3.5.8 \(\int (d+e x)^7 (a+c x^2)^4 \, dx\)

Optimal. Leaf size=278 \[ \frac {c^2 (d+e x)^{12} \left (3 a^2 e^4+30 a c d^2 e^2+35 c^2 d^4\right )}{6 e^9}+\frac {2 c^3 (d+e x)^{14} \left (a e^2+7 c d^2\right )}{7 e^9}-\frac {8 c^3 d (d+e x)^{13} \left (3 a e^2+7 c d^2\right )}{13 e^9}-\frac {8 c^2 d (d+e x)^{11} \left (a e^2+c d^2\right ) \left (3 a e^2+7 c d^2\right )}{11 e^9}+\frac {2 c (d+e x)^{10} \left (a e^2+c d^2\right )^2 \left (a e^2+7 c d^2\right )}{5 e^9}-\frac {8 c d (d+e x)^9 \left (a e^2+c d^2\right )^3}{9 e^9}+\frac {(d+e x)^8 \left (a e^2+c d^2\right )^4}{8 e^9}+\frac {c^4 (d+e x)^{16}}{16 e^9}-\frac {8 c^4 d (d+e x)^{15}}{15 e^9} \]

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Rubi [A]  time = 0.52, antiderivative size = 278, 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)^{12} \left (3 a^2 e^4+30 a c d^2 e^2+35 c^2 d^4\right )}{6 e^9}+\frac {2 c^3 (d+e x)^{14} \left (a e^2+7 c d^2\right )}{7 e^9}-\frac {8 c^3 d (d+e x)^{13} \left (3 a e^2+7 c d^2\right )}{13 e^9}-\frac {8 c^2 d (d+e x)^{11} \left (a e^2+c d^2\right ) \left (3 a e^2+7 c d^2\right )}{11 e^9}+\frac {2 c (d+e x)^{10} \left (a e^2+c d^2\right )^2 \left (a e^2+7 c d^2\right )}{5 e^9}-\frac {8 c d (d+e x)^9 \left (a e^2+c d^2\right )^3}{9 e^9}+\frac {(d+e x)^8 \left (a e^2+c d^2\right )^4}{8 e^9}+\frac {c^4 (d+e x)^{16}}{16 e^9}-\frac {8 c^4 d (d+e x)^{15}}{15 e^9} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^7*(a + c*x^2)^4,x]

[Out]

((c*d^2 + a*e^2)^4*(d + e*x)^8)/(8*e^9) - (8*c*d*(c*d^2 + a*e^2)^3*(d + e*x)^9)/(9*e^9) + (2*c*(c*d^2 + a*e^2)
^2*(7*c*d^2 + a*e^2)*(d + e*x)^10)/(5*e^9) - (8*c^2*d*(c*d^2 + a*e^2)*(7*c*d^2 + 3*a*e^2)*(d + e*x)^11)/(11*e^
9) + (c^2*(35*c^2*d^4 + 30*a*c*d^2*e^2 + 3*a^2*e^4)*(d + e*x)^12)/(6*e^9) - (8*c^3*d*(7*c*d^2 + 3*a*e^2)*(d +
e*x)^13)/(13*e^9) + (2*c^3*(7*c*d^2 + a*e^2)*(d + e*x)^14)/(7*e^9) - (8*c^4*d*(d + e*x)^15)/(15*e^9) + (c^4*(d
+ e*x)^16)/(16*e^9)

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)^7 \left (a+c x^2\right )^4 \, dx &=\int \left (\frac {\left (c d^2+a e^2\right )^4 (d+e x)^7}{e^8}-\frac {8 c d \left (c d^2+a e^2\right )^3 (d+e x)^8}{e^8}+\frac {4 c \left (c d^2+a e^2\right )^2 \left (7 c d^2+a e^2\right ) (d+e x)^9}{e^8}+\frac {8 c^2 d \left (-7 c d^2-3 a e^2\right ) \left (c d^2+a e^2\right ) (d+e x)^{10}}{e^8}+\frac {2 c^2 \left (35 c^2 d^4+30 a c d^2 e^2+3 a^2 e^4\right ) (d+e x)^{11}}{e^8}-\frac {8 c^3 d \left (7 c d^2+3 a e^2\right ) (d+e x)^{12}}{e^8}+\frac {4 c^3 \left (7 c d^2+a e^2\right ) (d+e x)^{13}}{e^8}-\frac {8 c^4 d (d+e x)^{14}}{e^8}+\frac {c^4 (d+e x)^{15}}{e^8}\right ) \, dx\\ &=\frac {\left (c d^2+a e^2\right )^4 (d+e x)^8}{8 e^9}-\frac {8 c d \left (c d^2+a e^2\right )^3 (d+e x)^9}{9 e^9}+\frac {2 c \left (c d^2+a e^2\right )^2 \left (7 c d^2+a e^2\right ) (d+e x)^{10}}{5 e^9}-\frac {8 c^2 d \left (c d^2+a e^2\right ) \left (7 c d^2+3 a e^2\right ) (d+e x)^{11}}{11 e^9}+\frac {c^2 \left (35 c^2 d^4+30 a c d^2 e^2+3 a^2 e^4\right ) (d+e x)^{12}}{6 e^9}-\frac {8 c^3 d \left (7 c d^2+3 a e^2\right ) (d+e x)^{13}}{13 e^9}+\frac {2 c^3 \left (7 c d^2+a e^2\right ) (d+e x)^{14}}{7 e^9}-\frac {8 c^4 d (d+e x)^{15}}{15 e^9}+\frac {c^4 (d+e x)^{16}}{16 e^9}\\ \end {align*}

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Mathematica [A]  time = 0.08, size = 423, normalized size = 1.52 \begin {gather*} \frac {1}{8} a^4 x \left (8 d^7+28 d^6 e x+56 d^5 e^2 x^2+70 d^4 e^3 x^3+56 d^3 e^4 x^4+28 d^2 e^5 x^5+8 d e^6 x^6+e^7 x^7\right )+\frac {1}{90} a^3 c x^3 \left (120 d^7+630 d^6 e x+1512 d^5 e^2 x^2+2100 d^4 e^3 x^3+1800 d^3 e^4 x^4+945 d^2 e^5 x^5+280 d e^6 x^6+36 e^7 x^7\right )+\frac {1}{660} a^2 c^2 x^5 \left (792 d^7+4620 d^6 e x+11880 d^5 e^2 x^2+17325 d^4 e^3 x^3+15400 d^3 e^4 x^4+8316 d^2 e^5 x^5+2520 d e^6 x^6+330 e^7 x^7\right )+\frac {a c^3 x^7 \left (3432 d^7+21021 d^6 e x+56056 d^5 e^2 x^2+84084 d^4 e^3 x^3+76440 d^3 e^4 x^4+42042 d^2 e^5 x^5+12936 d e^6 x^6+1716 e^7 x^7\right )}{6006}+\frac {c^4 x^9 \left (11440 d^7+72072 d^6 e x+196560 d^5 e^2 x^2+300300 d^4 e^3 x^3+277200 d^3 e^4 x^4+154440 d^2 e^5 x^5+48048 d e^6 x^6+6435 e^7 x^7\right )}{102960} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^7*(a + c*x^2)^4,x]

[Out]

(a^4*x*(8*d^7 + 28*d^6*e*x + 56*d^5*e^2*x^2 + 70*d^4*e^3*x^3 + 56*d^3*e^4*x^4 + 28*d^2*e^5*x^5 + 8*d*e^6*x^6 +
e^7*x^7))/8 + (a^3*c*x^3*(120*d^7 + 630*d^6*e*x + 1512*d^5*e^2*x^2 + 2100*d^4*e^3*x^3 + 1800*d^3*e^4*x^4 + 94
5*d^2*e^5*x^5 + 280*d*e^6*x^6 + 36*e^7*x^7))/90 + (a^2*c^2*x^5*(792*d^7 + 4620*d^6*e*x + 11880*d^5*e^2*x^2 + 1
7325*d^4*e^3*x^3 + 15400*d^3*e^4*x^4 + 8316*d^2*e^5*x^5 + 2520*d*e^6*x^6 + 330*e^7*x^7))/660 + (a*c^3*x^7*(343
2*d^7 + 21021*d^6*e*x + 56056*d^5*e^2*x^2 + 84084*d^4*e^3*x^3 + 76440*d^3*e^4*x^4 + 42042*d^2*e^5*x^5 + 12936*
d*e^6*x^6 + 1716*e^7*x^7))/6006 + (c^4*x^9*(11440*d^7 + 72072*d^6*e*x + 196560*d^5*e^2*x^2 + 300300*d^4*e^3*x^
3 + 277200*d^3*e^4*x^4 + 154440*d^2*e^5*x^5 + 48048*d*e^6*x^6 + 6435*e^7*x^7))/102960

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int (d+e x)^7 \left (a+c x^2\right )^4 \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(d + e*x)^7*(a + c*x^2)^4,x]

[Out]

IntegrateAlgebraic[(d + e*x)^7*(a + c*x^2)^4, x]

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fricas [B]  time = 0.36, size = 547, normalized size = 1.97 \begin {gather*} \frac {1}{16} x^{16} e^{7} c^{4} + \frac {7}{15} x^{15} e^{6} d c^{4} + \frac {3}{2} x^{14} e^{5} d^{2} c^{4} + \frac {2}{7} x^{14} e^{7} c^{3} a + \frac {35}{13} x^{13} e^{4} d^{3} c^{4} + \frac {28}{13} x^{13} e^{6} d c^{3} a + \frac {35}{12} x^{12} e^{3} d^{4} c^{4} + 7 x^{12} e^{5} d^{2} c^{3} a + \frac {1}{2} x^{12} e^{7} c^{2} a^{2} + \frac {21}{11} x^{11} e^{2} d^{5} c^{4} + \frac {140}{11} x^{11} e^{4} d^{3} c^{3} a + \frac {42}{11} x^{11} e^{6} d c^{2} a^{2} + \frac {7}{10} x^{10} e d^{6} c^{4} + 14 x^{10} e^{3} d^{4} c^{3} a + \frac {63}{5} x^{10} e^{5} d^{2} c^{2} a^{2} + \frac {2}{5} x^{10} e^{7} c a^{3} + \frac {1}{9} x^{9} d^{7} c^{4} + \frac {28}{3} x^{9} e^{2} d^{5} c^{3} a + \frac {70}{3} x^{9} e^{4} d^{3} c^{2} a^{2} + \frac {28}{9} x^{9} e^{6} d c a^{3} + \frac {7}{2} x^{8} e d^{6} c^{3} a + \frac {105}{4} x^{8} e^{3} d^{4} c^{2} a^{2} + \frac {21}{2} x^{8} e^{5} d^{2} c a^{3} + \frac {1}{8} x^{8} e^{7} a^{4} + \frac {4}{7} x^{7} d^{7} c^{3} a + 18 x^{7} e^{2} d^{5} c^{2} a^{2} + 20 x^{7} e^{4} d^{3} c a^{3} + x^{7} e^{6} d a^{4} + 7 x^{6} e d^{6} c^{2} a^{2} + \frac {70}{3} x^{6} e^{3} d^{4} c a^{3} + \frac {7}{2} x^{6} e^{5} d^{2} a^{4} + \frac {6}{5} x^{5} d^{7} c^{2} a^{2} + \frac {84}{5} x^{5} e^{2} d^{5} c a^{3} + 7 x^{5} e^{4} d^{3} a^{4} + 7 x^{4} e d^{6} c a^{3} + \frac {35}{4} x^{4} e^{3} d^{4} a^{4} + \frac {4}{3} x^{3} d^{7} c a^{3} + 7 x^{3} e^{2} d^{5} a^{4} + \frac {7}{2} x^{2} e d^{6} a^{4} + x d^{7} a^{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^7*(c*x^2+a)^4,x, algorithm="fricas")

[Out]

1/16*x^16*e^7*c^4 + 7/15*x^15*e^6*d*c^4 + 3/2*x^14*e^5*d^2*c^4 + 2/7*x^14*e^7*c^3*a + 35/13*x^13*e^4*d^3*c^4 +
28/13*x^13*e^6*d*c^3*a + 35/12*x^12*e^3*d^4*c^4 + 7*x^12*e^5*d^2*c^3*a + 1/2*x^12*e^7*c^2*a^2 + 21/11*x^11*e^
2*d^5*c^4 + 140/11*x^11*e^4*d^3*c^3*a + 42/11*x^11*e^6*d*c^2*a^2 + 7/10*x^10*e*d^6*c^4 + 14*x^10*e^3*d^4*c^3*a
+ 63/5*x^10*e^5*d^2*c^2*a^2 + 2/5*x^10*e^7*c*a^3 + 1/9*x^9*d^7*c^4 + 28/3*x^9*e^2*d^5*c^3*a + 70/3*x^9*e^4*d^
3*c^2*a^2 + 28/9*x^9*e^6*d*c*a^3 + 7/2*x^8*e*d^6*c^3*a + 105/4*x^8*e^3*d^4*c^2*a^2 + 21/2*x^8*e^5*d^2*c*a^3 +
1/8*x^8*e^7*a^4 + 4/7*x^7*d^7*c^3*a + 18*x^7*e^2*d^5*c^2*a^2 + 20*x^7*e^4*d^3*c*a^3 + x^7*e^6*d*a^4 + 7*x^6*e*
d^6*c^2*a^2 + 70/3*x^6*e^3*d^4*c*a^3 + 7/2*x^6*e^5*d^2*a^4 + 6/5*x^5*d^7*c^2*a^2 + 84/5*x^5*e^2*d^5*c*a^3 + 7*
x^5*e^4*d^3*a^4 + 7*x^4*e*d^6*c*a^3 + 35/4*x^4*e^3*d^4*a^4 + 4/3*x^3*d^7*c*a^3 + 7*x^3*e^2*d^5*a^4 + 7/2*x^2*e
*d^6*a^4 + x*d^7*a^4

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giac [B]  time = 0.22, size = 522, normalized size = 1.88 \begin {gather*} \frac {1}{16} \, c^{4} x^{16} e^{7} + \frac {7}{15} \, c^{4} d x^{15} e^{6} + \frac {3}{2} \, c^{4} d^{2} x^{14} e^{5} + \frac {35}{13} \, c^{4} d^{3} x^{13} e^{4} + \frac {35}{12} \, c^{4} d^{4} x^{12} e^{3} + \frac {21}{11} \, c^{4} d^{5} x^{11} e^{2} + \frac {7}{10} \, c^{4} d^{6} x^{10} e + \frac {1}{9} \, c^{4} d^{7} x^{9} + \frac {2}{7} \, a c^{3} x^{14} e^{7} + \frac {28}{13} \, a c^{3} d x^{13} e^{6} + 7 \, a c^{3} d^{2} x^{12} e^{5} + \frac {140}{11} \, a c^{3} d^{3} x^{11} e^{4} + 14 \, a c^{3} d^{4} x^{10} e^{3} + \frac {28}{3} \, a c^{3} d^{5} x^{9} e^{2} + \frac {7}{2} \, a c^{3} d^{6} x^{8} e + \frac {4}{7} \, a c^{3} d^{7} x^{7} + \frac {1}{2} \, a^{2} c^{2} x^{12} e^{7} + \frac {42}{11} \, a^{2} c^{2} d x^{11} e^{6} + \frac {63}{5} \, a^{2} c^{2} d^{2} x^{10} e^{5} + \frac {70}{3} \, a^{2} c^{2} d^{3} x^{9} e^{4} + \frac {105}{4} \, a^{2} c^{2} d^{4} x^{8} e^{3} + 18 \, a^{2} c^{2} d^{5} x^{7} e^{2} + 7 \, a^{2} c^{2} d^{6} x^{6} e + \frac {6}{5} \, a^{2} c^{2} d^{7} x^{5} + \frac {2}{5} \, a^{3} c x^{10} e^{7} + \frac {28}{9} \, a^{3} c d x^{9} e^{6} + \frac {21}{2} \, a^{3} c d^{2} x^{8} e^{5} + 20 \, a^{3} c d^{3} x^{7} e^{4} + \frac {70}{3} \, a^{3} c d^{4} x^{6} e^{3} + \frac {84}{5} \, a^{3} c d^{5} x^{5} e^{2} + 7 \, a^{3} c d^{6} x^{4} e + \frac {4}{3} \, a^{3} c d^{7} x^{3} + \frac {1}{8} \, a^{4} x^{8} e^{7} + a^{4} d x^{7} e^{6} + \frac {7}{2} \, a^{4} d^{2} x^{6} e^{5} + 7 \, a^{4} d^{3} x^{5} e^{4} + \frac {35}{4} \, a^{4} d^{4} x^{4} e^{3} + 7 \, a^{4} d^{5} x^{3} e^{2} + \frac {7}{2} \, a^{4} d^{6} x^{2} e + a^{4} d^{7} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^7*(c*x^2+a)^4,x, algorithm="giac")

[Out]

1/16*c^4*x^16*e^7 + 7/15*c^4*d*x^15*e^6 + 3/2*c^4*d^2*x^14*e^5 + 35/13*c^4*d^3*x^13*e^4 + 35/12*c^4*d^4*x^12*e
^3 + 21/11*c^4*d^5*x^11*e^2 + 7/10*c^4*d^6*x^10*e + 1/9*c^4*d^7*x^9 + 2/7*a*c^3*x^14*e^7 + 28/13*a*c^3*d*x^13*
e^6 + 7*a*c^3*d^2*x^12*e^5 + 140/11*a*c^3*d^3*x^11*e^4 + 14*a*c^3*d^4*x^10*e^3 + 28/3*a*c^3*d^5*x^9*e^2 + 7/2*
a*c^3*d^6*x^8*e + 4/7*a*c^3*d^7*x^7 + 1/2*a^2*c^2*x^12*e^7 + 42/11*a^2*c^2*d*x^11*e^6 + 63/5*a^2*c^2*d^2*x^10*
e^5 + 70/3*a^2*c^2*d^3*x^9*e^4 + 105/4*a^2*c^2*d^4*x^8*e^3 + 18*a^2*c^2*d^5*x^7*e^2 + 7*a^2*c^2*d^6*x^6*e + 6/
5*a^2*c^2*d^7*x^5 + 2/5*a^3*c*x^10*e^7 + 28/9*a^3*c*d*x^9*e^6 + 21/2*a^3*c*d^2*x^8*e^5 + 20*a^3*c*d^3*x^7*e^4
+ 70/3*a^3*c*d^4*x^6*e^3 + 84/5*a^3*c*d^5*x^5*e^2 + 7*a^3*c*d^6*x^4*e + 4/3*a^3*c*d^7*x^3 + 1/8*a^4*x^8*e^7 +
a^4*d*x^7*e^6 + 7/2*a^4*d^2*x^6*e^5 + 7*a^4*d^3*x^5*e^4 + 35/4*a^4*d^4*x^4*e^3 + 7*a^4*d^5*x^3*e^2 + 7/2*a^4*d
^6*x^2*e + a^4*d^7*x

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maple [A]  time = 0.04, size = 511, normalized size = 1.84 \begin {gather*} \frac {c^{4} e^{7} x^{16}}{16}+\frac {7 c^{4} d \,e^{6} x^{15}}{15}+\frac {\left (4 e^{7} a \,c^{3}+21 d^{2} e^{5} c^{4}\right ) x^{14}}{14}+\frac {7 a^{4} d^{6} e \,x^{2}}{2}+\frac {\left (28 d \,e^{6} a \,c^{3}+35 d^{3} e^{4} c^{4}\right ) x^{13}}{13}+a^{4} d^{7} x +\frac {\left (6 e^{7} a^{2} c^{2}+84 d^{2} e^{5} a \,c^{3}+35 d^{4} e^{3} c^{4}\right ) x^{12}}{12}+\frac {\left (42 d \,e^{6} a^{2} c^{2}+140 d^{3} e^{4} a \,c^{3}+21 d^{5} e^{2} c^{4}\right ) x^{11}}{11}+\frac {\left (4 e^{7} a^{3} c +126 d^{2} e^{5} a^{2} c^{2}+140 d^{4} e^{3} a \,c^{3}+7 d^{6} e \,c^{4}\right ) x^{10}}{10}+\frac {\left (28 d \,e^{6} a^{3} c +210 d^{3} e^{4} a^{2} c^{2}+84 d^{5} e^{2} a \,c^{3}+d^{7} c^{4}\right ) x^{9}}{9}+\frac {\left (e^{7} a^{4}+84 d^{2} e^{5} a^{3} c +210 d^{4} e^{3} a^{2} c^{2}+28 d^{6} e a \,c^{3}\right ) x^{8}}{8}+\frac {\left (7 d \,e^{6} a^{4}+140 d^{3} e^{4} a^{3} c +126 d^{5} e^{2} a^{2} c^{2}+4 d^{7} a \,c^{3}\right ) x^{7}}{7}+\frac {\left (21 d^{2} e^{5} a^{4}+140 d^{4} e^{3} a^{3} c +42 d^{6} e \,a^{2} c^{2}\right ) x^{6}}{6}+\frac {\left (35 d^{3} e^{4} a^{4}+84 d^{5} e^{2} a^{3} c +6 d^{7} a^{2} c^{2}\right ) x^{5}}{5}+\frac {\left (35 d^{4} e^{3} a^{4}+28 d^{6} e \,a^{3} c \right ) x^{4}}{4}+\frac {\left (21 d^{5} e^{2} a^{4}+4 d^{7} a^{3} c \right ) x^{3}}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^7*(c*x^2+a)^4,x)

[Out]

1/16*e^7*c^4*x^16+7/15*d*e^6*c^4*x^15+1/14*(4*a*c^3*e^7+21*c^4*d^2*e^5)*x^14+1/13*(28*a*c^3*d*e^6+35*c^4*d^3*e
^4)*x^13+1/12*(6*a^2*c^2*e^7+84*a*c^3*d^2*e^5+35*c^4*d^4*e^3)*x^12+1/11*(42*a^2*c^2*d*e^6+140*a*c^3*d^3*e^4+21
*c^4*d^5*e^2)*x^11+1/10*(4*a^3*c*e^7+126*a^2*c^2*d^2*e^5+140*a*c^3*d^4*e^3+7*c^4*d^6*e)*x^10+1/9*(28*a^3*c*d*e
^6+210*a^2*c^2*d^3*e^4+84*a*c^3*d^5*e^2+c^4*d^7)*x^9+1/8*(a^4*e^7+84*a^3*c*d^2*e^5+210*a^2*c^2*d^4*e^3+28*a*c^
3*d^6*e)*x^8+1/7*(7*a^4*d*e^6+140*a^3*c*d^3*e^4+126*a^2*c^2*d^5*e^2+4*a*c^3*d^7)*x^7+1/6*(21*a^4*d^2*e^5+140*a
^3*c*d^4*e^3+42*a^2*c^2*d^6*e)*x^6+1/5*(35*a^4*d^3*e^4+84*a^3*c*d^5*e^2+6*a^2*c^2*d^7)*x^5+1/4*(35*a^4*d^4*e^3
+28*a^3*c*d^6*e)*x^4+1/3*(21*a^4*d^5*e^2+4*a^3*c*d^7)*x^3+7/2*d^6*e*a^4*x^2+d^7*a^4*x

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maxima [A]  time = 1.38, size = 510, normalized size = 1.83 \begin {gather*} \frac {1}{16} \, c^{4} e^{7} x^{16} + \frac {7}{15} \, c^{4} d e^{6} x^{15} + \frac {1}{14} \, {\left (21 \, c^{4} d^{2} e^{5} + 4 \, a c^{3} e^{7}\right )} x^{14} + \frac {7}{13} \, {\left (5 \, c^{4} d^{3} e^{4} + 4 \, a c^{3} d e^{6}\right )} x^{13} + \frac {7}{2} \, a^{4} d^{6} e x^{2} + \frac {1}{12} \, {\left (35 \, c^{4} d^{4} e^{3} + 84 \, a c^{3} d^{2} e^{5} + 6 \, a^{2} c^{2} e^{7}\right )} x^{12} + a^{4} d^{7} x + \frac {7}{11} \, {\left (3 \, c^{4} d^{5} e^{2} + 20 \, a c^{3} d^{3} e^{4} + 6 \, a^{2} c^{2} d e^{6}\right )} x^{11} + \frac {1}{10} \, {\left (7 \, c^{4} d^{6} e + 140 \, a c^{3} d^{4} e^{3} + 126 \, a^{2} c^{2} d^{2} e^{5} + 4 \, a^{3} c e^{7}\right )} x^{10} + \frac {1}{9} \, {\left (c^{4} d^{7} + 84 \, a c^{3} d^{5} e^{2} + 210 \, a^{2} c^{2} d^{3} e^{4} + 28 \, a^{3} c d e^{6}\right )} x^{9} + \frac {1}{8} \, {\left (28 \, a c^{3} d^{6} e + 210 \, a^{2} c^{2} d^{4} e^{3} + 84 \, a^{3} c d^{2} e^{5} + a^{4} e^{7}\right )} x^{8} + \frac {1}{7} \, {\left (4 \, a c^{3} d^{7} + 126 \, a^{2} c^{2} d^{5} e^{2} + 140 \, a^{3} c d^{3} e^{4} + 7 \, a^{4} d e^{6}\right )} x^{7} + \frac {7}{6} \, {\left (6 \, a^{2} c^{2} d^{6} e + 20 \, a^{3} c d^{4} e^{3} + 3 \, a^{4} d^{2} e^{5}\right )} x^{6} + \frac {1}{5} \, {\left (6 \, a^{2} c^{2} d^{7} + 84 \, a^{3} c d^{5} e^{2} + 35 \, a^{4} d^{3} e^{4}\right )} x^{5} + \frac {7}{4} \, {\left (4 \, a^{3} c d^{6} e + 5 \, a^{4} d^{4} e^{3}\right )} x^{4} + \frac {1}{3} \, {\left (4 \, a^{3} c d^{7} + 21 \, a^{4} d^{5} e^{2}\right )} x^{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^7*(c*x^2+a)^4,x, algorithm="maxima")

[Out]

1/16*c^4*e^7*x^16 + 7/15*c^4*d*e^6*x^15 + 1/14*(21*c^4*d^2*e^5 + 4*a*c^3*e^7)*x^14 + 7/13*(5*c^4*d^3*e^4 + 4*a
*c^3*d*e^6)*x^13 + 7/2*a^4*d^6*e*x^2 + 1/12*(35*c^4*d^4*e^3 + 84*a*c^3*d^2*e^5 + 6*a^2*c^2*e^7)*x^12 + a^4*d^7
*x + 7/11*(3*c^4*d^5*e^2 + 20*a*c^3*d^3*e^4 + 6*a^2*c^2*d*e^6)*x^11 + 1/10*(7*c^4*d^6*e + 140*a*c^3*d^4*e^3 +
126*a^2*c^2*d^2*e^5 + 4*a^3*c*e^7)*x^10 + 1/9*(c^4*d^7 + 84*a*c^3*d^5*e^2 + 210*a^2*c^2*d^3*e^4 + 28*a^3*c*d*e
^6)*x^9 + 1/8*(28*a*c^3*d^6*e + 210*a^2*c^2*d^4*e^3 + 84*a^3*c*d^2*e^5 + a^4*e^7)*x^8 + 1/7*(4*a*c^3*d^7 + 126
*a^2*c^2*d^5*e^2 + 140*a^3*c*d^3*e^4 + 7*a^4*d*e^6)*x^7 + 7/6*(6*a^2*c^2*d^6*e + 20*a^3*c*d^4*e^3 + 3*a^4*d^2*
e^5)*x^6 + 1/5*(6*a^2*c^2*d^7 + 84*a^3*c*d^5*e^2 + 35*a^4*d^3*e^4)*x^5 + 7/4*(4*a^3*c*d^6*e + 5*a^4*d^4*e^3)*x
^4 + 1/3*(4*a^3*c*d^7 + 21*a^4*d^5*e^2)*x^3

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mupad [B]  time = 0.49, size = 495, normalized size = 1.78 \begin {gather*} x^3\,\left (7\,a^4\,d^5\,e^2+\frac {4\,c\,a^3\,d^7}{3}\right )+x^{14}\,\left (\frac {3\,c^4\,d^2\,e^5}{2}+\frac {2\,a\,c^3\,e^7}{7}\right )+x^7\,\left (a^4\,d\,e^6+20\,a^3\,c\,d^3\,e^4+18\,a^2\,c^2\,d^5\,e^2+\frac {4\,a\,c^3\,d^7}{7}\right )+x^8\,\left (\frac {a^4\,e^7}{8}+\frac {21\,a^3\,c\,d^2\,e^5}{2}+\frac {105\,a^2\,c^2\,d^4\,e^3}{4}+\frac {7\,a\,c^3\,d^6\,e}{2}\right )+x^9\,\left (\frac {28\,a^3\,c\,d\,e^6}{9}+\frac {70\,a^2\,c^2\,d^3\,e^4}{3}+\frac {28\,a\,c^3\,d^5\,e^2}{3}+\frac {c^4\,d^7}{9}\right )+x^{10}\,\left (\frac {2\,a^3\,c\,e^7}{5}+\frac {63\,a^2\,c^2\,d^2\,e^5}{5}+14\,a\,c^3\,d^4\,e^3+\frac {7\,c^4\,d^6\,e}{10}\right )+x^5\,\left (7\,a^4\,d^3\,e^4+\frac {84\,a^3\,c\,d^5\,e^2}{5}+\frac {6\,a^2\,c^2\,d^7}{5}\right )+x^{12}\,\left (\frac {a^2\,c^2\,e^7}{2}+7\,a\,c^3\,d^2\,e^5+\frac {35\,c^4\,d^4\,e^3}{12}\right )+a^4\,d^7\,x+\frac {c^4\,e^7\,x^{16}}{16}+\frac {7\,a^4\,d^6\,e\,x^2}{2}+\frac {7\,c^4\,d\,e^6\,x^{15}}{15}+\frac {7\,a^3\,d^4\,e\,x^4\,\left (4\,c\,d^2+5\,a\,e^2\right )}{4}+\frac {7\,c^3\,d\,e^4\,x^{13}\,\left (5\,c\,d^2+4\,a\,e^2\right )}{13}+\frac {7\,a^2\,d^2\,e\,x^6\,\left (3\,a^2\,e^4+20\,a\,c\,d^2\,e^2+6\,c^2\,d^4\right )}{6}+\frac {7\,c^2\,d\,e^2\,x^{11}\,\left (6\,a^2\,e^4+20\,a\,c\,d^2\,e^2+3\,c^2\,d^4\right )}{11} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + c*x^2)^4*(d + e*x)^7,x)

[Out]

x^3*((4*a^3*c*d^7)/3 + 7*a^4*d^5*e^2) + x^14*((2*a*c^3*e^7)/7 + (3*c^4*d^2*e^5)/2) + x^7*((4*a*c^3*d^7)/7 + a^
4*d*e^6 + 20*a^3*c*d^3*e^4 + 18*a^2*c^2*d^5*e^2) + x^8*((a^4*e^7)/8 + (21*a^3*c*d^2*e^5)/2 + (105*a^2*c^2*d^4*
e^3)/4 + (7*a*c^3*d^6*e)/2) + x^9*((c^4*d^7)/9 + (28*a*c^3*d^5*e^2)/3 + (70*a^2*c^2*d^3*e^4)/3 + (28*a^3*c*d*e
^6)/9) + x^10*((2*a^3*c*e^7)/5 + (7*c^4*d^6*e)/10 + 14*a*c^3*d^4*e^3 + (63*a^2*c^2*d^2*e^5)/5) + x^5*((6*a^2*c
^2*d^7)/5 + 7*a^4*d^3*e^4 + (84*a^3*c*d^5*e^2)/5) + x^12*((a^2*c^2*e^7)/2 + (35*c^4*d^4*e^3)/12 + 7*a*c^3*d^2*
e^5) + a^4*d^7*x + (c^4*e^7*x^16)/16 + (7*a^4*d^6*e*x^2)/2 + (7*c^4*d*e^6*x^15)/15 + (7*a^3*d^4*e*x^4*(5*a*e^2
+ 4*c*d^2))/4 + (7*c^3*d*e^4*x^13*(4*a*e^2 + 5*c*d^2))/13 + (7*a^2*d^2*e*x^6*(3*a^2*e^4 + 6*c^2*d^4 + 20*a*c*
d^2*e^2))/6 + (7*c^2*d*e^2*x^11*(6*a^2*e^4 + 3*c^2*d^4 + 20*a*c*d^2*e^2))/11

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sympy [B]  time = 0.16, size = 571, normalized size = 2.05 \begin {gather*} a^{4} d^{7} x + \frac {7 a^{4} d^{6} e x^{2}}{2} + \frac {7 c^{4} d e^{6} x^{15}}{15} + \frac {c^{4} e^{7} x^{16}}{16} + x^{14} \left (\frac {2 a c^{3} e^{7}}{7} + \frac {3 c^{4} d^{2} e^{5}}{2}\right ) + x^{13} \left (\frac {28 a c^{3} d e^{6}}{13} + \frac {35 c^{4} d^{3} e^{4}}{13}\right ) + x^{12} \left (\frac {a^{2} c^{2} e^{7}}{2} + 7 a c^{3} d^{2} e^{5} + \frac {35 c^{4} d^{4} e^{3}}{12}\right ) + x^{11} \left (\frac {42 a^{2} c^{2} d e^{6}}{11} + \frac {140 a c^{3} d^{3} e^{4}}{11} + \frac {21 c^{4} d^{5} e^{2}}{11}\right ) + x^{10} \left (\frac {2 a^{3} c e^{7}}{5} + \frac {63 a^{2} c^{2} d^{2} e^{5}}{5} + 14 a c^{3} d^{4} e^{3} + \frac {7 c^{4} d^{6} e}{10}\right ) + x^{9} \left (\frac {28 a^{3} c d e^{6}}{9} + \frac {70 a^{2} c^{2} d^{3} e^{4}}{3} + \frac {28 a c^{3} d^{5} e^{2}}{3} + \frac {c^{4} d^{7}}{9}\right ) + x^{8} \left (\frac {a^{4} e^{7}}{8} + \frac {21 a^{3} c d^{2} e^{5}}{2} + \frac {105 a^{2} c^{2} d^{4} e^{3}}{4} + \frac {7 a c^{3} d^{6} e}{2}\right ) + x^{7} \left (a^{4} d e^{6} + 20 a^{3} c d^{3} e^{4} + 18 a^{2} c^{2} d^{5} e^{2} + \frac {4 a c^{3} d^{7}}{7}\right ) + x^{6} \left (\frac {7 a^{4} d^{2} e^{5}}{2} + \frac {70 a^{3} c d^{4} e^{3}}{3} + 7 a^{2} c^{2} d^{6} e\right ) + x^{5} \left (7 a^{4} d^{3} e^{4} + \frac {84 a^{3} c d^{5} e^{2}}{5} + \frac {6 a^{2} c^{2} d^{7}}{5}\right ) + x^{4} \left (\frac {35 a^{4} d^{4} e^{3}}{4} + 7 a^{3} c d^{6} e\right ) + x^{3} \left (7 a^{4} d^{5} e^{2} + \frac {4 a^{3} c d^{7}}{3}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**7*(c*x**2+a)**4,x)

[Out]

a**4*d**7*x + 7*a**4*d**6*e*x**2/2 + 7*c**4*d*e**6*x**15/15 + c**4*e**7*x**16/16 + x**14*(2*a*c**3*e**7/7 + 3*
c**4*d**2*e**5/2) + x**13*(28*a*c**3*d*e**6/13 + 35*c**4*d**3*e**4/13) + x**12*(a**2*c**2*e**7/2 + 7*a*c**3*d*
*2*e**5 + 35*c**4*d**4*e**3/12) + x**11*(42*a**2*c**2*d*e**6/11 + 140*a*c**3*d**3*e**4/11 + 21*c**4*d**5*e**2/
11) + x**10*(2*a**3*c*e**7/5 + 63*a**2*c**2*d**2*e**5/5 + 14*a*c**3*d**4*e**3 + 7*c**4*d**6*e/10) + x**9*(28*a
**3*c*d*e**6/9 + 70*a**2*c**2*d**3*e**4/3 + 28*a*c**3*d**5*e**2/3 + c**4*d**7/9) + x**8*(a**4*e**7/8 + 21*a**3
*c*d**2*e**5/2 + 105*a**2*c**2*d**4*e**3/4 + 7*a*c**3*d**6*e/2) + x**7*(a**4*d*e**6 + 20*a**3*c*d**3*e**4 + 18
*a**2*c**2*d**5*e**2 + 4*a*c**3*d**7/7) + x**6*(7*a**4*d**2*e**5/2 + 70*a**3*c*d**4*e**3/3 + 7*a**2*c**2*d**6*
e) + x**5*(7*a**4*d**3*e**4 + 84*a**3*c*d**5*e**2/5 + 6*a**2*c**2*d**7/5) + x**4*(35*a**4*d**4*e**3/4 + 7*a**3
*c*d**6*e) + x**3*(7*a**4*d**5*e**2 + 4*a**3*c*d**7/3)

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