3.473 \(\int (d+e x)^5 (a+c x^2)^3 \, dx\)

Optimal. Leaf size=190 \[ \frac {3 c^2 (d+e x)^{10} \left (a e^2+5 c d^2\right )}{10 e^7}-\frac {4 c^2 d (d+e x)^9 \left (3 a e^2+5 c d^2\right )}{9 e^7}+\frac {3 c (d+e x)^8 \left (a e^2+c d^2\right ) \left (a e^2+5 c d^2\right )}{8 e^7}-\frac {6 c d (d+e x)^7 \left (a e^2+c d^2\right )^2}{7 e^7}+\frac {(d+e x)^6 \left (a e^2+c d^2\right )^3}{6 e^7}+\frac {c^3 (d+e x)^{12}}{12 e^7}-\frac {6 c^3 d (d+e x)^{11}}{11 e^7} \]

[Out]

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

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Rubi [A]  time = 0.26, antiderivative size = 190, 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} \[ \frac {3 c^2 (d+e x)^{10} \left (a e^2+5 c d^2\right )}{10 e^7}-\frac {4 c^2 d (d+e x)^9 \left (3 a e^2+5 c d^2\right )}{9 e^7}+\frac {3 c (d+e x)^8 \left (a e^2+c d^2\right ) \left (a e^2+5 c d^2\right )}{8 e^7}-\frac {6 c d (d+e x)^7 \left (a e^2+c d^2\right )^2}{7 e^7}+\frac {(d+e x)^6 \left (a e^2+c d^2\right )^3}{6 e^7}+\frac {c^3 (d+e x)^{12}}{12 e^7}-\frac {6 c^3 d (d+e x)^{11}}{11 e^7} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^5*(a + c*x^2)^3,x]

[Out]

((c*d^2 + a*e^2)^3*(d + e*x)^6)/(6*e^7) - (6*c*d*(c*d^2 + a*e^2)^2*(d + e*x)^7)/(7*e^7) + (3*c*(c*d^2 + a*e^2)
*(5*c*d^2 + a*e^2)*(d + e*x)^8)/(8*e^7) - (4*c^2*d*(5*c*d^2 + 3*a*e^2)*(d + e*x)^9)/(9*e^7) + (3*c^2*(5*c*d^2
+ a*e^2)*(d + e*x)^10)/(10*e^7) - (6*c^3*d*(d + e*x)^11)/(11*e^7) + (c^3*(d + e*x)^12)/(12*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)^5 \left (a+c x^2\right )^3 \, dx &=\int \left (\frac {\left (c d^2+a e^2\right )^3 (d+e x)^5}{e^6}-\frac {6 c d \left (c d^2+a e^2\right )^2 (d+e x)^6}{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)^7}{e^6}-\frac {4 c^2 d \left (5 c d^2+3 a e^2\right ) (d+e x)^8}{e^6}+\frac {3 c^2 \left (5 c d^2+a e^2\right ) (d+e x)^9}{e^6}-\frac {6 c^3 d (d+e x)^{10}}{e^6}+\frac {c^3 (d+e x)^{11}}{e^6}\right ) \, dx\\ &=\frac {\left (c d^2+a e^2\right )^3 (d+e x)^6}{6 e^7}-\frac {6 c d \left (c d^2+a e^2\right )^2 (d+e x)^7}{7 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)^8}{8 e^7}-\frac {4 c^2 d \left (5 c d^2+3 a e^2\right ) (d+e x)^9}{9 e^7}+\frac {3 c^2 \left (5 c d^2+a e^2\right ) (d+e x)^{10}}{10 e^7}-\frac {6 c^3 d (d+e x)^{11}}{11 e^7}+\frac {c^3 (d+e x)^{12}}{12 e^7}\\ \end {align*}

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Mathematica [A]  time = 0.05, size = 252, normalized size = 1.33 \[ \frac {1}{6} a^3 x \left (6 d^5+15 d^4 e x+20 d^3 e^2 x^2+15 d^2 e^3 x^3+6 d e^4 x^4+e^5 x^5\right )+a^2 c \left (d^5 x^3+\frac {15}{4} d^4 e x^4+6 d^3 e^2 x^5+5 d^2 e^3 x^6+\frac {15}{7} d e^4 x^7+\frac {3 e^5 x^8}{8}\right )+\frac {1}{420} a c^2 x^5 \left (252 d^5+1050 d^4 e x+1800 d^3 e^2 x^2+1575 d^2 e^3 x^3+700 d e^4 x^4+126 e^5 x^5\right )+\frac {c^3 x^7 \left (792 d^5+3465 d^4 e x+6160 d^3 e^2 x^2+5544 d^2 e^3 x^3+2520 d e^4 x^4+462 e^5 x^5\right )}{5544} \]

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^5*(a + c*x^2)^3,x]

[Out]

(a^3*x*(6*d^5 + 15*d^4*e*x + 20*d^3*e^2*x^2 + 15*d^2*e^3*x^3 + 6*d*e^4*x^4 + e^5*x^5))/6 + (a*c^2*x^5*(252*d^5
+ 1050*d^4*e*x + 1800*d^3*e^2*x^2 + 1575*d^2*e^3*x^3 + 700*d*e^4*x^4 + 126*e^5*x^5))/420 + (c^3*x^7*(792*d^5
+ 3465*d^4*e*x + 6160*d^3*e^2*x^2 + 5544*d^2*e^3*x^3 + 2520*d*e^4*x^4 + 462*e^5*x^5))/5544 + a^2*c*(d^5*x^3 +
(15*d^4*e*x^4)/4 + 6*d^3*e^2*x^5 + 5*d^2*e^3*x^6 + (15*d*e^4*x^7)/7 + (3*e^5*x^8)/8)

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fricas [A]  time = 0.95, size = 303, normalized size = 1.59 \[ \frac {1}{12} x^{12} e^{5} c^{3} + \frac {5}{11} x^{11} e^{4} d c^{3} + x^{10} e^{3} d^{2} c^{3} + \frac {3}{10} x^{10} e^{5} c^{2} a + \frac {10}{9} x^{9} e^{2} d^{3} c^{3} + \frac {5}{3} x^{9} e^{4} d c^{2} a + \frac {5}{8} x^{8} e d^{4} c^{3} + \frac {15}{4} x^{8} e^{3} d^{2} c^{2} a + \frac {3}{8} x^{8} e^{5} c a^{2} + \frac {1}{7} x^{7} d^{5} c^{3} + \frac {30}{7} x^{7} e^{2} d^{3} c^{2} a + \frac {15}{7} x^{7} e^{4} d c a^{2} + \frac {5}{2} x^{6} e d^{4} c^{2} a + 5 x^{6} e^{3} d^{2} c a^{2} + \frac {1}{6} x^{6} e^{5} a^{3} + \frac {3}{5} x^{5} d^{5} c^{2} a + 6 x^{5} e^{2} d^{3} c a^{2} + x^{5} e^{4} d a^{3} + \frac {15}{4} x^{4} e d^{4} c a^{2} + \frac {5}{2} x^{4} e^{3} d^{2} a^{3} + x^{3} d^{5} c a^{2} + \frac {10}{3} x^{3} e^{2} d^{3} a^{3} + \frac {5}{2} x^{2} e d^{4} a^{3} + x d^{5} a^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^5*(c*x^2+a)^3,x, algorithm="fricas")

[Out]

1/12*x^12*e^5*c^3 + 5/11*x^11*e^4*d*c^3 + x^10*e^3*d^2*c^3 + 3/10*x^10*e^5*c^2*a + 10/9*x^9*e^2*d^3*c^3 + 5/3*
x^9*e^4*d*c^2*a + 5/8*x^8*e*d^4*c^3 + 15/4*x^8*e^3*d^2*c^2*a + 3/8*x^8*e^5*c*a^2 + 1/7*x^7*d^5*c^3 + 30/7*x^7*
e^2*d^3*c^2*a + 15/7*x^7*e^4*d*c*a^2 + 5/2*x^6*e*d^4*c^2*a + 5*x^6*e^3*d^2*c*a^2 + 1/6*x^6*e^5*a^3 + 3/5*x^5*d
^5*c^2*a + 6*x^5*e^2*d^3*c*a^2 + x^5*e^4*d*a^3 + 15/4*x^4*e*d^4*c*a^2 + 5/2*x^4*e^3*d^2*a^3 + x^3*d^5*c*a^2 +
10/3*x^3*e^2*d^3*a^3 + 5/2*x^2*e*d^4*a^3 + x*d^5*a^3

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giac [A]  time = 0.17, size = 291, normalized size = 1.53 \[ \frac {1}{12} \, c^{3} x^{12} e^{5} + \frac {5}{11} \, c^{3} d x^{11} e^{4} + c^{3} d^{2} x^{10} e^{3} + \frac {10}{9} \, c^{3} d^{3} x^{9} e^{2} + \frac {5}{8} \, c^{3} d^{4} x^{8} e + \frac {1}{7} \, c^{3} d^{5} x^{7} + \frac {3}{10} \, a c^{2} x^{10} e^{5} + \frac {5}{3} \, a c^{2} d x^{9} e^{4} + \frac {15}{4} \, a c^{2} d^{2} x^{8} e^{3} + \frac {30}{7} \, a c^{2} d^{3} x^{7} e^{2} + \frac {5}{2} \, a c^{2} d^{4} x^{6} e + \frac {3}{5} \, a c^{2} d^{5} x^{5} + \frac {3}{8} \, a^{2} c x^{8} e^{5} + \frac {15}{7} \, a^{2} c d x^{7} e^{4} + 5 \, a^{2} c d^{2} x^{6} e^{3} + 6 \, a^{2} c d^{3} x^{5} e^{2} + \frac {15}{4} \, a^{2} c d^{4} x^{4} e + a^{2} c d^{5} x^{3} + \frac {1}{6} \, a^{3} x^{6} e^{5} + a^{3} d x^{5} e^{4} + \frac {5}{2} \, a^{3} d^{2} x^{4} e^{3} + \frac {10}{3} \, a^{3} d^{3} x^{3} e^{2} + \frac {5}{2} \, a^{3} d^{4} x^{2} e + a^{3} d^{5} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^5*(c*x^2+a)^3,x, algorithm="giac")

[Out]

1/12*c^3*x^12*e^5 + 5/11*c^3*d*x^11*e^4 + c^3*d^2*x^10*e^3 + 10/9*c^3*d^3*x^9*e^2 + 5/8*c^3*d^4*x^8*e + 1/7*c^
3*d^5*x^7 + 3/10*a*c^2*x^10*e^5 + 5/3*a*c^2*d*x^9*e^4 + 15/4*a*c^2*d^2*x^8*e^3 + 30/7*a*c^2*d^3*x^7*e^2 + 5/2*
a*c^2*d^4*x^6*e + 3/5*a*c^2*d^5*x^5 + 3/8*a^2*c*x^8*e^5 + 15/7*a^2*c*d*x^7*e^4 + 5*a^2*c*d^2*x^6*e^3 + 6*a^2*c
*d^3*x^5*e^2 + 15/4*a^2*c*d^4*x^4*e + a^2*c*d^5*x^3 + 1/6*a^3*x^6*e^5 + a^3*d*x^5*e^4 + 5/2*a^3*d^2*x^4*e^3 +
10/3*a^3*d^3*x^3*e^2 + 5/2*a^3*d^4*x^2*e + a^3*d^5*x

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maple [A]  time = 0.04, size = 293, normalized size = 1.54 \[ \frac {c^{3} e^{5} x^{12}}{12}+\frac {5 c^{3} d \,e^{4} x^{11}}{11}+\frac {5 a^{3} d^{4} e \,x^{2}}{2}+\frac {\left (3 e^{5} a \,c^{2}+10 d^{2} e^{3} c^{3}\right ) x^{10}}{10}+a^{3} d^{5} x +\frac {\left (15 d \,e^{4} a \,c^{2}+10 d^{3} e^{2} c^{3}\right ) x^{9}}{9}+\frac {\left (3 e^{5} a^{2} c +30 d^{2} e^{3} a \,c^{2}+5 d^{4} e \,c^{3}\right ) x^{8}}{8}+\frac {\left (15 d \,e^{4} a^{2} c +30 d^{3} e^{2} a \,c^{2}+d^{5} c^{3}\right ) x^{7}}{7}+\frac {\left (a^{3} e^{5}+30 d^{2} e^{3} a^{2} c +15 d^{4} e a \,c^{2}\right ) x^{6}}{6}+\frac {\left (5 d \,e^{4} a^{3}+30 d^{3} e^{2} a^{2} c +3 d^{5} a \,c^{2}\right ) x^{5}}{5}+\frac {\left (10 d^{2} e^{3} a^{3}+15 d^{4} e \,a^{2} c \right ) x^{4}}{4}+\frac {\left (10 d^{3} e^{2} a^{3}+3 d^{5} a^{2} c \right ) x^{3}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^5*(c*x^2+a)^3,x)

[Out]

1/12*e^5*c^3*x^12+5/11*d*e^4*c^3*x^11+1/10*(3*a*c^2*e^5+10*c^3*d^2*e^3)*x^10+1/9*(15*a*c^2*d*e^4+10*c^3*d^3*e^
2)*x^9+1/8*(3*a^2*c*e^5+30*a*c^2*d^2*e^3+5*c^3*d^4*e)*x^8+1/7*(15*a^2*c*d*e^4+30*a*c^2*d^3*e^2+c^3*d^5)*x^7+1/
6*(a^3*e^5+30*a^2*c*d^2*e^3+15*a*c^2*d^4*e)*x^6+1/5*(5*a^3*d*e^4+30*a^2*c*d^3*e^2+3*a*c^2*d^5)*x^5+1/4*(10*a^3
*d^2*e^3+15*a^2*c*d^4*e)*x^4+1/3*(10*a^3*d^3*e^2+3*a^2*c*d^5)*x^3+5/2*d^4*e*a^3*x^2+d^5*a^3*x

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maxima [A]  time = 1.32, size = 292, normalized size = 1.54 \[ \frac {1}{12} \, c^{3} e^{5} x^{12} + \frac {5}{11} \, c^{3} d e^{4} x^{11} + \frac {1}{10} \, {\left (10 \, c^{3} d^{2} e^{3} + 3 \, a c^{2} e^{5}\right )} x^{10} + \frac {5}{2} \, a^{3} d^{4} e x^{2} + \frac {5}{9} \, {\left (2 \, c^{3} d^{3} e^{2} + 3 \, a c^{2} d e^{4}\right )} x^{9} + a^{3} d^{5} x + \frac {1}{8} \, {\left (5 \, c^{3} d^{4} e + 30 \, a c^{2} d^{2} e^{3} + 3 \, a^{2} c e^{5}\right )} x^{8} + \frac {1}{7} \, {\left (c^{3} d^{5} + 30 \, a c^{2} d^{3} e^{2} + 15 \, a^{2} c d e^{4}\right )} x^{7} + \frac {1}{6} \, {\left (15 \, a c^{2} d^{4} e + 30 \, a^{2} c d^{2} e^{3} + a^{3} e^{5}\right )} x^{6} + \frac {1}{5} \, {\left (3 \, a c^{2} d^{5} + 30 \, a^{2} c d^{3} e^{2} + 5 \, a^{3} d e^{4}\right )} x^{5} + \frac {5}{4} \, {\left (3 \, a^{2} c d^{4} e + 2 \, a^{3} d^{2} e^{3}\right )} x^{4} + \frac {1}{3} \, {\left (3 \, a^{2} c d^{5} + 10 \, a^{3} d^{3} e^{2}\right )} x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^5*(c*x^2+a)^3,x, algorithm="maxima")

[Out]

1/12*c^3*e^5*x^12 + 5/11*c^3*d*e^4*x^11 + 1/10*(10*c^3*d^2*e^3 + 3*a*c^2*e^5)*x^10 + 5/2*a^3*d^4*e*x^2 + 5/9*(
2*c^3*d^3*e^2 + 3*a*c^2*d*e^4)*x^9 + a^3*d^5*x + 1/8*(5*c^3*d^4*e + 30*a*c^2*d^2*e^3 + 3*a^2*c*e^5)*x^8 + 1/7*
(c^3*d^5 + 30*a*c^2*d^3*e^2 + 15*a^2*c*d*e^4)*x^7 + 1/6*(15*a*c^2*d^4*e + 30*a^2*c*d^2*e^3 + a^3*e^5)*x^6 + 1/
5*(3*a*c^2*d^5 + 30*a^2*c*d^3*e^2 + 5*a^3*d*e^4)*x^5 + 5/4*(3*a^2*c*d^4*e + 2*a^3*d^2*e^3)*x^4 + 1/3*(3*a^2*c*
d^5 + 10*a^3*d^3*e^2)*x^3

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mupad [B]  time = 0.36, size = 281, normalized size = 1.48 \[ x^5\,\left (a^3\,d\,e^4+6\,a^2\,c\,d^3\,e^2+\frac {3\,a\,c^2\,d^5}{5}\right )+x^6\,\left (\frac {a^3\,e^5}{6}+5\,a^2\,c\,d^2\,e^3+\frac {5\,a\,c^2\,d^4\,e}{2}\right )+x^7\,\left (\frac {15\,a^2\,c\,d\,e^4}{7}+\frac {30\,a\,c^2\,d^3\,e^2}{7}+\frac {c^3\,d^5}{7}\right )+x^8\,\left (\frac {3\,a^2\,c\,e^5}{8}+\frac {15\,a\,c^2\,d^2\,e^3}{4}+\frac {5\,c^3\,d^4\,e}{8}\right )+x^3\,\left (\frac {10\,a^3\,d^3\,e^2}{3}+c\,a^2\,d^5\right )+x^{10}\,\left (c^3\,d^2\,e^3+\frac {3\,a\,c^2\,e^5}{10}\right )+a^3\,d^5\,x+\frac {c^3\,e^5\,x^{12}}{12}+\frac {5\,a^3\,d^4\,e\,x^2}{2}+\frac {5\,c^3\,d\,e^4\,x^{11}}{11}+\frac {5\,a^2\,d^2\,e\,x^4\,\left (3\,c\,d^2+2\,a\,e^2\right )}{4}+\frac {5\,c^2\,d\,e^2\,x^9\,\left (2\,c\,d^2+3\,a\,e^2\right )}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + c*x^2)^3*(d + e*x)^5,x)

[Out]

x^5*((3*a*c^2*d^5)/5 + a^3*d*e^4 + 6*a^2*c*d^3*e^2) + x^6*((a^3*e^5)/6 + 5*a^2*c*d^2*e^3 + (5*a*c^2*d^4*e)/2)
+ x^7*((c^3*d^5)/7 + (30*a*c^2*d^3*e^2)/7 + (15*a^2*c*d*e^4)/7) + x^8*((3*a^2*c*e^5)/8 + (5*c^3*d^4*e)/8 + (15
*a*c^2*d^2*e^3)/4) + x^3*(a^2*c*d^5 + (10*a^3*d^3*e^2)/3) + x^10*((3*a*c^2*e^5)/10 + c^3*d^2*e^3) + a^3*d^5*x
+ (c^3*e^5*x^12)/12 + (5*a^3*d^4*e*x^2)/2 + (5*c^3*d*e^4*x^11)/11 + (5*a^2*d^2*e*x^4*(2*a*e^2 + 3*c*d^2))/4 +
(5*c^2*d*e^2*x^9*(3*a*e^2 + 2*c*d^2))/9

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sympy [A]  time = 0.12, size = 321, normalized size = 1.69 \[ a^{3} d^{5} x + \frac {5 a^{3} d^{4} e x^{2}}{2} + \frac {5 c^{3} d e^{4} x^{11}}{11} + \frac {c^{3} e^{5} x^{12}}{12} + x^{10} \left (\frac {3 a c^{2} e^{5}}{10} + c^{3} d^{2} e^{3}\right ) + x^{9} \left (\frac {5 a c^{2} d e^{4}}{3} + \frac {10 c^{3} d^{3} e^{2}}{9}\right ) + x^{8} \left (\frac {3 a^{2} c e^{5}}{8} + \frac {15 a c^{2} d^{2} e^{3}}{4} + \frac {5 c^{3} d^{4} e}{8}\right ) + x^{7} \left (\frac {15 a^{2} c d e^{4}}{7} + \frac {30 a c^{2} d^{3} e^{2}}{7} + \frac {c^{3} d^{5}}{7}\right ) + x^{6} \left (\frac {a^{3} e^{5}}{6} + 5 a^{2} c d^{2} e^{3} + \frac {5 a c^{2} d^{4} e}{2}\right ) + x^{5} \left (a^{3} d e^{4} + 6 a^{2} c d^{3} e^{2} + \frac {3 a c^{2} d^{5}}{5}\right ) + x^{4} \left (\frac {5 a^{3} d^{2} e^{3}}{2} + \frac {15 a^{2} c d^{4} e}{4}\right ) + x^{3} \left (\frac {10 a^{3} d^{3} e^{2}}{3} + a^{2} c d^{5}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**5*(c*x**2+a)**3,x)

[Out]

a**3*d**5*x + 5*a**3*d**4*e*x**2/2 + 5*c**3*d*e**4*x**11/11 + c**3*e**5*x**12/12 + x**10*(3*a*c**2*e**5/10 + c
**3*d**2*e**3) + x**9*(5*a*c**2*d*e**4/3 + 10*c**3*d**3*e**2/9) + x**8*(3*a**2*c*e**5/8 + 15*a*c**2*d**2*e**3/
4 + 5*c**3*d**4*e/8) + x**7*(15*a**2*c*d*e**4/7 + 30*a*c**2*d**3*e**2/7 + c**3*d**5/7) + x**6*(a**3*e**5/6 + 5
*a**2*c*d**2*e**3 + 5*a*c**2*d**4*e/2) + x**5*(a**3*d*e**4 + 6*a**2*c*d**3*e**2 + 3*a*c**2*d**5/5) + x**4*(5*a
**3*d**2*e**3/2 + 15*a**2*c*d**4*e/4) + x**3*(10*a**3*d**3*e**2/3 + a**2*c*d**5)

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