^2-4 a c}\right ) F_1\left (\frac{5}{4};\frac{1}{2},\frac{3}{2};\frac{9}{4};-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}},\frac{2 c x^2}{\sqrt{b^2-4 a c}-b}\right )+\left (b-\sqrt{b^2-4 a c}\right ) F_1\left (\frac{5}{4};\frac{3}{2},\frac{1}{2};\frac{9}{4};-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}},\frac{2 c x^2}{\sqrt{b^2-4 a c}-b}\right )\right )}+\frac{90 c x^2 \left (2 c x^2+b-\sqrt{b^2-4 a c}\right ) \left (2 c x^2+b+\sqrt{b^2-4 a c}\right ) F_1\left (\frac{5}{4};\frac{1}{2},\frac{1}{2};\frac{9}{4};-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}},\frac{2 c x^2}{\sqrt{b^2-4 a c}-b}\right ) a^2}{9 a F_1\left (\frac{5}{4};\frac{1}{2},\frac{1}{2};\frac{9}{4};-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}},\frac{2 c x^2}{\sqrt{b^2-4 a c}-b}\right )-x^2 \left (\left (b+\sqrt{b^2-4 a c}\right ) F_1\left (\frac{9}{4};\frac{1}{2},\frac{3}{2};\frac{13}{4};-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}},\frac{2 c x^2}{\sqrt{b^2-4 a c}-b}\right )+\left (b-\sqrt{b^2-4 a c}\right ) F_1\left (\frac{9}{4};\frac{3}{2},\frac{1}{2};\frac{13}{4};-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}},\frac{2 c x^2}{\sqrt{b^2-4 a c}-b}\right )\right )}+\frac{27 b^2 x^2 \left (2 c x^2+b-\sqrt{b^2-4 a c}\right ) \left (2 c x^2+b+\sqrt{b^2-4 a c}\right ) F_1\left (\frac{5}{4};\frac{1}{2},\frac{1}{2};\frac{9}{4};-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}},\frac{2 c x^2}{\sqrt{b^2-4 a c}-b}\right ) a}{x^2 \left (\left (b+\sqrt{b^2-4 a c}\right ) F_1\left (\frac{9}{4};\frac{1}{2},\frac{3}{2};\frac{13}{4};-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}},\frac{2 c x^2}{\sqrt{b^2-4 a c}-b}\right )+\left (b-\sqrt{b^2-4 a c}\right ) F_1\left (\frac{9}{4};\frac{3}{2},\frac{1}{2};\frac{13}{4};-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}},\frac{2 c x^2}{\sqrt{b^2-4 a c}-b}\right )\right )-9 a F_1\left (\frac{5}{4};\frac{1}{2},\frac{1}{2};\frac{9}{4};-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}},\frac{2 c x^2}{\sqrt{b^2-4 a c}-b}\right )}+10 c \left (5 c x^2+2 b\right ) \left (c x^4+b x^2+a\right )^2\right )}{225 c^2 \left (c x^4+b x^2+a\right )^{3/2}} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[(d*x)^(3/2)*Sqrt[a + b*x^2 + c*x^4],x]

[Out]

(d*Sqrt[d*x]*(10*c*(2*b + 5*c*x^2)*(a + b*x^2 + c*x^4)^2 - (25*a^2*b*(b - Sqrt[b
^2 - 4*a*c] + 2*c*x^2)*(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)*AppellF1[1/4, 1/2, 1/2,
5/4, (-2*c*x^2)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^2)/(-b + Sqrt[b^2 - 4*a*c])])/(
5*a*AppellF1[1/4, 1/2, 1/2, 5/4, (-2*c*x^2)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^2)/(
-b + Sqrt[b^2 - 4*a*c])] - x^2*((b + Sqrt[b^2 - 4*a*c])*AppellF1[5/4, 1/2, 3/2,
9/4, (-2*c*x^2)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^2)/(-b + Sqrt[b^2 - 4*a*c])] + (
b - Sqrt[b^2 - 4*a*c])*AppellF1[5/4, 3/2, 1/2, 9/4, (-2*c*x^2)/(b + Sqrt[b^2 - 4
*a*c]), (2*c*x^2)/(-b + Sqrt[b^2 - 4*a*c])])) + (90*a^2*c*x^2*(b - Sqrt[b^2 - 4*
a*c] + 2*c*x^2)*(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)*AppellF1[5/4, 1/2, 1/2, 9/4, (
-2*c*x^2)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^2)/(-b + Sqrt[b^2 - 4*a*c])])/(9*a*App
ellF1[5/4, 1/2, 1/2, 9/4, (-2*c*x^2)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^2)/(-b + Sq
rt[b^2 - 4*a*c])] - x^2*((b + Sqrt[b^2 - 4*a*c])*AppellF1[9/4, 1/2, 3/2, 13/4, (
-2*c*x^2)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^2)/(-b + Sqrt[b^2 - 4*a*c])] + (b - Sq
rt[b^2 - 4*a*c])*AppellF1[9/4, 3/2, 1/2, 13/4, (-2*c*x^2)/(b + Sqrt[b^2 - 4*a*c]
), (2*c*x^2)/(-b + Sqrt[b^2 - 4*a*c])])) + (27*a*b^2*x^2*(b - Sqrt[b^2 - 4*a*c]
+ 2*c*x^2)*(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)*AppellF1[5/4, 1/2, 1/2, 9/4, (-2*c*
x^2)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^2)/(-b + Sqrt[b^2 - 4*a*c])])/(-9*a*AppellF
1[5/4, 1/2, 1/2, 9/4, (-2*c*x^2)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^2)/(-b + Sqrt[b
^2 - 4*a*c])] + x^2*((b + Sqrt[b^2 - 4*a*c])*AppellF1[9/4, 1/2, 3/2, 13/4, (-2*c
*x^2)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^2)/(-b + Sqrt[b^2 - 4*a*c])] + (b - Sqrt[b
^2 - 4*a*c])*AppellF1[9/4, 3/2, 1/2, 13/4, (-2*c*x^2)/(b + Sqrt[b^2 - 4*a*c]), (
2*c*x^2)/(-b + Sqrt[b^2 - 4*a*c])]))))/(225*c^2*(a + b*x^2 + c*x^4)^(3/2))

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Maple [F]  time = 0.063, size = 0, normalized size = 0. \[ \int \left ( dx \right ) ^{{\frac{3}{2}}}\sqrt{c{x}^{4}+b{x}^{2}+a}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x)^(3/2)*(c*x^4+b*x^2+a)^(1/2),x)

[Out]

int((d*x)^(3/2)*(c*x^4+b*x^2+a)^(1/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \sqrt{c x^{4} + b x^{2} + a} \left (d x\right )^{\frac{3}{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^4 + b*x^2 + a)*(d*x)^(3/2),x, algorithm="maxima")

[Out]

integrate(sqrt(c*x^4 + b*x^2 + a)*(d*x)^(3/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^4 + b*x^2 + a)*(d*x)^(3/2),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \left (d x\right )^{\frac{3}{2}} \sqrt{a + b x^{2} + c x^{4}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x)**(3/2)*(c*x**4+b*x**2+a)**(1/2),x)

[Out]

Integral((d*x)**(3/2)*sqrt(a + b*x**2 + c*x**4), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \sqrt{c x^{4} + b x^{2} + a} \left (d x\right )^{\frac{3}{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^4 + b*x^2 + a)*(d*x)^(3/2),x, algorithm="giac")

[Out]

integrate(sqrt(c*x^4 + b*x^2 + a)*(d*x)^(3/2), x)