3.621 \(\int \frac{(c x)^{7/2}}{\left (a+b x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=153 \[ -\frac{5 a^{3/4} c^{7/2} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a} \sqrt{c}}\right )|\frac{1}{2}\right )}{6 b^{9/4} \sqrt{a+b x^2}}+\frac{5 c^3 \sqrt{c x} \sqrt{a+b x^2}}{3 b^2}-\frac{c (c x)^{5/2}}{b \sqrt{a+b x^2}} \]

[Out]

-((c*(c*x)^(5/2))/(b*Sqrt[a + b*x^2])) + (5*c^3*Sqrt[c*x]*Sqrt[a + b*x^2])/(3*b^
2) - (5*a^(3/4)*c^(7/2)*(Sqrt[a] + Sqrt[b]*x)*Sqrt[(a + b*x^2)/(Sqrt[a] + Sqrt[b
]*x)^2]*EllipticF[2*ArcTan[(b^(1/4)*Sqrt[c*x])/(a^(1/4)*Sqrt[c])], 1/2])/(6*b^(9
/4)*Sqrt[a + b*x^2])

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Rubi [A]  time = 0.251851, antiderivative size = 153, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21 \[ -\frac{5 a^{3/4} c^{7/2} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a} \sqrt{c}}\right )|\frac{1}{2}\right )}{6 b^{9/4} \sqrt{a+b x^2}}+\frac{5 c^3 \sqrt{c x} \sqrt{a+b x^2}}{3 b^2}-\frac{c (c x)^{5/2}}{b \sqrt{a+b x^2}} \]

Antiderivative was successfully verified.

[In]  Int[(c*x)^(7/2)/(a + b*x^2)^(3/2),x]

[Out]

-((c*(c*x)^(5/2))/(b*Sqrt[a + b*x^2])) + (5*c^3*Sqrt[c*x]*Sqrt[a + b*x^2])/(3*b^
2) - (5*a^(3/4)*c^(7/2)*(Sqrt[a] + Sqrt[b]*x)*Sqrt[(a + b*x^2)/(Sqrt[a] + Sqrt[b
]*x)^2]*EllipticF[2*ArcTan[(b^(1/4)*Sqrt[c*x])/(a^(1/4)*Sqrt[c])], 1/2])/(6*b^(9
/4)*Sqrt[a + b*x^2])

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Rubi in Sympy [A]  time = 24.5488, size = 139, normalized size = 0.91 \[ - \frac{5 a^{\frac{3}{4}} c^{\frac{7}{2}} \sqrt{\frac{a + b x^{2}}{\left (\sqrt{a} + \sqrt{b} x\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a} \sqrt{c}} \right )}\middle | \frac{1}{2}\right )}{6 b^{\frac{9}{4}} \sqrt{a + b x^{2}}} - \frac{c \left (c x\right )^{\frac{5}{2}}}{b \sqrt{a + b x^{2}}} + \frac{5 c^{3} \sqrt{c x} \sqrt{a + b x^{2}}}{3 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x)**(7/2)/(b*x**2+a)**(3/2),x)

[Out]

-5*a**(3/4)*c**(7/2)*sqrt((a + b*x**2)/(sqrt(a) + sqrt(b)*x)**2)*(sqrt(a) + sqrt
(b)*x)*elliptic_f(2*atan(b**(1/4)*sqrt(c*x)/(a**(1/4)*sqrt(c))), 1/2)/(6*b**(9/4
)*sqrt(a + b*x**2)) - c*(c*x)**(5/2)/(b*sqrt(a + b*x**2)) + 5*c**3*sqrt(c*x)*sqr
t(a + b*x**2)/(3*b**2)

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Mathematica [C]  time = 0.127496, size = 131, normalized size = 0.86 \[ \frac{c^3 \sqrt{c x} \left (\sqrt{\frac{i \sqrt{a}}{\sqrt{b}}} \left (5 a+2 b x^2\right )-5 i a \sqrt{x} \sqrt{\frac{a}{b x^2}+1} F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{a}}{\sqrt{b}}}}{\sqrt{x}}\right )\right |-1\right )\right )}{3 b^2 \sqrt{\frac{i \sqrt{a}}{\sqrt{b}}} \sqrt{a+b x^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(c*x)^(7/2)/(a + b*x^2)^(3/2),x]

[Out]

(c^3*Sqrt[c*x]*(Sqrt[(I*Sqrt[a])/Sqrt[b]]*(5*a + 2*b*x^2) - (5*I)*a*Sqrt[1 + a/(
b*x^2)]*Sqrt[x]*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[a])/Sqrt[b]]/Sqrt[x]], -1]))/(3
*Sqrt[(I*Sqrt[a])/Sqrt[b]]*b^2*Sqrt[a + b*x^2])

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Maple [A]  time = 0.048, size = 128, normalized size = 0.8 \[ -{\frac{{c}^{3}}{6\,{b}^{3}x}\sqrt{cx} \left ( 5\,\sqrt{-ab}\sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{{\frac{-bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-{\frac{bx}{\sqrt{-ab}}}}{\it EllipticF} \left ( \sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}},1/2\,\sqrt{2} \right ) \sqrt{2}a-4\,{b}^{2}{x}^{3}-10\,abx \right ){\frac{1}{\sqrt{b{x}^{2}+a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x)^(7/2)/(b*x^2+a)^(3/2),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (c x\right )^{\frac{7}{2}}}{{\left (b x^{2} + a\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x)^(7/2)/(b*x^2 + a)^(3/2),x, algorithm="maxima")

[Out]

integrate((c*x)^(7/2)/(b*x^2 + a)^(3/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{\sqrt{c x} c^{3} x^{3}}{{\left (b x^{2} + a\right )}^{\frac{3}{2}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x)^(7/2)/(b*x^2 + a)^(3/2),x, algorithm="fricas")

[Out]

integral(sqrt(c*x)*c^3*x^3/(b*x^2 + a)^(3/2), 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((c*x)**(7/2)/(b*x**2+a)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (c x\right )^{\frac{7}{2}}}{{\left (b x^{2} + a\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x)^(7/2)/(b*x^2 + a)^(3/2),x, algorithm="giac")

[Out]

integrate((c*x)^(7/2)/(b*x^2 + a)^(3/2), x)