3.3.48 \(\int (d x)^m (a+b x^3+c x^6)^2 \, dx\) [248]

Optimal. Leaf size=101 \[ \frac {a^2 (d x)^{1+m}}{d (1+m)}+\frac {2 a b (d x)^{4+m}}{d^4 (4+m)}+\frac {\left (b^2+2 a c\right ) (d x)^{7+m}}{d^7 (7+m)}+\frac {2 b c (d x)^{10+m}}{d^{10} (10+m)}+\frac {c^2 (d x)^{13+m}}{d^{13} (13+m)} \]

[Out]

a^2*(d*x)^(1+m)/d/(1+m)+2*a*b*(d*x)^(4+m)/d^4/(4+m)+(2*a*c+b^2)*(d*x)^(7+m)/d^7/(7+m)+2*b*c*(d*x)^(10+m)/d^10/
(10+m)+c^2*(d*x)^(13+m)/d^13/(13+m)

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Rubi [A]
time = 0.04, antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {1367} \begin {gather*} \frac {a^2 (d x)^{m+1}}{d (m+1)}+\frac {\left (2 a c+b^2\right ) (d x)^{m+7}}{d^7 (m+7)}+\frac {2 a b (d x)^{m+4}}{d^4 (m+4)}+\frac {2 b c (d x)^{m+10}}{d^{10} (m+10)}+\frac {c^2 (d x)^{m+13}}{d^{13} (m+13)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d*x)^m*(a + b*x^3 + c*x^6)^2,x]

[Out]

(a^2*(d*x)^(1 + m))/(d*(1 + m)) + (2*a*b*(d*x)^(4 + m))/(d^4*(4 + m)) + ((b^2 + 2*a*c)*(d*x)^(7 + m))/(d^7*(7
+ m)) + (2*b*c*(d*x)^(10 + m))/(d^10*(10 + m)) + (c^2*(d*x)^(13 + m))/(d^13*(13 + m))

Rule 1367

Int[((d_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d
*x)^m*(a + b*x^n + c*x^(2*n))^p, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && EqQ[n2, 2*n] && IGtQ[p, 0] &&  !Int
egerQ[Simplify[(m + 1)/n]]

Rubi steps

\begin {align*} \int (d x)^m \left (a+b x^3+c x^6\right )^2 \, dx &=\int \left (a^2 (d x)^m+\frac {2 a b (d x)^{3+m}}{d^3}+\frac {\left (b^2+2 a c\right ) (d x)^{6+m}}{d^6}+\frac {2 b c (d x)^{9+m}}{d^9}+\frac {c^2 (d x)^{12+m}}{d^{12}}\right ) \, dx\\ &=\frac {a^2 (d x)^{1+m}}{d (1+m)}+\frac {2 a b (d x)^{4+m}}{d^4 (4+m)}+\frac {\left (b^2+2 a c\right ) (d x)^{7+m}}{d^7 (7+m)}+\frac {2 b c (d x)^{10+m}}{d^{10} (10+m)}+\frac {c^2 (d x)^{13+m}}{d^{13} (13+m)}\\ \end {align*}

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Mathematica [A]
time = 0.54, size = 70, normalized size = 0.69 \begin {gather*} (d x)^m \left (\frac {a^2 x}{1+m}+\frac {2 a b x^4}{4+m}+\frac {\left (b^2+2 a c\right ) x^7}{7+m}+\frac {2 b c x^{10}}{10+m}+\frac {c^2 x^{13}}{13+m}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d*x)^m*(a + b*x^3 + c*x^6)^2,x]

[Out]

(d*x)^m*((a^2*x)/(1 + m) + (2*a*b*x^4)/(4 + m) + ((b^2 + 2*a*c)*x^7)/(7 + m) + (2*b*c*x^10)/(10 + m) + (c^2*x^
13)/(13 + m))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(300\) vs. \(2(101)=202\).
time = 0.01, size = 301, normalized size = 2.98

method	result	size
gosper	\(\frac {x \left (c^{2} m^{4} x^{12}+22 c^{2} m^{3} x^{12}+159 c^{2} m^{2} x^{12}+2 b c \,m^{4} x^{9}+418 m \,x^{12} c^{2}+50 b c \,m^{3} x^{9}+280 c^{2} x^{12}+390 b c \,m^{2} x^{9}+2 a c \,m^{4} x^{6}+b^{2} m^{4} x^{6}+1070 m \,x^{9} b c +56 a c \,m^{3} x^{6}+28 b^{2} m^{3} x^{6}+728 b c \,x^{9}+498 a c \,m^{2} x^{6}+249 b^{2} m^{2} x^{6}+2 a b \,m^{4} x^{3}+1484 a c \,x^{6} m +742 b^{2} x^{6} m +62 a b \,m^{3} x^{3}+1040 a c \,x^{6}+520 b^{2} x^{6}+642 a b \,m^{2} x^{3}+a^{2} m^{4}+2402 a b \,x^{3} m +34 a^{2} m^{3}+1820 a b \,x^{3}+411 a^{2} m^{2}+2074 a^{2} m +3640 a^{2}\right ) \left (d x \right )^{m}}{\left (13+m \right ) \left (10+m \right ) \left (7+m \right ) \left (4+m \right ) \left (1+m \right )}\)	\(301\)
risch	\(\frac {x \left (c^{2} m^{4} x^{12}+22 c^{2} m^{3} x^{12}+159 c^{2} m^{2} x^{12}+2 b c \,m^{4} x^{9}+418 m \,x^{12} c^{2}+50 b c \,m^{3} x^{9}+280 c^{2} x^{12}+390 b c \,m^{2} x^{9}+2 a c \,m^{4} x^{6}+b^{2} m^{4} x^{6}+1070 m \,x^{9} b c +56 a c \,m^{3} x^{6}+28 b^{2} m^{3} x^{6}+728 b c \,x^{9}+498 a c \,m^{2} x^{6}+249 b^{2} m^{2} x^{6}+2 a b \,m^{4} x^{3}+1484 a c \,x^{6} m +742 b^{2} x^{6} m +62 a b \,m^{3} x^{3}+1040 a c \,x^{6}+520 b^{2} x^{6}+642 a b \,m^{2} x^{3}+a^{2} m^{4}+2402 a b \,x^{3} m +34 a^{2} m^{3}+1820 a b \,x^{3}+411 a^{2} m^{2}+2074 a^{2} m +3640 a^{2}\right ) \left (d x \right )^{m}}{\left (13+m \right ) \left (10+m \right ) \left (7+m \right ) \left (4+m \right ) \left (1+m \right )}\)	\(301\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x)^m*(c*x^6+b*x^3+a)^2,x,method=_RETURNVERBOSE)

[Out]

x*(c^2*m^4*x^12+22*c^2*m^3*x^12+159*c^2*m^2*x^12+2*b*c*m^4*x^9+418*c^2*m*x^12+50*b*c*m^3*x^9+280*c^2*x^12+390*
b*c*m^2*x^9+2*a*c*m^4*x^6+b^2*m^4*x^6+1070*b*c*m*x^9+56*a*c*m^3*x^6+28*b^2*m^3*x^6+728*b*c*x^9+498*a*c*m^2*x^6
+249*b^2*m^2*x^6+2*a*b*m^4*x^3+1484*a*c*m*x^6+742*b^2*m*x^6+62*a*b*m^3*x^3+1040*a*c*x^6+520*b^2*x^6+642*a*b*m^
2*x^3+a^2*m^4+2402*a*b*m*x^3+34*a^2*m^3+1820*a*b*x^3+411*a^2*m^2+2074*a^2*m+3640*a^2)*(d*x)^m/(13+m)/(10+m)/(7
+m)/(4+m)/(1+m)

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Maxima [A]
time = 0.34, size = 110, normalized size = 1.09 \begin {gather*} \frac {c^{2} d^{m} x^{13} x^{m}}{m + 13} + \frac {2 \, b c d^{m} x^{10} x^{m}}{m + 10} + \frac {b^{2} d^{m} x^{7} x^{m}}{m + 7} + \frac {2 \, a c d^{m} x^{7} x^{m}}{m + 7} + \frac {2 \, a b d^{m} x^{4} x^{m}}{m + 4} + \frac {\left (d x\right )^{m + 1} a^{2}}{d {\left (m + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^m*(c*x^6+b*x^3+a)^2,x, algorithm="maxima")

[Out]

c^2*d^m*x^13*x^m/(m + 13) + 2*b*c*d^m*x^10*x^m/(m + 10) + b^2*d^m*x^7*x^m/(m + 7) + 2*a*c*d^m*x^7*x^m/(m + 7)
+ 2*a*b*d^m*x^4*x^m/(m + 4) + (d*x)^(m + 1)*a^2/(d*(m + 1))

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 241 vs. \(2 (101) = 202\).
time = 0.41, size = 241, normalized size = 2.39 \begin {gather*} \frac {{\left ({\left (c^{2} m^{4} + 22 \, c^{2} m^{3} + 159 \, c^{2} m^{2} + 418 \, c^{2} m + 280 \, c^{2}\right )} x^{13} + 2 \, {\left (b c m^{4} + 25 \, b c m^{3} + 195 \, b c m^{2} + 535 \, b c m + 364 \, b c\right )} x^{10} + {\left ({\left (b^{2} + 2 \, a c\right )} m^{4} + 28 \, {\left (b^{2} + 2 \, a c\right )} m^{3} + 249 \, {\left (b^{2} + 2 \, a c\right )} m^{2} + 520 \, b^{2} + 1040 \, a c + 742 \, {\left (b^{2} + 2 \, a c\right )} m\right )} x^{7} + 2 \, {\left (a b m^{4} + 31 \, a b m^{3} + 321 \, a b m^{2} + 1201 \, a b m + 910 \, a b\right )} x^{4} + {\left (a^{2} m^{4} + 34 \, a^{2} m^{3} + 411 \, a^{2} m^{2} + 2074 \, a^{2} m + 3640 \, a^{2}\right )} x\right )} \left (d x\right )^{m}}{m^{5} + 35 \, m^{4} + 445 \, m^{3} + 2485 \, m^{2} + 5714 \, m + 3640} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^m*(c*x^6+b*x^3+a)^2,x, algorithm="fricas")

[Out]

((c^2*m^4 + 22*c^2*m^3 + 159*c^2*m^2 + 418*c^2*m + 280*c^2)*x^13 + 2*(b*c*m^4 + 25*b*c*m^3 + 195*b*c*m^2 + 535
*b*c*m + 364*b*c)*x^10 + ((b^2 + 2*a*c)*m^4 + 28*(b^2 + 2*a*c)*m^3 + 249*(b^2 + 2*a*c)*m^2 + 520*b^2 + 1040*a*
c + 742*(b^2 + 2*a*c)*m)*x^7 + 2*(a*b*m^4 + 31*a*b*m^3 + 321*a*b*m^2 + 1201*a*b*m + 910*a*b)*x^4 + (a^2*m^4 +
34*a^2*m^3 + 411*a^2*m^2 + 2074*a^2*m + 3640*a^2)*x)*(d*x)^m/(m^5 + 35*m^4 + 445*m^3 + 2485*m^2 + 5714*m + 364
0)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 1459 vs. \(2 (90) = 180\).
time = 0.81, size = 1459, normalized size = 14.45 \begin {gather*} \begin {cases} \frac {- \frac {a^{2}}{12 x^{12}} - \frac {2 a b}{9 x^{9}} - \frac {a c}{3 x^{6}} - \frac {b^{2}}{6 x^{6}} - \frac {2 b c}{3 x^{3}} + c^{2} \log {\left (x \right )}}{d^{13}} & \text {for}\: m = -13 \\\frac {- \frac {a^{2}}{9 x^{9}} - \frac {a b}{3 x^{6}} - \frac {2 a c}{3 x^{3}} - \frac {b^{2}}{3 x^{3}} + 2 b c \log {\left (x \right )} + \frac {c^{2} x^{3}}{3}}{d^{10}} & \text {for}\: m = -10 \\\frac {- \frac {a^{2}}{6 x^{6}} - \frac {2 a b}{3 x^{3}} + 2 a c \log {\left (x \right )} + b^{2} \log {\left (x \right )} + \frac {2 b c x^{3}}{3} + \frac {c^{2} x^{6}}{6}}{d^{7}} & \text {for}\: m = -7 \\\frac {- \frac {a^{2}}{3 x^{3}} + 2 a b \log {\left (x \right )} + \frac {2 a c x^{3}}{3} + \frac {b^{2} x^{3}}{3} + \frac {b c x^{6}}{3} + \frac {c^{2} x^{9}}{9}}{d^{4}} & \text {for}\: m = -4 \\\frac {a^{2} \log {\left (x \right )} + \frac {2 a b x^{3}}{3} + \frac {a c x^{6}}{3} + \frac {b^{2} x^{6}}{6} + \frac {2 b c x^{9}}{9} + \frac {c^{2} x^{12}}{12}}{d} & \text {for}\: m = -1 \\\frac {a^{2} m^{4} x \left (d x\right )^{m}}{m^{5} + 35 m^{4} + 445 m^{3} + 2485 m^{2} + 5714 m + 3640} + \frac {34 a^{2} m^{3} x \left (d x\right )^{m}}{m^{5} + 35 m^{4} + 445 m^{3} + 2485 m^{2} + 5714 m + 3640} + \frac {411 a^{2} m^{2} x \left (d x\right )^{m}}{m^{5} + 35 m^{4} + 445 m^{3} + 2485 m^{2} + 5714 m + 3640} + \frac {2074 a^{2} m x \left (d x\right )^{m}}{m^{5} + 35 m^{4} + 445 m^{3} + 2485 m^{2} + 5714 m + 3640} + \frac {3640 a^{2} x \left (d x\right )^{m}}{m^{5} + 35 m^{4} + 445 m^{3} + 2485 m^{2} + 5714 m + 3640} + \frac {2 a b m^{4} x^{4} \left (d x\right )^{m}}{m^{5} + 35 m^{4} + 445 m^{3} + 2485 m^{2} + 5714 m + 3640} + \frac {62 a b m^{3} x^{4} \left (d x\right )^{m}}{m^{5} + 35 m^{4} + 445 m^{3} + 2485 m^{2} + 5714 m + 3640} + \frac {642 a b m^{2} x^{4} \left (d x\right )^{m}}{m^{5} + 35 m^{4} + 445 m^{3} + 2485 m^{2} + 5714 m + 3640} + \frac {2402 a b m x^{4} \left (d x\right )^{m}}{m^{5} + 35 m^{4} + 445 m^{3} + 2485 m^{2} + 5714 m + 3640} + \frac {1820 a b x^{4} \left (d x\right )^{m}}{m^{5} + 35 m^{4} + 445 m^{3} + 2485 m^{2} + 5714 m + 3640} + \frac {2 a c m^{4} x^{7} \left (d x\right )^{m}}{m^{5} + 35 m^{4} + 445 m^{3} + 2485 m^{2} + 5714 m + 3640} + \frac {56 a c m^{3} x^{7} \left (d x\right )^{m}}{m^{5} + 35 m^{4} + 445 m^{3} + 2485 m^{2} + 5714 m + 3640} + \frac {498 a c m^{2} x^{7} \left (d x\right )^{m}}{m^{5} + 35 m^{4} + 445 m^{3} + 2485 m^{2} + 5714 m + 3640} + \frac {1484 a c m x^{7} \left (d x\right )^{m}}{m^{5} + 35 m^{4} + 445 m^{3} + 2485 m^{2} + 5714 m + 3640} + \frac {1040 a c x^{7} \left (d x\right )^{m}}{m^{5} + 35 m^{4} + 445 m^{3} + 2485 m^{2} + 5714 m + 3640} + \frac {b^{2} m^{4} x^{7} \left (d x\right )^{m}}{m^{5} + 35 m^{4} + 445 m^{3} + 2485 m^{2} + 5714 m + 3640} + \frac {28 b^{2} m^{3} x^{7} \left (d x\right )^{m}}{m^{5} + 35 m^{4} + 445 m^{3} + 2485 m^{2} + 5714 m + 3640} + \frac {249 b^{2} m^{2} x^{7} \left (d x\right )^{m}}{m^{5} + 35 m^{4} + 445 m^{3} + 2485 m^{2} + 5714 m + 3640} + \frac {742 b^{2} m x^{7} \left (d x\right )^{m}}{m^{5} + 35 m^{4} + 445 m^{3} + 2485 m^{2} + 5714 m + 3640} + \frac {520 b^{2} x^{7} \left (d x\right )^{m}}{m^{5} + 35 m^{4} + 445 m^{3} + 2485 m^{2} + 5714 m + 3640} + \frac {2 b c m^{4} x^{10} \left (d x\right )^{m}}{m^{5} + 35 m^{4} + 445 m^{3} + 2485 m^{2} + 5714 m + 3640} + \frac {50 b c m^{3} x^{10} \left (d x\right )^{m}}{m^{5} + 35 m^{4} + 445 m^{3} + 2485 m^{2} + 5714 m + 3640} + \frac {390 b c m^{2} x^{10} \left (d x\right )^{m}}{m^{5} + 35 m^{4} + 445 m^{3} + 2485 m^{2} + 5714 m + 3640} + \frac {1070 b c m x^{10} \left (d x\right )^{m}}{m^{5} + 35 m^{4} + 445 m^{3} + 2485 m^{2} + 5714 m + 3640} + \frac {728 b c x^{10} \left (d x\right )^{m}}{m^{5} + 35 m^{4} + 445 m^{3} + 2485 m^{2} + 5714 m + 3640} + \frac {c^{2} m^{4} x^{13} \left (d x\right )^{m}}{m^{5} + 35 m^{4} + 445 m^{3} + 2485 m^{2} + 5714 m + 3640} + \frac {22 c^{2} m^{3} x^{13} \left (d x\right )^{m}}{m^{5} + 35 m^{4} + 445 m^{3} + 2485 m^{2} + 5714 m + 3640} + \frac {159 c^{2} m^{2} x^{13} \left (d x\right )^{m}}{m^{5} + 35 m^{4} + 445 m^{3} + 2485 m^{2} + 5714 m + 3640} + \frac {418 c^{2} m x^{13} \left (d x\right )^{m}}{m^{5} + 35 m^{4} + 445 m^{3} + 2485 m^{2} + 5714 m + 3640} + \frac {280 c^{2} x^{13} \left (d x\right )^{m}}{m^{5} + 35 m^{4} + 445 m^{3} + 2485 m^{2} + 5714 m + 3640} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)**m*(c*x**6+b*x**3+a)**2,x)

[Out]

Piecewise(((-a**2/(12*x**12) - 2*a*b/(9*x**9) - a*c/(3*x**6) - b**2/(6*x**6) - 2*b*c/(3*x**3) + c**2*log(x))/d
**13, Eq(m, -13)), ((-a**2/(9*x**9) - a*b/(3*x**6) - 2*a*c/(3*x**3) - b**2/(3*x**3) + 2*b*c*log(x) + c**2*x**3
/3)/d**10, Eq(m, -10)), ((-a**2/(6*x**6) - 2*a*b/(3*x**3) + 2*a*c*log(x) + b**2*log(x) + 2*b*c*x**3/3 + c**2*x
**6/6)/d**7, Eq(m, -7)), ((-a**2/(3*x**3) + 2*a*b*log(x) + 2*a*c*x**3/3 + b**2*x**3/3 + b*c*x**6/3 + c**2*x**9
/9)/d**4, Eq(m, -4)), ((a**2*log(x) + 2*a*b*x**3/3 + a*c*x**6/3 + b**2*x**6/6 + 2*b*c*x**9/9 + c**2*x**12/12)/
d, Eq(m, -1)), (a**2*m**4*x*(d*x)**m/(m**5 + 35*m**4 + 445*m**3 + 2485*m**2 + 5714*m + 3640) + 34*a**2*m**3*x*
(d*x)**m/(m**5 + 35*m**4 + 445*m**3 + 2485*m**2 + 5714*m + 3640) + 411*a**2*m**2*x*(d*x)**m/(m**5 + 35*m**4 +
445*m**3 + 2485*m**2 + 5714*m + 3640) + 2074*a**2*m*x*(d*x)**m/(m**5 + 35*m**4 + 445*m**3 + 2485*m**2 + 5714*m
+ 3640) + 3640*a**2*x*(d*x)**m/(m**5 + 35*m**4 + 445*m**3 + 2485*m**2 + 5714*m + 3640) + 2*a*b*m**4*x**4*(d*x
)**m/(m**5 + 35*m**4 + 445*m**3 + 2485*m**2 + 5714*m + 3640) + 62*a*b*m**3*x**4*(d*x)**m/(m**5 + 35*m**4 + 445
*m**3 + 2485*m**2 + 5714*m + 3640) + 642*a*b*m**2*x**4*(d*x)**m/(m**5 + 35*m**4 + 445*m**3 + 2485*m**2 + 5714*
m + 3640) + 2402*a*b*m*x**4*(d*x)**m/(m**5 + 35*m**4 + 445*m**3 + 2485*m**2 + 5714*m + 3640) + 1820*a*b*x**4*(
d*x)**m/(m**5 + 35*m**4 + 445*m**3 + 2485*m**2 + 5714*m + 3640) + 2*a*c*m**4*x**7*(d*x)**m/(m**5 + 35*m**4 + 4
45*m**3 + 2485*m**2 + 5714*m + 3640) + 56*a*c*m**3*x**7*(d*x)**m/(m**5 + 35*m**4 + 445*m**3 + 2485*m**2 + 5714
*m + 3640) + 498*a*c*m**2*x**7*(d*x)**m/(m**5 + 35*m**4 + 445*m**3 + 2485*m**2 + 5714*m + 3640) + 1484*a*c*m*x
**7*(d*x)**m/(m**5 + 35*m**4 + 445*m**3 + 2485*m**2 + 5714*m + 3640) + 1040*a*c*x**7*(d*x)**m/(m**5 + 35*m**4
+ 445*m**3 + 2485*m**2 + 5714*m + 3640) + b**2*m**4*x**7*(d*x)**m/(m**5 + 35*m**4 + 445*m**3 + 2485*m**2 + 571
4*m + 3640) + 28*b**2*m**3*x**7*(d*x)**m/(m**5 + 35*m**4 + 445*m**3 + 2485*m**2 + 5714*m + 3640) + 249*b**2*m*
*2*x**7*(d*x)**m/(m**5 + 35*m**4 + 445*m**3 + 2485*m**2 + 5714*m + 3640) + 742*b**2*m*x**7*(d*x)**m/(m**5 + 35
*m**4 + 445*m**3 + 2485*m**2 + 5714*m + 3640) + 520*b**2*x**7*(d*x)**m/(m**5 + 35*m**4 + 445*m**3 + 2485*m**2
+ 5714*m + 3640) + 2*b*c*m**4*x**10*(d*x)**m/(m**5 + 35*m**4 + 445*m**3 + 2485*m**2 + 5714*m + 3640) + 50*b*c*
m**3*x**10*(d*x)**m/(m**5 + 35*m**4 + 445*m**3 + 2485*m**2 + 5714*m + 3640) + 390*b*c*m**2*x**10*(d*x)**m/(m**
5 + 35*m**4 + 445*m**3 + 2485*m**2 + 5714*m + 3640) + 1070*b*c*m*x**10*(d*x)**m/(m**5 + 35*m**4 + 445*m**3 + 2
485*m**2 + 5714*m + 3640) + 728*b*c*x**10*(d*x)**m/(m**5 + 35*m**4 + 445*m**3 + 2485*m**2 + 5714*m + 3640) + c
**2*m**4*x**13*(d*x)**m/(m**5 + 35*m**4 + 445*m**3 + 2485*m**2 + 5714*m + 3640) + 22*c**2*m**3*x**13*(d*x)**m/
(m**5 + 35*m**4 + 445*m**3 + 2485*m**2 + 5714*m + 3640) + 159*c**2*m**2*x**13*(d*x)**m/(m**5 + 35*m**4 + 445*m
**3 + 2485*m**2 + 5714*m + 3640) + 418*c**2*m*x**13*(d*x)**m/(m**5 + 35*m**4 + 445*m**3 + 2485*m**2 + 5714*m +
3640) + 280*c**2*x**13*(d*x)**m/(m**5 + 35*m**4 + 445*m**3 + 2485*m**2 + 5714*m + 3640), True))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 449 vs. \(2 (101) = 202\).
time = 4.34, size = 449, normalized size = 4.45 \begin {gather*} \frac {\left (d x\right )^{m} c^{2} m^{4} x^{13} + 22 \, \left (d x\right )^{m} c^{2} m^{3} x^{13} + 159 \, \left (d x\right )^{m} c^{2} m^{2} x^{13} + 2 \, \left (d x\right )^{m} b c m^{4} x^{10} + 418 \, \left (d x\right )^{m} c^{2} m x^{13} + 50 \, \left (d x\right )^{m} b c m^{3} x^{10} + 280 \, \left (d x\right )^{m} c^{2} x^{13} + 390 \, \left (d x\right )^{m} b c m^{2} x^{10} + \left (d x\right )^{m} b^{2} m^{4} x^{7} + 2 \, \left (d x\right )^{m} a c m^{4} x^{7} + 1070 \, \left (d x\right )^{m} b c m x^{10} + 28 \, \left (d x\right )^{m} b^{2} m^{3} x^{7} + 56 \, \left (d x\right )^{m} a c m^{3} x^{7} + 728 \, \left (d x\right )^{m} b c x^{10} + 249 \, \left (d x\right )^{m} b^{2} m^{2} x^{7} + 498 \, \left (d x\right )^{m} a c m^{2} x^{7} + 2 \, \left (d x\right )^{m} a b m^{4} x^{4} + 742 \, \left (d x\right )^{m} b^{2} m x^{7} + 1484 \, \left (d x\right )^{m} a c m x^{7} + 62 \, \left (d x\right )^{m} a b m^{3} x^{4} + 520 \, \left (d x\right )^{m} b^{2} x^{7} + 1040 \, \left (d x\right )^{m} a c x^{7} + 642 \, \left (d x\right )^{m} a b m^{2} x^{4} + \left (d x\right )^{m} a^{2} m^{4} x + 2402 \, \left (d x\right )^{m} a b m x^{4} + 34 \, \left (d x\right )^{m} a^{2} m^{3} x + 1820 \, \left (d x\right )^{m} a b x^{4} + 411 \, \left (d x\right )^{m} a^{2} m^{2} x + 2074 \, \left (d x\right )^{m} a^{2} m x + 3640 \, \left (d x\right )^{m} a^{2} x}{m^{5} + 35 \, m^{4} + 445 \, m^{3} + 2485 \, m^{2} + 5714 \, m + 3640} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^m*(c*x^6+b*x^3+a)^2,x, algorithm="giac")

[Out]

((d*x)^m*c^2*m^4*x^13 + 22*(d*x)^m*c^2*m^3*x^13 + 159*(d*x)^m*c^2*m^2*x^13 + 2*(d*x)^m*b*c*m^4*x^10 + 418*(d*x
)^m*c^2*m*x^13 + 50*(d*x)^m*b*c*m^3*x^10 + 280*(d*x)^m*c^2*x^13 + 390*(d*x)^m*b*c*m^2*x^10 + (d*x)^m*b^2*m^4*x
^7 + 2*(d*x)^m*a*c*m^4*x^7 + 1070*(d*x)^m*b*c*m*x^10 + 28*(d*x)^m*b^2*m^3*x^7 + 56*(d*x)^m*a*c*m^3*x^7 + 728*(
d*x)^m*b*c*x^10 + 249*(d*x)^m*b^2*m^2*x^7 + 498*(d*x)^m*a*c*m^2*x^7 + 2*(d*x)^m*a*b*m^4*x^4 + 742*(d*x)^m*b^2*
m*x^7 + 1484*(d*x)^m*a*c*m*x^7 + 62*(d*x)^m*a*b*m^3*x^4 + 520*(d*x)^m*b^2*x^7 + 1040*(d*x)^m*a*c*x^7 + 642*(d*
x)^m*a*b*m^2*x^4 + (d*x)^m*a^2*m^4*x + 2402*(d*x)^m*a*b*m*x^4 + 34*(d*x)^m*a^2*m^3*x + 1820*(d*x)^m*a*b*x^4 +
411*(d*x)^m*a^2*m^2*x + 2074*(d*x)^m*a^2*m*x + 3640*(d*x)^m*a^2*x)/(m^5 + 35*m^4 + 445*m^3 + 2485*m^2 + 5714*m
+ 3640)

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Mupad [B]
time = 1.52, size = 260, normalized size = 2.57 \begin {gather*} {\left (d\,x\right )}^m\,\left (\frac {c^2\,x^{13}\,\left (m^4+22\,m^3+159\,m^2+418\,m+280\right )}{m^5+35\,m^4+445\,m^3+2485\,m^2+5714\,m+3640}+\frac {x^7\,\left (b^2+2\,a\,c\right )\,\left (m^4+28\,m^3+249\,m^2+742\,m+520\right )}{m^5+35\,m^4+445\,m^3+2485\,m^2+5714\,m+3640}+\frac {a^2\,x\,\left (m^4+34\,m^3+411\,m^2+2074\,m+3640\right )}{m^5+35\,m^4+445\,m^3+2485\,m^2+5714\,m+3640}+\frac {2\,a\,b\,x^4\,\left (m^4+31\,m^3+321\,m^2+1201\,m+910\right )}{m^5+35\,m^4+445\,m^3+2485\,m^2+5714\,m+3640}+\frac {2\,b\,c\,x^{10}\,\left (m^4+25\,m^3+195\,m^2+535\,m+364\right )}{m^5+35\,m^4+445\,m^3+2485\,m^2+5714\,m+3640}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x)^m*(a + b*x^3 + c*x^6)^2,x)

[Out]

(d*x)^m*((c^2*x^13*(418*m + 159*m^2 + 22*m^3 + m^4 + 280))/(5714*m + 2485*m^2 + 445*m^3 + 35*m^4 + m^5 + 3640)
+ (x^7*(2*a*c + b^2)*(742*m + 249*m^2 + 28*m^3 + m^4 + 520))/(5714*m + 2485*m^2 + 445*m^3 + 35*m^4 + m^5 + 36
40) + (a^2*x*(2074*m + 411*m^2 + 34*m^3 + m^4 + 3640))/(5714*m + 2485*m^2 + 445*m^3 + 35*m^4 + m^5 + 3640) + (
2*a*b*x^4*(1201*m + 321*m^2 + 31*m^3 + m^4 + 910))/(5714*m + 2485*m^2 + 445*m^3 + 35*m^4 + m^5 + 3640) + (2*b*
c*x^10*(535*m + 195*m^2 + 25*m^3 + m^4 + 364))/(5714*m + 2485*m^2 + 445*m^3 + 35*m^4 + m^5 + 3640))

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