3.944 \(\int \frac{(d+e x)^m \left (a+b x+c x^2\right )}{(e+f x)^{3/2}} \, dx\)

Optimal. Leaf size=237 \[ \frac{2 \sqrt{e+f x} (d+e x)^m \left (-\frac{f (d+e x)}{e^2-d f}\right )^{-m} \, _2F_1\left (\frac{1}{2},-m;\frac{3}{2};\frac{e (e+f x)}{e^2-d f}\right ) \left (c \left (d^2 f^2+4 d e^2 f (m+1)-4 e^4 \left (m^2+3 m+2\right )\right )-e f (2 m+3) \left (a e f (2 m+1)+b \left (d f-2 e^2 (m+1)\right )\right )\right )}{e f^3 (2 m+3) \left (e^2-d f\right )}+\frac{2 (d+e x)^{m+1} \left (a+\frac{e (c e-b f)}{f^2}\right )}{\left (e^2-d f\right ) \sqrt{e+f x}}+\frac{2 c \sqrt{e+f x} (d+e x)^{m+1}}{e f^2 (2 m+3)} \]

[Out]

(2*(a + (e*(c*e - b*f))/f^2)*(d + e*x)^(1 + m))/((e^2 - d*f)*Sqrt[e + f*x]) + (2
*c*(d + e*x)^(1 + m)*Sqrt[e + f*x])/(e*f^2*(3 + 2*m)) + (2*(c*(d^2*f^2 + 4*d*e^2
*f*(1 + m) - 4*e^4*(2 + 3*m + m^2)) - e*f*(3 + 2*m)*(a*e*f*(1 + 2*m) + b*(d*f -
2*e^2*(1 + m))))*(d + e*x)^m*Sqrt[e + f*x]*Hypergeometric2F1[1/2, -m, 3/2, (e*(e
+ f*x))/(e^2 - d*f)])/(e*f^3*(e^2 - d*f)*(3 + 2*m)*(-((f*(d + e*x))/(e^2 - d*f)
))^m)

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Rubi [A]  time = 0.838316, antiderivative size = 230, normalized size of antiderivative = 0.97, number of steps used = 4, number of rules used = 4, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.148 \[ -\frac{2 \sqrt{e+f x} (d+e x)^m \left (-\frac{f (d+e x)}{e^2-d f}\right )^{-m} \, _2F_1\left (\frac{1}{2},-m;\frac{3}{2};\frac{e (e+f x)}{e^2-d f}\right ) \left (f \left (a e f (2 m+1)+b d f-2 b e^2 (m+1)\right )-\frac{c \left (d^2 f^2+4 d e^2 f (m+1)-4 e^4 \left (m^2+3 m+2\right )\right )}{e (2 m+3)}\right )}{f^3 \left (e^2-d f\right )}+\frac{2 (d+e x)^{m+1} \left (a+\frac{e (c e-b f)}{f^2}\right )}{\left (e^2-d f\right ) \sqrt{e+f x}}+\frac{2 c \sqrt{e+f x} (d+e x)^{m+1}}{e f^2 (2 m+3)} \]

Antiderivative was successfully verified.

[In]  Int[((d + e*x)^m*(a + b*x + c*x^2))/(e + f*x)^(3/2),x]

[Out]

(2*(a + (e*(c*e - b*f))/f^2)*(d + e*x)^(1 + m))/((e^2 - d*f)*Sqrt[e + f*x]) + (2
*c*(d + e*x)^(1 + m)*Sqrt[e + f*x])/(e*f^2*(3 + 2*m)) - (2*(f*(b*d*f - 2*b*e^2*(
1 + m) + a*e*f*(1 + 2*m)) - (c*(d^2*f^2 + 4*d*e^2*f*(1 + m) - 4*e^4*(2 + 3*m + m
^2)))/(e*(3 + 2*m)))*(d + e*x)^m*Sqrt[e + f*x]*Hypergeometric2F1[1/2, -m, 3/2, (
e*(e + f*x))/(e^2 - d*f)])/(f^3*(e^2 - d*f)*(-((f*(d + e*x))/(e^2 - d*f)))^m)

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Rubi in Sympy [A]  time = 70.8945, size = 201, normalized size = 0.85 \[ \frac{2 c \left (\frac{f \left (d + e x\right )}{d f - e^{2}}\right )^{- m} \left (d + e x\right )^{m} \left (e + f x\right )^{\frac{3}{2}}{{}_{2}F_{1}\left (\begin{matrix} - m, \frac{3}{2} \\ \frac{5}{2} \end{matrix}\middle |{\frac{e \left (- e - f x\right )}{d f - e^{2}}} \right )}}{3 f^{3}} + \frac{2 \left (\frac{f \left (d + e x\right )}{d f - e^{2}}\right )^{- m} \left (d + e x\right )^{m} \sqrt{e + f x} \left (b f - 2 c e\right ){{}_{2}F_{1}\left (\begin{matrix} - m, \frac{1}{2} \\ \frac{3}{2} \end{matrix}\middle |{\frac{e \left (- e - f x\right )}{d f - e^{2}}} \right )}}{f^{3}} - \frac{2 \left (\frac{f \left (d + e x\right )}{d f - e^{2}}\right )^{- m} \left (d + e x\right )^{m} \left (a f^{2} - b e f + c e^{2}\right ){{}_{2}F_{1}\left (\begin{matrix} - m, - \frac{1}{2} \\ \frac{1}{2} \end{matrix}\middle |{\frac{e \left (- e - f x\right )}{d f - e^{2}}} \right )}}{f^{3} \sqrt{e + f x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**m*(c*x**2+b*x+a)/(f*x+e)**(3/2),x)

[Out]

2*c*(f*(d + e*x)/(d*f - e**2))**(-m)*(d + e*x)**m*(e + f*x)**(3/2)*hyper((-m, 3/
2), (5/2,), e*(-e - f*x)/(d*f - e**2))/(3*f**3) + 2*(f*(d + e*x)/(d*f - e**2))**
(-m)*(d + e*x)**m*sqrt(e + f*x)*(b*f - 2*c*e)*hyper((-m, 1/2), (3/2,), e*(-e - f
*x)/(d*f - e**2))/f**3 - 2*(f*(d + e*x)/(d*f - e**2))**(-m)*(d + e*x)**m*(a*f**2
- b*e*f + c*e**2)*hyper((-m, -1/2), (1/2,), e*(-e - f*x)/(d*f - e**2))/(f**3*sq
rt(e + f*x))

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Mathematica [A]  time = 0.28262, size = 171, normalized size = 0.72 \[ \frac{2 (d+e x)^m \left (\frac{f (d+e x)}{d f-e^2}\right )^{-m} \left (-3 \left (f (a f-b e)+c e^2\right ) \, _2F_1\left (-\frac{1}{2},-m;\frac{1}{2};\frac{e (e+f x)}{e^2-d f}\right )-(e+f x) \left ((6 c e-3 b f) \, _2F_1\left (\frac{1}{2},-m;\frac{3}{2};\frac{e (e+f x)}{e^2-d f}\right )-c (e+f x) \, _2F_1\left (\frac{3}{2},-m;\frac{5}{2};\frac{e (e+f x)}{e^2-d f}\right )\right )\right )}{3 f^3 \sqrt{e+f x}} \]

Antiderivative was successfully verified.

[In]  Integrate[((d + e*x)^m*(a + b*x + c*x^2))/(e + f*x)^(3/2),x]

[Out]

(2*(d + e*x)^m*(-3*(c*e^2 + f*(-(b*e) + a*f))*Hypergeometric2F1[-1/2, -m, 1/2, (
e*(e + f*x))/(e^2 - d*f)] - (e + f*x)*((6*c*e - 3*b*f)*Hypergeometric2F1[1/2, -m
, 3/2, (e*(e + f*x))/(e^2 - d*f)] - c*(e + f*x)*Hypergeometric2F1[3/2, -m, 5/2,
(e*(e + f*x))/(e^2 - d*f)])))/(3*f^3*((f*(d + e*x))/(-e^2 + d*f))^m*Sqrt[e + f*x
])

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Maple [F]  time = 0.048, size = 0, normalized size = 0. \[ \int{ \left ( ex+d \right ) ^{m} \left ( c{x}^{2}+bx+a \right ) \left ( fx+e \right ) ^{-{\frac{3}{2}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^m*(c*x^2+b*x+a)/(f*x+e)^(3/2),x)

[Out]

int((e*x+d)^m*(c*x^2+b*x+a)/(f*x+e)^(3/2),x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((c*x^2 + b*x + a)*(e*x + d)^m/(f*x + e)^(3/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((c*x^2 + b*x + a)*(e*x + d)^m/(f*x + e)^(3/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)**m*(c*x**2+b*x+a)/(f*x+e)**(3/2),x)

[Out]

Integral((d + e*x)**m*(a + b*x + c*x**2)/(e + f*x)**(3/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((c*x^2 + b*x + a)*(e*x + d)^m/(f*x + e)^(3/2), x)