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

Optimal. Leaf size=212 \[ \frac{4 a^{15/4} c^{7/2} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a} \sqrt{c}}\right )|\frac{1}{2}\right )}{231 b^{9/4} \sqrt{a+b x^2}}-\frac{8 a^3 c^3 \sqrt{c x} \sqrt{a+b x^2}}{231 b^2}+\frac{8 a^2 c (c x)^{5/2} \sqrt{a+b x^2}}{385 b}+\frac{2 (c x)^{9/2} \left (a+b x^2\right )^{3/2}}{15 c}+\frac{4 a (c x)^{9/2} \sqrt{a+b x^2}}{55 c} \]

[Out]

(-8*a^3*c^3*Sqrt[c*x]*Sqrt[a + b*x^2])/(231*b^2) + (8*a^2*c*(c*x)^(5/2)*Sqrt[a +
b*x^2])/(385*b) + (4*a*(c*x)^(9/2)*Sqrt[a + b*x^2])/(55*c) + (2*(c*x)^(9/2)*(a
+ b*x^2)^(3/2))/(15*c) + (4*a^(15/4)*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)*Sqr
t[c])], 1/2])/(231*b^(9/4)*Sqrt[a + b*x^2])

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Rubi [A]  time = 0.366531, antiderivative size = 212, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21 \[ \frac{4 a^{15/4} c^{7/2} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a} \sqrt{c}}\right )|\frac{1}{2}\right )}{231 b^{9/4} \sqrt{a+b x^2}}-\frac{8 a^3 c^3 \sqrt{c x} \sqrt{a+b x^2}}{231 b^2}+\frac{8 a^2 c (c x)^{5/2} \sqrt{a+b x^2}}{385 b}+\frac{2 (c x)^{9/2} \left (a+b x^2\right )^{3/2}}{15 c}+\frac{4 a (c x)^{9/2} \sqrt{a+b x^2}}{55 c} \]

Antiderivative was successfully verified.

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

[Out]

(-8*a^3*c^3*Sqrt[c*x]*Sqrt[a + b*x^2])/(231*b^2) + (8*a^2*c*(c*x)^(5/2)*Sqrt[a +
b*x^2])/(385*b) + (4*a*(c*x)^(9/2)*Sqrt[a + b*x^2])/(55*c) + (2*(c*x)^(9/2)*(a
+ b*x^2)^(3/2))/(15*c) + (4*a^(15/4)*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)*Sqr
t[c])], 1/2])/(231*b^(9/4)*Sqrt[a + b*x^2])

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Rubi in Sympy [A]  time = 37.1248, size = 196, normalized size = 0.92 \[ \frac{4 a^{\frac{15}{4}} c^{\frac{7}{2}} \sqrt{\frac{a + b x^{2}}{\left (\sqrt{a} + \sqrt{b} x\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a} \sqrt{c}} \right )}\middle | \frac{1}{2}\right )}{231 b^{\frac{9}{4}} \sqrt{a + b x^{2}}} - \frac{8 a^{3} c^{3} \sqrt{c x} \sqrt{a + b x^{2}}}{231 b^{2}} + \frac{8 a^{2} c \left (c x\right )^{\frac{5}{2}} \sqrt{a + b x^{2}}}{385 b} + \frac{4 a \left (c x\right )^{\frac{9}{2}} \sqrt{a + b x^{2}}}{55 c} + \frac{2 \left (c x\right )^{\frac{9}{2}} \left (a + b x^{2}\right )^{\frac{3}{2}}}{15 c} \]

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]

4*a**(15/4)*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)/(231*b**(9
/4)*sqrt(a + b*x**2)) - 8*a**3*c**3*sqrt(c*x)*sqrt(a + b*x**2)/(231*b**2) + 8*a*
*2*c*(c*x)**(5/2)*sqrt(a + b*x**2)/(385*b) + 4*a*(c*x)**(9/2)*sqrt(a + b*x**2)/(
55*c) + 2*(c*x)**(9/2)*(a + b*x**2)**(3/2)/(15*c)

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Mathematica [C]  time = 0.203836, size = 166, normalized size = 0.78 \[ \frac{2 c^3 \sqrt{c x} \left (20 i a^4 \sqrt{x} \sqrt{\frac{a}{b x^2}+1} F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{a}}{\sqrt{b}}}}{\sqrt{x}}\right )\right |-1\right )+\sqrt{\frac{i \sqrt{a}}{\sqrt{b}}} \left (-20 a^4-8 a^3 b x^2+131 a^2 b^2 x^4+196 a b^3 x^6+77 b^4 x^8\right )\right )}{1155 b^2 \sqrt{\frac{i \sqrt{a}}{\sqrt{b}}} \sqrt{a+b x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*c^3*Sqrt[c*x]*(Sqrt[(I*Sqrt[a])/Sqrt[b]]*(-20*a^4 - 8*a^3*b*x^2 + 131*a^2*b^2
*x^4 + 196*a*b^3*x^6 + 77*b^4*x^8) + (20*I)*a^4*Sqrt[1 + a/(b*x^2)]*Sqrt[x]*Elli
pticF[I*ArcSinh[Sqrt[(I*Sqrt[a])/Sqrt[b]]/Sqrt[x]], -1]))/(1155*Sqrt[(I*Sqrt[a])
/Sqrt[b]]*b^2*Sqrt[a + b*x^2])

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Maple [A]  time = 0.034, size = 163, normalized size = 0.8 \[{\frac{2\,{c}^{3}}{1155\,{b}^{3}x}\sqrt{cx} \left ( 77\,{b}^{5}{x}^{9}+196\,a{b}^{4}{x}^{7}+10\,\sqrt{-ab}\sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{{\frac{-bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-{\frac{bx}{\sqrt{-ab}}}}{\it EllipticF} \left ( \sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}},1/2\,\sqrt{2} \right ) \sqrt{2}{a}^{4}+131\,{a}^{2}{b}^{3}{x}^{5}-8\,{a}^{3}{b}^{2}{x}^{3}-20\,{a}^{4}bx \right ){\frac{1}{\sqrt{b{x}^{2}+a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/1155*c^3/x*(c*x)^(1/2)/(b*x^2+a)^(1/2)*(77*b^5*x^9+196*a*b^4*x^7+10*(-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+131*a^2*b^3*x^5-8*a^3*b^2*x^3-20*a^4*b*x)/b^3

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((b*c^3*x^5 + a*c^3*x^3)*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((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. \[ \int{\left (b x^{2} + a\right )}^{\frac{3}{2}} \left (c x\right )^{\frac{7}{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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