3.712 \(\int (d+e x)^m \left (a+c x^2\right ) \, dx\)

Optimal. Leaf size=70 \[ \frac{\left (a e^2+c d^2\right ) (d+e x)^{m+1}}{e^3 (m+1)}-\frac{2 c d (d+e x)^{m+2}}{e^3 (m+2)}+\frac{c (d+e x)^{m+3}}{e^3 (m+3)} \]

[Out]

((c*d^2 + a*e^2)*(d + e*x)^(1 + m))/(e^3*(1 + m)) - (2*c*d*(d + e*x)^(2 + m))/(e
^3*(2 + m)) + (c*(d + e*x)^(3 + m))/(e^3*(3 + m))

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Rubi [A]  time = 0.0878331, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067 \[ \frac{\left (a e^2+c d^2\right ) (d+e x)^{m+1}}{e^3 (m+1)}-\frac{2 c d (d+e x)^{m+2}}{e^3 (m+2)}+\frac{c (d+e x)^{m+3}}{e^3 (m+3)} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^m*(a + c*x^2),x]

[Out]

((c*d^2 + a*e^2)*(d + e*x)^(1 + m))/(e^3*(1 + m)) - (2*c*d*(d + e*x)^(2 + m))/(e
^3*(2 + m)) + (c*(d + e*x)^(3 + m))/(e^3*(3 + m))

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Rubi in Sympy [A]  time = 16.7234, size = 61, normalized size = 0.87 \[ - \frac{2 c d \left (d + e x\right )^{m + 2}}{e^{3} \left (m + 2\right )} + \frac{c \left (d + e x\right )^{m + 3}}{e^{3} \left (m + 3\right )} + \frac{\left (d + e x\right )^{m + 1} \left (a e^{2} + c d^{2}\right )}{e^{3} \left (m + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*c*d*(d + e*x)**(m + 2)/(e**3*(m + 2)) + c*(d + e*x)**(m + 3)/(e**3*(m + 3)) +
(d + e*x)**(m + 1)*(a*e**2 + c*d**2)/(e**3*(m + 1))

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Mathematica [A]  time = 0.0658794, size = 73, normalized size = 1.04 \[ \frac{(d+e x)^{m+1} \left (a e^2 \left (m^2+5 m+6\right )+c \left (2 d^2-2 d e (m+1) x+e^2 \left (m^2+3 m+2\right ) x^2\right )\right )}{e^3 (m+1) (m+2) (m+3)} \]

Antiderivative was successfully verified.

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

[Out]

((d + e*x)^(1 + m)*(a*e^2*(6 + 5*m + m^2) + c*(2*d^2 - 2*d*e*(1 + m)*x + e^2*(2
+ 3*m + m^2)*x^2)))/(e^3*(1 + m)*(2 + m)*(3 + m))

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Maple [A]  time = 0.006, size = 100, normalized size = 1.4 \[{\frac{ \left ( ex+d \right ) ^{1+m} \left ( c{e}^{2}{m}^{2}{x}^{2}+3\,c{e}^{2}m{x}^{2}+a{e}^{2}{m}^{2}-2\,cdemx+2\,c{e}^{2}{x}^{2}+5\,a{e}^{2}m-2\,cdex+6\,a{e}^{2}+2\,c{d}^{2} \right ) }{{e}^{3} \left ({m}^{3}+6\,{m}^{2}+11\,m+6 \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^m*(c*x^2+a),x)

[Out]

(e*x+d)^(1+m)*(c*e^2*m^2*x^2+3*c*e^2*m*x^2+a*e^2*m^2-2*c*d*e*m*x+2*c*e^2*x^2+5*a
*e^2*m-2*c*d*e*x+6*a*e^2+2*c*d^2)/e^3/(m^3+6*m^2+11*m+6)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.240606, size = 201, normalized size = 2.87 \[ \frac{{\left (a d e^{2} m^{2} + 5 \, a d e^{2} m + 2 \, c d^{3} + 6 \, a d e^{2} +{\left (c e^{3} m^{2} + 3 \, c e^{3} m + 2 \, c e^{3}\right )} x^{3} +{\left (c d e^{2} m^{2} + c d e^{2} m\right )} x^{2} +{\left (a e^{3} m^{2} + 6 \, a e^{3} -{\left (2 \, c d^{2} e - 5 \, a e^{3}\right )} m\right )} x\right )}{\left (e x + d\right )}^{m}}{e^{3} m^{3} + 6 \, e^{3} m^{2} + 11 \, e^{3} m + 6 \, e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(a*d*e^2*m^2 + 5*a*d*e^2*m + 2*c*d^3 + 6*a*d*e^2 + (c*e^3*m^2 + 3*c*e^3*m + 2*c*
e^3)*x^3 + (c*d*e^2*m^2 + c*d*e^2*m)*x^2 + (a*e^3*m^2 + 6*a*e^3 - (2*c*d^2*e - 5
*a*e^3)*m)*x)*(e*x + d)^m/(e^3*m^3 + 6*e^3*m^2 + 11*e^3*m + 6*e^3)

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Sympy [A]  time = 4.32045, size = 978, normalized size = 13.97 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((d**m*(a*x + c*x**3/3), Eq(e, 0)), (-a*e**2/(2*d**2*e**3 + 4*d*e**4*x
+ 2*e**5*x**2) + 2*c*d**2*log(d/e + x)/(2*d**2*e**3 + 4*d*e**4*x + 2*e**5*x**2)
+ c*d**2/(2*d**2*e**3 + 4*d*e**4*x + 2*e**5*x**2) + 4*c*d*e*x*log(d/e + x)/(2*d*
*2*e**3 + 4*d*e**4*x + 2*e**5*x**2) + 2*c*e**2*x**2*log(d/e + x)/(2*d**2*e**3 +
4*d*e**4*x + 2*e**5*x**2) - 2*c*e**2*x**2/(2*d**2*e**3 + 4*d*e**4*x + 2*e**5*x**
2), Eq(m, -3)), (a*e**3*x/(d**2*e**3 + d*e**4*x) - 2*c*d**3*log(d/e + x)/(d**2*e
**3 + d*e**4*x) - 2*c*d**2*e*x*log(d/e + x)/(d**2*e**3 + d*e**4*x) + 2*c*d**2*e*
x/(d**2*e**3 + d*e**4*x) + c*d*e**2*x**2/(d**2*e**3 + d*e**4*x), Eq(m, -2)), (a*
log(d/e + x)/e + c*d**2*log(d/e + x)/e**3 - c*d*x/e**2 + c*x**2/(2*e), Eq(m, -1)
), (a*d*e**2*m**2*(d + e*x)**m/(e**3*m**3 + 6*e**3*m**2 + 11*e**3*m + 6*e**3) +
5*a*d*e**2*m*(d + e*x)**m/(e**3*m**3 + 6*e**3*m**2 + 11*e**3*m + 6*e**3) + 6*a*d
*e**2*(d + e*x)**m/(e**3*m**3 + 6*e**3*m**2 + 11*e**3*m + 6*e**3) + a*e**3*m**2*
x*(d + e*x)**m/(e**3*m**3 + 6*e**3*m**2 + 11*e**3*m + 6*e**3) + 5*a*e**3*m*x*(d
+ e*x)**m/(e**3*m**3 + 6*e**3*m**2 + 11*e**3*m + 6*e**3) + 6*a*e**3*x*(d + e*x)*
*m/(e**3*m**3 + 6*e**3*m**2 + 11*e**3*m + 6*e**3) + 2*c*d**3*(d + e*x)**m/(e**3*
m**3 + 6*e**3*m**2 + 11*e**3*m + 6*e**3) - 2*c*d**2*e*m*x*(d + e*x)**m/(e**3*m**
3 + 6*e**3*m**2 + 11*e**3*m + 6*e**3) + c*d*e**2*m**2*x**2*(d + e*x)**m/(e**3*m*
*3 + 6*e**3*m**2 + 11*e**3*m + 6*e**3) + c*d*e**2*m*x**2*(d + e*x)**m/(e**3*m**3
+ 6*e**3*m**2 + 11*e**3*m + 6*e**3) + c*e**3*m**2*x**3*(d + e*x)**m/(e**3*m**3
+ 6*e**3*m**2 + 11*e**3*m + 6*e**3) + 3*c*e**3*m*x**3*(d + e*x)**m/(e**3*m**3 +
6*e**3*m**2 + 11*e**3*m + 6*e**3) + 2*c*e**3*x**3*(d + e*x)**m/(e**3*m**3 + 6*e*
*3*m**2 + 11*e**3*m + 6*e**3), True))

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GIAC/XCAS [A]  time = 0.217643, size = 354, normalized size = 5.06 \[ \frac{c m^{2} x^{3} e^{\left (m{\rm ln}\left (x e + d\right ) + 3\right )} + c d m^{2} x^{2} e^{\left (m{\rm ln}\left (x e + d\right ) + 2\right )} + 3 \, c m x^{3} e^{\left (m{\rm ln}\left (x e + d\right ) + 3\right )} + c d m x^{2} e^{\left (m{\rm ln}\left (x e + d\right ) + 2\right )} - 2 \, c d^{2} m x e^{\left (m{\rm ln}\left (x e + d\right ) + 1\right )} + a m^{2} x e^{\left (m{\rm ln}\left (x e + d\right ) + 3\right )} + 2 \, c x^{3} e^{\left (m{\rm ln}\left (x e + d\right ) + 3\right )} + a d m^{2} e^{\left (m{\rm ln}\left (x e + d\right ) + 2\right )} + 2 \, c d^{3} e^{\left (m{\rm ln}\left (x e + d\right )\right )} + 5 \, a m x e^{\left (m{\rm ln}\left (x e + d\right ) + 3\right )} + 5 \, a d m e^{\left (m{\rm ln}\left (x e + d\right ) + 2\right )} + 6 \, a x e^{\left (m{\rm ln}\left (x e + d\right ) + 3\right )} + 6 \, a d e^{\left (m{\rm ln}\left (x e + d\right ) + 2\right )}}{m^{3} e^{3} + 6 \, m^{2} e^{3} + 11 \, m e^{3} + 6 \, e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(c*m^2*x^3*e^(m*ln(x*e + d) + 3) + c*d*m^2*x^2*e^(m*ln(x*e + d) + 2) + 3*c*m*x^3
*e^(m*ln(x*e + d) + 3) + c*d*m*x^2*e^(m*ln(x*e + d) + 2) - 2*c*d^2*m*x*e^(m*ln(x
*e + d) + 1) + a*m^2*x*e^(m*ln(x*e + d) + 3) + 2*c*x^3*e^(m*ln(x*e + d) + 3) + a
*d*m^2*e^(m*ln(x*e + d) + 2) + 2*c*d^3*e^(m*ln(x*e + d)) + 5*a*m*x*e^(m*ln(x*e +
d) + 3) + 5*a*d*m*e^(m*ln(x*e + d) + 2) + 6*a*x*e^(m*ln(x*e + d) + 3) + 6*a*d*e
^(m*ln(x*e + d) + 2))/(m^3*e^3 + 6*m^2*e^3 + 11*m*e^3 + 6*e^3)