3.13.96 \(\int \frac {(c+d x)^7}{(a+b x)^{14}} \, dx\) [1296]

Optimal. Leaf size=198 \[ -\frac {(b c-a d)^7}{13 b^8 (a+b x)^{13}}-\frac {7 d (b c-a d)^6}{12 b^8 (a+b x)^{12}}-\frac {21 d^2 (b c-a d)^5}{11 b^8 (a+b x)^{11}}-\frac {7 d^3 (b c-a d)^4}{2 b^8 (a+b x)^{10}}-\frac {35 d^4 (b c-a d)^3}{9 b^8 (a+b x)^9}-\frac {21 d^5 (b c-a d)^2}{8 b^8 (a+b x)^8}-\frac {d^6 (b c-a d)}{b^8 (a+b x)^7}-\frac {d^7}{6 b^8 (a+b x)^6} \]

[Out]

-1/13*(-a*d+b*c)^7/b^8/(b*x+a)^13-7/12*d*(-a*d+b*c)^6/b^8/(b*x+a)^12-21/11*d^2*(-a*d+b*c)^5/b^8/(b*x+a)^11-7/2
*d^3*(-a*d+b*c)^4/b^8/(b*x+a)^10-35/9*d^4*(-a*d+b*c)^3/b^8/(b*x+a)^9-21/8*d^5*(-a*d+b*c)^2/b^8/(b*x+a)^8-d^6*(
-a*d+b*c)/b^8/(b*x+a)^7-1/6*d^7/b^8/(b*x+a)^6

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Rubi [A]
time = 0.11, antiderivative size = 198, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {45} \begin {gather*} -\frac {d^6 (b c-a d)}{b^8 (a+b x)^7}-\frac {21 d^5 (b c-a d)^2}{8 b^8 (a+b x)^8}-\frac {35 d^4 (b c-a d)^3}{9 b^8 (a+b x)^9}-\frac {7 d^3 (b c-a d)^4}{2 b^8 (a+b x)^{10}}-\frac {21 d^2 (b c-a d)^5}{11 b^8 (a+b x)^{11}}-\frac {7 d (b c-a d)^6}{12 b^8 (a+b x)^{12}}-\frac {(b c-a d)^7}{13 b^8 (a+b x)^{13}}-\frac {d^7}{6 b^8 (a+b x)^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^7/(a + b*x)^14,x]

[Out]

-1/13*(b*c - a*d)^7/(b^8*(a + b*x)^13) - (7*d*(b*c - a*d)^6)/(12*b^8*(a + b*x)^12) - (21*d^2*(b*c - a*d)^5)/(1
1*b^8*(a + b*x)^11) - (7*d^3*(b*c - a*d)^4)/(2*b^8*(a + b*x)^10) - (35*d^4*(b*c - a*d)^3)/(9*b^8*(a + b*x)^9)
- (21*d^5*(b*c - a*d)^2)/(8*b^8*(a + b*x)^8) - (d^6*(b*c - a*d))/(b^8*(a + b*x)^7) - d^7/(6*b^8*(a + b*x)^6)

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin {align*} \int \frac {(c+d x)^7}{(a+b x)^{14}} \, dx &=\int \left (\frac {(b c-a d)^7}{b^7 (a+b x)^{14}}+\frac {7 d (b c-a d)^6}{b^7 (a+b x)^{13}}+\frac {21 d^2 (b c-a d)^5}{b^7 (a+b x)^{12}}+\frac {35 d^3 (b c-a d)^4}{b^7 (a+b x)^{11}}+\frac {35 d^4 (b c-a d)^3}{b^7 (a+b x)^{10}}+\frac {21 d^5 (b c-a d)^2}{b^7 (a+b x)^9}+\frac {7 d^6 (b c-a d)}{b^7 (a+b x)^8}+\frac {d^7}{b^7 (a+b x)^7}\right ) \, dx\\ &=-\frac {(b c-a d)^7}{13 b^8 (a+b x)^{13}}-\frac {7 d (b c-a d)^6}{12 b^8 (a+b x)^{12}}-\frac {21 d^2 (b c-a d)^5}{11 b^8 (a+b x)^{11}}-\frac {7 d^3 (b c-a d)^4}{2 b^8 (a+b x)^{10}}-\frac {35 d^4 (b c-a d)^3}{9 b^8 (a+b x)^9}-\frac {21 d^5 (b c-a d)^2}{8 b^8 (a+b x)^8}-\frac {d^6 (b c-a d)}{b^8 (a+b x)^7}-\frac {d^7}{6 b^8 (a+b x)^6}\\ \end {align*}

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Mathematica [A]
time = 0.08, size = 369, normalized size = 1.86 \begin {gather*} -\frac {a^7 d^7+a^6 b d^6 (6 c+13 d x)+3 a^5 b^2 d^5 \left (7 c^2+26 c d x+26 d^2 x^2\right )+a^4 b^3 d^4 \left (56 c^3+273 c^2 d x+468 c d^2 x^2+286 d^3 x^3\right )+a^3 b^4 d^3 \left (126 c^4+728 c^3 d x+1638 c^2 d^2 x^2+1716 c d^3 x^3+715 d^4 x^4\right )+3 a^2 b^5 d^2 \left (84 c^5+546 c^4 d x+1456 c^3 d^2 x^2+2002 c^2 d^3 x^3+1430 c d^4 x^4+429 d^5 x^5\right )+a b^6 d \left (462 c^6+3276 c^5 d x+9828 c^4 d^2 x^2+16016 c^3 d^3 x^3+15015 c^2 d^4 x^4+7722 c d^5 x^5+1716 d^6 x^6\right )+b^7 \left (792 c^7+6006 c^6 d x+19656 c^5 d^2 x^2+36036 c^4 d^3 x^3+40040 c^3 d^4 x^4+27027 c^2 d^5 x^5+10296 c d^6 x^6+1716 d^7 x^7\right )}{10296 b^8 (a+b x)^{13}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)^7/(a + b*x)^14,x]

[Out]

-1/10296*(a^7*d^7 + a^6*b*d^6*(6*c + 13*d*x) + 3*a^5*b^2*d^5*(7*c^2 + 26*c*d*x + 26*d^2*x^2) + a^4*b^3*d^4*(56
*c^3 + 273*c^2*d*x + 468*c*d^2*x^2 + 286*d^3*x^3) + a^3*b^4*d^3*(126*c^4 + 728*c^3*d*x + 1638*c^2*d^2*x^2 + 17
16*c*d^3*x^3 + 715*d^4*x^4) + 3*a^2*b^5*d^2*(84*c^5 + 546*c^4*d*x + 1456*c^3*d^2*x^2 + 2002*c^2*d^3*x^3 + 1430
*c*d^4*x^4 + 429*d^5*x^5) + a*b^6*d*(462*c^6 + 3276*c^5*d*x + 9828*c^4*d^2*x^2 + 16016*c^3*d^3*x^3 + 15015*c^2
*d^4*x^4 + 7722*c*d^5*x^5 + 1716*d^6*x^6) + b^7*(792*c^7 + 6006*c^6*d*x + 19656*c^5*d^2*x^2 + 36036*c^4*d^3*x^
3 + 40040*c^3*d^4*x^4 + 27027*c^2*d^5*x^5 + 10296*c*d^6*x^6 + 1716*d^7*x^7))/(b^8*(a + b*x)^13)

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Mathics [F(-1)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

mathics('Integrate[(c + d*x)^7/(a + b*x)^14,x]')

[Out]

Timed out

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(462\) vs. \(2(184)=368\).
time = 0.17, size = 463, normalized size = 2.34

method	result	size
risch	\(\frac {-\frac {d^{7} x^{7}}{6 b}-\frac {d^{6} \left (a d +6 b c \right ) x^{6}}{6 b^{2}}-\frac {d^{5} \left (a^{2} d^{2}+6 a b c d +21 b^{2} c^{2}\right ) x^{5}}{8 b^{3}}-\frac {5 d^{4} \left (a^{3} d^{3}+6 a^{2} b c \,d^{2}+21 a \,b^{2} c^{2} d +56 b^{3} c^{3}\right ) x^{4}}{72 b^{4}}-\frac {d^{3} \left (a^{4} d^{4}+6 a^{3} b c \,d^{3}+21 a^{2} b^{2} c^{2} d^{2}+56 a \,b^{3} c^{3} d +126 b^{4} c^{4}\right ) x^{3}}{36 b^{5}}-\frac {d^{2} \left (a^{5} d^{5}+6 a^{4} b c \,d^{4}+21 a^{3} b^{2} c^{2} d^{3}+56 a^{2} b^{3} c^{3} d^{2}+126 a \,b^{4} c^{4} d +252 b^{5} c^{5}\right ) x^{2}}{132 b^{6}}-\frac {d \left (a^{6} d^{6}+6 a^{5} b c \,d^{5}+21 a^{4} b^{2} c^{2} d^{4}+56 a^{3} b^{3} c^{3} d^{3}+126 a^{2} b^{4} c^{4} d^{2}+252 a \,b^{5} c^{5} d +462 b^{6} c^{6}\right ) x}{792 b^{7}}-\frac {a^{7} d^{7}+6 a^{6} b c \,d^{6}+21 a^{5} b^{2} c^{2} d^{5}+56 a^{4} b^{3} c^{3} d^{4}+126 a^{3} b^{4} c^{4} d^{3}+252 a^{2} b^{5} c^{5} d^{2}+462 a \,b^{6} c^{6} d +792 b^{7} c^{7}}{10296 b^{8}}}{\left (b x +a \right )^{13}}\)	\(438\)
default	\(-\frac {-a^{7} d^{7}+7 a^{6} b c \,d^{6}-21 a^{5} b^{2} c^{2} d^{5}+35 a^{4} b^{3} c^{3} d^{4}-35 a^{3} b^{4} c^{4} d^{3}+21 a^{2} b^{5} c^{5} d^{2}-7 a \,b^{6} c^{6} d +b^{7} c^{7}}{13 b^{8} \left (b x +a \right )^{13}}-\frac {7 d \left (a^{6} d^{6}-6 a^{5} b c \,d^{5}+15 a^{4} b^{2} c^{2} d^{4}-20 a^{3} b^{3} c^{3} d^{3}+15 a^{2} b^{4} c^{4} d^{2}-6 a \,b^{5} c^{5} d +b^{6} c^{6}\right )}{12 b^{8} \left (b x +a \right )^{12}}+\frac {35 d^{4} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}{9 b^{8} \left (b x +a \right )^{9}}-\frac {21 d^{5} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}{8 b^{8} \left (b x +a \right )^{8}}+\frac {d^{6} \left (a d -b c \right )}{b^{8} \left (b x +a \right )^{7}}-\frac {7 d^{3} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}{2 b^{8} \left (b x +a \right )^{10}}-\frac {d^{7}}{6 b^{8} \left (b x +a \right )^{6}}+\frac {21 d^{2} \left (a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -b^{5} c^{5}\right )}{11 b^{8} \left (b x +a \right )^{11}}\)	\(463\)
norman	\(\frac {-\frac {d^{7} x^{7}}{6 b}+\frac {\left (-a \,b^{5} d^{7}-6 b^{6} c \,d^{6}\right ) x^{6}}{6 b^{7}}+\frac {\left (-a^{2} b^{5} d^{7}-6 a \,b^{6} c \,d^{6}-21 b^{7} c^{2} d^{5}\right ) x^{5}}{8 b^{8}}+\frac {5 \left (-a^{3} b^{5} d^{7}-6 a^{2} b^{6} c \,d^{6}-21 a \,b^{7} c^{2} d^{5}-56 b^{8} c^{3} d^{4}\right ) x^{4}}{72 b^{9}}+\frac {\left (-a^{4} b^{5} d^{7}-6 a^{3} b^{6} c \,d^{6}-21 a^{2} b^{7} c^{2} d^{5}-56 a \,b^{8} c^{3} d^{4}-126 b^{9} c^{4} d^{3}\right ) x^{3}}{36 b^{10}}+\frac {\left (-a^{5} b^{5} d^{7}-6 a^{4} b^{6} c \,d^{6}-21 a^{3} b^{7} c^{2} d^{5}-56 a^{2} b^{8} c^{3} d^{4}-126 a \,b^{9} c^{4} d^{3}-252 b^{10} c^{5} d^{2}\right ) x^{2}}{132 b^{11}}+\frac {\left (-a^{6} b^{5} d^{7}-6 a^{5} b^{6} c \,d^{6}-21 a^{4} b^{7} c^{2} d^{5}-56 a^{3} b^{8} c^{3} d^{4}-126 a^{2} b^{9} c^{4} d^{3}-252 a \,b^{10} c^{5} d^{2}-462 b^{11} c^{6} d \right ) x}{792 b^{12}}+\frac {-a^{7} b^{5} d^{7}-6 a^{6} b^{6} c \,d^{6}-21 a^{5} b^{7} c^{2} d^{5}-56 a^{4} b^{8} c^{3} d^{4}-126 a^{3} b^{9} c^{4} d^{3}-252 a^{2} b^{10} c^{5} d^{2}-462 a \,b^{11} c^{6} d -792 b^{12} c^{7}}{10296 b^{13}}}{\left (b x +a \right )^{13}}\)	\(492\)
gosper	\(-\frac {1716 d^{7} x^{7} b^{7}+1716 a \,b^{6} d^{7} x^{6}+10296 b^{7} c \,d^{6} x^{6}+1287 a^{2} b^{5} d^{7} x^{5}+7722 a \,b^{6} c \,d^{6} x^{5}+27027 b^{7} c^{2} d^{5} x^{5}+715 a^{3} b^{4} d^{7} x^{4}+4290 a^{2} b^{5} c \,d^{6} x^{4}+15015 a \,b^{6} c^{2} d^{5} x^{4}+40040 b^{7} c^{3} d^{4} x^{4}+286 a^{4} b^{3} d^{7} x^{3}+1716 a^{3} b^{4} c \,d^{6} x^{3}+6006 a^{2} b^{5} c^{2} d^{5} x^{3}+16016 a \,b^{6} c^{3} d^{4} x^{3}+36036 b^{7} c^{4} d^{3} x^{3}+78 a^{5} b^{2} d^{7} x^{2}+468 a^{4} b^{3} c \,d^{6} x^{2}+1638 a^{3} b^{4} c^{2} d^{5} x^{2}+4368 a^{2} b^{5} c^{3} d^{4} x^{2}+9828 a \,b^{6} c^{4} d^{3} x^{2}+19656 b^{7} c^{5} d^{2} x^{2}+13 a^{6} b \,d^{7} x +78 a^{5} b^{2} c \,d^{6} x +273 a^{4} b^{3} c^{2} d^{5} x +728 a^{3} b^{4} c^{3} d^{4} x +1638 a^{2} b^{5} c^{4} d^{3} x +3276 a \,b^{6} c^{5} d^{2} x +6006 b^{7} c^{6} d x +a^{7} d^{7}+6 a^{6} b c \,d^{6}+21 a^{5} b^{2} c^{2} d^{5}+56 a^{4} b^{3} c^{3} d^{4}+126 a^{3} b^{4} c^{4} d^{3}+252 a^{2} b^{5} c^{5} d^{2}+462 a \,b^{6} c^{6} d +792 b^{7} c^{7}}{10296 b^{8} \left (b x +a \right )^{13}}\)	\(497\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)^7/(b*x+a)^14,x,method=_RETURNVERBOSE)

[Out]

-1/13*(-a^7*d^7+7*a^6*b*c*d^6-21*a^5*b^2*c^2*d^5+35*a^4*b^3*c^3*d^4-35*a^3*b^4*c^4*d^3+21*a^2*b^5*c^5*d^2-7*a*
b^6*c^6*d+b^7*c^7)/b^8/(b*x+a)^13-7/12/b^8*d*(a^6*d^6-6*a^5*b*c*d^5+15*a^4*b^2*c^2*d^4-20*a^3*b^3*c^3*d^3+15*a
^2*b^4*c^4*d^2-6*a*b^5*c^5*d+b^6*c^6)/(b*x+a)^12+35/9/b^8*d^4*(a^3*d^3-3*a^2*b*c*d^2+3*a*b^2*c^2*d-b^3*c^3)/(b
*x+a)^9-21/8/b^8*d^5*(a^2*d^2-2*a*b*c*d+b^2*c^2)/(b*x+a)^8+1/b^8*d^6*(a*d-b*c)/(b*x+a)^7-7/2/b^8*d^3*(a^4*d^4-
4*a^3*b*c*d^3+6*a^2*b^2*c^2*d^2-4*a*b^3*c^3*d+b^4*c^4)/(b*x+a)^10-1/6*d^7/b^8/(b*x+a)^6+21/11/b^8*d^2*(a^5*d^5
-5*a^4*b*c*d^4+10*a^3*b^2*c^2*d^3-10*a^2*b^3*c^3*d^2+5*a*b^4*c^4*d-b^5*c^5)/(b*x+a)^11

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 592 vs. \(2 (184) = 368\).
time = 0.31, size = 592, normalized size = 2.99 \begin {gather*} -\frac {1716 \, b^{7} d^{7} x^{7} + 792 \, b^{7} c^{7} + 462 \, a b^{6} c^{6} d + 252 \, a^{2} b^{5} c^{5} d^{2} + 126 \, a^{3} b^{4} c^{4} d^{3} + 56 \, a^{4} b^{3} c^{3} d^{4} + 21 \, a^{5} b^{2} c^{2} d^{5} + 6 \, a^{6} b c d^{6} + a^{7} d^{7} + 1716 \, {\left (6 \, b^{7} c d^{6} + a b^{6} d^{7}\right )} x^{6} + 1287 \, {\left (21 \, b^{7} c^{2} d^{5} + 6 \, a b^{6} c d^{6} + a^{2} b^{5} d^{7}\right )} x^{5} + 715 \, {\left (56 \, b^{7} c^{3} d^{4} + 21 \, a b^{6} c^{2} d^{5} + 6 \, a^{2} b^{5} c d^{6} + a^{3} b^{4} d^{7}\right )} x^{4} + 286 \, {\left (126 \, b^{7} c^{4} d^{3} + 56 \, a b^{6} c^{3} d^{4} + 21 \, a^{2} b^{5} c^{2} d^{5} + 6 \, a^{3} b^{4} c d^{6} + a^{4} b^{3} d^{7}\right )} x^{3} + 78 \, {\left (252 \, b^{7} c^{5} d^{2} + 126 \, a b^{6} c^{4} d^{3} + 56 \, a^{2} b^{5} c^{3} d^{4} + 21 \, a^{3} b^{4} c^{2} d^{5} + 6 \, a^{4} b^{3} c d^{6} + a^{5} b^{2} d^{7}\right )} x^{2} + 13 \, {\left (462 \, b^{7} c^{6} d + 252 \, a b^{6} c^{5} d^{2} + 126 \, a^{2} b^{5} c^{4} d^{3} + 56 \, a^{3} b^{4} c^{3} d^{4} + 21 \, a^{4} b^{3} c^{2} d^{5} + 6 \, a^{5} b^{2} c d^{6} + a^{6} b d^{7}\right )} x}{10296 \, {\left (b^{21} x^{13} + 13 \, a b^{20} x^{12} + 78 \, a^{2} b^{19} x^{11} + 286 \, a^{3} b^{18} x^{10} + 715 \, a^{4} b^{17} x^{9} + 1287 \, a^{5} b^{16} x^{8} + 1716 \, a^{6} b^{15} x^{7} + 1716 \, a^{7} b^{14} x^{6} + 1287 \, a^{8} b^{13} x^{5} + 715 \, a^{9} b^{12} x^{4} + 286 \, a^{10} b^{11} x^{3} + 78 \, a^{11} b^{10} x^{2} + 13 \, a^{12} b^{9} x + a^{13} b^{8}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^7/(b*x+a)^14,x, algorithm="maxima")

[Out]

-1/10296*(1716*b^7*d^7*x^7 + 792*b^7*c^7 + 462*a*b^6*c^6*d + 252*a^2*b^5*c^5*d^2 + 126*a^3*b^4*c^4*d^3 + 56*a^
4*b^3*c^3*d^4 + 21*a^5*b^2*c^2*d^5 + 6*a^6*b*c*d^6 + a^7*d^7 + 1716*(6*b^7*c*d^6 + a*b^6*d^7)*x^6 + 1287*(21*b
^7*c^2*d^5 + 6*a*b^6*c*d^6 + a^2*b^5*d^7)*x^5 + 715*(56*b^7*c^3*d^4 + 21*a*b^6*c^2*d^5 + 6*a^2*b^5*c*d^6 + a^3
*b^4*d^7)*x^4 + 286*(126*b^7*c^4*d^3 + 56*a*b^6*c^3*d^4 + 21*a^2*b^5*c^2*d^5 + 6*a^3*b^4*c*d^6 + a^4*b^3*d^7)*
x^3 + 78*(252*b^7*c^5*d^2 + 126*a*b^6*c^4*d^3 + 56*a^2*b^5*c^3*d^4 + 21*a^3*b^4*c^2*d^5 + 6*a^4*b^3*c*d^6 + a^
5*b^2*d^7)*x^2 + 13*(462*b^7*c^6*d + 252*a*b^6*c^5*d^2 + 126*a^2*b^5*c^4*d^3 + 56*a^3*b^4*c^3*d^4 + 21*a^4*b^3
*c^2*d^5 + 6*a^5*b^2*c*d^6 + a^6*b*d^7)*x)/(b^21*x^13 + 13*a*b^20*x^12 + 78*a^2*b^19*x^11 + 286*a^3*b^18*x^10
+ 715*a^4*b^17*x^9 + 1287*a^5*b^16*x^8 + 1716*a^6*b^15*x^7 + 1716*a^7*b^14*x^6 + 1287*a^8*b^13*x^5 + 715*a^9*b
^12*x^4 + 286*a^10*b^11*x^3 + 78*a^11*b^10*x^2 + 13*a^12*b^9*x + a^13*b^8)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 592 vs. \(2 (184) = 368\).
time = 0.30, size = 592, normalized size = 2.99 \begin {gather*} -\frac {1716 \, b^{7} d^{7} x^{7} + 792 \, b^{7} c^{7} + 462 \, a b^{6} c^{6} d + 252 \, a^{2} b^{5} c^{5} d^{2} + 126 \, a^{3} b^{4} c^{4} d^{3} + 56 \, a^{4} b^{3} c^{3} d^{4} + 21 \, a^{5} b^{2} c^{2} d^{5} + 6 \, a^{6} b c d^{6} + a^{7} d^{7} + 1716 \, {\left (6 \, b^{7} c d^{6} + a b^{6} d^{7}\right )} x^{6} + 1287 \, {\left (21 \, b^{7} c^{2} d^{5} + 6 \, a b^{6} c d^{6} + a^{2} b^{5} d^{7}\right )} x^{5} + 715 \, {\left (56 \, b^{7} c^{3} d^{4} + 21 \, a b^{6} c^{2} d^{5} + 6 \, a^{2} b^{5} c d^{6} + a^{3} b^{4} d^{7}\right )} x^{4} + 286 \, {\left (126 \, b^{7} c^{4} d^{3} + 56 \, a b^{6} c^{3} d^{4} + 21 \, a^{2} b^{5} c^{2} d^{5} + 6 \, a^{3} b^{4} c d^{6} + a^{4} b^{3} d^{7}\right )} x^{3} + 78 \, {\left (252 \, b^{7} c^{5} d^{2} + 126 \, a b^{6} c^{4} d^{3} + 56 \, a^{2} b^{5} c^{3} d^{4} + 21 \, a^{3} b^{4} c^{2} d^{5} + 6 \, a^{4} b^{3} c d^{6} + a^{5} b^{2} d^{7}\right )} x^{2} + 13 \, {\left (462 \, b^{7} c^{6} d + 252 \, a b^{6} c^{5} d^{2} + 126 \, a^{2} b^{5} c^{4} d^{3} + 56 \, a^{3} b^{4} c^{3} d^{4} + 21 \, a^{4} b^{3} c^{2} d^{5} + 6 \, a^{5} b^{2} c d^{6} + a^{6} b d^{7}\right )} x}{10296 \, {\left (b^{21} x^{13} + 13 \, a b^{20} x^{12} + 78 \, a^{2} b^{19} x^{11} + 286 \, a^{3} b^{18} x^{10} + 715 \, a^{4} b^{17} x^{9} + 1287 \, a^{5} b^{16} x^{8} + 1716 \, a^{6} b^{15} x^{7} + 1716 \, a^{7} b^{14} x^{6} + 1287 \, a^{8} b^{13} x^{5} + 715 \, a^{9} b^{12} x^{4} + 286 \, a^{10} b^{11} x^{3} + 78 \, a^{11} b^{10} x^{2} + 13 \, a^{12} b^{9} x + a^{13} b^{8}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^7/(b*x+a)^14,x, algorithm="fricas")

[Out]

-1/10296*(1716*b^7*d^7*x^7 + 792*b^7*c^7 + 462*a*b^6*c^6*d + 252*a^2*b^5*c^5*d^2 + 126*a^3*b^4*c^4*d^3 + 56*a^
4*b^3*c^3*d^4 + 21*a^5*b^2*c^2*d^5 + 6*a^6*b*c*d^6 + a^7*d^7 + 1716*(6*b^7*c*d^6 + a*b^6*d^7)*x^6 + 1287*(21*b
^7*c^2*d^5 + 6*a*b^6*c*d^6 + a^2*b^5*d^7)*x^5 + 715*(56*b^7*c^3*d^4 + 21*a*b^6*c^2*d^5 + 6*a^2*b^5*c*d^6 + a^3
*b^4*d^7)*x^4 + 286*(126*b^7*c^4*d^3 + 56*a*b^6*c^3*d^4 + 21*a^2*b^5*c^2*d^5 + 6*a^3*b^4*c*d^6 + a^4*b^3*d^7)*
x^3 + 78*(252*b^7*c^5*d^2 + 126*a*b^6*c^4*d^3 + 56*a^2*b^5*c^3*d^4 + 21*a^3*b^4*c^2*d^5 + 6*a^4*b^3*c*d^6 + a^
5*b^2*d^7)*x^2 + 13*(462*b^7*c^6*d + 252*a*b^6*c^5*d^2 + 126*a^2*b^5*c^4*d^3 + 56*a^3*b^4*c^3*d^4 + 21*a^4*b^3
*c^2*d^5 + 6*a^5*b^2*c*d^6 + a^6*b*d^7)*x)/(b^21*x^13 + 13*a*b^20*x^12 + 78*a^2*b^19*x^11 + 286*a^3*b^18*x^10
+ 715*a^4*b^17*x^9 + 1287*a^5*b^16*x^8 + 1716*a^6*b^15*x^7 + 1716*a^7*b^14*x^6 + 1287*a^8*b^13*x^5 + 715*a^9*b
^12*x^4 + 286*a^10*b^11*x^3 + 78*a^11*b^10*x^2 + 13*a^12*b^9*x + a^13*b^8)

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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.

[In]

integrate((d*x+c)**7/(b*x+a)**14,x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 496 vs. \(2 (184) = 368\).
time = 0.00, size = 534, normalized size = 2.70 \begin {gather*} \frac {-1716 x^{7} d^{7} b^{7}-1716 x^{6} d^{7} b^{6} a-10296 x^{6} d^{6} c b^{7}-1287 x^{5} d^{7} b^{5} a^{2}-7722 x^{5} d^{6} c b^{6} a-27027 x^{5} d^{5} c^{2} b^{7}-715 x^{4} d^{7} b^{4} a^{3}-4290 x^{4} d^{6} c b^{5} a^{2}-15015 x^{4} d^{5} c^{2} b^{6} a-40040 x^{4} d^{4} c^{3} b^{7}-286 x^{3} d^{7} b^{3} a^{4}-1716 x^{3} d^{6} c b^{4} a^{3}-6006 x^{3} d^{5} c^{2} b^{5} a^{2}-16016 x^{3} d^{4} c^{3} b^{6} a-36036 x^{3} d^{3} c^{4} b^{7}-78 x^{2} d^{7} b^{2} a^{5}-468 x^{2} d^{6} c b^{3} a^{4}-1638 x^{2} d^{5} c^{2} b^{4} a^{3}-4368 x^{2} d^{4} c^{3} b^{5} a^{2}-9828 x^{2} d^{3} c^{4} b^{6} a-19656 x^{2} d^{2} c^{5} b^{7}-13 x d^{7} b a^{6}-78 x d^{6} c b^{2} a^{5}-273 x d^{5} c^{2} b^{3} a^{4}-728 x d^{4} c^{3} b^{4} a^{3}-1638 x d^{3} c^{4} b^{5} a^{2}-3276 x d^{2} c^{5} b^{6} a-6006 x d c^{6} b^{7}-d^{7} a^{7}-6 d^{6} c b a^{6}-21 d^{5} c^{2} b^{2} a^{5}-56 d^{4} c^{3} b^{3} a^{4}-126 d^{3} c^{4} b^{4} a^{3}-252 d^{2} c^{5} b^{5} a^{2}-462 d c^{6} b^{6} a-792 c^{7} b^{7}}{10296 b^{8} \left (x b+a\right )^{13}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^7/(b*x+a)^14,x)

[Out]

-1/10296*(1716*b^7*d^7*x^7 + 10296*b^7*c*d^6*x^6 + 1716*a*b^6*d^7*x^6 + 27027*b^7*c^2*d^5*x^5 + 7722*a*b^6*c*d
^6*x^5 + 1287*a^2*b^5*d^7*x^5 + 40040*b^7*c^3*d^4*x^4 + 15015*a*b^6*c^2*d^5*x^4 + 4290*a^2*b^5*c*d^6*x^4 + 715
*a^3*b^4*d^7*x^4 + 36036*b^7*c^4*d^3*x^3 + 16016*a*b^6*c^3*d^4*x^3 + 6006*a^2*b^5*c^2*d^5*x^3 + 1716*a^3*b^4*c
*d^6*x^3 + 286*a^4*b^3*d^7*x^3 + 19656*b^7*c^5*d^2*x^2 + 9828*a*b^6*c^4*d^3*x^2 + 4368*a^2*b^5*c^3*d^4*x^2 + 1
638*a^3*b^4*c^2*d^5*x^2 + 468*a^4*b^3*c*d^6*x^2 + 78*a^5*b^2*d^7*x^2 + 6006*b^7*c^6*d*x + 3276*a*b^6*c^5*d^2*x
+ 1638*a^2*b^5*c^4*d^3*x + 728*a^3*b^4*c^3*d^4*x + 273*a^4*b^3*c^2*d^5*x + 78*a^5*b^2*c*d^6*x + 13*a^6*b*d^7*
x + 792*b^7*c^7 + 462*a*b^6*c^6*d + 252*a^2*b^5*c^5*d^2 + 126*a^3*b^4*c^4*d^3 + 56*a^4*b^3*c^3*d^4 + 21*a^5*b^
2*c^2*d^5 + 6*a^6*b*c*d^6 + a^7*d^7)/((b*x + a)^13*b^8)

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Mupad [B]
time = 0.40, size = 570, normalized size = 2.88 \begin {gather*} -\frac {\frac {a^7\,d^7+6\,a^6\,b\,c\,d^6+21\,a^5\,b^2\,c^2\,d^5+56\,a^4\,b^3\,c^3\,d^4+126\,a^3\,b^4\,c^4\,d^3+252\,a^2\,b^5\,c^5\,d^2+462\,a\,b^6\,c^6\,d+792\,b^7\,c^7}{10296\,b^8}+\frac {d^7\,x^7}{6\,b}+\frac {d^2\,x^2\,\left (a^5\,d^5+6\,a^4\,b\,c\,d^4+21\,a^3\,b^2\,c^2\,d^3+56\,a^2\,b^3\,c^3\,d^2+126\,a\,b^4\,c^4\,d+252\,b^5\,c^5\right )}{132\,b^6}+\frac {5\,d^4\,x^4\,\left (a^3\,d^3+6\,a^2\,b\,c\,d^2+21\,a\,b^2\,c^2\,d+56\,b^3\,c^3\right )}{72\,b^4}+\frac {d^6\,x^6\,\left (a\,d+6\,b\,c\right )}{6\,b^2}+\frac {d^3\,x^3\,\left (a^4\,d^4+6\,a^3\,b\,c\,d^3+21\,a^2\,b^2\,c^2\,d^2+56\,a\,b^3\,c^3\,d+126\,b^4\,c^4\right )}{36\,b^5}+\frac {d\,x\,\left (a^6\,d^6+6\,a^5\,b\,c\,d^5+21\,a^4\,b^2\,c^2\,d^4+56\,a^3\,b^3\,c^3\,d^3+126\,a^2\,b^4\,c^4\,d^2+252\,a\,b^5\,c^5\,d+462\,b^6\,c^6\right )}{792\,b^7}+\frac {d^5\,x^5\,\left (a^2\,d^2+6\,a\,b\,c\,d+21\,b^2\,c^2\right )}{8\,b^3}}{a^{13}+13\,a^{12}\,b\,x+78\,a^{11}\,b^2\,x^2+286\,a^{10}\,b^3\,x^3+715\,a^9\,b^4\,x^4+1287\,a^8\,b^5\,x^5+1716\,a^7\,b^6\,x^6+1716\,a^6\,b^7\,x^7+1287\,a^5\,b^8\,x^8+715\,a^4\,b^9\,x^9+286\,a^3\,b^{10}\,x^{10}+78\,a^2\,b^{11}\,x^{11}+13\,a\,b^{12}\,x^{12}+b^{13}\,x^{13}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x)^7/(a + b*x)^14,x)

[Out]

-((a^7*d^7 + 792*b^7*c^7 + 252*a^2*b^5*c^5*d^2 + 126*a^3*b^4*c^4*d^3 + 56*a^4*b^3*c^3*d^4 + 21*a^5*b^2*c^2*d^5
+ 462*a*b^6*c^6*d + 6*a^6*b*c*d^6)/(10296*b^8) + (d^7*x^7)/(6*b) + (d^2*x^2*(a^5*d^5 + 252*b^5*c^5 + 56*a^2*b
^3*c^3*d^2 + 21*a^3*b^2*c^2*d^3 + 126*a*b^4*c^4*d + 6*a^4*b*c*d^4))/(132*b^6) + (5*d^4*x^4*(a^3*d^3 + 56*b^3*c
^3 + 21*a*b^2*c^2*d + 6*a^2*b*c*d^2))/(72*b^4) + (d^6*x^6*(a*d + 6*b*c))/(6*b^2) + (d^3*x^3*(a^4*d^4 + 126*b^4
*c^4 + 21*a^2*b^2*c^2*d^2 + 56*a*b^3*c^3*d + 6*a^3*b*c*d^3))/(36*b^5) + (d*x*(a^6*d^6 + 462*b^6*c^6 + 126*a^2*
b^4*c^4*d^2 + 56*a^3*b^3*c^3*d^3 + 21*a^4*b^2*c^2*d^4 + 252*a*b^5*c^5*d + 6*a^5*b*c*d^5))/(792*b^7) + (d^5*x^5
*(a^2*d^2 + 21*b^2*c^2 + 6*a*b*c*d))/(8*b^3))/(a^13 + b^13*x^13 + 13*a*b^12*x^12 + 78*a^11*b^2*x^2 + 286*a^10*
b^3*x^3 + 715*a^9*b^4*x^4 + 1287*a^8*b^5*x^5 + 1716*a^7*b^6*x^6 + 1716*a^6*b^7*x^7 + 1287*a^5*b^8*x^8 + 715*a^
4*b^9*x^9 + 286*a^3*b^10*x^10 + 78*a^2*b^11*x^11 + 13*a^12*b*x)

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