3.4.94 \(\int (d+e x)^4 (a+c x^2)^3 \, dx\)

Optimal. Leaf size=188 \[ \frac {c^2 (d+e x)^9 \left (a e^2+5 c d^2\right )}{3 e^7}-\frac {c^2 d (d+e x)^8 \left (3 a e^2+5 c d^2\right )}{2 e^7}+\frac {3 c (d+e x)^7 \left (a e^2+c d^2\right ) \left (a e^2+5 c d^2\right )}{7 e^7}-\frac {c d (d+e x)^6 \left (a e^2+c d^2\right )^2}{e^7}+\frac {(d+e x)^5 \left (a e^2+c d^2\right )^3}{5 e^7}+\frac {c^3 (d+e x)^{11}}{11 e^7}-\frac {3 c^3 d (d+e x)^{10}}{5 e^7} \]

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Rubi [A]  time = 0.23, antiderivative size = 188, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {697} \begin {gather*} \frac {c^2 (d+e x)^9 \left (a e^2+5 c d^2\right )}{3 e^7}-\frac {c^2 d (d+e x)^8 \left (3 a e^2+5 c d^2\right )}{2 e^7}+\frac {3 c (d+e x)^7 \left (a e^2+c d^2\right ) \left (a e^2+5 c d^2\right )}{7 e^7}-\frac {c d (d+e x)^6 \left (a e^2+c d^2\right )^2}{e^7}+\frac {(d+e x)^5 \left (a e^2+c d^2\right )^3}{5 e^7}+\frac {c^3 (d+e x)^{11}}{11 e^7}-\frac {3 c^3 d (d+e x)^{10}}{5 e^7} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^4*(a + c*x^2)^3,x]

[Out]

((c*d^2 + a*e^2)^3*(d + e*x)^5)/(5*e^7) - (c*d*(c*d^2 + a*e^2)^2*(d + e*x)^6)/e^7 + (3*c*(c*d^2 + a*e^2)*(5*c*
d^2 + a*e^2)*(d + e*x)^7)/(7*e^7) - (c^2*d*(5*c*d^2 + 3*a*e^2)*(d + e*x)^8)/(2*e^7) + (c^2*(5*c*d^2 + a*e^2)*(
d + e*x)^9)/(3*e^7) - (3*c^3*d*(d + e*x)^10)/(5*e^7) + (c^3*(d + e*x)^11)/(11*e^7)

Rule 697

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(a + c*
x^2)^p, x], x] /; FreeQ[{a, c, d, e, m}, x] && NeQ[c*d^2 + a*e^2, 0] && IGtQ[p, 0]

Rubi steps

\begin {align*} \int (d+e x)^4 \left (a+c x^2\right )^3 \, dx &=\int \left (\frac {\left (c d^2+a e^2\right )^3 (d+e x)^4}{e^6}-\frac {6 c d \left (c d^2+a e^2\right )^2 (d+e x)^5}{e^6}+\frac {3 c \left (c d^2+a e^2\right ) \left (5 c d^2+a e^2\right ) (d+e x)^6}{e^6}-\frac {4 c^2 d \left (5 c d^2+3 a e^2\right ) (d+e x)^7}{e^6}+\frac {3 c^2 \left (5 c d^2+a e^2\right ) (d+e x)^8}{e^6}-\frac {6 c^3 d (d+e x)^9}{e^6}+\frac {c^3 (d+e x)^{10}}{e^6}\right ) \, dx\\ &=\frac {\left (c d^2+a e^2\right )^3 (d+e x)^5}{5 e^7}-\frac {c d \left (c d^2+a e^2\right )^2 (d+e x)^6}{e^7}+\frac {3 c \left (c d^2+a e^2\right ) \left (5 c d^2+a e^2\right ) (d+e x)^7}{7 e^7}-\frac {c^2 d \left (5 c d^2+3 a e^2\right ) (d+e x)^8}{2 e^7}+\frac {c^2 \left (5 c d^2+a e^2\right ) (d+e x)^9}{3 e^7}-\frac {3 c^3 d (d+e x)^{10}}{5 e^7}+\frac {c^3 (d+e x)^{11}}{11 e^7}\\ \end {align*}

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Mathematica [A]  time = 0.05, size = 206, normalized size = 1.10 \begin {gather*} a^3 \left (d^4 x+2 d^3 e x^2+2 d^2 e^2 x^3+d e^3 x^4+\frac {e^4 x^5}{5}\right )+a^2 c \left (d^4 x^3+3 d^3 e x^4+\frac {18}{5} d^2 e^2 x^5+2 d e^3 x^6+\frac {3 e^4 x^7}{7}\right )+\frac {1}{210} a c^2 x^5 \left (126 d^4+420 d^3 e x+540 d^2 e^2 x^2+315 d e^3 x^3+70 e^4 x^4\right )+\frac {c^3 x^7 \left (330 d^4+1155 d^3 e x+1540 d^2 e^2 x^2+924 d e^3 x^3+210 e^4 x^4\right )}{2310} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^4*(a + c*x^2)^3,x]

[Out]

(a*c^2*x^5*(126*d^4 + 420*d^3*e*x + 540*d^2*e^2*x^2 + 315*d*e^3*x^3 + 70*e^4*x^4))/210 + (c^3*x^7*(330*d^4 + 1
155*d^3*e*x + 1540*d^2*e^2*x^2 + 924*d*e^3*x^3 + 210*e^4*x^4))/2310 + a^3*(d^4*x + 2*d^3*e*x^2 + 2*d^2*e^2*x^3
+ d*e^3*x^4 + (e^4*x^5)/5) + a^2*c*(d^4*x^3 + 3*d^3*e*x^4 + (18*d^2*e^2*x^5)/5 + 2*d*e^3*x^6 + (3*e^4*x^7)/7)

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int (d+e x)^4 \left (a+c x^2\right )^3 \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(d + e*x)^4*(a + c*x^2)^3,x]

[Out]

IntegrateAlgebraic[(d + e*x)^4*(a + c*x^2)^3, x]

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fricas [A]  time = 0.37, size = 246, normalized size = 1.31 \begin {gather*} \frac {1}{11} x^{11} e^{4} c^{3} + \frac {2}{5} x^{10} e^{3} d c^{3} + \frac {2}{3} x^{9} e^{2} d^{2} c^{3} + \frac {1}{3} x^{9} e^{4} c^{2} a + \frac {1}{2} x^{8} e d^{3} c^{3} + \frac {3}{2} x^{8} e^{3} d c^{2} a + \frac {1}{7} x^{7} d^{4} c^{3} + \frac {18}{7} x^{7} e^{2} d^{2} c^{2} a + \frac {3}{7} x^{7} e^{4} c a^{2} + 2 x^{6} e d^{3} c^{2} a + 2 x^{6} e^{3} d c a^{2} + \frac {3}{5} x^{5} d^{4} c^{2} a + \frac {18}{5} x^{5} e^{2} d^{2} c a^{2} + \frac {1}{5} x^{5} e^{4} a^{3} + 3 x^{4} e d^{3} c a^{2} + x^{4} e^{3} d a^{3} + x^{3} d^{4} c a^{2} + 2 x^{3} e^{2} d^{2} a^{3} + 2 x^{2} e d^{3} a^{3} + x d^{4} a^{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^4*(c*x^2+a)^3,x, algorithm="fricas")

[Out]

1/11*x^11*e^4*c^3 + 2/5*x^10*e^3*d*c^3 + 2/3*x^9*e^2*d^2*c^3 + 1/3*x^9*e^4*c^2*a + 1/2*x^8*e*d^3*c^3 + 3/2*x^8
*e^3*d*c^2*a + 1/7*x^7*d^4*c^3 + 18/7*x^7*e^2*d^2*c^2*a + 3/7*x^7*e^4*c*a^2 + 2*x^6*e*d^3*c^2*a + 2*x^6*e^3*d*
c*a^2 + 3/5*x^5*d^4*c^2*a + 18/5*x^5*e^2*d^2*c*a^2 + 1/5*x^5*e^4*a^3 + 3*x^4*e*d^3*c*a^2 + x^4*e^3*d*a^3 + x^3
*d^4*c*a^2 + 2*x^3*e^2*d^2*a^3 + 2*x^2*e*d^3*a^3 + x*d^4*a^3

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giac [A]  time = 0.16, size = 238, normalized size = 1.27 \begin {gather*} \frac {1}{11} \, c^{3} x^{11} e^{4} + \frac {2}{5} \, c^{3} d x^{10} e^{3} + \frac {2}{3} \, c^{3} d^{2} x^{9} e^{2} + \frac {1}{2} \, c^{3} d^{3} x^{8} e + \frac {1}{7} \, c^{3} d^{4} x^{7} + \frac {1}{3} \, a c^{2} x^{9} e^{4} + \frac {3}{2} \, a c^{2} d x^{8} e^{3} + \frac {18}{7} \, a c^{2} d^{2} x^{7} e^{2} + 2 \, a c^{2} d^{3} x^{6} e + \frac {3}{5} \, a c^{2} d^{4} x^{5} + \frac {3}{7} \, a^{2} c x^{7} e^{4} + 2 \, a^{2} c d x^{6} e^{3} + \frac {18}{5} \, a^{2} c d^{2} x^{5} e^{2} + 3 \, a^{2} c d^{3} x^{4} e + a^{2} c d^{4} x^{3} + \frac {1}{5} \, a^{3} x^{5} e^{4} + a^{3} d x^{4} e^{3} + 2 \, a^{3} d^{2} x^{3} e^{2} + 2 \, a^{3} d^{3} x^{2} e + a^{3} d^{4} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^4*(c*x^2+a)^3,x, algorithm="giac")

[Out]

1/11*c^3*x^11*e^4 + 2/5*c^3*d*x^10*e^3 + 2/3*c^3*d^2*x^9*e^2 + 1/2*c^3*d^3*x^8*e + 1/7*c^3*d^4*x^7 + 1/3*a*c^2
*x^9*e^4 + 3/2*a*c^2*d*x^8*e^3 + 18/7*a*c^2*d^2*x^7*e^2 + 2*a*c^2*d^3*x^6*e + 3/5*a*c^2*d^4*x^5 + 3/7*a^2*c*x^
7*e^4 + 2*a^2*c*d*x^6*e^3 + 18/5*a^2*c*d^2*x^5*e^2 + 3*a^2*c*d^3*x^4*e + a^2*c*d^4*x^3 + 1/5*a^3*x^5*e^4 + a^3
*d*x^4*e^3 + 2*a^3*d^2*x^3*e^2 + 2*a^3*d^3*x^2*e + a^3*d^4*x

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maple [A]  time = 0.04, size = 241, normalized size = 1.28 \begin {gather*} \frac {c^{3} e^{4} x^{11}}{11}+\frac {2 c^{3} d \,e^{3} x^{10}}{5}+2 a^{3} d^{3} e \,x^{2}+\frac {\left (3 e^{4} a \,c^{2}+6 d^{2} e^{2} c^{3}\right ) x^{9}}{9}+a^{3} d^{4} x +\frac {\left (12 d \,e^{3} a \,c^{2}+4 d^{3} e \,c^{3}\right ) x^{8}}{8}+\frac {\left (3 e^{4} a^{2} c +18 d^{2} e^{2} a \,c^{2}+d^{4} c^{3}\right ) x^{7}}{7}+\frac {\left (12 d \,e^{3} a^{2} c +12 d^{3} e a \,c^{2}\right ) x^{6}}{6}+\frac {\left (e^{4} a^{3}+18 d^{2} e^{2} a^{2} c +3 d^{4} a \,c^{2}\right ) x^{5}}{5}+\frac {\left (4 d \,e^{3} a^{3}+12 d^{3} e \,a^{2} c \right ) x^{4}}{4}+\frac {\left (6 d^{2} e^{2} a^{3}+3 d^{4} a^{2} c \right ) x^{3}}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^4*(c*x^2+a)^3,x)

[Out]

1/11*c^3*e^4*x^11+2/5*d*e^3*c^3*x^10+1/9*(3*a*c^2*e^4+6*c^3*d^2*e^2)*x^9+1/8*(12*a*c^2*d*e^3+4*c^3*d^3*e)*x^8+
1/7*(3*a^2*c*e^4+18*a*c^2*d^2*e^2+c^3*d^4)*x^7+1/6*(12*a^2*c*d*e^3+12*a*c^2*d^3*e)*x^6+1/5*(a^3*e^4+18*a^2*c*d
^2*e^2+3*a*c^2*d^4)*x^5+1/4*(4*a^3*d*e^3+12*a^2*c*d^3*e)*x^4+1/3*(6*a^3*d^2*e^2+3*a^2*c*d^4)*x^3+2*d^3*e*a^3*x
^2+d^4*a^3*x

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maxima [A]  time = 1.41, size = 232, normalized size = 1.23 \begin {gather*} \frac {1}{11} \, c^{3} e^{4} x^{11} + \frac {2}{5} \, c^{3} d e^{3} x^{10} + \frac {1}{3} \, {\left (2 \, c^{3} d^{2} e^{2} + a c^{2} e^{4}\right )} x^{9} + 2 \, a^{3} d^{3} e x^{2} + \frac {1}{2} \, {\left (c^{3} d^{3} e + 3 \, a c^{2} d e^{3}\right )} x^{8} + a^{3} d^{4} x + \frac {1}{7} \, {\left (c^{3} d^{4} + 18 \, a c^{2} d^{2} e^{2} + 3 \, a^{2} c e^{4}\right )} x^{7} + 2 \, {\left (a c^{2} d^{3} e + a^{2} c d e^{3}\right )} x^{6} + \frac {1}{5} \, {\left (3 \, a c^{2} d^{4} + 18 \, a^{2} c d^{2} e^{2} + a^{3} e^{4}\right )} x^{5} + {\left (3 \, a^{2} c d^{3} e + a^{3} d e^{3}\right )} x^{4} + {\left (a^{2} c d^{4} + 2 \, a^{3} d^{2} e^{2}\right )} x^{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^4*(c*x^2+a)^3,x, algorithm="maxima")

[Out]

1/11*c^3*e^4*x^11 + 2/5*c^3*d*e^3*x^10 + 1/3*(2*c^3*d^2*e^2 + a*c^2*e^4)*x^9 + 2*a^3*d^3*e*x^2 + 1/2*(c^3*d^3*
e + 3*a*c^2*d*e^3)*x^8 + a^3*d^4*x + 1/7*(c^3*d^4 + 18*a*c^2*d^2*e^2 + 3*a^2*c*e^4)*x^7 + 2*(a*c^2*d^3*e + a^2
*c*d*e^3)*x^6 + 1/5*(3*a*c^2*d^4 + 18*a^2*c*d^2*e^2 + a^3*e^4)*x^5 + (3*a^2*c*d^3*e + a^3*d*e^3)*x^4 + (a^2*c*
d^4 + 2*a^3*d^2*e^2)*x^3

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mupad [B]  time = 0.32, size = 224, normalized size = 1.19 \begin {gather*} x^3\,\left (2\,a^3\,d^2\,e^2+c\,a^2\,d^4\right )+x^9\,\left (\frac {2\,c^3\,d^2\,e^2}{3}+\frac {a\,c^2\,e^4}{3}\right )+x^5\,\left (\frac {a^3\,e^4}{5}+\frac {18\,a^2\,c\,d^2\,e^2}{5}+\frac {3\,a\,c^2\,d^4}{5}\right )+x^7\,\left (\frac {3\,a^2\,c\,e^4}{7}+\frac {18\,a\,c^2\,d^2\,e^2}{7}+\frac {c^3\,d^4}{7}\right )+a^3\,d^4\,x+\frac {c^3\,e^4\,x^{11}}{11}+2\,a^3\,d^3\,e\,x^2+\frac {2\,c^3\,d\,e^3\,x^{10}}{5}+a^2\,d\,e\,x^4\,\left (3\,c\,d^2+a\,e^2\right )+\frac {c^2\,d\,e\,x^8\,\left (c\,d^2+3\,a\,e^2\right )}{2}+2\,a\,c\,d\,e\,x^6\,\left (c\,d^2+a\,e^2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + c*x^2)^3*(d + e*x)^4,x)

[Out]

x^3*(a^2*c*d^4 + 2*a^3*d^2*e^2) + x^9*((a*c^2*e^4)/3 + (2*c^3*d^2*e^2)/3) + x^5*((a^3*e^4)/5 + (3*a*c^2*d^4)/5
+ (18*a^2*c*d^2*e^2)/5) + x^7*((c^3*d^4)/7 + (3*a^2*c*e^4)/7 + (18*a*c^2*d^2*e^2)/7) + a^3*d^4*x + (c^3*e^4*x
^11)/11 + 2*a^3*d^3*e*x^2 + (2*c^3*d*e^3*x^10)/5 + a^2*d*e*x^4*(a*e^2 + 3*c*d^2) + (c^2*d*e*x^8*(3*a*e^2 + c*d
^2))/2 + 2*a*c*d*e*x^6*(a*e^2 + c*d^2)

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sympy [A]  time = 0.11, size = 255, normalized size = 1.36 \begin {gather*} a^{3} d^{4} x + 2 a^{3} d^{3} e x^{2} + \frac {2 c^{3} d e^{3} x^{10}}{5} + \frac {c^{3} e^{4} x^{11}}{11} + x^{9} \left (\frac {a c^{2} e^{4}}{3} + \frac {2 c^{3} d^{2} e^{2}}{3}\right ) + x^{8} \left (\frac {3 a c^{2} d e^{3}}{2} + \frac {c^{3} d^{3} e}{2}\right ) + x^{7} \left (\frac {3 a^{2} c e^{4}}{7} + \frac {18 a c^{2} d^{2} e^{2}}{7} + \frac {c^{3} d^{4}}{7}\right ) + x^{6} \left (2 a^{2} c d e^{3} + 2 a c^{2} d^{3} e\right ) + x^{5} \left (\frac {a^{3} e^{4}}{5} + \frac {18 a^{2} c d^{2} e^{2}}{5} + \frac {3 a c^{2} d^{4}}{5}\right ) + x^{4} \left (a^{3} d e^{3} + 3 a^{2} c d^{3} e\right ) + x^{3} \left (2 a^{3} d^{2} e^{2} + a^{2} c d^{4}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**4*(c*x**2+a)**3,x)

[Out]

a**3*d**4*x + 2*a**3*d**3*e*x**2 + 2*c**3*d*e**3*x**10/5 + c**3*e**4*x**11/11 + x**9*(a*c**2*e**4/3 + 2*c**3*d
**2*e**2/3) + x**8*(3*a*c**2*d*e**3/2 + c**3*d**3*e/2) + x**7*(3*a**2*c*e**4/7 + 18*a*c**2*d**2*e**2/7 + c**3*
d**4/7) + x**6*(2*a**2*c*d*e**3 + 2*a*c**2*d**3*e) + x**5*(a**3*e**4/5 + 18*a**2*c*d**2*e**2/5 + 3*a*c**2*d**4
/5) + x**4*(a**3*d*e**3 + 3*a**2*c*d**3*e) + x**3*(2*a**3*d**2*e**2 + a**2*c*d**4)

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