3.12.91 \(\int \frac {(c+d x)^7}{(a+b x)^{15}} \, dx\)

Optimal. Leaf size=200 \[ -\frac {7 d^6 (b c-a d)}{8 b^8 (a+b x)^8}-\frac {7 d^5 (b c-a d)^2}{3 b^8 (a+b x)^9}-\frac {7 d^4 (b c-a d)^3}{2 b^8 (a+b x)^{10}}-\frac {35 d^3 (b c-a d)^4}{11 b^8 (a+b x)^{11}}-\frac {7 d^2 (b c-a d)^5}{4 b^8 (a+b x)^{12}}-\frac {7 d (b c-a d)^6}{13 b^8 (a+b x)^{13}}-\frac {(b c-a d)^7}{14 b^8 (a+b x)^{14}}-\frac {d^7}{7 b^8 (a+b x)^7} \]

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Rubi [A]  time = 0.14, antiderivative size = 200, 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 = {43} \begin {gather*} -\frac {7 d^6 (b c-a d)}{8 b^8 (a+b x)^8}-\frac {7 d^5 (b c-a d)^2}{3 b^8 (a+b x)^9}-\frac {7 d^4 (b c-a d)^3}{2 b^8 (a+b x)^{10}}-\frac {35 d^3 (b c-a d)^4}{11 b^8 (a+b x)^{11}}-\frac {7 d^2 (b c-a d)^5}{4 b^8 (a+b x)^{12}}-\frac {7 d (b c-a d)^6}{13 b^8 (a+b x)^{13}}-\frac {(b c-a d)^7}{14 b^8 (a+b x)^{14}}-\frac {d^7}{7 b^8 (a+b x)^7} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^7/(a + b*x)^15,x]

[Out]

-(b*c - a*d)^7/(14*b^8*(a + b*x)^14) - (7*d*(b*c - a*d)^6)/(13*b^8*(a + b*x)^13) - (7*d^2*(b*c - a*d)^5)/(4*b^
8*(a + b*x)^12) - (35*d^3*(b*c - a*d)^4)/(11*b^8*(a + b*x)^11) - (7*d^4*(b*c - a*d)^3)/(2*b^8*(a + b*x)^10) -
(7*d^5*(b*c - a*d)^2)/(3*b^8*(a + b*x)^9) - (7*d^6*(b*c - a*d))/(8*b^8*(a + b*x)^8) - d^7/(7*b^8*(a + b*x)^7)

Rule 43

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)^{15}} \, dx &=\int \left (\frac {(b c-a d)^7}{b^7 (a+b x)^{15}}+\frac {7 d (b c-a d)^6}{b^7 (a+b x)^{14}}+\frac {21 d^2 (b c-a d)^5}{b^7 (a+b x)^{13}}+\frac {35 d^3 (b c-a d)^4}{b^7 (a+b x)^{12}}+\frac {35 d^4 (b c-a d)^3}{b^7 (a+b x)^{11}}+\frac {21 d^5 (b c-a d)^2}{b^7 (a+b x)^{10}}+\frac {7 d^6 (b c-a d)}{b^7 (a+b x)^9}+\frac {d^7}{b^7 (a+b x)^8}\right ) \, dx\\ &=-\frac {(b c-a d)^7}{14 b^8 (a+b x)^{14}}-\frac {7 d (b c-a d)^6}{13 b^8 (a+b x)^{13}}-\frac {7 d^2 (b c-a d)^5}{4 b^8 (a+b x)^{12}}-\frac {35 d^3 (b c-a d)^4}{11 b^8 (a+b x)^{11}}-\frac {7 d^4 (b c-a d)^3}{2 b^8 (a+b x)^{10}}-\frac {7 d^5 (b c-a d)^2}{3 b^8 (a+b x)^9}-\frac {7 d^6 (b c-a d)}{8 b^8 (a+b x)^8}-\frac {d^7}{7 b^8 (a+b x)^7}\\ \end {align*}

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Mathematica [A]  time = 0.13, size = 371, normalized size = 1.86 \begin {gather*} -\frac {a^7 d^7+7 a^6 b d^6 (c+2 d x)+7 a^5 b^2 d^5 \left (4 c^2+14 c d x+13 d^2 x^2\right )+7 a^4 b^3 d^4 \left (12 c^3+56 c^2 d x+91 c d^2 x^2+52 d^3 x^3\right )+7 a^3 b^4 d^3 \left (30 c^4+168 c^3 d x+364 c^2 d^2 x^2+364 c d^3 x^3+143 d^4 x^4\right )+7 a^2 b^5 d^2 \left (66 c^5+420 c^4 d x+1092 c^3 d^2 x^2+1456 c^2 d^3 x^3+1001 c d^4 x^4+286 d^5 x^5\right )+7 a b^6 d \left (132 c^6+924 c^5 d x+2730 c^4 d^2 x^2+4368 c^3 d^3 x^3+4004 c^2 d^4 x^4+2002 c d^5 x^5+429 d^6 x^6\right )+b^7 \left (1716 c^7+12936 c^6 d x+42042 c^5 d^2 x^2+76440 c^4 d^3 x^3+84084 c^3 d^4 x^4+56056 c^2 d^5 x^5+21021 c d^6 x^6+3432 d^7 x^7\right )}{24024 b^8 (a+b x)^{14}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)^7/(a + b*x)^15,x]

[Out]

-1/24024*(a^7*d^7 + 7*a^6*b*d^6*(c + 2*d*x) + 7*a^5*b^2*d^5*(4*c^2 + 14*c*d*x + 13*d^2*x^2) + 7*a^4*b^3*d^4*(1
2*c^3 + 56*c^2*d*x + 91*c*d^2*x^2 + 52*d^3*x^3) + 7*a^3*b^4*d^3*(30*c^4 + 168*c^3*d*x + 364*c^2*d^2*x^2 + 364*
c*d^3*x^3 + 143*d^4*x^4) + 7*a^2*b^5*d^2*(66*c^5 + 420*c^4*d*x + 1092*c^3*d^2*x^2 + 1456*c^2*d^3*x^3 + 1001*c*
d^4*x^4 + 286*d^5*x^5) + 7*a*b^6*d*(132*c^6 + 924*c^5*d*x + 2730*c^4*d^2*x^2 + 4368*c^3*d^3*x^3 + 4004*c^2*d^4
*x^4 + 2002*c*d^5*x^5 + 429*d^6*x^6) + b^7*(1716*c^7 + 12936*c^6*d*x + 42042*c^5*d^2*x^2 + 76440*c^4*d^3*x^3 +
84084*c^3*d^4*x^4 + 56056*c^2*d^5*x^5 + 21021*c*d^6*x^6 + 3432*d^7*x^7))/(b^8*(a + b*x)^14)

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(c+d x)^7}{(a+b x)^{15}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(c + d*x)^7/(a + b*x)^15,x]

[Out]

IntegrateAlgebraic[(c + d*x)^7/(a + b*x)^15, x]

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fricas [B]  time = 1.35, size = 603, normalized size = 3.02 \begin {gather*} -\frac {3432 \, b^{7} d^{7} x^{7} + 1716 \, b^{7} c^{7} + 924 \, a b^{6} c^{6} d + 462 \, a^{2} b^{5} c^{5} d^{2} + 210 \, a^{3} b^{4} c^{4} d^{3} + 84 \, a^{4} b^{3} c^{3} d^{4} + 28 \, a^{5} b^{2} c^{2} d^{5} + 7 \, a^{6} b c d^{6} + a^{7} d^{7} + 3003 \, {\left (7 \, b^{7} c d^{6} + a b^{6} d^{7}\right )} x^{6} + 2002 \, {\left (28 \, b^{7} c^{2} d^{5} + 7 \, a b^{6} c d^{6} + a^{2} b^{5} d^{7}\right )} x^{5} + 1001 \, {\left (84 \, b^{7} c^{3} d^{4} + 28 \, a b^{6} c^{2} d^{5} + 7 \, a^{2} b^{5} c d^{6} + a^{3} b^{4} d^{7}\right )} x^{4} + 364 \, {\left (210 \, b^{7} c^{4} d^{3} + 84 \, a b^{6} c^{3} d^{4} + 28 \, a^{2} b^{5} c^{2} d^{5} + 7 \, a^{3} b^{4} c d^{6} + a^{4} b^{3} d^{7}\right )} x^{3} + 91 \, {\left (462 \, b^{7} c^{5} d^{2} + 210 \, a b^{6} c^{4} d^{3} + 84 \, a^{2} b^{5} c^{3} d^{4} + 28 \, a^{3} b^{4} c^{2} d^{5} + 7 \, a^{4} b^{3} c d^{6} + a^{5} b^{2} d^{7}\right )} x^{2} + 14 \, {\left (924 \, b^{7} c^{6} d + 462 \, a b^{6} c^{5} d^{2} + 210 \, a^{2} b^{5} c^{4} d^{3} + 84 \, a^{3} b^{4} c^{3} d^{4} + 28 \, a^{4} b^{3} c^{2} d^{5} + 7 \, a^{5} b^{2} c d^{6} + a^{6} b d^{7}\right )} x}{24024 \, {\left (b^{22} x^{14} + 14 \, a b^{21} x^{13} + 91 \, a^{2} b^{20} x^{12} + 364 \, a^{3} b^{19} x^{11} + 1001 \, a^{4} b^{18} x^{10} + 2002 \, a^{5} b^{17} x^{9} + 3003 \, a^{6} b^{16} x^{8} + 3432 \, a^{7} b^{15} x^{7} + 3003 \, a^{8} b^{14} x^{6} + 2002 \, a^{9} b^{13} x^{5} + 1001 \, a^{10} b^{12} x^{4} + 364 \, a^{11} b^{11} x^{3} + 91 \, a^{12} b^{10} x^{2} + 14 \, a^{13} b^{9} x + a^{14} b^{8}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^7/(b*x+a)^15,x, algorithm="fricas")

[Out]

-1/24024*(3432*b^7*d^7*x^7 + 1716*b^7*c^7 + 924*a*b^6*c^6*d + 462*a^2*b^5*c^5*d^2 + 210*a^3*b^4*c^4*d^3 + 84*a
^4*b^3*c^3*d^4 + 28*a^5*b^2*c^2*d^5 + 7*a^6*b*c*d^6 + a^7*d^7 + 3003*(7*b^7*c*d^6 + a*b^6*d^7)*x^6 + 2002*(28*
b^7*c^2*d^5 + 7*a*b^6*c*d^6 + a^2*b^5*d^7)*x^5 + 1001*(84*b^7*c^3*d^4 + 28*a*b^6*c^2*d^5 + 7*a^2*b^5*c*d^6 + a
^3*b^4*d^7)*x^4 + 364*(210*b^7*c^4*d^3 + 84*a*b^6*c^3*d^4 + 28*a^2*b^5*c^2*d^5 + 7*a^3*b^4*c*d^6 + a^4*b^3*d^7
)*x^3 + 91*(462*b^7*c^5*d^2 + 210*a*b^6*c^4*d^3 + 84*a^2*b^5*c^3*d^4 + 28*a^3*b^4*c^2*d^5 + 7*a^4*b^3*c*d^6 +
a^5*b^2*d^7)*x^2 + 14*(924*b^7*c^6*d + 462*a*b^6*c^5*d^2 + 210*a^2*b^5*c^4*d^3 + 84*a^3*b^4*c^3*d^4 + 28*a^4*b
^3*c^2*d^5 + 7*a^5*b^2*c*d^6 + a^6*b*d^7)*x)/(b^22*x^14 + 14*a*b^21*x^13 + 91*a^2*b^20*x^12 + 364*a^3*b^19*x^1
1 + 1001*a^4*b^18*x^10 + 2002*a^5*b^17*x^9 + 3003*a^6*b^16*x^8 + 3432*a^7*b^15*x^7 + 3003*a^8*b^14*x^6 + 2002*
a^9*b^13*x^5 + 1001*a^10*b^12*x^4 + 364*a^11*b^11*x^3 + 91*a^12*b^10*x^2 + 14*a^13*b^9*x + a^14*b^8)

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giac [B]  time = 1.28, size = 496, normalized size = 2.48 \begin {gather*} -\frac {3432 \, b^{7} d^{7} x^{7} + 21021 \, b^{7} c d^{6} x^{6} + 3003 \, a b^{6} d^{7} x^{6} + 56056 \, b^{7} c^{2} d^{5} x^{5} + 14014 \, a b^{6} c d^{6} x^{5} + 2002 \, a^{2} b^{5} d^{7} x^{5} + 84084 \, b^{7} c^{3} d^{4} x^{4} + 28028 \, a b^{6} c^{2} d^{5} x^{4} + 7007 \, a^{2} b^{5} c d^{6} x^{4} + 1001 \, a^{3} b^{4} d^{7} x^{4} + 76440 \, b^{7} c^{4} d^{3} x^{3} + 30576 \, a b^{6} c^{3} d^{4} x^{3} + 10192 \, a^{2} b^{5} c^{2} d^{5} x^{3} + 2548 \, a^{3} b^{4} c d^{6} x^{3} + 364 \, a^{4} b^{3} d^{7} x^{3} + 42042 \, b^{7} c^{5} d^{2} x^{2} + 19110 \, a b^{6} c^{4} d^{3} x^{2} + 7644 \, a^{2} b^{5} c^{3} d^{4} x^{2} + 2548 \, a^{3} b^{4} c^{2} d^{5} x^{2} + 637 \, a^{4} b^{3} c d^{6} x^{2} + 91 \, a^{5} b^{2} d^{7} x^{2} + 12936 \, b^{7} c^{6} d x + 6468 \, a b^{6} c^{5} d^{2} x + 2940 \, a^{2} b^{5} c^{4} d^{3} x + 1176 \, a^{3} b^{4} c^{3} d^{4} x + 392 \, a^{4} b^{3} c^{2} d^{5} x + 98 \, a^{5} b^{2} c d^{6} x + 14 \, a^{6} b d^{7} x + 1716 \, b^{7} c^{7} + 924 \, a b^{6} c^{6} d + 462 \, a^{2} b^{5} c^{5} d^{2} + 210 \, a^{3} b^{4} c^{4} d^{3} + 84 \, a^{4} b^{3} c^{3} d^{4} + 28 \, a^{5} b^{2} c^{2} d^{5} + 7 \, a^{6} b c d^{6} + a^{7} d^{7}}{24024 \, {\left (b x + a\right )}^{14} b^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^7/(b*x+a)^15,x, algorithm="giac")

[Out]

-1/24024*(3432*b^7*d^7*x^7 + 21021*b^7*c*d^6*x^6 + 3003*a*b^6*d^7*x^6 + 56056*b^7*c^2*d^5*x^5 + 14014*a*b^6*c*
d^6*x^5 + 2002*a^2*b^5*d^7*x^5 + 84084*b^7*c^3*d^4*x^4 + 28028*a*b^6*c^2*d^5*x^4 + 7007*a^2*b^5*c*d^6*x^4 + 10
01*a^3*b^4*d^7*x^4 + 76440*b^7*c^4*d^3*x^3 + 30576*a*b^6*c^3*d^4*x^3 + 10192*a^2*b^5*c^2*d^5*x^3 + 2548*a^3*b^
4*c*d^6*x^3 + 364*a^4*b^3*d^7*x^3 + 42042*b^7*c^5*d^2*x^2 + 19110*a*b^6*c^4*d^3*x^2 + 7644*a^2*b^5*c^3*d^4*x^2
+ 2548*a^3*b^4*c^2*d^5*x^2 + 637*a^4*b^3*c*d^6*x^2 + 91*a^5*b^2*d^7*x^2 + 12936*b^7*c^6*d*x + 6468*a*b^6*c^5*
d^2*x + 2940*a^2*b^5*c^4*d^3*x + 1176*a^3*b^4*c^3*d^4*x + 392*a^4*b^3*c^2*d^5*x + 98*a^5*b^2*c*d^6*x + 14*a^6*
b*d^7*x + 1716*b^7*c^7 + 924*a*b^6*c^6*d + 462*a^2*b^5*c^5*d^2 + 210*a^3*b^4*c^4*d^3 + 84*a^4*b^3*c^3*d^4 + 28
*a^5*b^2*c^2*d^5 + 7*a^6*b*c*d^6 + a^7*d^7)/((b*x + a)^14*b^8)

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maple [B]  time = 0.01, size = 464, normalized size = 2.32 \begin {gather*} -\frac {d^{7}}{7 \left (b x +a \right )^{7} b^{8}}+\frac {7 \left (a d -b c \right ) d^{6}}{8 \left (b x +a \right )^{8} b^{8}}-\frac {7 \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) d^{5}}{3 \left (b x +a \right )^{9} b^{8}}+\frac {7 \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) d^{4}}{2 \left (b x +a \right )^{10} b^{8}}-\frac {35 \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 ) d^{3}}{11 \left (b x +a \right )^{11} b^{8}}+\frac {7 \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 ) d^{2}}{4 \left (b x +a \right )^{12} b^{8}}-\frac {7 \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 ) d}{13 \left (b x +a \right )^{13} b^{8}}-\frac {-a^{7} d^{7}+7 a^{6} b c \,d^{6}-21 a^{5} b^{2} c^{2} d^{5}+35 a^{4} c^{3} d^{4} b^{3}-35 a^{3} b^{4} c^{4} d^{3}+21 a^{2} c^{5} d^{2} b^{5}-7 a \,b^{6} c^{6} d +b^{7} c^{7}}{14 \left (b x +a \right )^{14} b^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)^7/(b*x+a)^15,x)

[Out]

7/8*d^6*(a*d-b*c)/b^8/(b*x+a)^8-7/13*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^8/(b*x+a)^13-1/7*d^7/b^8/(b*x+a)^7-7/3*d^5*(a^2*d^2-2*a*b*c*d+b^2*c^2)/b^8/(
b*x+a)^9-35/11*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^8/(b*x+a)^11+7/4*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^8/(b*x+a)^12+7/2*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^8/(b*x+a)^10-1/14*(-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)^14

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maxima [B]  time = 1.83, size = 603, normalized size = 3.02 \begin {gather*} -\frac {3432 \, b^{7} d^{7} x^{7} + 1716 \, b^{7} c^{7} + 924 \, a b^{6} c^{6} d + 462 \, a^{2} b^{5} c^{5} d^{2} + 210 \, a^{3} b^{4} c^{4} d^{3} + 84 \, a^{4} b^{3} c^{3} d^{4} + 28 \, a^{5} b^{2} c^{2} d^{5} + 7 \, a^{6} b c d^{6} + a^{7} d^{7} + 3003 \, {\left (7 \, b^{7} c d^{6} + a b^{6} d^{7}\right )} x^{6} + 2002 \, {\left (28 \, b^{7} c^{2} d^{5} + 7 \, a b^{6} c d^{6} + a^{2} b^{5} d^{7}\right )} x^{5} + 1001 \, {\left (84 \, b^{7} c^{3} d^{4} + 28 \, a b^{6} c^{2} d^{5} + 7 \, a^{2} b^{5} c d^{6} + a^{3} b^{4} d^{7}\right )} x^{4} + 364 \, {\left (210 \, b^{7} c^{4} d^{3} + 84 \, a b^{6} c^{3} d^{4} + 28 \, a^{2} b^{5} c^{2} d^{5} + 7 \, a^{3} b^{4} c d^{6} + a^{4} b^{3} d^{7}\right )} x^{3} + 91 \, {\left (462 \, b^{7} c^{5} d^{2} + 210 \, a b^{6} c^{4} d^{3} + 84 \, a^{2} b^{5} c^{3} d^{4} + 28 \, a^{3} b^{4} c^{2} d^{5} + 7 \, a^{4} b^{3} c d^{6} + a^{5} b^{2} d^{7}\right )} x^{2} + 14 \, {\left (924 \, b^{7} c^{6} d + 462 \, a b^{6} c^{5} d^{2} + 210 \, a^{2} b^{5} c^{4} d^{3} + 84 \, a^{3} b^{4} c^{3} d^{4} + 28 \, a^{4} b^{3} c^{2} d^{5} + 7 \, a^{5} b^{2} c d^{6} + a^{6} b d^{7}\right )} x}{24024 \, {\left (b^{22} x^{14} + 14 \, a b^{21} x^{13} + 91 \, a^{2} b^{20} x^{12} + 364 \, a^{3} b^{19} x^{11} + 1001 \, a^{4} b^{18} x^{10} + 2002 \, a^{5} b^{17} x^{9} + 3003 \, a^{6} b^{16} x^{8} + 3432 \, a^{7} b^{15} x^{7} + 3003 \, a^{8} b^{14} x^{6} + 2002 \, a^{9} b^{13} x^{5} + 1001 \, a^{10} b^{12} x^{4} + 364 \, a^{11} b^{11} x^{3} + 91 \, a^{12} b^{10} x^{2} + 14 \, a^{13} b^{9} x + a^{14} b^{8}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^7/(b*x+a)^15,x, algorithm="maxima")

[Out]

-1/24024*(3432*b^7*d^7*x^7 + 1716*b^7*c^7 + 924*a*b^6*c^6*d + 462*a^2*b^5*c^5*d^2 + 210*a^3*b^4*c^4*d^3 + 84*a
^4*b^3*c^3*d^4 + 28*a^5*b^2*c^2*d^5 + 7*a^6*b*c*d^6 + a^7*d^7 + 3003*(7*b^7*c*d^6 + a*b^6*d^7)*x^6 + 2002*(28*
b^7*c^2*d^5 + 7*a*b^6*c*d^6 + a^2*b^5*d^7)*x^5 + 1001*(84*b^7*c^3*d^4 + 28*a*b^6*c^2*d^5 + 7*a^2*b^5*c*d^6 + a
^3*b^4*d^7)*x^4 + 364*(210*b^7*c^4*d^3 + 84*a*b^6*c^3*d^4 + 28*a^2*b^5*c^2*d^5 + 7*a^3*b^4*c*d^6 + a^4*b^3*d^7
)*x^3 + 91*(462*b^7*c^5*d^2 + 210*a*b^6*c^4*d^3 + 84*a^2*b^5*c^3*d^4 + 28*a^3*b^4*c^2*d^5 + 7*a^4*b^3*c*d^6 +
a^5*b^2*d^7)*x^2 + 14*(924*b^7*c^6*d + 462*a*b^6*c^5*d^2 + 210*a^2*b^5*c^4*d^3 + 84*a^3*b^4*c^3*d^4 + 28*a^4*b
^3*c^2*d^5 + 7*a^5*b^2*c*d^6 + a^6*b*d^7)*x)/(b^22*x^14 + 14*a*b^21*x^13 + 91*a^2*b^20*x^12 + 364*a^3*b^19*x^1
1 + 1001*a^4*b^18*x^10 + 2002*a^5*b^17*x^9 + 3003*a^6*b^16*x^8 + 3432*a^7*b^15*x^7 + 3003*a^8*b^14*x^6 + 2002*
a^9*b^13*x^5 + 1001*a^10*b^12*x^4 + 364*a^11*b^11*x^3 + 91*a^12*b^10*x^2 + 14*a^13*b^9*x + a^14*b^8)

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mupad [B]  time = 1.24, size = 581, normalized size = 2.90 \begin {gather*} -\frac {\frac {a^7\,d^7+7\,a^6\,b\,c\,d^6+28\,a^5\,b^2\,c^2\,d^5+84\,a^4\,b^3\,c^3\,d^4+210\,a^3\,b^4\,c^4\,d^3+462\,a^2\,b^5\,c^5\,d^2+924\,a\,b^6\,c^6\,d+1716\,b^7\,c^7}{24024\,b^8}+\frac {d^7\,x^7}{7\,b}+\frac {d^2\,x^2\,\left (a^5\,d^5+7\,a^4\,b\,c\,d^4+28\,a^3\,b^2\,c^2\,d^3+84\,a^2\,b^3\,c^3\,d^2+210\,a\,b^4\,c^4\,d+462\,b^5\,c^5\right )}{264\,b^6}+\frac {d^4\,x^4\,\left (a^3\,d^3+7\,a^2\,b\,c\,d^2+28\,a\,b^2\,c^2\,d+84\,b^3\,c^3\right )}{24\,b^4}+\frac {d^6\,x^6\,\left (a\,d+7\,b\,c\right )}{8\,b^2}+\frac {d^3\,x^3\,\left (a^4\,d^4+7\,a^3\,b\,c\,d^3+28\,a^2\,b^2\,c^2\,d^2+84\,a\,b^3\,c^3\,d+210\,b^4\,c^4\right )}{66\,b^5}+\frac {d\,x\,\left (a^6\,d^6+7\,a^5\,b\,c\,d^5+28\,a^4\,b^2\,c^2\,d^4+84\,a^3\,b^3\,c^3\,d^3+210\,a^2\,b^4\,c^4\,d^2+462\,a\,b^5\,c^5\,d+924\,b^6\,c^6\right )}{1716\,b^7}+\frac {d^5\,x^5\,\left (a^2\,d^2+7\,a\,b\,c\,d+28\,b^2\,c^2\right )}{12\,b^3}}{a^{14}+14\,a^{13}\,b\,x+91\,a^{12}\,b^2\,x^2+364\,a^{11}\,b^3\,x^3+1001\,a^{10}\,b^4\,x^4+2002\,a^9\,b^5\,x^5+3003\,a^8\,b^6\,x^6+3432\,a^7\,b^7\,x^7+3003\,a^6\,b^8\,x^8+2002\,a^5\,b^9\,x^9+1001\,a^4\,b^{10}\,x^{10}+364\,a^3\,b^{11}\,x^{11}+91\,a^2\,b^{12}\,x^{12}+14\,a\,b^{13}\,x^{13}+b^{14}\,x^{14}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x)^7/(a + b*x)^15,x)

[Out]

-((a^7*d^7 + 1716*b^7*c^7 + 462*a^2*b^5*c^5*d^2 + 210*a^3*b^4*c^4*d^3 + 84*a^4*b^3*c^3*d^4 + 28*a^5*b^2*c^2*d^
5 + 924*a*b^6*c^6*d + 7*a^6*b*c*d^6)/(24024*b^8) + (d^7*x^7)/(7*b) + (d^2*x^2*(a^5*d^5 + 462*b^5*c^5 + 84*a^2*
b^3*c^3*d^2 + 28*a^3*b^2*c^2*d^3 + 210*a*b^4*c^4*d + 7*a^4*b*c*d^4))/(264*b^6) + (d^4*x^4*(a^3*d^3 + 84*b^3*c^
3 + 28*a*b^2*c^2*d + 7*a^2*b*c*d^2))/(24*b^4) + (d^6*x^6*(a*d + 7*b*c))/(8*b^2) + (d^3*x^3*(a^4*d^4 + 210*b^4*
c^4 + 28*a^2*b^2*c^2*d^2 + 84*a*b^3*c^3*d + 7*a^3*b*c*d^3))/(66*b^5) + (d*x*(a^6*d^6 + 924*b^6*c^6 + 210*a^2*b
^4*c^4*d^2 + 84*a^3*b^3*c^3*d^3 + 28*a^4*b^2*c^2*d^4 + 462*a*b^5*c^5*d + 7*a^5*b*c*d^5))/(1716*b^7) + (d^5*x^5
*(a^2*d^2 + 28*b^2*c^2 + 7*a*b*c*d))/(12*b^3))/(a^14 + b^14*x^14 + 14*a*b^13*x^13 + 91*a^12*b^2*x^2 + 364*a^11
*b^3*x^3 + 1001*a^10*b^4*x^4 + 2002*a^9*b^5*x^5 + 3003*a^8*b^6*x^6 + 3432*a^7*b^7*x^7 + 3003*a^6*b^8*x^8 + 200
2*a^5*b^9*x^9 + 1001*a^4*b^10*x^10 + 364*a^3*b^11*x^11 + 91*a^2*b^12*x^12 + 14*a^13*b*x)

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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)**15,x)

[Out]

Timed out

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