3.13.95 \(\int \frac {(c+d x)^7}{(a+b x)^{13}} \, dx\) [1295]

Optimal. Leaf size=151 \[ -\frac {(c+d x)^8}{12 (b c-a d) (a+b x)^{12}}+\frac {d (c+d x)^8}{33 (b c-a d)^2 (a+b x)^{11}}-\frac {d^2 (c+d x)^8}{110 (b c-a d)^3 (a+b x)^{10}}+\frac {d^3 (c+d x)^8}{495 (b c-a d)^4 (a+b x)^9}-\frac {d^4 (c+d x)^8}{3960 (b c-a d)^5 (a+b x)^8} \]

[Out]

-1/12*(d*x+c)^8/(-a*d+b*c)/(b*x+a)^12+1/33*d*(d*x+c)^8/(-a*d+b*c)^2/(b*x+a)^11-1/110*d^2*(d*x+c)^8/(-a*d+b*c)^
3/(b*x+a)^10+1/495*d^3*(d*x+c)^8/(-a*d+b*c)^4/(b*x+a)^9-1/3960*d^4*(d*x+c)^8/(-a*d+b*c)^5/(b*x+a)^8

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Rubi [A]
time = 0.03, antiderivative size = 151, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {47, 37} \begin {gather*} -\frac {d^4 (c+d x)^8}{3960 (a+b x)^8 (b c-a d)^5}+\frac {d^3 (c+d x)^8}{495 (a+b x)^9 (b c-a d)^4}-\frac {d^2 (c+d x)^8}{110 (a+b x)^{10} (b c-a d)^3}+\frac {d (c+d x)^8}{33 (a+b x)^{11} (b c-a d)^2}-\frac {(c+d x)^8}{12 (a+b x)^{12} (b c-a d)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^7/(a + b*x)^13,x]

[Out]

-1/12*(c + d*x)^8/((b*c - a*d)*(a + b*x)^12) + (d*(c + d*x)^8)/(33*(b*c - a*d)^2*(a + b*x)^11) - (d^2*(c + d*x
)^8)/(110*(b*c - a*d)^3*(a + b*x)^10) + (d^3*(c + d*x)^8)/(495*(b*c - a*d)^4*(a + b*x)^9) - (d^4*(c + d*x)^8)/
(3960*(b*c - a*d)^5*(a + b*x)^8)

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n +
1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rule 47

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n + 1
)/((b*c - a*d)*(m + 1))), x] - Dist[d*(Simplify[m + n + 2]/((b*c - a*d)*(m + 1))), Int[(a + b*x)^Simplify[m +
1]*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[Simplify[m + n + 2], 0] &&
NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (
SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 1])

Rubi steps

\begin {align*} \int \frac {(c+d x)^7}{(a+b x)^{13}} \, dx &=-\frac {(c+d x)^8}{12 (b c-a d) (a+b x)^{12}}-\frac {d \int \frac {(c+d x)^7}{(a+b x)^{12}} \, dx}{3 (b c-a d)}\\ &=-\frac {(c+d x)^8}{12 (b c-a d) (a+b x)^{12}}+\frac {d (c+d x)^8}{33 (b c-a d)^2 (a+b x)^{11}}+\frac {d^2 \int \frac {(c+d x)^7}{(a+b x)^{11}} \, dx}{11 (b c-a d)^2}\\ &=-\frac {(c+d x)^8}{12 (b c-a d) (a+b x)^{12}}+\frac {d (c+d x)^8}{33 (b c-a d)^2 (a+b x)^{11}}-\frac {d^2 (c+d x)^8}{110 (b c-a d)^3 (a+b x)^{10}}-\frac {d^3 \int \frac {(c+d x)^7}{(a+b x)^{10}} \, dx}{55 (b c-a d)^3}\\ &=-\frac {(c+d x)^8}{12 (b c-a d) (a+b x)^{12}}+\frac {d (c+d x)^8}{33 (b c-a d)^2 (a+b x)^{11}}-\frac {d^2 (c+d x)^8}{110 (b c-a d)^3 (a+b x)^{10}}+\frac {d^3 (c+d x)^8}{495 (b c-a d)^4 (a+b x)^9}+\frac {d^4 \int \frac {(c+d x)^7}{(a+b x)^9} \, dx}{495 (b c-a d)^4}\\ &=-\frac {(c+d x)^8}{12 (b c-a d) (a+b x)^{12}}+\frac {d (c+d x)^8}{33 (b c-a d)^2 (a+b x)^{11}}-\frac {d^2 (c+d x)^8}{110 (b c-a d)^3 (a+b x)^{10}}+\frac {d^3 (c+d x)^8}{495 (b c-a d)^4 (a+b x)^9}-\frac {d^4 (c+d x)^8}{3960 (b c-a d)^5 (a+b x)^8}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(371\) vs. \(2(151)=302\).
time = 0.08, size = 371, normalized size = 2.46 \begin {gather*} -\frac {a^7 d^7+a^6 b d^6 (5 c+12 d x)+3 a^5 b^2 d^5 \left (5 c^2+20 c d x+22 d^2 x^2\right )+5 a^4 b^3 d^4 \left (7 c^3+36 c^2 d x+66 c d^2 x^2+44 d^3 x^3\right )+5 a^3 b^4 d^3 \left (14 c^4+84 c^3 d x+198 c^2 d^2 x^2+220 c d^3 x^3+99 d^4 x^4\right )+3 a^2 b^5 d^2 \left (42 c^5+280 c^4 d x+770 c^3 d^2 x^2+1100 c^2 d^3 x^3+825 c d^4 x^4+264 d^5 x^5\right )+a b^6 d \left (210 c^6+1512 c^5 d x+4620 c^4 d^2 x^2+7700 c^3 d^3 x^3+7425 c^2 d^4 x^4+3960 c d^5 x^5+924 d^6 x^6\right )+b^7 \left (330 c^7+2520 c^6 d x+8316 c^5 d^2 x^2+15400 c^4 d^3 x^3+17325 c^3 d^4 x^4+11880 c^2 d^5 x^5+4620 c d^6 x^6+792 d^7 x^7\right )}{3960 b^8 (a+b x)^{12}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)^7/(a + b*x)^13,x]

[Out]

-1/3960*(a^7*d^7 + a^6*b*d^6*(5*c + 12*d*x) + 3*a^5*b^2*d^5*(5*c^2 + 20*c*d*x + 22*d^2*x^2) + 5*a^4*b^3*d^4*(7
*c^3 + 36*c^2*d*x + 66*c*d^2*x^2 + 44*d^3*x^3) + 5*a^3*b^4*d^3*(14*c^4 + 84*c^3*d*x + 198*c^2*d^2*x^2 + 220*c*
d^3*x^3 + 99*d^4*x^4) + 3*a^2*b^5*d^2*(42*c^5 + 280*c^4*d*x + 770*c^3*d^2*x^2 + 1100*c^2*d^3*x^3 + 825*c*d^4*x
^4 + 264*d^5*x^5) + a*b^6*d*(210*c^6 + 1512*c^5*d*x + 4620*c^4*d^2*x^2 + 7700*c^3*d^3*x^3 + 7425*c^2*d^4*x^4 +
3960*c*d^5*x^5 + 924*d^6*x^6) + b^7*(330*c^7 + 2520*c^6*d*x + 8316*c^5*d^2*x^2 + 15400*c^4*d^3*x^3 + 17325*c^
3*d^4*x^4 + 11880*c^2*d^5*x^5 + 4620*c*d^6*x^6 + 792*d^7*x^7))/(b^8*(a + b*x)^12)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(463\) vs. \(2(141)=282\).
time = 0.14, size = 464, normalized size = 3.07

method	result	size
risch	\(\frac {-\frac {d^{7} x^{7}}{5 b}-\frac {7 d^{6} \left (a d +5 b c \right ) x^{6}}{30 b^{2}}-\frac {d^{5} \left (a^{2} d^{2}+5 a b c d +15 b^{2} c^{2}\right ) x^{5}}{5 b^{3}}-\frac {d^{4} \left (a^{3} d^{3}+5 a^{2} b c \,d^{2}+15 a \,b^{2} c^{2} d +35 b^{3} c^{3}\right ) x^{4}}{8 b^{4}}-\frac {d^{3} \left (a^{4} d^{4}+5 a^{3} b c \,d^{3}+15 a^{2} b^{2} c^{2} d^{2}+35 a \,b^{3} c^{3} d +70 b^{4} c^{4}\right ) x^{3}}{18 b^{5}}-\frac {d^{2} \left (a^{5} d^{5}+5 a^{4} b c \,d^{4}+15 a^{3} b^{2} c^{2} d^{3}+35 a^{2} b^{3} c^{3} d^{2}+70 a \,b^{4} c^{4} d +126 b^{5} c^{5}\right ) x^{2}}{60 b^{6}}-\frac {d \left (a^{6} d^{6}+5 a^{5} b c \,d^{5}+15 a^{4} b^{2} c^{2} d^{4}+35 a^{3} b^{3} c^{3} d^{3}+70 a^{2} b^{4} c^{4} d^{2}+126 a \,b^{5} c^{5} d +210 b^{6} c^{6}\right ) x}{330 b^{7}}-\frac {a^{7} d^{7}+5 a^{6} b c \,d^{6}+15 a^{5} b^{2} c^{2} d^{5}+35 a^{4} b^{3} c^{3} d^{4}+70 a^{3} b^{4} c^{4} d^{3}+126 a^{2} b^{5} c^{5} d^{2}+210 a \,b^{6} c^{6} d +330 b^{7} c^{7}}{3960 b^{8}}}{\left (b x +a \right )^{12}}\)	\(438\)
default	\(-\frac {-a^{7} d^{7}+7 a^{6} b c \,d^{6}-21 a^{5} b^{2} c^{2} d^{5}+35 a^{4} b^{3} c^{3} d^{4}-35 a^{3} b^{4} c^{4} d^{3}+21 a^{2} b^{5} c^{5} d^{2}-7 a \,b^{6} c^{6} d +b^{7} c^{7}}{12 b^{8} \left (b x +a \right )^{12}}-\frac {35 d^{3} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}{9 b^{8} \left (b x +a \right )^{9}}+\frac {35 d^{4} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}{8 b^{8} \left (b x +a \right )^{8}}-\frac {d^{7}}{5 b^{8} \left (b x +a \right )^{5}}-\frac {3 d^{5} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}{b^{8} \left (b x +a \right )^{7}}+\frac {21 d^{2} \left (a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -b^{5} c^{5}\right )}{10 b^{8} \left (b x +a \right )^{10}}+\frac {7 d^{6} \left (a d -b c \right )}{6 b^{8} \left (b x +a \right )^{6}}-\frac {7 d \left (a^{6} d^{6}-6 a^{5} b c \,d^{5}+15 a^{4} b^{2} c^{2} d^{4}-20 a^{3} b^{3} c^{3} d^{3}+15 a^{2} b^{4} c^{4} d^{2}-6 a \,b^{5} c^{5} d +b^{6} c^{6}\right )}{11 b^{8} \left (b x +a \right )^{11}}\)	\(464\)
norman	\(\frac {-\frac {d^{7} x^{7}}{5 b}+\frac {7 \left (-a \,b^{4} d^{7}-5 b^{5} c \,d^{6}\right ) x^{6}}{30 b^{6}}+\frac {\left (-a^{2} b^{4} d^{7}-5 a \,b^{5} c \,d^{6}-15 b^{6} c^{2} d^{5}\right ) x^{5}}{5 b^{7}}+\frac {\left (-a^{3} b^{4} d^{7}-5 a^{2} b^{5} c \,d^{6}-15 a \,b^{6} c^{2} d^{5}-35 b^{7} c^{3} d^{4}\right ) x^{4}}{8 b^{8}}+\frac {\left (-a^{4} b^{4} d^{7}-5 a^{3} b^{5} c \,d^{6}-15 a^{2} b^{6} c^{2} d^{5}-35 a \,b^{7} c^{3} d^{4}-70 b^{8} c^{4} d^{3}\right ) x^{3}}{18 b^{9}}+\frac {\left (-a^{5} b^{4} d^{7}-5 a^{4} b^{5} c \,d^{6}-15 a^{3} b^{6} c^{2} d^{5}-35 a^{2} b^{7} c^{3} d^{4}-70 a \,b^{8} c^{4} d^{3}-126 b^{9} c^{5} d^{2}\right ) x^{2}}{60 b^{10}}+\frac {\left (-a^{6} b^{4} d^{7}-5 a^{5} b^{5} c \,d^{6}-15 a^{4} b^{6} c^{2} d^{5}-35 a^{3} b^{7} c^{3} d^{4}-70 a^{2} b^{8} c^{4} d^{3}-126 a \,b^{9} c^{5} d^{2}-210 b^{10} c^{6} d \right ) x}{330 b^{11}}+\frac {-a^{7} b^{4} d^{7}-5 a^{6} b^{5} c \,d^{6}-15 a^{5} b^{6} c^{2} d^{5}-35 a^{4} b^{7} c^{3} d^{4}-70 a^{3} b^{8} c^{4} d^{3}-126 a^{2} b^{9} c^{5} d^{2}-210 a \,b^{10} c^{6} d -330 b^{11} c^{7}}{3960 b^{12}}}{\left (b x +a \right )^{12}}\)	\(492\)
gosper	\(-\frac {792 d^{7} x^{7} b^{7}+924 a \,b^{6} d^{7} x^{6}+4620 b^{7} c \,d^{6} x^{6}+792 a^{2} b^{5} d^{7} x^{5}+3960 a \,b^{6} c \,d^{6} x^{5}+11880 b^{7} c^{2} d^{5} x^{5}+495 a^{3} b^{4} d^{7} x^{4}+2475 a^{2} b^{5} c \,d^{6} x^{4}+7425 a \,b^{6} c^{2} d^{5} x^{4}+17325 b^{7} c^{3} d^{4} x^{4}+220 a^{4} b^{3} d^{7} x^{3}+1100 a^{3} b^{4} c \,d^{6} x^{3}+3300 a^{2} b^{5} c^{2} d^{5} x^{3}+7700 a \,b^{6} c^{3} d^{4} x^{3}+15400 b^{7} c^{4} d^{3} x^{3}+66 a^{5} b^{2} d^{7} x^{2}+330 a^{4} b^{3} c \,d^{6} x^{2}+990 a^{3} b^{4} c^{2} d^{5} x^{2}+2310 a^{2} b^{5} c^{3} d^{4} x^{2}+4620 a \,b^{6} c^{4} d^{3} x^{2}+8316 b^{7} c^{5} d^{2} x^{2}+12 a^{6} b \,d^{7} x +60 a^{5} b^{2} c \,d^{6} x +180 a^{4} b^{3} c^{2} d^{5} x +420 a^{3} b^{4} c^{3} d^{4} x +840 a^{2} b^{5} c^{4} d^{3} x +1512 a \,b^{6} c^{5} d^{2} x +2520 b^{7} c^{6} d x +a^{7} d^{7}+5 a^{6} b c \,d^{6}+15 a^{5} b^{2} c^{2} d^{5}+35 a^{4} b^{3} c^{3} d^{4}+70 a^{3} b^{4} c^{4} d^{3}+126 a^{2} b^{5} c^{5} d^{2}+210 a \,b^{6} c^{6} d +330 b^{7} c^{7}}{3960 b^{8} \left (b x +a \right )^{12}}\)	\(497\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)^7/(b*x+a)^13,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 581 vs. \(2 (141) = 282\).
time = 0.31, size = 581, normalized size = 3.85 \begin {gather*} -\frac {792 \, b^{7} d^{7} x^{7} + 330 \, b^{7} c^{7} + 210 \, a b^{6} c^{6} d + 126 \, a^{2} b^{5} c^{5} d^{2} + 70 \, a^{3} b^{4} c^{4} d^{3} + 35 \, a^{4} b^{3} c^{3} d^{4} + 15 \, a^{5} b^{2} c^{2} d^{5} + 5 \, a^{6} b c d^{6} + a^{7} d^{7} + 924 \, {\left (5 \, b^{7} c d^{6} + a b^{6} d^{7}\right )} x^{6} + 792 \, {\left (15 \, b^{7} c^{2} d^{5} + 5 \, a b^{6} c d^{6} + a^{2} b^{5} d^{7}\right )} x^{5} + 495 \, {\left (35 \, b^{7} c^{3} d^{4} + 15 \, a b^{6} c^{2} d^{5} + 5 \, a^{2} b^{5} c d^{6} + a^{3} b^{4} d^{7}\right )} x^{4} + 220 \, {\left (70 \, b^{7} c^{4} d^{3} + 35 \, a b^{6} c^{3} d^{4} + 15 \, a^{2} b^{5} c^{2} d^{5} + 5 \, a^{3} b^{4} c d^{6} + a^{4} b^{3} d^{7}\right )} x^{3} + 66 \, {\left (126 \, b^{7} c^{5} d^{2} + 70 \, a b^{6} c^{4} d^{3} + 35 \, a^{2} b^{5} c^{3} d^{4} + 15 \, a^{3} b^{4} c^{2} d^{5} + 5 \, a^{4} b^{3} c d^{6} + a^{5} b^{2} d^{7}\right )} x^{2} + 12 \, {\left (210 \, b^{7} c^{6} d + 126 \, a b^{6} c^{5} d^{2} + 70 \, a^{2} b^{5} c^{4} d^{3} + 35 \, a^{3} b^{4} c^{3} d^{4} + 15 \, a^{4} b^{3} c^{2} d^{5} + 5 \, a^{5} b^{2} c d^{6} + a^{6} b d^{7}\right )} x}{3960 \, {\left (b^{20} x^{12} + 12 \, a b^{19} x^{11} + 66 \, a^{2} b^{18} x^{10} + 220 \, a^{3} b^{17} x^{9} + 495 \, a^{4} b^{16} x^{8} + 792 \, a^{5} b^{15} x^{7} + 924 \, a^{6} b^{14} x^{6} + 792 \, a^{7} b^{13} x^{5} + 495 \, a^{8} b^{12} x^{4} + 220 \, a^{9} b^{11} x^{3} + 66 \, a^{10} b^{10} x^{2} + 12 \, a^{11} b^{9} x + a^{12} b^{8}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^7/(b*x+a)^13,x, algorithm="maxima")

[Out]

-1/3960*(792*b^7*d^7*x^7 + 330*b^7*c^7 + 210*a*b^6*c^6*d + 126*a^2*b^5*c^5*d^2 + 70*a^3*b^4*c^4*d^3 + 35*a^4*b
^3*c^3*d^4 + 15*a^5*b^2*c^2*d^5 + 5*a^6*b*c*d^6 + a^7*d^7 + 924*(5*b^7*c*d^6 + a*b^6*d^7)*x^6 + 792*(15*b^7*c^
2*d^5 + 5*a*b^6*c*d^6 + a^2*b^5*d^7)*x^5 + 495*(35*b^7*c^3*d^4 + 15*a*b^6*c^2*d^5 + 5*a^2*b^5*c*d^6 + a^3*b^4*
d^7)*x^4 + 220*(70*b^7*c^4*d^3 + 35*a*b^6*c^3*d^4 + 15*a^2*b^5*c^2*d^5 + 5*a^3*b^4*c*d^6 + a^4*b^3*d^7)*x^3 +
66*(126*b^7*c^5*d^2 + 70*a*b^6*c^4*d^3 + 35*a^2*b^5*c^3*d^4 + 15*a^3*b^4*c^2*d^5 + 5*a^4*b^3*c*d^6 + a^5*b^2*d
^7)*x^2 + 12*(210*b^7*c^6*d + 126*a*b^6*c^5*d^2 + 70*a^2*b^5*c^4*d^3 + 35*a^3*b^4*c^3*d^4 + 15*a^4*b^3*c^2*d^5
+ 5*a^5*b^2*c*d^6 + a^6*b*d^7)*x)/(b^20*x^12 + 12*a*b^19*x^11 + 66*a^2*b^18*x^10 + 220*a^3*b^17*x^9 + 495*a^4
*b^16*x^8 + 792*a^5*b^15*x^7 + 924*a^6*b^14*x^6 + 792*a^7*b^13*x^5 + 495*a^8*b^12*x^4 + 220*a^9*b^11*x^3 + 66*
a^10*b^10*x^2 + 12*a^11*b^9*x + a^12*b^8)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 581 vs. \(2 (141) = 282\).
time = 0.45, size = 581, normalized size = 3.85 \begin {gather*} -\frac {792 \, b^{7} d^{7} x^{7} + 330 \, b^{7} c^{7} + 210 \, a b^{6} c^{6} d + 126 \, a^{2} b^{5} c^{5} d^{2} + 70 \, a^{3} b^{4} c^{4} d^{3} + 35 \, a^{4} b^{3} c^{3} d^{4} + 15 \, a^{5} b^{2} c^{2} d^{5} + 5 \, a^{6} b c d^{6} + a^{7} d^{7} + 924 \, {\left (5 \, b^{7} c d^{6} + a b^{6} d^{7}\right )} x^{6} + 792 \, {\left (15 \, b^{7} c^{2} d^{5} + 5 \, a b^{6} c d^{6} + a^{2} b^{5} d^{7}\right )} x^{5} + 495 \, {\left (35 \, b^{7} c^{3} d^{4} + 15 \, a b^{6} c^{2} d^{5} + 5 \, a^{2} b^{5} c d^{6} + a^{3} b^{4} d^{7}\right )} x^{4} + 220 \, {\left (70 \, b^{7} c^{4} d^{3} + 35 \, a b^{6} c^{3} d^{4} + 15 \, a^{2} b^{5} c^{2} d^{5} + 5 \, a^{3} b^{4} c d^{6} + a^{4} b^{3} d^{7}\right )} x^{3} + 66 \, {\left (126 \, b^{7} c^{5} d^{2} + 70 \, a b^{6} c^{4} d^{3} + 35 \, a^{2} b^{5} c^{3} d^{4} + 15 \, a^{3} b^{4} c^{2} d^{5} + 5 \, a^{4} b^{3} c d^{6} + a^{5} b^{2} d^{7}\right )} x^{2} + 12 \, {\left (210 \, b^{7} c^{6} d + 126 \, a b^{6} c^{5} d^{2} + 70 \, a^{2} b^{5} c^{4} d^{3} + 35 \, a^{3} b^{4} c^{3} d^{4} + 15 \, a^{4} b^{3} c^{2} d^{5} + 5 \, a^{5} b^{2} c d^{6} + a^{6} b d^{7}\right )} x}{3960 \, {\left (b^{20} x^{12} + 12 \, a b^{19} x^{11} + 66 \, a^{2} b^{18} x^{10} + 220 \, a^{3} b^{17} x^{9} + 495 \, a^{4} b^{16} x^{8} + 792 \, a^{5} b^{15} x^{7} + 924 \, a^{6} b^{14} x^{6} + 792 \, a^{7} b^{13} x^{5} + 495 \, a^{8} b^{12} x^{4} + 220 \, a^{9} b^{11} x^{3} + 66 \, a^{10} b^{10} x^{2} + 12 \, a^{11} b^{9} x + a^{12} b^{8}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^7/(b*x+a)^13,x, algorithm="fricas")

[Out]

-1/3960*(792*b^7*d^7*x^7 + 330*b^7*c^7 + 210*a*b^6*c^6*d + 126*a^2*b^5*c^5*d^2 + 70*a^3*b^4*c^4*d^3 + 35*a^4*b
^3*c^3*d^4 + 15*a^5*b^2*c^2*d^5 + 5*a^6*b*c*d^6 + a^7*d^7 + 924*(5*b^7*c*d^6 + a*b^6*d^7)*x^6 + 792*(15*b^7*c^
2*d^5 + 5*a*b^6*c*d^6 + a^2*b^5*d^7)*x^5 + 495*(35*b^7*c^3*d^4 + 15*a*b^6*c^2*d^5 + 5*a^2*b^5*c*d^6 + a^3*b^4*
d^7)*x^4 + 220*(70*b^7*c^4*d^3 + 35*a*b^6*c^3*d^4 + 15*a^2*b^5*c^2*d^5 + 5*a^3*b^4*c*d^6 + a^4*b^3*d^7)*x^3 +
66*(126*b^7*c^5*d^2 + 70*a*b^6*c^4*d^3 + 35*a^2*b^5*c^3*d^4 + 15*a^3*b^4*c^2*d^5 + 5*a^4*b^3*c*d^6 + a^5*b^2*d
^7)*x^2 + 12*(210*b^7*c^6*d + 126*a*b^6*c^5*d^2 + 70*a^2*b^5*c^4*d^3 + 35*a^3*b^4*c^3*d^4 + 15*a^4*b^3*c^2*d^5
+ 5*a^5*b^2*c*d^6 + a^6*b*d^7)*x)/(b^20*x^12 + 12*a*b^19*x^11 + 66*a^2*b^18*x^10 + 220*a^3*b^17*x^9 + 495*a^4
*b^16*x^8 + 792*a^5*b^15*x^7 + 924*a^6*b^14*x^6 + 792*a^7*b^13*x^5 + 495*a^8*b^12*x^4 + 220*a^9*b^11*x^3 + 66*
a^10*b^10*x^2 + 12*a^11*b^9*x + a^12*b^8)

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Sympy [F(-1)] Timed out
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)**13,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. \(2 (141) = 282\).
time = 1.10, size = 496, normalized size = 3.28 \begin {gather*} -\frac {792 \, b^{7} d^{7} x^{7} + 4620 \, b^{7} c d^{6} x^{6} + 924 \, a b^{6} d^{7} x^{6} + 11880 \, b^{7} c^{2} d^{5} x^{5} + 3960 \, a b^{6} c d^{6} x^{5} + 792 \, a^{2} b^{5} d^{7} x^{5} + 17325 \, b^{7} c^{3} d^{4} x^{4} + 7425 \, a b^{6} c^{2} d^{5} x^{4} + 2475 \, a^{2} b^{5} c d^{6} x^{4} + 495 \, a^{3} b^{4} d^{7} x^{4} + 15400 \, b^{7} c^{4} d^{3} x^{3} + 7700 \, a b^{6} c^{3} d^{4} x^{3} + 3300 \, a^{2} b^{5} c^{2} d^{5} x^{3} + 1100 \, a^{3} b^{4} c d^{6} x^{3} + 220 \, a^{4} b^{3} d^{7} x^{3} + 8316 \, b^{7} c^{5} d^{2} x^{2} + 4620 \, a b^{6} c^{4} d^{3} x^{2} + 2310 \, a^{2} b^{5} c^{3} d^{4} x^{2} + 990 \, a^{3} b^{4} c^{2} d^{5} x^{2} + 330 \, a^{4} b^{3} c d^{6} x^{2} + 66 \, a^{5} b^{2} d^{7} x^{2} + 2520 \, b^{7} c^{6} d x + 1512 \, a b^{6} c^{5} d^{2} x + 840 \, a^{2} b^{5} c^{4} d^{3} x + 420 \, a^{3} b^{4} c^{3} d^{4} x + 180 \, a^{4} b^{3} c^{2} d^{5} x + 60 \, a^{5} b^{2} c d^{6} x + 12 \, a^{6} b d^{7} x + 330 \, b^{7} c^{7} + 210 \, a b^{6} c^{6} d + 126 \, a^{2} b^{5} c^{5} d^{2} + 70 \, a^{3} b^{4} c^{4} d^{3} + 35 \, a^{4} b^{3} c^{3} d^{4} + 15 \, a^{5} b^{2} c^{2} d^{5} + 5 \, a^{6} b c d^{6} + a^{7} d^{7}}{3960 \, {\left (b x + a\right )}^{12} b^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^7/(b*x+a)^13,x, algorithm="giac")

[Out]

-1/3960*(792*b^7*d^7*x^7 + 4620*b^7*c*d^6*x^6 + 924*a*b^6*d^7*x^6 + 11880*b^7*c^2*d^5*x^5 + 3960*a*b^6*c*d^6*x
^5 + 792*a^2*b^5*d^7*x^5 + 17325*b^7*c^3*d^4*x^4 + 7425*a*b^6*c^2*d^5*x^4 + 2475*a^2*b^5*c*d^6*x^4 + 495*a^3*b
^4*d^7*x^4 + 15400*b^7*c^4*d^3*x^3 + 7700*a*b^6*c^3*d^4*x^3 + 3300*a^2*b^5*c^2*d^5*x^3 + 1100*a^3*b^4*c*d^6*x^
3 + 220*a^4*b^3*d^7*x^3 + 8316*b^7*c^5*d^2*x^2 + 4620*a*b^6*c^4*d^3*x^2 + 2310*a^2*b^5*c^3*d^4*x^2 + 990*a^3*b
^4*c^2*d^5*x^2 + 330*a^4*b^3*c*d^6*x^2 + 66*a^5*b^2*d^7*x^2 + 2520*b^7*c^6*d*x + 1512*a*b^6*c^5*d^2*x + 840*a^
2*b^5*c^4*d^3*x + 420*a^3*b^4*c^3*d^4*x + 180*a^4*b^3*c^2*d^5*x + 60*a^5*b^2*c*d^6*x + 12*a^6*b*d^7*x + 330*b^
7*c^7 + 210*a*b^6*c^6*d + 126*a^2*b^5*c^5*d^2 + 70*a^3*b^4*c^4*d^3 + 35*a^4*b^3*c^3*d^4 + 15*a^5*b^2*c^2*d^5 +
5*a^6*b*c*d^6 + a^7*d^7)/((b*x + a)^12*b^8)

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Mupad [B]
time = 0.23, size = 559, normalized size = 3.70 \begin {gather*} -\frac {\frac {a^7\,d^7+5\,a^6\,b\,c\,d^6+15\,a^5\,b^2\,c^2\,d^5+35\,a^4\,b^3\,c^3\,d^4+70\,a^3\,b^4\,c^4\,d^3+126\,a^2\,b^5\,c^5\,d^2+210\,a\,b^6\,c^6\,d+330\,b^7\,c^7}{3960\,b^8}+\frac {d^7\,x^7}{5\,b}+\frac {d^2\,x^2\,\left (a^5\,d^5+5\,a^4\,b\,c\,d^4+15\,a^3\,b^2\,c^2\,d^3+35\,a^2\,b^3\,c^3\,d^2+70\,a\,b^4\,c^4\,d+126\,b^5\,c^5\right )}{60\,b^6}+\frac {d^4\,x^4\,\left (a^3\,d^3+5\,a^2\,b\,c\,d^2+15\,a\,b^2\,c^2\,d+35\,b^3\,c^3\right )}{8\,b^4}+\frac {7\,d^6\,x^6\,\left (a\,d+5\,b\,c\right )}{30\,b^2}+\frac {d^3\,x^3\,\left (a^4\,d^4+5\,a^3\,b\,c\,d^3+15\,a^2\,b^2\,c^2\,d^2+35\,a\,b^3\,c^3\,d+70\,b^4\,c^4\right )}{18\,b^5}+\frac {d\,x\,\left (a^6\,d^6+5\,a^5\,b\,c\,d^5+15\,a^4\,b^2\,c^2\,d^4+35\,a^3\,b^3\,c^3\,d^3+70\,a^2\,b^4\,c^4\,d^2+126\,a\,b^5\,c^5\,d+210\,b^6\,c^6\right )}{330\,b^7}+\frac {d^5\,x^5\,\left (a^2\,d^2+5\,a\,b\,c\,d+15\,b^2\,c^2\right )}{5\,b^3}}{a^{12}+12\,a^{11}\,b\,x+66\,a^{10}\,b^2\,x^2+220\,a^9\,b^3\,x^3+495\,a^8\,b^4\,x^4+792\,a^7\,b^5\,x^5+924\,a^6\,b^6\,x^6+792\,a^5\,b^7\,x^7+495\,a^4\,b^8\,x^8+220\,a^3\,b^9\,x^9+66\,a^2\,b^{10}\,x^{10}+12\,a\,b^{11}\,x^{11}+b^{12}\,x^{12}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x)^7/(a + b*x)^13,x)

[Out]

-((a^7*d^7 + 330*b^7*c^7 + 126*a^2*b^5*c^5*d^2 + 70*a^3*b^4*c^4*d^3 + 35*a^4*b^3*c^3*d^4 + 15*a^5*b^2*c^2*d^5
+ 210*a*b^6*c^6*d + 5*a^6*b*c*d^6)/(3960*b^8) + (d^7*x^7)/(5*b) + (d^2*x^2*(a^5*d^5 + 126*b^5*c^5 + 35*a^2*b^3
*c^3*d^2 + 15*a^3*b^2*c^2*d^3 + 70*a*b^4*c^4*d + 5*a^4*b*c*d^4))/(60*b^6) + (d^4*x^4*(a^3*d^3 + 35*b^3*c^3 + 1
5*a*b^2*c^2*d + 5*a^2*b*c*d^2))/(8*b^4) + (7*d^6*x^6*(a*d + 5*b*c))/(30*b^2) + (d^3*x^3*(a^4*d^4 + 70*b^4*c^4
+ 15*a^2*b^2*c^2*d^2 + 35*a*b^3*c^3*d + 5*a^3*b*c*d^3))/(18*b^5) + (d*x*(a^6*d^6 + 210*b^6*c^6 + 70*a^2*b^4*c^
4*d^2 + 35*a^3*b^3*c^3*d^3 + 15*a^4*b^2*c^2*d^4 + 126*a*b^5*c^5*d + 5*a^5*b*c*d^5))/(330*b^7) + (d^5*x^5*(a^2*
d^2 + 15*b^2*c^2 + 5*a*b*c*d))/(5*b^3))/(a^12 + b^12*x^12 + 12*a*b^11*x^11 + 66*a^10*b^2*x^2 + 220*a^9*b^3*x^3
+ 495*a^8*b^4*x^4 + 792*a^7*b^5*x^5 + 924*a^6*b^6*x^6 + 792*a^5*b^7*x^7 + 495*a^4*b^8*x^8 + 220*a^3*b^9*x^9 +
66*a^2*b^10*x^10 + 12*a^11*b*x)

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