Optimal. Leaf size=223 \[ -\frac {4 a d^2 \left (5 a^2 d+3 b^2 c\right ) (a+b x)^{n+4}}{b^7 (n+4)}+\frac {3 d^2 \left (5 a^2 d+b^2 c\right ) (a+b x)^{n+5}}{b^7 (n+5)}+\frac {\left (a^2 d+b^2 c\right )^3 (a+b x)^{n+1}}{b^7 (n+1)}-\frac {6 a d \left (a^2 d+b^2 c\right )^2 (a+b x)^{n+2}}{b^7 (n+2)}+\frac {3 d \left (a^2 d+b^2 c\right ) \left (5 a^2 d+b^2 c\right ) (a+b x)^{n+3}}{b^7 (n+3)}-\frac {6 a d^3 (a+b x)^{n+6}}{b^7 (n+6)}+\frac {d^3 (a+b x)^{n+7}}{b^7 (n+7)} \]
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Rubi [A] time = 0.13, antiderivative size = 223, 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 {4 a d^2 \left (5 a^2 d+3 b^2 c\right ) (a+b x)^{n+4}}{b^7 (n+4)}+\frac {3 d^2 \left (5 a^2 d+b^2 c\right ) (a+b x)^{n+5}}{b^7 (n+5)}+\frac {\left (a^2 d+b^2 c\right )^3 (a+b x)^{n+1}}{b^7 (n+1)}-\frac {6 a d \left (a^2 d+b^2 c\right )^2 (a+b x)^{n+2}}{b^7 (n+2)}+\frac {3 d \left (a^2 d+b^2 c\right ) \left (5 a^2 d+b^2 c\right ) (a+b x)^{n+3}}{b^7 (n+3)}-\frac {6 a d^3 (a+b x)^{n+6}}{b^7 (n+6)}+\frac {d^3 (a+b x)^{n+7}}{b^7 (n+7)} \end {gather*}
Antiderivative was successfully verified.
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Rule 697
Rubi steps
\begin {align*} \int (a+b x)^n \left (c+d x^2\right )^3 \, dx &=\int \left (\frac {\left (b^2 c+a^2 d\right )^3 (a+b x)^n}{b^6}-\frac {6 a d \left (b^2 c+a^2 d\right )^2 (a+b x)^{1+n}}{b^6}+\frac {3 d \left (b^2 c+a^2 d\right ) \left (b^2 c+5 a^2 d\right ) (a+b x)^{2+n}}{b^6}-\frac {4 a d^2 \left (3 b^2 c+5 a^2 d\right ) (a+b x)^{3+n}}{b^6}+\frac {3 d^2 \left (b^2 c+5 a^2 d\right ) (a+b x)^{4+n}}{b^6}-\frac {6 a d^3 (a+b x)^{5+n}}{b^6}+\frac {d^3 (a+b x)^{6+n}}{b^6}\right ) \, dx\\ &=\frac {\left (b^2 c+a^2 d\right )^3 (a+b x)^{1+n}}{b^7 (1+n)}-\frac {6 a d \left (b^2 c+a^2 d\right )^2 (a+b x)^{2+n}}{b^7 (2+n)}+\frac {3 d \left (b^2 c+a^2 d\right ) \left (b^2 c+5 a^2 d\right ) (a+b x)^{3+n}}{b^7 (3+n)}-\frac {4 a d^2 \left (3 b^2 c+5 a^2 d\right ) (a+b x)^{4+n}}{b^7 (4+n)}+\frac {3 d^2 \left (b^2 c+5 a^2 d\right ) (a+b x)^{5+n}}{b^7 (5+n)}-\frac {6 a d^3 (a+b x)^{6+n}}{b^7 (6+n)}+\frac {d^3 (a+b x)^{7+n}}{b^7 (7+n)}\\ \end {align*}
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Mathematica [A] time = 0.50, size = 347, normalized size = 1.56 \begin {gather*} \frac {(a+b x)^{n+1} \left (\frac {6 \left ((n+6) \left (a^2 d+b^2 c\right ) \left (4 (n+4) \left (a^2 d+b^2 c\right ) \left (2 a^2 d-2 a b d (n+1) x+b^2 (n+2) \left (c (n+3)+d (n+1) x^2\right )\right )-4 a d (n+1) (a+b x) \left (2 a^2 d-2 a b d (n+2) x+b^2 (n+3) \left (c (n+4)+d (n+2) x^2\right )\right )+b^4 (n+1) (n+2) (n+3) (n+4) \left (c+d x^2\right )^2\right )-a d (n+1) (a+b x) \left (4 (n+5) \left (a^2 d+b^2 c\right ) \left (2 a^2 d-2 a b d (n+2) x+b^2 (n+3) \left (c (n+4)+d (n+2) x^2\right )\right )-4 a d (n+2) (a+b x) \left (2 a^2 d-2 a b d (n+3) x+b^2 (n+4) \left (c (n+5)+d (n+3) x^2\right )\right )+b^4 (n+2) (n+3) (n+4) (n+5) \left (c+d x^2\right )^2\right )\right )}{b^6 (n+1) (n+2) (n+3) (n+4) (n+5) (n+6)}+\left (c+d x^2\right )^3\right )}{b (n+7)} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.07, size = 0, normalized size = 0.00 \begin {gather*} \int (a+b x)^n \left (c+d x^2\right )^3 \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 0.41, size = 1244, normalized size = 5.58
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.24, size = 2085, normalized size = 9.35
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 1140, normalized size = 5.11 \begin {gather*} \frac {\left (b^{6} d^{3} n^{6} x^{6}+21 b^{6} d^{3} n^{5} x^{6}-6 a \,b^{5} d^{3} n^{5} x^{5}+3 b^{6} c \,d^{2} n^{6} x^{4}+175 b^{6} d^{3} n^{4} x^{6}-90 a \,b^{5} d^{3} n^{4} x^{5}+69 b^{6} c \,d^{2} n^{5} x^{4}+735 b^{6} d^{3} n^{3} x^{6}+30 a^{2} b^{4} d^{3} n^{4} x^{4}-12 a \,b^{5} c \,d^{2} n^{5} x^{3}-510 a \,b^{5} d^{3} n^{3} x^{5}+3 b^{6} c^{2} d \,n^{6} x^{2}+621 b^{6} c \,d^{2} n^{4} x^{4}+1624 b^{6} d^{3} n^{2} x^{6}+300 a^{2} b^{4} d^{3} n^{3} x^{4}-228 a \,b^{5} c \,d^{2} n^{4} x^{3}-1350 a \,b^{5} d^{3} n^{2} x^{5}+75 b^{6} c^{2} d \,n^{5} x^{2}+2775 b^{6} c \,d^{2} n^{3} x^{4}+1764 b^{6} d^{3} n \,x^{6}-120 a^{3} b^{3} d^{3} n^{3} x^{3}+36 a^{2} b^{4} c \,d^{2} n^{4} x^{2}+1050 a^{2} b^{4} d^{3} n^{2} x^{4}-6 a \,b^{5} c^{2} d \,n^{5} x -1572 a \,b^{5} c \,d^{2} n^{3} x^{3}-1644 a \,b^{5} d^{3} n \,x^{5}+b^{6} c^{3} n^{6}+741 b^{6} c^{2} d \,n^{4} x^{2}+6432 b^{6} c \,d^{2} n^{2} x^{4}+720 d^{3} x^{6} b^{6}-720 a^{3} b^{3} d^{3} n^{2} x^{3}+576 a^{2} b^{4} c \,d^{2} n^{3} x^{2}+1500 a^{2} b^{4} d^{3} n \,x^{4}-138 a \,b^{5} c^{2} d \,n^{4} x -4812 a \,b^{5} c \,d^{2} n^{2} x^{3}-720 a \,d^{3} x^{5} b^{5}+27 b^{6} c^{3} n^{5}+3657 b^{6} c^{2} d \,n^{3} x^{2}+7236 b^{6} c \,d^{2} n \,x^{4}+360 a^{4} b^{2} d^{3} n^{2} x^{2}-72 a^{3} b^{3} c \,d^{2} n^{3} x -1320 a^{3} b^{3} d^{3} n \,x^{3}+6 a^{2} b^{4} c^{2} d \,n^{4}+2988 a^{2} b^{4} c \,d^{2} n^{2} x^{2}+720 a^{2} b^{4} d^{3} x^{4}-1206 a \,b^{5} c^{2} d \,n^{3} x -6480 a \,b^{5} c \,d^{2} n \,x^{3}+295 b^{6} c^{3} n^{4}+9336 b^{6} c^{2} d \,n^{2} x^{2}+3024 b^{6} c \,d^{2} x^{4}+1080 a^{4} b^{2} d^{3} n \,x^{2}-1008 a^{3} b^{3} c \,d^{2} n^{2} x -720 a^{3} b^{3} d^{3} x^{3}+132 a^{2} b^{4} c^{2} d \,n^{3}+5472 a^{2} b^{4} c \,d^{2} n \,x^{2}-4902 a \,b^{5} c^{2} d \,n^{2} x -3024 a \,b^{5} c \,d^{2} x^{3}+1665 b^{6} c^{3} n^{3}+11388 b^{6} c^{2} d n \,x^{2}-720 a^{5} b \,d^{3} n x +72 a^{4} b^{2} c \,d^{2} n^{2}+720 a^{4} b^{2} d^{3} x^{2}-3960 a^{3} b^{3} c \,d^{2} n x +1074 a^{2} b^{4} c^{2} d \,n^{2}+3024 a^{2} b^{4} c \,d^{2} x^{2}-8868 a \,b^{5} c^{2} d n x +5104 b^{6} c^{3} n^{2}+5040 b^{6} c^{2} d \,x^{2}-720 a^{5} b \,d^{3} x +936 a^{4} b^{2} c \,d^{2} n -3024 a^{3} b^{3} c \,d^{2} x +3828 a^{2} b^{4} c^{2} d n -5040 a \,b^{5} c^{2} d x +8028 b^{6} c^{3} n +720 a^{6} d^{3}+3024 a^{4} b^{2} c \,d^{2}+5040 a^{2} b^{4} c^{2} d +5040 b^{6} c^{3}\right ) \left (b x +a \right )^{n +1}}{\left (n^{7}+28 n^{6}+322 n^{5}+1960 n^{4}+6769 n^{3}+13132 n^{2}+13068 n +5040\right ) b^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.51, size = 472, normalized size = 2.12 \begin {gather*} \frac {{\left (b x + a\right )}^{n + 1} c^{3}}{b {\left (n + 1\right )}} + \frac {3 \, {\left ({\left (n^{2} + 3 \, n + 2\right )} b^{3} x^{3} + {\left (n^{2} + n\right )} a b^{2} x^{2} - 2 \, a^{2} b n x + 2 \, a^{3}\right )} {\left (b x + a\right )}^{n} c^{2} d}{{\left (n^{3} + 6 \, n^{2} + 11 \, n + 6\right )} b^{3}} + \frac {3 \, {\left ({\left (n^{4} + 10 \, n^{3} + 35 \, n^{2} + 50 \, n + 24\right )} b^{5} x^{5} + {\left (n^{4} + 6 \, n^{3} + 11 \, n^{2} + 6 \, n\right )} a b^{4} x^{4} - 4 \, {\left (n^{3} + 3 \, n^{2} + 2 \, n\right )} a^{2} b^{3} x^{3} + 12 \, {\left (n^{2} + n\right )} a^{3} b^{2} x^{2} - 24 \, a^{4} b n x + 24 \, a^{5}\right )} {\left (b x + a\right )}^{n} c d^{2}}{{\left (n^{5} + 15 \, n^{4} + 85 \, n^{3} + 225 \, n^{2} + 274 \, n + 120\right )} b^{5}} + \frac {{\left ({\left (n^{6} + 21 \, n^{5} + 175 \, n^{4} + 735 \, n^{3} + 1624 \, n^{2} + 1764 \, n + 720\right )} b^{7} x^{7} + {\left (n^{6} + 15 \, n^{5} + 85 \, n^{4} + 225 \, n^{3} + 274 \, n^{2} + 120 \, n\right )} a b^{6} x^{6} - 6 \, {\left (n^{5} + 10 \, n^{4} + 35 \, n^{3} + 50 \, n^{2} + 24 \, n\right )} a^{2} b^{5} x^{5} + 30 \, {\left (n^{4} + 6 \, n^{3} + 11 \, n^{2} + 6 \, n\right )} a^{3} b^{4} x^{4} - 120 \, {\left (n^{3} + 3 \, n^{2} + 2 \, n\right )} a^{4} b^{3} x^{3} + 360 \, {\left (n^{2} + n\right )} a^{5} b^{2} x^{2} - 720 \, a^{6} b n x + 720 \, a^{7}\right )} {\left (b x + a\right )}^{n} d^{3}}{{\left (n^{7} + 28 \, n^{6} + 322 \, n^{5} + 1960 \, n^{4} + 6769 \, n^{3} + 13132 \, n^{2} + 13068 \, n + 5040\right )} b^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.16, size = 1144, normalized size = 5.13 \begin {gather*} \frac {{\left (a+b\,x\right )}^n\,\left (720\,a^7\,d^3+72\,a^5\,b^2\,c\,d^2\,n^2+936\,a^5\,b^2\,c\,d^2\,n+3024\,a^5\,b^2\,c\,d^2+6\,a^3\,b^4\,c^2\,d\,n^4+132\,a^3\,b^4\,c^2\,d\,n^3+1074\,a^3\,b^4\,c^2\,d\,n^2+3828\,a^3\,b^4\,c^2\,d\,n+5040\,a^3\,b^4\,c^2\,d+a\,b^6\,c^3\,n^6+27\,a\,b^6\,c^3\,n^5+295\,a\,b^6\,c^3\,n^4+1665\,a\,b^6\,c^3\,n^3+5104\,a\,b^6\,c^3\,n^2+8028\,a\,b^6\,c^3\,n+5040\,a\,b^6\,c^3\right )}{b^7\,\left (n^7+28\,n^6+322\,n^5+1960\,n^4+6769\,n^3+13132\,n^2+13068\,n+5040\right )}-\frac {x\,{\left (a+b\,x\right )}^n\,\left (720\,a^6\,b\,d^3\,n+72\,a^4\,b^3\,c\,d^2\,n^3+936\,a^4\,b^3\,c\,d^2\,n^2+3024\,a^4\,b^3\,c\,d^2\,n+6\,a^2\,b^5\,c^2\,d\,n^5+132\,a^2\,b^5\,c^2\,d\,n^4+1074\,a^2\,b^5\,c^2\,d\,n^3+3828\,a^2\,b^5\,c^2\,d\,n^2+5040\,a^2\,b^5\,c^2\,d\,n-b^7\,c^3\,n^6-27\,b^7\,c^3\,n^5-295\,b^7\,c^3\,n^4-1665\,b^7\,c^3\,n^3-5104\,b^7\,c^3\,n^2-8028\,b^7\,c^3\,n-5040\,b^7\,c^3\right )}{b^7\,\left (n^7+28\,n^6+322\,n^5+1960\,n^4+6769\,n^3+13132\,n^2+13068\,n+5040\right )}+\frac {d^3\,x^7\,{\left (a+b\,x\right )}^n\,\left (n^6+21\,n^5+175\,n^4+735\,n^3+1624\,n^2+1764\,n+720\right )}{n^7+28\,n^6+322\,n^5+1960\,n^4+6769\,n^3+13132\,n^2+13068\,n+5040}+\frac {3\,d^2\,x^5\,{\left (a+b\,x\right )}^n\,\left (-2\,d\,a^2\,n+c\,b^2\,n^2+13\,c\,b^2\,n+42\,c\,b^2\right )\,\left (n^4+10\,n^3+35\,n^2+50\,n+24\right )}{b^2\,\left (n^7+28\,n^6+322\,n^5+1960\,n^4+6769\,n^3+13132\,n^2+13068\,n+5040\right )}+\frac {3\,d\,x^3\,{\left (a+b\,x\right )}^n\,\left (n^2+3\,n+2\right )\,\left (-40\,a^4\,d^2\,n-4\,a^2\,b^2\,c\,d\,n^3-52\,a^2\,b^2\,c\,d\,n^2-168\,a^2\,b^2\,c\,d\,n+b^4\,c^2\,n^4+22\,b^4\,c^2\,n^3+179\,b^4\,c^2\,n^2+638\,b^4\,c^2\,n+840\,b^4\,c^2\right )}{b^4\,\left (n^7+28\,n^6+322\,n^5+1960\,n^4+6769\,n^3+13132\,n^2+13068\,n+5040\right )}+\frac {a\,d^3\,n\,x^6\,{\left (a+b\,x\right )}^n\,\left (n^5+15\,n^4+85\,n^3+225\,n^2+274\,n+120\right )}{b\,\left (n^7+28\,n^6+322\,n^5+1960\,n^4+6769\,n^3+13132\,n^2+13068\,n+5040\right )}+\frac {3\,a\,d^2\,n\,x^4\,{\left (a+b\,x\right )}^n\,\left (n^3+6\,n^2+11\,n+6\right )\,\left (10\,d\,a^2+c\,b^2\,n^2+13\,c\,b^2\,n+42\,c\,b^2\right )}{b^3\,\left (n^7+28\,n^6+322\,n^5+1960\,n^4+6769\,n^3+13132\,n^2+13068\,n+5040\right )}+\frac {3\,a\,d\,n\,x^2\,\left (n+1\right )\,{\left (a+b\,x\right )}^n\,\left (120\,a^4\,d^2+12\,a^2\,b^2\,c\,d\,n^2+156\,a^2\,b^2\,c\,d\,n+504\,a^2\,b^2\,c\,d+b^4\,c^2\,n^4+22\,b^4\,c^2\,n^3+179\,b^4\,c^2\,n^2+638\,b^4\,c^2\,n+840\,b^4\,c^2\right )}{b^5\,\left (n^7+28\,n^6+322\,n^5+1960\,n^4+6769\,n^3+13132\,n^2+13068\,n+5040\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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