3.662 \(\int \left (\frac{a (2+m) x^{1+m}}{\sqrt{a+b x^2}}+\frac{b (3+m) x^{3+m}}{\sqrt{a+b x^2}}\right ) \, dx\)

Optimal. Leaf size=17 \[ x^{m+2} \sqrt{a+b x^2} \]

[Out]

x^(2 + m)*Sqrt[a + b*x^2]

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Rubi [C]  time = 0.188891, antiderivative size = 127, normalized size of antiderivative = 7.47, number of steps used = 5, number of rules used = 2, integrand size = 43, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.047 \[ \frac{a x^{m+2} \sqrt{\frac{b x^2}{a}+1} \, _2F_1\left (\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};-\frac{b x^2}{a}\right )}{\sqrt{a+b x^2}}+\frac{b (m+3) x^{m+4} \sqrt{\frac{b x^2}{a}+1} \, _2F_1\left (\frac{1}{2},\frac{m+4}{2};\frac{m+6}{2};-\frac{b x^2}{a}\right )}{(m+4) \sqrt{a+b x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(a*x^(2 + m)*Sqrt[1 + (b*x^2)/a]*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, -(
(b*x^2)/a)])/Sqrt[a + b*x^2] + (b*(3 + m)*x^(4 + m)*Sqrt[1 + (b*x^2)/a]*Hypergeo
metric2F1[1/2, (4 + m)/2, (6 + m)/2, -((b*x^2)/a)])/((4 + m)*Sqrt[a + b*x^2])

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Rubi in Sympy [A]  time = 19.3719, size = 100, normalized size = 5.88 \[ \frac{x^{m + 2} \sqrt{a + b x^{2}}{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{m}{2} + 1 \\ \frac{m}{2} + 2 \end{matrix}\middle |{- \frac{b x^{2}}{a}} \right )}}{\sqrt{1 + \frac{b x^{2}}{a}}} + \frac{b x^{m + 4} \sqrt{a + b x^{2}} \left (m + 3\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{m}{2} + 2 \\ \frac{m}{2} + 3 \end{matrix}\middle |{- \frac{b x^{2}}{a}} \right )}}{a \sqrt{1 + \frac{b x^{2}}{a}} \left (m + 4\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**(m + 2)*sqrt(a + b*x**2)*hyper((1/2, m/2 + 1), (m/2 + 2,), -b*x**2/a)/sqrt(1
+ b*x**2/a) + b*x**(m + 4)*sqrt(a + b*x**2)*(m + 3)*hyper((1/2, m/2 + 2), (m/2 +
3,), -b*x**2/a)/(a*sqrt(1 + b*x**2/a)*(m + 4))

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Mathematica [C]  time = 0.0467591, size = 97, normalized size = 5.71 \[ \frac{x^{m+2} \sqrt{a+b x^2} \left ((m+3) \, _2F_1\left (-\frac{1}{2},\frac{m}{2}+1;\frac{m}{2}+2;-\frac{b x^2}{a}\right )-\, _2F_1\left (\frac{1}{2},\frac{m}{2}+1;\frac{m}{2}+2;-\frac{b x^2}{a}\right )\right )}{(m+2) \sqrt{\frac{b x^2}{a}+1}} \]

Antiderivative was successfully verified.

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

[Out]

(x^(2 + m)*Sqrt[a + b*x^2]*((3 + m)*Hypergeometric2F1[-1/2, 1 + m/2, 2 + m/2, -(
(b*x^2)/a)] - Hypergeometric2F1[1/2, 1 + m/2, 2 + m/2, -((b*x^2)/a)]))/((2 + m)*
Sqrt[1 + (b*x^2)/a])

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Maple [F]  time = 0.041, size = 0, normalized size = 0. \[ \int{a \left ( 2+m \right ){x}^{1+m}{\frac{1}{\sqrt{b{x}^{2}+a}}}}+{b \left ( 3+m \right ){x}^{3+m}{\frac{1}{\sqrt{b{x}^{2}+a}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

int(a*(2+m)*x^(1+m)/(b*x^2+a)^(1/2)+b*(3+m)*x^(3+m)/(b*x^2+a)^(1/2),x)

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Maxima [A]  time = 1.49329, size = 22, normalized size = 1.29 \[ \sqrt{b x^{2} + a} x^{2} x^{m} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(b*x^2 + a)*x^2*x^m

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Fricas [A]  time = 0.27008, size = 24, normalized size = 1.41 \[ \frac{\sqrt{b x^{2} + a} x^{m + 3}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(b*x^2 + a)*x^(m + 3)/x

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Sympy [A]  time = 49.1256, size = 105, normalized size = 6.18 \[ \frac{\sqrt{a} x^{2} x^{m} \left (m + 2\right ) \Gamma \left (\frac{m}{2} + 1\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{m}{2} + 1 \\ \frac{m}{2} + 2 \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (\frac{m}{2} + 2\right )} + \frac{b x^{4} x^{m} \left (m + 3\right ) \Gamma \left (\frac{m}{2} + 2\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{m}{2} + 2 \\ \frac{m}{2} + 3 \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{2 \sqrt{a} \Gamma \left (\frac{m}{2} + 3\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(a)*x**2*x**m*(m + 2)*gamma(m/2 + 1)*hyper((1/2, m/2 + 1), (m/2 + 2,), b*x**
2*exp_polar(I*pi)/a)/(2*gamma(m/2 + 2)) + b*x**4*x**m*(m + 3)*gamma(m/2 + 2)*hyp
er((1/2, m/2 + 2), (m/2 + 3,), b*x**2*exp_polar(I*pi)/a)/(2*sqrt(a)*gamma(m/2 +
3))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(b*(m + 3)*x^(m + 3)/sqrt(b*x^2 + a) + a*(m + 2)*x^(m + 1)/sqrt(b*x^2 +
a), x)