3.492 \(\int (e x)^m (A+B x) \left (a+c x^2\right )^{5/2} \, dx\)

Optimal. Leaf size=145 \[ \frac{a^2 A \sqrt{a+c x^2} (e x)^{m+1} \, _2F_1\left (-\frac{5}{2},\frac{m+1}{2};\frac{m+3}{2};-\frac{c x^2}{a}\right )}{e (m+1) \sqrt{\frac{c x^2}{a}+1}}+\frac{a^2 B \sqrt{a+c x^2} (e x)^{m+2} \, _2F_1\left (-\frac{5}{2},\frac{m+2}{2};\frac{m+4}{2};-\frac{c x^2}{a}\right )}{e^2 (m+2) \sqrt{\frac{c x^2}{a}+1}} \]

[Out]

(a^2*A*(e*x)^(1 + m)*Sqrt[a + c*x^2]*Hypergeometric2F1[-5/2, (1 + m)/2, (3 + m)/
2, -((c*x^2)/a)])/(e*(1 + m)*Sqrt[1 + (c*x^2)/a]) + (a^2*B*(e*x)^(2 + m)*Sqrt[a
+ c*x^2]*Hypergeometric2F1[-5/2, (2 + m)/2, (4 + m)/2, -((c*x^2)/a)])/(e^2*(2 +
m)*Sqrt[1 + (c*x^2)/a])

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Rubi [A]  time = 0.214471, antiderivative size = 145, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136 \[ \frac{a^2 A \sqrt{a+c x^2} (e x)^{m+1} \, _2F_1\left (-\frac{5}{2},\frac{m+1}{2};\frac{m+3}{2};-\frac{c x^2}{a}\right )}{e (m+1) \sqrt{\frac{c x^2}{a}+1}}+\frac{a^2 B \sqrt{a+c x^2} (e x)^{m+2} \, _2F_1\left (-\frac{5}{2},\frac{m+2}{2};\frac{m+4}{2};-\frac{c x^2}{a}\right )}{e^2 (m+2) \sqrt{\frac{c x^2}{a}+1}} \]

Antiderivative was successfully verified.

[In]  Int[(e*x)^m*(A + B*x)*(a + c*x^2)^(5/2),x]

[Out]

(a^2*A*(e*x)^(1 + m)*Sqrt[a + c*x^2]*Hypergeometric2F1[-5/2, (1 + m)/2, (3 + m)/
2, -((c*x^2)/a)])/(e*(1 + m)*Sqrt[1 + (c*x^2)/a]) + (a^2*B*(e*x)^(2 + m)*Sqrt[a
+ c*x^2]*Hypergeometric2F1[-5/2, (2 + m)/2, (4 + m)/2, -((c*x^2)/a)])/(e^2*(2 +
m)*Sqrt[1 + (c*x^2)/a])

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Rubi in Sympy [A]  time = 21.1839, size = 122, normalized size = 0.84 \[ \frac{A a^{2} \left (e x\right )^{m + 1} \sqrt{a + c x^{2}}{{}_{2}F_{1}\left (\begin{matrix} - \frac{5}{2}, \frac{m}{2} + \frac{1}{2} \\ \frac{m}{2} + \frac{3}{2} \end{matrix}\middle |{- \frac{c x^{2}}{a}} \right )}}{e \sqrt{1 + \frac{c x^{2}}{a}} \left (m + 1\right )} + \frac{B a^{2} \left (e x\right )^{m + 2} \sqrt{a + c x^{2}}{{}_{2}F_{1}\left (\begin{matrix} - \frac{5}{2}, \frac{m}{2} + 1 \\ \frac{m}{2} + 2 \end{matrix}\middle |{- \frac{c x^{2}}{a}} \right )}}{e^{2} \sqrt{1 + \frac{c x^{2}}{a}} \left (m + 2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x)**m*(B*x+A)*(c*x**2+a)**(5/2),x)

[Out]

A*a**2*(e*x)**(m + 1)*sqrt(a + c*x**2)*hyper((-5/2, m/2 + 1/2), (m/2 + 3/2,), -c
*x**2/a)/(e*sqrt(1 + c*x**2/a)*(m + 1)) + B*a**2*(e*x)**(m + 2)*sqrt(a + c*x**2)
*hyper((-5/2, m/2 + 1), (m/2 + 2,), -c*x**2/a)/(e**2*sqrt(1 + c*x**2/a)*(m + 2))

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Mathematica [A]  time = 0.472329, size = 268, normalized size = 1.85 \[ \frac{x \sqrt{a+c x^2} (e x)^m \left (\frac{a^2 A \, _2F_1\left (-\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};-\frac{c x^2}{a}\right )}{m+1}+\frac{a^2 B x \, _2F_1\left (-\frac{1}{2},\frac{m}{2}+1;\frac{m}{2}+2;-\frac{c x^2}{a}\right )}{m+2}+\frac{A c^2 x^4 \, _2F_1\left (-\frac{1}{2},\frac{m+5}{2};\frac{m+7}{2};-\frac{c x^2}{a}\right )}{m+5}+\frac{2 a A c x^2 \, _2F_1\left (-\frac{1}{2},\frac{m+3}{2};\frac{m+5}{2};-\frac{c x^2}{a}\right )}{m+3}+\frac{B c^2 x^5 \, _2F_1\left (-\frac{1}{2},\frac{m}{2}+3;\frac{m}{2}+4;-\frac{c x^2}{a}\right )}{m+6}+\frac{2 a B c x^3 \, _2F_1\left (-\frac{1}{2},\frac{m}{2}+2;\frac{m}{2}+3;-\frac{c x^2}{a}\right )}{m+4}\right )}{\sqrt{\frac{c x^2}{a}+1}} \]

Antiderivative was successfully verified.

[In]  Integrate[(e*x)^m*(A + B*x)*(a + c*x^2)^(5/2),x]

[Out]

(x*(e*x)^m*Sqrt[a + c*x^2]*((a^2*B*x*Hypergeometric2F1[-1/2, 1 + m/2, 2 + m/2, -
((c*x^2)/a)])/(2 + m) + (2*a*B*c*x^3*Hypergeometric2F1[-1/2, 2 + m/2, 3 + m/2, -
((c*x^2)/a)])/(4 + m) + (B*c^2*x^5*Hypergeometric2F1[-1/2, 3 + m/2, 4 + m/2, -((
c*x^2)/a)])/(6 + m) + (a^2*A*Hypergeometric2F1[-1/2, (1 + m)/2, (3 + m)/2, -((c*
x^2)/a)])/(1 + m) + (2*a*A*c*x^2*Hypergeometric2F1[-1/2, (3 + m)/2, (5 + m)/2, -
((c*x^2)/a)])/(3 + m) + (A*c^2*x^4*Hypergeometric2F1[-1/2, (5 + m)/2, (7 + m)/2,
-((c*x^2)/a)])/(5 + m)))/Sqrt[1 + (c*x^2)/a]

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Maple [F]  time = 0.033, size = 0, normalized size = 0. \[ \int \left ( ex \right ) ^{m} \left ( Bx+A \right ) \left ( c{x}^{2}+a \right ) ^{{\frac{5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x)^m*(B*x+A)*(c*x^2+a)^(5/2),x)

[Out]

int((e*x)^m*(B*x+A)*(c*x^2+a)^(5/2),x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^(5/2)*(B*x + A)*(e*x)^m,x, algorithm="maxima")

[Out]

integrate((c*x^2 + a)^(5/2)*(B*x + A)*(e*x)^m, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (B c^{2} x^{5} + A c^{2} x^{4} + 2 \, B a c x^{3} + 2 \, A a c x^{2} + B a^{2} x + A a^{2}\right )} \sqrt{c x^{2} + a} \left (e x\right )^{m}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^(5/2)*(B*x + A)*(e*x)^m,x, algorithm="fricas")

[Out]

integral((B*c^2*x^5 + A*c^2*x^4 + 2*B*a*c*x^3 + 2*A*a*c*x^2 + B*a^2*x + A*a^2)*s
qrt(c*x^2 + a)*(e*x)^m, x)

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Sympy [A]  time = 172.879, size = 360, normalized size = 2.48 \[ \frac{A a^{\frac{5}{2}} e^{m} x x^{m} \Gamma \left (\frac{m}{2} + \frac{1}{2}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{m}{2} + \frac{1}{2} \\ \frac{m}{2} + \frac{3}{2} \end{matrix}\middle |{\frac{c x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (\frac{m}{2} + \frac{3}{2}\right )} + \frac{A a^{\frac{3}{2}} c e^{m} x^{3} x^{m} \Gamma \left (\frac{m}{2} + \frac{3}{2}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{m}{2} + \frac{3}{2} \\ \frac{m}{2} + \frac{5}{2} \end{matrix}\middle |{\frac{c x^{2} e^{i \pi }}{a}} \right )}}{\Gamma \left (\frac{m}{2} + \frac{5}{2}\right )} + \frac{A \sqrt{a} c^{2} e^{m} x^{5} x^{m} \Gamma \left (\frac{m}{2} + \frac{5}{2}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{m}{2} + \frac{5}{2} \\ \frac{m}{2} + \frac{7}{2} \end{matrix}\middle |{\frac{c x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (\frac{m}{2} + \frac{7}{2}\right )} + \frac{B a^{\frac{5}{2}} e^{m} x^{2} x^{m} \Gamma \left (\frac{m}{2} + 1\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{m}{2} + 1 \\ \frac{m}{2} + 2 \end{matrix}\middle |{\frac{c x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (\frac{m}{2} + 2\right )} + \frac{B a^{\frac{3}{2}} c e^{m} x^{4} x^{m} \Gamma \left (\frac{m}{2} + 2\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{m}{2} + 2 \\ \frac{m}{2} + 3 \end{matrix}\middle |{\frac{c x^{2} e^{i \pi }}{a}} \right )}}{\Gamma \left (\frac{m}{2} + 3\right )} + \frac{B \sqrt{a} c^{2} e^{m} x^{6} x^{m} \Gamma \left (\frac{m}{2} + 3\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{m}{2} + 3 \\ \frac{m}{2} + 4 \end{matrix}\middle |{\frac{c x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (\frac{m}{2} + 4\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x)**m*(B*x+A)*(c*x**2+a)**(5/2),x)

[Out]

A*a**(5/2)*e**m*x*x**m*gamma(m/2 + 1/2)*hyper((-1/2, m/2 + 1/2), (m/2 + 3/2,), c
*x**2*exp_polar(I*pi)/a)/(2*gamma(m/2 + 3/2)) + A*a**(3/2)*c*e**m*x**3*x**m*gamm
a(m/2 + 3/2)*hyper((-1/2, m/2 + 3/2), (m/2 + 5/2,), c*x**2*exp_polar(I*pi)/a)/ga
mma(m/2 + 5/2) + A*sqrt(a)*c**2*e**m*x**5*x**m*gamma(m/2 + 5/2)*hyper((-1/2, m/2
+ 5/2), (m/2 + 7/2,), c*x**2*exp_polar(I*pi)/a)/(2*gamma(m/2 + 7/2)) + B*a**(5/
2)*e**m*x**2*x**m*gamma(m/2 + 1)*hyper((-1/2, m/2 + 1), (m/2 + 2,), c*x**2*exp_p
olar(I*pi)/a)/(2*gamma(m/2 + 2)) + B*a**(3/2)*c*e**m*x**4*x**m*gamma(m/2 + 2)*hy
per((-1/2, m/2 + 2), (m/2 + 3,), c*x**2*exp_polar(I*pi)/a)/gamma(m/2 + 3) + B*sq
rt(a)*c**2*e**m*x**6*x**m*gamma(m/2 + 3)*hyper((-1/2, m/2 + 3), (m/2 + 4,), c*x*
*2*exp_polar(I*pi)/a)/(2*gamma(m/2 + 4))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^(5/2)*(B*x + A)*(e*x)^m,x, algorithm="giac")

[Out]

integrate((c*x^2 + a)^(5/2)*(B*x + A)*(e*x)^m, x)