3.634 \(\int \frac{1}{(c x)^{3/2} \left (a+b x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=333 \[ \frac{7 \sqrt [4]{b} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a} \sqrt{c}}\right )|\frac{1}{2}\right )}{4 a^{11/4} c^{3/2} \sqrt{a+b x^2}}-\frac{7 \sqrt [4]{b} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a} \sqrt{c}}\right )|\frac{1}{2}\right )}{2 a^{11/4} c^{3/2} \sqrt{a+b x^2}}+\frac{7 \sqrt{b} \sqrt{c x} \sqrt{a+b x^2}}{2 a^3 c^2 \left (\sqrt{a}+\sqrt{b} x\right )}-\frac{7 \sqrt{a+b x^2}}{2 a^3 c \sqrt{c x}}+\frac{7}{6 a^2 c \sqrt{c x} \sqrt{a+b x^2}}+\frac{1}{3 a c \sqrt{c x} \left (a+b x^2\right )^{3/2}} \]

[Out]

1/(3*a*c*Sqrt[c*x]*(a + b*x^2)^(3/2)) + 7/(6*a^2*c*Sqrt[c*x]*Sqrt[a + b*x^2]) -
(7*Sqrt[a + b*x^2])/(2*a^3*c*Sqrt[c*x]) + (7*Sqrt[b]*Sqrt[c*x]*Sqrt[a + b*x^2])/
(2*a^3*c^2*(Sqrt[a] + Sqrt[b]*x)) - (7*b^(1/4)*(Sqrt[a] + Sqrt[b]*x)*Sqrt[(a + b
*x^2)/(Sqrt[a] + Sqrt[b]*x)^2]*EllipticE[2*ArcTan[(b^(1/4)*Sqrt[c*x])/(a^(1/4)*S
qrt[c])], 1/2])/(2*a^(11/4)*c^(3/2)*Sqrt[a + b*x^2]) + (7*b^(1/4)*(Sqrt[a] + Sqr
t[b]*x)*Sqrt[(a + b*x^2)/(Sqrt[a] + Sqrt[b]*x)^2]*EllipticF[2*ArcTan[(b^(1/4)*Sq
rt[c*x])/(a^(1/4)*Sqrt[c])], 1/2])/(4*a^(11/4)*c^(3/2)*Sqrt[a + b*x^2])

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Rubi [A]  time = 0.670973, antiderivative size = 333, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.316 \[ \frac{7 \sqrt [4]{b} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a} \sqrt{c}}\right )|\frac{1}{2}\right )}{4 a^{11/4} c^{3/2} \sqrt{a+b x^2}}-\frac{7 \sqrt [4]{b} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a} \sqrt{c}}\right )|\frac{1}{2}\right )}{2 a^{11/4} c^{3/2} \sqrt{a+b x^2}}+\frac{7 \sqrt{b} \sqrt{c x} \sqrt{a+b x^2}}{2 a^3 c^2 \left (\sqrt{a}+\sqrt{b} x\right )}-\frac{7 \sqrt{a+b x^2}}{2 a^3 c \sqrt{c x}}+\frac{7}{6 a^2 c \sqrt{c x} \sqrt{a+b x^2}}+\frac{1}{3 a c \sqrt{c x} \left (a+b x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[1/((c*x)^(3/2)*(a + b*x^2)^(5/2)),x]

[Out]

1/(3*a*c*Sqrt[c*x]*(a + b*x^2)^(3/2)) + 7/(6*a^2*c*Sqrt[c*x]*Sqrt[a + b*x^2]) -
(7*Sqrt[a + b*x^2])/(2*a^3*c*Sqrt[c*x]) + (7*Sqrt[b]*Sqrt[c*x]*Sqrt[a + b*x^2])/
(2*a^3*c^2*(Sqrt[a] + Sqrt[b]*x)) - (7*b^(1/4)*(Sqrt[a] + Sqrt[b]*x)*Sqrt[(a + b
*x^2)/(Sqrt[a] + Sqrt[b]*x)^2]*EllipticE[2*ArcTan[(b^(1/4)*Sqrt[c*x])/(a^(1/4)*S
qrt[c])], 1/2])/(2*a^(11/4)*c^(3/2)*Sqrt[a + b*x^2]) + (7*b^(1/4)*(Sqrt[a] + Sqr
t[b]*x)*Sqrt[(a + b*x^2)/(Sqrt[a] + Sqrt[b]*x)^2]*EllipticF[2*ArcTan[(b^(1/4)*Sq
rt[c*x])/(a^(1/4)*Sqrt[c])], 1/2])/(4*a^(11/4)*c^(3/2)*Sqrt[a + b*x^2])

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Rubi in Sympy [A]  time = 69.1994, size = 304, normalized size = 0.91 \[ \frac{1}{3 a c \sqrt{c x} \left (a + b x^{2}\right )^{\frac{3}{2}}} + \frac{7}{6 a^{2} c \sqrt{c x} \sqrt{a + b x^{2}}} + \frac{7 \sqrt{b} \sqrt{c x} \sqrt{a + b x^{2}}}{2 a^{3} c^{2} \left (\sqrt{a} + \sqrt{b} x\right )} - \frac{7 \sqrt{a + b x^{2}}}{2 a^{3} c \sqrt{c x}} - \frac{7 \sqrt [4]{b} \sqrt{\frac{a + b x^{2}}{\left (\sqrt{a} + \sqrt{b} x\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a} \sqrt{c}} \right )}\middle | \frac{1}{2}\right )}{2 a^{\frac{11}{4}} c^{\frac{3}{2}} \sqrt{a + b x^{2}}} + \frac{7 \sqrt [4]{b} \sqrt{\frac{a + b x^{2}}{\left (\sqrt{a} + \sqrt{b} x\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a} \sqrt{c}} \right )}\middle | \frac{1}{2}\right )}{4 a^{\frac{11}{4}} c^{\frac{3}{2}} \sqrt{a + b x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(c*x)**(3/2)/(b*x**2+a)**(5/2),x)

[Out]

1/(3*a*c*sqrt(c*x)*(a + b*x**2)**(3/2)) + 7/(6*a**2*c*sqrt(c*x)*sqrt(a + b*x**2)
) + 7*sqrt(b)*sqrt(c*x)*sqrt(a + b*x**2)/(2*a**3*c**2*(sqrt(a) + sqrt(b)*x)) - 7
*sqrt(a + b*x**2)/(2*a**3*c*sqrt(c*x)) - 7*b**(1/4)*sqrt((a + b*x**2)/(sqrt(a) +
sqrt(b)*x)**2)*(sqrt(a) + sqrt(b)*x)*elliptic_e(2*atan(b**(1/4)*sqrt(c*x)/(a**(
1/4)*sqrt(c))), 1/2)/(2*a**(11/4)*c**(3/2)*sqrt(a + b*x**2)) + 7*b**(1/4)*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)/(4*a**(11/4)*c**(3/2)*sqrt(a + b*x**2
))

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Mathematica [C]  time = 0.422165, size = 208, normalized size = 0.62 \[ \frac{x \left (-\sqrt{\frac{i \sqrt{b} x}{\sqrt{a}}} \left (12 a^2+35 a b x^2+21 b^2 x^4\right )-21 \sqrt{a} \sqrt{b} x \left (a+b x^2\right ) \sqrt{\frac{b x^2}{a}+1} F\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{b} x}{\sqrt{a}}}\right )\right |-1\right )+21 \sqrt{a} \sqrt{b} x \left (a+b x^2\right ) \sqrt{\frac{b x^2}{a}+1} E\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{b} x}{\sqrt{a}}}\right )\right |-1\right )\right )}{6 a^3 (c x)^{3/2} \sqrt{\frac{i \sqrt{b} x}{\sqrt{a}}} \left (a+b x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((c*x)^(3/2)*(a + b*x^2)^(5/2)),x]

[Out]

(x*(-(Sqrt[(I*Sqrt[b]*x)/Sqrt[a]]*(12*a^2 + 35*a*b*x^2 + 21*b^2*x^4)) + 21*Sqrt[
a]*Sqrt[b]*x*(a + b*x^2)*Sqrt[1 + (b*x^2)/a]*EllipticE[I*ArcSinh[Sqrt[(I*Sqrt[b]
*x)/Sqrt[a]]], -1] - 21*Sqrt[a]*Sqrt[b]*x*(a + b*x^2)*Sqrt[1 + (b*x^2)/a]*Ellipt
icF[I*ArcSinh[Sqrt[(I*Sqrt[b]*x)/Sqrt[a]]], -1]))/(6*a^3*Sqrt[(I*Sqrt[b]*x)/Sqrt
[a]]*(c*x)^(3/2)*(a + b*x^2)^(3/2))

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Maple [A]  time = 0.03, size = 384, normalized size = 1.2 \[{\frac{1}{12\,c{a}^{3}} \left ( 42\,\sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{{\frac{-bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-{\frac{bx}{\sqrt{-ab}}}}{\it EllipticE} \left ( \sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}},1/2\,\sqrt{2} \right ) \sqrt{2}{x}^{2}ab-21\,\sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{{\frac{-bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-{\frac{bx}{\sqrt{-ab}}}}{\it EllipticF} \left ( \sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}},1/2\,\sqrt{2} \right ) \sqrt{2}{x}^{2}ab+42\,\sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{{\frac{-bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-{\frac{bx}{\sqrt{-ab}}}}{\it EllipticE} \left ( \sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}},1/2\,\sqrt{2} \right ) \sqrt{2}{a}^{2}-21\,\sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{{\frac{-bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-{\frac{bx}{\sqrt{-ab}}}}{\it EllipticF} \left ( \sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}},1/2\,\sqrt{2} \right ) \sqrt{2}{a}^{2}-42\,{b}^{2}{x}^{4}-70\,ab{x}^{2}-24\,{a}^{2} \right ){\frac{1}{\sqrt{cx}}} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(c*x)^(3/2)/(b*x^2+a)^(5/2),x)

[Out]

1/12*(42*((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)*EllipticE(((b*x+(-a*b)^(1/2))/(-a*b)^(1/2))^
(1/2),1/2*2^(1/2))*2^(1/2)*x^2*a*b-21*((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)*x^2*a*b+42*((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)*EllipticE(((b*x+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2),1/2*2^(1/2))*2^(
1/2)*a^2-21*((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^2-42*b^2*x^4-70*a*b*x^2-24*a^2)/a^3/c/(c*x)^(1/2
)/(b*x^2+a)^(3/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{2} + a\right )}^{\frac{5}{2}} \left (c x\right )^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + a)^(5/2)*(c*x)^(3/2)),x, algorithm="maxima")

[Out]

integrate(1/((b*x^2 + a)^(5/2)*(c*x)^(3/2)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (b^{2} c x^{5} + 2 \, a b c x^{3} + a^{2} c x\right )} \sqrt{b x^{2} + a} \sqrt{c x}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + a)^(5/2)*(c*x)^(3/2)),x, algorithm="fricas")

[Out]

integral(1/((b^2*c*x^5 + 2*a*b*c*x^3 + a^2*c*x)*sqrt(b*x^2 + a)*sqrt(c*x)), 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(1/(c*x)**(3/2)/(b*x**2+a)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{2} + a\right )}^{\frac{5}{2}} \left (c x\right )^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + a)^(5/2)*(c*x)^(3/2)),x, algorithm="giac")

[Out]

integrate(1/((b*x^2 + a)^(5/2)*(c*x)^(3/2)), x)