3.13.87 \(\int \frac {(c+d x)^7}{(a+b x)^5} \, dx\) [1287]

Optimal. Leaf size=187 \[ \frac {21 d^5 (b c-a d)^2 x}{b^7}-\frac {(b c-a d)^7}{4 b^8 (a+b x)^4}-\frac {7 d (b c-a d)^6}{3 b^8 (a+b x)^3}-\frac {21 d^2 (b c-a d)^5}{2 b^8 (a+b x)^2}-\frac {35 d^3 (b c-a d)^4}{b^8 (a+b x)}+\frac {7 d^6 (b c-a d) (a+b x)^2}{2 b^8}+\frac {d^7 (a+b x)^3}{3 b^8}+\frac {35 d^4 (b c-a d)^3 \log (a+b x)}{b^8} \]

[Out]

21*d^5*(-a*d+b*c)^2*x/b^7-1/4*(-a*d+b*c)^7/b^8/(b*x+a)^4-7/3*d*(-a*d+b*c)^6/b^8/(b*x+a)^3-21/2*d^2*(-a*d+b*c)^
5/b^8/(b*x+a)^2-35*d^3*(-a*d+b*c)^4/b^8/(b*x+a)+7/2*d^6*(-a*d+b*c)*(b*x+a)^2/b^8+1/3*d^7*(b*x+a)^3/b^8+35*d^4*
(-a*d+b*c)^3*ln(b*x+a)/b^8

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Rubi [A]
time = 0.13, antiderivative size = 187, 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 {7 d^6 (a+b x)^2 (b c-a d)}{2 b^8}+\frac {35 d^4 (b c-a d)^3 \log (a+b x)}{b^8}-\frac {35 d^3 (b c-a d)^4}{b^8 (a+b x)}-\frac {21 d^2 (b c-a d)^5}{2 b^8 (a+b x)^2}-\frac {7 d (b c-a d)^6}{3 b^8 (a+b x)^3}-\frac {(b c-a d)^7}{4 b^8 (a+b x)^4}+\frac {d^7 (a+b x)^3}{3 b^8}+\frac {21 d^5 x (b c-a d)^2}{b^7} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^7/(a + b*x)^5,x]

[Out]

(21*d^5*(b*c - a*d)^2*x)/b^7 - (b*c - a*d)^7/(4*b^8*(a + b*x)^4) - (7*d*(b*c - a*d)^6)/(3*b^8*(a + b*x)^3) - (
21*d^2*(b*c - a*d)^5)/(2*b^8*(a + b*x)^2) - (35*d^3*(b*c - a*d)^4)/(b^8*(a + b*x)) + (7*d^6*(b*c - a*d)*(a + b
*x)^2)/(2*b^8) + (d^7*(a + b*x)^3)/(3*b^8) + (35*d^4*(b*c - a*d)^3*Log[a + b*x])/b^8

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

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Mathematica [A]
time = 0.07, size = 173, normalized size = 0.93 \begin {gather*} \frac {12 b d^5 \left (21 b^2 c^2-35 a b c d+15 a^2 d^2\right ) x+6 b^2 d^6 (7 b c-5 a d) x^2+4 b^3 d^7 x^3-\frac {3 (b c-a d)^7}{(a+b x)^4}-\frac {28 d (b c-a d)^6}{(a+b x)^3}+\frac {126 d^2 (-b c+a d)^5}{(a+b x)^2}-\frac {420 d^3 (b c-a d)^4}{a+b x}+420 d^4 (b c-a d)^3 \log (a+b x)}{12 b^8} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)^7/(a + b*x)^5,x]

[Out]

(12*b*d^5*(21*b^2*c^2 - 35*a*b*c*d + 15*a^2*d^2)*x + 6*b^2*d^6*(7*b*c - 5*a*d)*x^2 + 4*b^3*d^7*x^3 - (3*(b*c -
a*d)^7)/(a + b*x)^4 - (28*d*(b*c - a*d)^6)/(a + b*x)^3 + (126*d^2*(-(b*c) + a*d)^5)/(a + b*x)^2 - (420*d^3*(b
*c - a*d)^4)/(a + b*x) + 420*d^4*(b*c - a*d)^3*Log[a + b*x])/(12*b^8)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(452\) vs. \(2(177)=354\).
time = 0.14, size = 453, normalized size = 2.42

method	result	size
norman	\(\frac {-\frac {875 a^{7} d^{7}-2625 a^{6} b c \,d^{6}+2625 a^{5} b^{2} c^{2} d^{5}-875 a^{4} b^{3} c^{3} d^{4}+105 a^{3} b^{4} c^{4} d^{3}+21 a^{2} b^{5} c^{5} d^{2}+7 a \,b^{6} c^{6} d +3 b^{7} c^{7}}{12 b^{8}}+\frac {d^{7} x^{7}}{3 b}-\frac {\left (140 a^{4} d^{7}-420 a^{3} b c \,d^{6}+420 b^{2} a^{2} c^{2} d^{5}-140 a \,b^{3} c^{3} d^{4}+35 b^{4} c^{4} d^{3}\right ) x^{3}}{b^{5}}-\frac {3 \left (210 a^{5} d^{7}-630 a^{4} b c \,d^{6}+630 a^{3} b^{2} c^{2} d^{5}-210 a^{2} b^{3} c^{3} d^{4}+35 a \,b^{4} c^{4} d^{3}+7 b^{5} c^{5} d^{2}\right ) x^{2}}{2 b^{6}}-\frac {\left (770 a^{6} d^{7}-2310 a^{5} b c \,d^{6}+2310 a^{4} b^{2} c^{2} d^{5}-770 a^{3} b^{3} c^{3} d^{4}+105 a^{2} b^{4} c^{4} d^{3}+21 a \,b^{5} c^{5} d^{2}+7 b^{6} c^{6} d \right ) x}{3 b^{7}}+\frac {7 d^{5} \left (a^{2} d^{2}-3 a b c d +3 b^{2} c^{2}\right ) x^{5}}{b^{3}}-\frac {7 d^{6} \left (a d -3 b c \right ) x^{6}}{6 b^{2}}}{\left (b x +a \right )^{4}}-\frac {35 d^{4} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \ln \left (b x +a \right )}{b^{8}}\)	\(452\)
default	\(\frac {d^{5} \left (\frac {1}{3} d^{2} x^{3} b^{2}-\frac {5}{2} a b \,d^{2} x^{2}+\frac {7}{2} b^{2} c d \,x^{2}+15 a^{2} d^{2} x -35 a b c d x +21 b^{2} c^{2} x \right )}{b^{7}}-\frac {35 d^{3} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}{b^{8} \left (b x +a \right )}-\frac {-a^{7} d^{7}+7 a^{6} b c \,d^{6}-21 a^{5} b^{2} c^{2} d^{5}+35 a^{4} b^{3} c^{3} d^{4}-35 a^{3} b^{4} c^{4} d^{3}+21 a^{2} b^{5} c^{5} d^{2}-7 a \,b^{6} c^{6} d +b^{7} c^{7}}{4 b^{8} \left (b x +a \right )^{4}}+\frac {21 d^{2} \left (a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -b^{5} c^{5}\right )}{2 b^{8} \left (b x +a \right )^{2}}-\frac {35 d^{4} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \ln \left (b x +a \right )}{b^{8}}-\frac {7 d \left (a^{6} d^{6}-6 a^{5} b c \,d^{5}+15 a^{4} b^{2} c^{2} d^{4}-20 a^{3} b^{3} c^{3} d^{3}+15 a^{2} b^{4} c^{4} d^{2}-6 a \,b^{5} c^{5} d +b^{6} c^{6}\right )}{3 b^{8} \left (b x +a \right )^{3}}\)	\(453\)
risch	\(\frac {d^{7} x^{3}}{3 b^{5}}-\frac {5 d^{7} a \,x^{2}}{2 b^{6}}+\frac {7 d^{6} c \,x^{2}}{2 b^{5}}+\frac {15 d^{7} a^{2} x}{b^{7}}-\frac {35 d^{6} a c x}{b^{6}}+\frac {21 d^{5} c^{2} x}{b^{5}}+\frac {\left (-35 a^{4} b^{2} d^{7}+140 a^{3} b^{3} c \,d^{6}-210 a^{2} b^{4} c^{2} d^{5}+140 a \,b^{5} c^{3} d^{4}-35 b^{6} c^{4} d^{3}\right ) x^{3}-\frac {21 b \,d^{2} \left (9 a^{5} d^{5}-35 a^{4} b c \,d^{4}+50 a^{3} b^{2} c^{2} d^{3}-30 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d +b^{5} c^{5}\right ) x^{2}}{2}-\frac {7 d \left (37 a^{6} d^{6}-141 a^{5} b c \,d^{5}+195 a^{4} b^{2} c^{2} d^{4}-110 a^{3} b^{3} c^{3} d^{3}+15 a^{2} b^{4} c^{4} d^{2}+3 a \,b^{5} c^{5} d +b^{6} c^{6}\right ) x}{3}-\frac {319 a^{7} d^{7}-1197 a^{6} b c \,d^{6}+1617 a^{5} b^{2} c^{2} d^{5}-875 a^{4} b^{3} c^{3} d^{4}+105 a^{3} b^{4} c^{4} d^{3}+21 a^{2} b^{5} c^{5} d^{2}+7 a \,b^{6} c^{6} d +3 b^{7} c^{7}}{12 b}}{b^{7} \left (b x +a \right )^{4}}-\frac {35 d^{7} \ln \left (b x +a \right ) a^{3}}{b^{8}}+\frac {105 d^{6} \ln \left (b x +a \right ) a^{2} c}{b^{7}}-\frac {105 d^{5} \ln \left (b x +a \right ) a \,c^{2}}{b^{6}}+\frac {35 d^{4} \ln \left (b x +a \right ) c^{3}}{b^{5}}\)	\(472\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)^7/(b*x+a)^5,x,method=_RETURNVERBOSE)

[Out]

d^5/b^7*(1/3*d^2*x^3*b^2-5/2*a*b*d^2*x^2+7/2*b^2*c*d*x^2+15*a^2*d^2*x-35*a*b*c*d*x+21*b^2*c^2*x)-35/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)-1/4/b^8*(-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*x+a)^4+21/2/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)^2-35/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)*ln(b*x+a)-7/3/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)^3

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 494 vs. \(2 (177) = 354\).
time = 0.35, size = 494, normalized size = 2.64 \begin {gather*} -\frac {3 \, b^{7} c^{7} + 7 \, a b^{6} c^{6} d + 21 \, a^{2} b^{5} c^{5} d^{2} + 105 \, a^{3} b^{4} c^{4} d^{3} - 875 \, a^{4} b^{3} c^{3} d^{4} + 1617 \, a^{5} b^{2} c^{2} d^{5} - 1197 \, a^{6} b c d^{6} + 319 \, a^{7} d^{7} + 420 \, {\left (b^{7} c^{4} d^{3} - 4 \, a b^{6} c^{3} d^{4} + 6 \, a^{2} b^{5} c^{2} d^{5} - 4 \, a^{3} b^{4} c d^{6} + a^{4} b^{3} d^{7}\right )} x^{3} + 126 \, {\left (b^{7} c^{5} d^{2} + 5 \, a b^{6} c^{4} d^{3} - 30 \, a^{2} b^{5} c^{3} d^{4} + 50 \, a^{3} b^{4} c^{2} d^{5} - 35 \, a^{4} b^{3} c d^{6} + 9 \, a^{5} b^{2} d^{7}\right )} x^{2} + 28 \, {\left (b^{7} c^{6} d + 3 \, a b^{6} c^{5} d^{2} + 15 \, a^{2} b^{5} c^{4} d^{3} - 110 \, a^{3} b^{4} c^{3} d^{4} + 195 \, a^{4} b^{3} c^{2} d^{5} - 141 \, a^{5} b^{2} c d^{6} + 37 \, a^{6} b d^{7}\right )} x}{12 \, {\left (b^{12} x^{4} + 4 \, a b^{11} x^{3} + 6 \, a^{2} b^{10} x^{2} + 4 \, a^{3} b^{9} x + a^{4} b^{8}\right )}} + \frac {2 \, b^{2} d^{7} x^{3} + 3 \, {\left (7 \, b^{2} c d^{6} - 5 \, a b d^{7}\right )} x^{2} + 6 \, {\left (21 \, b^{2} c^{2} d^{5} - 35 \, a b c d^{6} + 15 \, a^{2} d^{7}\right )} x}{6 \, b^{7}} + \frac {35 \, {\left (b^{3} c^{3} d^{4} - 3 \, a b^{2} c^{2} d^{5} + 3 \, a^{2} b c d^{6} - a^{3} d^{7}\right )} \log \left (b x + a\right )}{b^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^7/(b*x+a)^5,x, algorithm="maxima")

[Out]

-1/12*(3*b^7*c^7 + 7*a*b^6*c^6*d + 21*a^2*b^5*c^5*d^2 + 105*a^3*b^4*c^4*d^3 - 875*a^4*b^3*c^3*d^4 + 1617*a^5*b
^2*c^2*d^5 - 1197*a^6*b*c*d^6 + 319*a^7*d^7 + 420*(b^7*c^4*d^3 - 4*a*b^6*c^3*d^4 + 6*a^2*b^5*c^2*d^5 - 4*a^3*b
^4*c*d^6 + a^4*b^3*d^7)*x^3 + 126*(b^7*c^5*d^2 + 5*a*b^6*c^4*d^3 - 30*a^2*b^5*c^3*d^4 + 50*a^3*b^4*c^2*d^5 - 3
5*a^4*b^3*c*d^6 + 9*a^5*b^2*d^7)*x^2 + 28*(b^7*c^6*d + 3*a*b^6*c^5*d^2 + 15*a^2*b^5*c^4*d^3 - 110*a^3*b^4*c^3*
d^4 + 195*a^4*b^3*c^2*d^5 - 141*a^5*b^2*c*d^6 + 37*a^6*b*d^7)*x)/(b^12*x^4 + 4*a*b^11*x^3 + 6*a^2*b^10*x^2 + 4
*a^3*b^9*x + a^4*b^8) + 1/6*(2*b^2*d^7*x^3 + 3*(7*b^2*c*d^6 - 5*a*b*d^7)*x^2 + 6*(21*b^2*c^2*d^5 - 35*a*b*c*d^
6 + 15*a^2*d^7)*x)/b^7 + 35*(b^3*c^3*d^4 - 3*a*b^2*c^2*d^5 + 3*a^2*b*c*d^6 - a^3*d^7)*log(b*x + a)/b^8

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 754 vs. \(2 (177) = 354\).
time = 0.90, size = 754, normalized size = 4.03 \begin {gather*} \frac {4 \, b^{7} d^{7} x^{7} - 3 \, b^{7} c^{7} - 7 \, a b^{6} c^{6} d - 21 \, a^{2} b^{5} c^{5} d^{2} - 105 \, a^{3} b^{4} c^{4} d^{3} + 875 \, a^{4} b^{3} c^{3} d^{4} - 1617 \, a^{5} b^{2} c^{2} d^{5} + 1197 \, a^{6} b c d^{6} - 319 \, a^{7} d^{7} + 14 \, {\left (3 \, b^{7} c d^{6} - a b^{6} d^{7}\right )} x^{6} + 84 \, {\left (3 \, b^{7} c^{2} d^{5} - 3 \, a b^{6} c d^{6} + a^{2} b^{5} d^{7}\right )} x^{5} + 4 \, {\left (252 \, a b^{6} c^{2} d^{5} - 357 \, a^{2} b^{5} c d^{6} + 139 \, a^{3} b^{4} d^{7}\right )} x^{4} - 4 \, {\left (105 \, b^{7} c^{4} d^{3} - 420 \, a b^{6} c^{3} d^{4} + 252 \, a^{2} b^{5} c^{2} d^{5} + 168 \, a^{3} b^{4} c d^{6} - 136 \, a^{4} b^{3} d^{7}\right )} x^{3} - 6 \, {\left (21 \, b^{7} c^{5} d^{2} + 105 \, a b^{6} c^{4} d^{3} - 630 \, a^{2} b^{5} c^{3} d^{4} + 882 \, a^{3} b^{4} c^{2} d^{5} - 462 \, a^{4} b^{3} c d^{6} + 74 \, a^{5} b^{2} d^{7}\right )} x^{2} - 4 \, {\left (7 \, b^{7} c^{6} d + 21 \, a b^{6} c^{5} d^{2} + 105 \, a^{2} b^{5} c^{4} d^{3} - 770 \, a^{3} b^{4} c^{3} d^{4} + 1302 \, a^{4} b^{3} c^{2} d^{5} - 882 \, a^{5} b^{2} c d^{6} + 214 \, a^{6} b d^{7}\right )} x + 420 \, {\left (a^{4} b^{3} c^{3} d^{4} - 3 \, a^{5} b^{2} c^{2} d^{5} + 3 \, a^{6} b c d^{6} - a^{7} d^{7} + {\left (b^{7} c^{3} d^{4} - 3 \, a b^{6} c^{2} d^{5} + 3 \, a^{2} b^{5} c d^{6} - a^{3} b^{4} d^{7}\right )} x^{4} + 4 \, {\left (a b^{6} c^{3} d^{4} - 3 \, a^{2} b^{5} c^{2} d^{5} + 3 \, a^{3} b^{4} c d^{6} - a^{4} b^{3} d^{7}\right )} x^{3} + 6 \, {\left (a^{2} b^{5} c^{3} d^{4} - 3 \, a^{3} b^{4} c^{2} d^{5} + 3 \, a^{4} b^{3} c d^{6} - a^{5} b^{2} d^{7}\right )} x^{2} + 4 \, {\left (a^{3} b^{4} c^{3} d^{4} - 3 \, a^{4} b^{3} c^{2} d^{5} + 3 \, a^{5} b^{2} c d^{6} - a^{6} b d^{7}\right )} x\right )} \log \left (b x + a\right )}{12 \, {\left (b^{12} x^{4} + 4 \, a b^{11} x^{3} + 6 \, a^{2} b^{10} x^{2} + 4 \, a^{3} b^{9} x + a^{4} b^{8}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^7/(b*x+a)^5,x, algorithm="fricas")

[Out]

1/12*(4*b^7*d^7*x^7 - 3*b^7*c^7 - 7*a*b^6*c^6*d - 21*a^2*b^5*c^5*d^2 - 105*a^3*b^4*c^4*d^3 + 875*a^4*b^3*c^3*d
^4 - 1617*a^5*b^2*c^2*d^5 + 1197*a^6*b*c*d^6 - 319*a^7*d^7 + 14*(3*b^7*c*d^6 - a*b^6*d^7)*x^6 + 84*(3*b^7*c^2*
d^5 - 3*a*b^6*c*d^6 + a^2*b^5*d^7)*x^5 + 4*(252*a*b^6*c^2*d^5 - 357*a^2*b^5*c*d^6 + 139*a^3*b^4*d^7)*x^4 - 4*(
105*b^7*c^4*d^3 - 420*a*b^6*c^3*d^4 + 252*a^2*b^5*c^2*d^5 + 168*a^3*b^4*c*d^6 - 136*a^4*b^3*d^7)*x^3 - 6*(21*b
^7*c^5*d^2 + 105*a*b^6*c^4*d^3 - 630*a^2*b^5*c^3*d^4 + 882*a^3*b^4*c^2*d^5 - 462*a^4*b^3*c*d^6 + 74*a^5*b^2*d^
7)*x^2 - 4*(7*b^7*c^6*d + 21*a*b^6*c^5*d^2 + 105*a^2*b^5*c^4*d^3 - 770*a^3*b^4*c^3*d^4 + 1302*a^4*b^3*c^2*d^5
- 882*a^5*b^2*c*d^6 + 214*a^6*b*d^7)*x + 420*(a^4*b^3*c^3*d^4 - 3*a^5*b^2*c^2*d^5 + 3*a^6*b*c*d^6 - a^7*d^7 +
(b^7*c^3*d^4 - 3*a*b^6*c^2*d^5 + 3*a^2*b^5*c*d^6 - a^3*b^4*d^7)*x^4 + 4*(a*b^6*c^3*d^4 - 3*a^2*b^5*c^2*d^5 + 3
*a^3*b^4*c*d^6 - a^4*b^3*d^7)*x^3 + 6*(a^2*b^5*c^3*d^4 - 3*a^3*b^4*c^2*d^5 + 3*a^4*b^3*c*d^6 - a^5*b^2*d^7)*x^
2 + 4*(a^3*b^4*c^3*d^4 - 3*a^4*b^3*c^2*d^5 + 3*a^5*b^2*c*d^6 - a^6*b*d^7)*x)*log(b*x + a))/(b^12*x^4 + 4*a*b^1
1*x^3 + 6*a^2*b^10*x^2 + 4*a^3*b^9*x + a^4*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)**5,x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 660 vs. \(2 (177) = 354\).
time = 0.61, size = 660, normalized size = 3.53 \begin {gather*} \frac {{\left (2 \, d^{7} + \frac {21 \, {\left (b^{2} c d^{6} - a b d^{7}\right )}}{{\left (b x + a\right )} b} + \frac {126 \, {\left (b^{4} c^{2} d^{5} - 2 \, a b^{3} c d^{6} + a^{2} b^{2} d^{7}\right )}}{{\left (b x + a\right )}^{2} b^{2}}\right )} {\left (b x + a\right )}^{3}}{6 \, b^{8}} - \frac {35 \, {\left (b^{3} c^{3} d^{4} - 3 \, a b^{2} c^{2} d^{5} + 3 \, a^{2} b c d^{6} - a^{3} d^{7}\right )} \log \left (\frac {{\left | b x + a \right |}}{{\left (b x + a\right )}^{2} {\left | b \right |}}\right )}{b^{8}} - \frac {\frac {3 \, b^{43} c^{7}}{{\left (b x + a\right )}^{4}} + \frac {28 \, b^{42} c^{6} d}{{\left (b x + a\right )}^{3}} - \frac {21 \, a b^{42} c^{6} d}{{\left (b x + a\right )}^{4}} + \frac {126 \, b^{41} c^{5} d^{2}}{{\left (b x + a\right )}^{2}} - \frac {168 \, a b^{41} c^{5} d^{2}}{{\left (b x + a\right )}^{3}} + \frac {63 \, a^{2} b^{41} c^{5} d^{2}}{{\left (b x + a\right )}^{4}} + \frac {420 \, b^{40} c^{4} d^{3}}{b x + a} - \frac {630 \, a b^{40} c^{4} d^{3}}{{\left (b x + a\right )}^{2}} + \frac {420 \, a^{2} b^{40} c^{4} d^{3}}{{\left (b x + a\right )}^{3}} - \frac {105 \, a^{3} b^{40} c^{4} d^{3}}{{\left (b x + a\right )}^{4}} - \frac {1680 \, a b^{39} c^{3} d^{4}}{b x + a} + \frac {1260 \, a^{2} b^{39} c^{3} d^{4}}{{\left (b x + a\right )}^{2}} - \frac {560 \, a^{3} b^{39} c^{3} d^{4}}{{\left (b x + a\right )}^{3}} + \frac {105 \, a^{4} b^{39} c^{3} d^{4}}{{\left (b x + a\right )}^{4}} + \frac {2520 \, a^{2} b^{38} c^{2} d^{5}}{b x + a} - \frac {1260 \, a^{3} b^{38} c^{2} d^{5}}{{\left (b x + a\right )}^{2}} + \frac {420 \, a^{4} b^{38} c^{2} d^{5}}{{\left (b x + a\right )}^{3}} - \frac {63 \, a^{5} b^{38} c^{2} d^{5}}{{\left (b x + a\right )}^{4}} - \frac {1680 \, a^{3} b^{37} c d^{6}}{b x + a} + \frac {630 \, a^{4} b^{37} c d^{6}}{{\left (b x + a\right )}^{2}} - \frac {168 \, a^{5} b^{37} c d^{6}}{{\left (b x + a\right )}^{3}} + \frac {21 \, a^{6} b^{37} c d^{6}}{{\left (b x + a\right )}^{4}} + \frac {420 \, a^{4} b^{36} d^{7}}{b x + a} - \frac {126 \, a^{5} b^{36} d^{7}}{{\left (b x + a\right )}^{2}} + \frac {28 \, a^{6} b^{36} d^{7}}{{\left (b x + a\right )}^{3}} - \frac {3 \, a^{7} b^{36} d^{7}}{{\left (b x + a\right )}^{4}}}{12 \, b^{44}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^7/(b*x+a)^5,x, algorithm="giac")

[Out]

1/6*(2*d^7 + 21*(b^2*c*d^6 - a*b*d^7)/((b*x + a)*b) + 126*(b^4*c^2*d^5 - 2*a*b^3*c*d^6 + a^2*b^2*d^7)/((b*x +
a)^2*b^2))*(b*x + a)^3/b^8 - 35*(b^3*c^3*d^4 - 3*a*b^2*c^2*d^5 + 3*a^2*b*c*d^6 - a^3*d^7)*log(abs(b*x + a)/((b
*x + a)^2*abs(b)))/b^8 - 1/12*(3*b^43*c^7/(b*x + a)^4 + 28*b^42*c^6*d/(b*x + a)^3 - 21*a*b^42*c^6*d/(b*x + a)^
4 + 126*b^41*c^5*d^2/(b*x + a)^2 - 168*a*b^41*c^5*d^2/(b*x + a)^3 + 63*a^2*b^41*c^5*d^2/(b*x + a)^4 + 420*b^40
*c^4*d^3/(b*x + a) - 630*a*b^40*c^4*d^3/(b*x + a)^2 + 420*a^2*b^40*c^4*d^3/(b*x + a)^3 - 105*a^3*b^40*c^4*d^3/
(b*x + a)^4 - 1680*a*b^39*c^3*d^4/(b*x + a) + 1260*a^2*b^39*c^3*d^4/(b*x + a)^2 - 560*a^3*b^39*c^3*d^4/(b*x +
a)^3 + 105*a^4*b^39*c^3*d^4/(b*x + a)^4 + 2520*a^2*b^38*c^2*d^5/(b*x + a) - 1260*a^3*b^38*c^2*d^5/(b*x + a)^2
+ 420*a^4*b^38*c^2*d^5/(b*x + a)^3 - 63*a^5*b^38*c^2*d^5/(b*x + a)^4 - 1680*a^3*b^37*c*d^6/(b*x + a) + 630*a^4
*b^37*c*d^6/(b*x + a)^2 - 168*a^5*b^37*c*d^6/(b*x + a)^3 + 21*a^6*b^37*c*d^6/(b*x + a)^4 + 420*a^4*b^36*d^7/(b
*x + a) - 126*a^5*b^36*d^7/(b*x + a)^2 + 28*a^6*b^36*d^7/(b*x + a)^3 - 3*a^7*b^36*d^7/(b*x + a)^4)/b^44

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Mupad [B]
time = 0.77, size = 512, normalized size = 2.74 \begin {gather*} x\,\left (\frac {5\,a\,\left (\frac {5\,a\,d^7}{b^6}-\frac {7\,c\,d^6}{b^5}\right )}{b}-\frac {10\,a^2\,d^7}{b^7}+\frac {21\,c^2\,d^5}{b^5}\right )-x^2\,\left (\frac {5\,a\,d^7}{2\,b^6}-\frac {7\,c\,d^6}{2\,b^5}\right )-\frac {\frac {319\,a^7\,d^7-1197\,a^6\,b\,c\,d^6+1617\,a^5\,b^2\,c^2\,d^5-875\,a^4\,b^3\,c^3\,d^4+105\,a^3\,b^4\,c^4\,d^3+21\,a^2\,b^5\,c^5\,d^2+7\,a\,b^6\,c^6\,d+3\,b^7\,c^7}{12\,b}+x\,\left (\frac {259\,a^6\,d^7}{3}-329\,a^5\,b\,c\,d^6+455\,a^4\,b^2\,c^2\,d^5-\frac {770\,a^3\,b^3\,c^3\,d^4}{3}+35\,a^2\,b^4\,c^4\,d^3+7\,a\,b^5\,c^5\,d^2+\frac {7\,b^6\,c^6\,d}{3}\right )+x^3\,\left (35\,a^4\,b^2\,d^7-140\,a^3\,b^3\,c\,d^6+210\,a^2\,b^4\,c^2\,d^5-140\,a\,b^5\,c^3\,d^4+35\,b^6\,c^4\,d^3\right )+x^2\,\left (\frac {189\,a^5\,b\,d^7}{2}-\frac {735\,a^4\,b^2\,c\,d^6}{2}+525\,a^3\,b^3\,c^2\,d^5-315\,a^2\,b^4\,c^3\,d^4+\frac {105\,a\,b^5\,c^4\,d^3}{2}+\frac {21\,b^6\,c^5\,d^2}{2}\right )}{a^4\,b^7+4\,a^3\,b^8\,x+6\,a^2\,b^9\,x^2+4\,a\,b^{10}\,x^3+b^{11}\,x^4}-\frac {\ln \left (a+b\,x\right )\,\left (35\,a^3\,d^7-105\,a^2\,b\,c\,d^6+105\,a\,b^2\,c^2\,d^5-35\,b^3\,c^3\,d^4\right )}{b^8}+\frac {d^7\,x^3}{3\,b^5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x)^7/(a + b*x)^5,x)

[Out]

x*((5*a*((5*a*d^7)/b^6 - (7*c*d^6)/b^5))/b - (10*a^2*d^7)/b^7 + (21*c^2*d^5)/b^5) - x^2*((5*a*d^7)/(2*b^6) - (
7*c*d^6)/(2*b^5)) - ((319*a^7*d^7 + 3*b^7*c^7 + 21*a^2*b^5*c^5*d^2 + 105*a^3*b^4*c^4*d^3 - 875*a^4*b^3*c^3*d^4
+ 1617*a^5*b^2*c^2*d^5 + 7*a*b^6*c^6*d - 1197*a^6*b*c*d^6)/(12*b) + x*((259*a^6*d^7)/3 + (7*b^6*c^6*d)/3 + 7*
a*b^5*c^5*d^2 + 35*a^2*b^4*c^4*d^3 - (770*a^3*b^3*c^3*d^4)/3 + 455*a^4*b^2*c^2*d^5 - 329*a^5*b*c*d^6) + x^3*(3
5*a^4*b^2*d^7 + 35*b^6*c^4*d^3 - 140*a*b^5*c^3*d^4 - 140*a^3*b^3*c*d^6 + 210*a^2*b^4*c^2*d^5) + x^2*((189*a^5*
b*d^7)/2 + (21*b^6*c^5*d^2)/2 + (105*a*b^5*c^4*d^3)/2 - (735*a^4*b^2*c*d^6)/2 - 315*a^2*b^4*c^3*d^4 + 525*a^3*
b^3*c^2*d^5))/(a^4*b^7 + b^11*x^4 + 4*a^3*b^8*x + 4*a*b^10*x^3 + 6*a^2*b^9*x^2) - (log(a + b*x)*(35*a^3*d^7 -
35*b^3*c^3*d^4 + 105*a*b^2*c^2*d^5 - 105*a^2*b*c*d^6))/b^8 + (d^7*x^3)/(3*b^5)

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