Optimal. Leaf size=235 \[ \frac{b^3 (a+b x)^7 (-11 a B e+4 A b e+7 b B d)}{9240 e (d+e x)^7 (b d-a e)^5}+\frac{b^2 (a+b x)^7 (-11 a B e+4 A b e+7 b B d)}{1320 e (d+e x)^8 (b d-a e)^4}+\frac{b (a+b x)^7 (-11 a B e+4 A b e+7 b B d)}{330 e (d+e x)^9 (b d-a e)^3}+\frac{(a+b x)^7 (-11 a B e+4 A b e+7 b B d)}{110 e (d+e x)^{10} (b d-a e)^2}-\frac{(a+b x)^7 (B d-A e)}{11 e (d+e x)^{11} (b d-a e)} \]
[Out]
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Rubi [A] time = 0.30075, antiderivative size = 235, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15 \[ \frac{b^3 (a+b x)^7 (-11 a B e+4 A b e+7 b B d)}{9240 e (d+e x)^7 (b d-a e)^5}+\frac{b^2 (a+b x)^7 (-11 a B e+4 A b e+7 b B d)}{1320 e (d+e x)^8 (b d-a e)^4}+\frac{b (a+b x)^7 (-11 a B e+4 A b e+7 b B d)}{330 e (d+e x)^9 (b d-a e)^3}+\frac{(a+b x)^7 (-11 a B e+4 A b e+7 b B d)}{110 e (d+e x)^{10} (b d-a e)^2}-\frac{(a+b x)^7 (B d-A e)}{11 e (d+e x)^{11} (b d-a e)} \]
Antiderivative was successfully verified.
[In] Int[((a + b*x)^6*(A + B*x))/(d + e*x)^12,x]
[Out]
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Rubi in Sympy [A] time = 50.5696, size = 221, normalized size = 0.94 \[ - \frac{b^{3} \left (a + b x\right )^{7} \left (4 A b e - 11 B a e + 7 B b d\right )}{9240 e \left (d + e x\right )^{7} \left (a e - b d\right )^{5}} + \frac{b^{2} \left (a + b x\right )^{7} \left (4 A b e - 11 B a e + 7 B b d\right )}{1320 e \left (d + e x\right )^{8} \left (a e - b d\right )^{4}} - \frac{b \left (a + b x\right )^{7} \left (4 A b e - 11 B a e + 7 B b d\right )}{330 e \left (d + e x\right )^{9} \left (a e - b d\right )^{3}} + \frac{\left (a + b x\right )^{7} \left (4 A b e - 11 B a e + 7 B b d\right )}{110 e \left (d + e x\right )^{10} \left (a e - b d\right )^{2}} - \frac{\left (a + b x\right )^{7} \left (A e - B d\right )}{11 e \left (d + e x\right )^{11} \left (a e - b d\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+a)**6*(B*x+A)/(e*x+d)**12,x)
[Out]
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Mathematica [B] time = 0.964981, size = 605, normalized size = 2.57 \[ -\frac{84 a^6 e^6 (10 A e+B (d+11 e x))+56 a^5 b e^5 \left (9 A e (d+11 e x)+2 B \left (d^2+11 d e x+55 e^2 x^2\right )\right )+35 a^4 b^2 e^4 \left (8 A e \left (d^2+11 d e x+55 e^2 x^2\right )+3 B \left (d^3+11 d^2 e x+55 d e^2 x^2+165 e^3 x^3\right )\right )+20 a^3 b^3 e^3 \left (7 A e \left (d^3+11 d^2 e x+55 d e^2 x^2+165 e^3 x^3\right )+4 B \left (d^4+11 d^3 e x+55 d^2 e^2 x^2+165 d e^3 x^3+330 e^4 x^4\right )\right )+10 a^2 b^4 e^2 \left (6 A e \left (d^4+11 d^3 e x+55 d^2 e^2 x^2+165 d e^3 x^3+330 e^4 x^4\right )+5 B \left (d^5+11 d^4 e x+55 d^3 e^2 x^2+165 d^2 e^3 x^3+330 d e^4 x^4+462 e^5 x^5\right )\right )+4 a b^5 e \left (5 A e \left (d^5+11 d^4 e x+55 d^3 e^2 x^2+165 d^2 e^3 x^3+330 d e^4 x^4+462 e^5 x^5\right )+6 B \left (d^6+11 d^5 e x+55 d^4 e^2 x^2+165 d^3 e^3 x^3+330 d^2 e^4 x^4+462 d e^5 x^5+462 e^6 x^6\right )\right )+b^6 \left (4 A e \left (d^6+11 d^5 e x+55 d^4 e^2 x^2+165 d^3 e^3 x^3+330 d^2 e^4 x^4+462 d e^5 x^5+462 e^6 x^6\right )+7 B \left (d^7+11 d^6 e x+55 d^5 e^2 x^2+165 d^4 e^3 x^3+330 d^3 e^4 x^4+462 d^2 e^5 x^5+462 d e^6 x^6+330 e^7 x^7\right )\right )}{9240 e^8 (d+e x)^{11}} \]
Antiderivative was successfully verified.
[In] Integrate[((a + b*x)^6*(A + B*x))/(d + e*x)^12,x]
[Out]
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Maple [B] time = 0.013, size = 814, normalized size = 3.5 \[ -{\frac{b \left ( 5\,A{a}^{4}b{e}^{5}-20\,A{a}^{3}{b}^{2}d{e}^{4}+30\,A{a}^{2}{b}^{3}{d}^{2}{e}^{3}-20\,Aa{b}^{4}{d}^{3}{e}^{2}+5\,A{b}^{5}{d}^{4}e+2\,B{a}^{5}{e}^{5}-15\,B{a}^{4}bd{e}^{4}+40\,B{a}^{3}{b}^{2}{d}^{2}{e}^{3}-50\,B{a}^{2}{b}^{3}{d}^{3}{e}^{2}+30\,Ba{b}^{4}{d}^{4}e-7\,B{b}^{5}{d}^{5} \right ) }{3\,{e}^{8} \left ( ex+d \right ) ^{9}}}-{\frac{6\,A{a}^{5}b{e}^{6}-30\,Ad{a}^{4}{b}^{2}{e}^{5}+60\,A{d}^{2}{a}^{3}{b}^{3}{e}^{4}-60\,A{d}^{3}{a}^{2}{b}^{4}{e}^{3}+30\,A{d}^{4}a{b}^{5}{e}^{2}-6\,A{d}^{5}{b}^{6}e+{a}^{6}B{e}^{6}-12\,Bd{a}^{5}b{e}^{5}+45\,B{d}^{2}{a}^{4}{b}^{2}{e}^{4}-80\,B{d}^{3}{a}^{3}{b}^{3}{e}^{3}+75\,B{d}^{4}{a}^{2}{b}^{4}{e}^{2}-36\,B{d}^{5}a{b}^{5}e+7\,{b}^{6}B{d}^{6}}{10\,{e}^{8} \left ( ex+d \right ) ^{10}}}-{\frac{{b}^{4} \left ( 2\,Aab{e}^{2}-2\,Ad{b}^{2}e+5\,B{a}^{2}{e}^{2}-12\,Bdabe+7\,{b}^{2}B{d}^{2} \right ) }{2\,{e}^{8} \left ( ex+d \right ) ^{6}}}-{\frac{5\,{b}^{2} \left ( 4\,A{a}^{3}b{e}^{4}-12\,A{a}^{2}{b}^{2}d{e}^{3}+12\,Aa{b}^{3}{d}^{2}{e}^{2}-4\,A{b}^{4}{d}^{3}e+3\,B{a}^{4}{e}^{4}-16\,B{a}^{3}bd{e}^{3}+30\,B{a}^{2}{b}^{2}{d}^{2}{e}^{2}-24\,Ba{b}^{3}{d}^{3}e+7\,B{b}^{4}{d}^{4} \right ) }{8\,{e}^{8} \left ( ex+d \right ) ^{8}}}-{\frac{5\,{b}^{3} \left ( 3\,A{a}^{2}b{e}^{3}-6\,Ada{b}^{2}{e}^{2}+3\,A{b}^{3}{d}^{2}e+4\,B{a}^{3}{e}^{3}-15\,B{a}^{2}bd{e}^{2}+18\,B{d}^{2}a{b}^{2}e-7\,{b}^{3}B{d}^{3} \right ) }{7\,{e}^{8} \left ( ex+d \right ) ^{7}}}-{\frac{{a}^{6}A{e}^{7}-6\,Ad{a}^{5}b{e}^{6}+15\,A{d}^{2}{a}^{4}{b}^{2}{e}^{5}-20\,A{d}^{3}{a}^{3}{b}^{3}{e}^{4}+15\,A{d}^{4}{a}^{2}{b}^{4}{e}^{3}-6\,A{d}^{5}a{b}^{5}{e}^{2}+A{d}^{6}{b}^{6}e-Bd{a}^{6}{e}^{6}+6\,B{d}^{2}{a}^{5}b{e}^{5}-15\,B{d}^{3}{a}^{4}{b}^{2}{e}^{4}+20\,B{d}^{4}{a}^{3}{b}^{3}{e}^{3}-15\,B{d}^{5}{a}^{2}{b}^{4}{e}^{2}+6\,B{d}^{6}a{b}^{5}e-{b}^{6}B{d}^{7}}{11\,{e}^{8} \left ( ex+d \right ) ^{11}}}-{\frac{B{b}^{6}}{4\,{e}^{8} \left ( ex+d \right ) ^{4}}}-{\frac{{b}^{5} \left ( Abe+6\,Bae-7\,Bbd \right ) }{5\,{e}^{8} \left ( ex+d \right ) ^{5}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)^6*(B*x+A)/(e*x+d)^12,x)
[Out]
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Maxima [A] time = 1.42132, size = 1192, normalized size = 5.07 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*(b*x + a)^6/(e*x + d)^12,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.209802, size = 1192, normalized size = 5.07 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*(b*x + a)^6/(e*x + d)^12,x, algorithm="fricas")
[Out]
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x+a)**6*(B*x+A)/(e*x+d)**12,x)
[Out]
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GIAC/XCAS [A] time = 0.230363, size = 1, normalized size = 0. \[ \mathit{Done} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*(b*x + a)^6/(e*x + d)^12,x, algorithm="giac")
[Out]