3.1286 \(\int \frac{(c+d x)^7}{(a+b x)^4} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.21263, antiderivative size = 187, normalized size of antiderivative = 1., 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} \[ \frac{7 d^6 (a+b x)^3 (b c-a d)}{3 b^8}+\frac{21 d^5 (a+b x)^2 (b c-a d)^2}{2 b^8}+\frac{35 d^4 x (b c-a d)^3}{b^7}-\frac{21 d^2 (b c-a d)^5}{b^8 (a+b x)}+\frac{35 d^3 (b c-a d)^4 \log (a+b x)}{b^8}-\frac{7 d (b c-a d)^6}{2 b^8 (a+b x)^2}-\frac{(b c-a d)^7}{3 b^8 (a+b x)^3}+\frac{d^7 (a+b x)^4}{4 b^8} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^7/(a + b*x)^4,x]

[Out]

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

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

Mathematica [A]  time = 0.10457, size = 199, normalized size = 1.06 \[ \frac{6 b^2 d^5 x^2 \left (10 a^2 d^2-28 a b c d+21 b^2 c^2\right )+12 b d^4 x \left (70 a^2 b c d^2-20 a^3 d^3-84 a b^2 c^2 d+35 b^3 c^3\right )+4 b^3 d^6 x^3 (7 b c-4 a d)+\frac{252 d^2 (a d-b c)^5}{a+b x}+420 d^3 (b c-a d)^4 \log (a+b x)-\frac{42 d (b c-a d)^6}{(a+b x)^2}-\frac{4 (b c-a d)^7}{(a+b x)^3}+3 b^4 d^7 x^4}{12 b^8} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)^7/(a + b*x)^4,x]

[Out]

(12*b*d^4*(35*b^3*c^3 - 84*a*b^2*c^2*d + 70*a^2*b*c*d^2 - 20*a^3*d^3)*x + 6*b^2*d^5*(21*b^2*c^2 - 28*a*b*c*d +
10*a^2*d^2)*x^2 + 4*b^3*d^6*(7*b*c - 4*a*d)*x^3 + 3*b^4*d^7*x^4 - (4*(b*c - a*d)^7)/(a + b*x)^3 - (42*d*(b*c
- a*d)^6)/(a + b*x)^2 + (252*d^2*(-(b*c) + a*d)^5)/(a + b*x) + 420*d^3*(b*c - a*d)^4*Log[a + b*x])/(12*b^8)

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Maple [B]  time = 0.011, size = 622, normalized size = 3.3 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)^7/(b*x+a)^4,x)

[Out]

-1/3/b/(b*x+a)^3*c^7+1/4*d^7/b^4*x^4-4/3*d^7/b^5*x^3*a+7/3*d^6/b^4*x^3*c+5*d^7/b^6*x^2*a^2+21/2*d^5/b^4*x^2*c^
2-20*d^7/b^7*a^3*x+35*d^4/b^4*c^3*x+21/b^8*d^7/(b*x+a)*a^5-21/b^3*d^2/(b*x+a)*c^5-7/2/b^8*d^7/(b*x+a)^2*a^6-7/
2/b^2*d/(b*x+a)^2*c^6+1/3/b^8/(b*x+a)^3*a^7*d^7+35/b^8*d^7*ln(b*x+a)*a^4+35/b^4*d^3*ln(b*x+a)*c^4+21/b^3*d^2/(
b*x+a)^2*a*c^5-7/3/b^7/(b*x+a)^3*a^6*c*d^6-140/b^5*d^4*ln(b*x+a)*a*c^3-7/b^3/(b*x+a)^3*a^2*c^5*d^2+7/3/b^2/(b*
x+a)^3*a*c^6*d-35/3/b^5/(b*x+a)^3*a^4*c^3*d^4+35/3/b^4/(b*x+a)^3*a^3*c^4*d^3+7/b^6/(b*x+a)^3*a^5*c^2*d^5-14*d^
6/b^5*x^2*a*c+70*d^6/b^6*a^2*c*x-84*d^5/b^5*a*c^2*x-105/b^7*d^6/(b*x+a)*a^4*c+210/b^6*d^5/(b*x+a)*a^3*c^2-210/
b^5*d^4/(b*x+a)*a^2*c^3+105/b^4*d^3/(b*x+a)*a*c^4+21/b^7*d^6/(b*x+a)^2*a^5*c-105/2/b^6*d^5/(b*x+a)^2*a^4*c^2+7
0/b^5*d^4/(b*x+a)^2*a^3*c^3-105/2/b^4*d^3/(b*x+a)^2*a^2*c^4-140/b^7*d^6*ln(b*x+a)*a^3*c+210/b^6*d^5*ln(b*x+a)*
a^2*c^2

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Maxima [B]  time = 1.02643, size = 653, normalized size = 3.49 \begin{align*} -\frac{2 \, b^{7} c^{7} + 7 \, a b^{6} c^{6} d + 42 \, a^{2} b^{5} c^{5} d^{2} - 385 \, a^{3} b^{4} c^{4} d^{3} + 910 \, a^{4} b^{3} c^{3} d^{4} - 987 \, a^{5} b^{2} c^{2} d^{5} + 518 \, a^{6} b c d^{6} - 107 \, a^{7} d^{7} + 126 \,{\left (b^{7} c^{5} d^{2} - 5 \, a b^{6} c^{4} d^{3} + 10 \, a^{2} b^{5} c^{3} d^{4} - 10 \, 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} + 21 \,{\left (b^{7} c^{6} d + 6 \, a b^{6} c^{5} d^{2} - 45 \, a^{2} b^{5} c^{4} d^{3} + 100 \, a^{3} b^{4} c^{3} d^{4} - 105 \, a^{4} b^{3} c^{2} d^{5} + 54 \, a^{5} b^{2} c d^{6} - 11 \, a^{6} b d^{7}\right )} x}{6 \,{\left (b^{11} x^{3} + 3 \, a b^{10} x^{2} + 3 \, a^{2} b^{9} x + a^{3} b^{8}\right )}} + \frac{3 \, b^{3} d^{7} x^{4} + 4 \,{\left (7 \, b^{3} c d^{6} - 4 \, a b^{2} d^{7}\right )} x^{3} + 6 \,{\left (21 \, b^{3} c^{2} d^{5} - 28 \, a b^{2} c d^{6} + 10 \, a^{2} b d^{7}\right )} x^{2} + 12 \,{\left (35 \, b^{3} c^{3} d^{4} - 84 \, a b^{2} c^{2} d^{5} + 70 \, a^{2} b c d^{6} - 20 \, a^{3} d^{7}\right )} x}{12 \, b^{7}} + \frac{35 \,{\left (b^{4} c^{4} d^{3} - 4 \, a b^{3} c^{3} d^{4} + 6 \, a^{2} b^{2} c^{2} d^{5} - 4 \, a^{3} b c d^{6} + a^{4} d^{7}\right )} \log \left (b x + a\right )}{b^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^7/(b*x+a)^4,x, algorithm="maxima")

[Out]

-1/6*(2*b^7*c^7 + 7*a*b^6*c^6*d + 42*a^2*b^5*c^5*d^2 - 385*a^3*b^4*c^4*d^3 + 910*a^4*b^3*c^3*d^4 - 987*a^5*b^2
*c^2*d^5 + 518*a^6*b*c*d^6 - 107*a^7*d^7 + 126*(b^7*c^5*d^2 - 5*a*b^6*c^4*d^3 + 10*a^2*b^5*c^3*d^4 - 10*a^3*b^
4*c^2*d^5 + 5*a^4*b^3*c*d^6 - a^5*b^2*d^7)*x^2 + 21*(b^7*c^6*d + 6*a*b^6*c^5*d^2 - 45*a^2*b^5*c^4*d^3 + 100*a^
3*b^4*c^3*d^4 - 105*a^4*b^3*c^2*d^5 + 54*a^5*b^2*c*d^6 - 11*a^6*b*d^7)*x)/(b^11*x^3 + 3*a*b^10*x^2 + 3*a^2*b^9
*x + a^3*b^8) + 1/12*(3*b^3*d^7*x^4 + 4*(7*b^3*c*d^6 - 4*a*b^2*d^7)*x^3 + 6*(21*b^3*c^2*d^5 - 28*a*b^2*c*d^6 +
10*a^2*b*d^7)*x^2 + 12*(35*b^3*c^3*d^4 - 84*a*b^2*c^2*d^5 + 70*a^2*b*c*d^6 - 20*a^3*d^7)*x)/b^7 + 35*(b^4*c^4
*d^3 - 4*a*b^3*c^3*d^4 + 6*a^2*b^2*c^2*d^5 - 4*a^3*b*c*d^6 + a^4*d^7)*log(b*x + a)/b^8

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Fricas [B]  time = 2.33662, size = 1520, normalized size = 8.13 \begin{align*} \frac{3 \, b^{7} d^{7} x^{7} - 4 \, b^{7} c^{7} - 14 \, a b^{6} c^{6} d - 84 \, a^{2} b^{5} c^{5} d^{2} + 770 \, a^{3} b^{4} c^{4} d^{3} - 1820 \, a^{4} b^{3} c^{3} d^{4} + 1974 \, a^{5} b^{2} c^{2} d^{5} - 1036 \, a^{6} b c d^{6} + 214 \, a^{7} d^{7} + 7 \,{\left (4 \, b^{7} c d^{6} - a b^{6} d^{7}\right )} x^{6} + 21 \,{\left (6 \, b^{7} c^{2} d^{5} - 4 \, a b^{6} c d^{6} + a^{2} b^{5} d^{7}\right )} x^{5} + 105 \,{\left (4 \, b^{7} c^{3} d^{4} - 6 \, a b^{6} c^{2} d^{5} + 4 \, a^{2} b^{5} c d^{6} - a^{3} b^{4} d^{7}\right )} x^{4} + 2 \,{\left (630 \, a b^{6} c^{3} d^{4} - 1323 \, a^{2} b^{5} c^{2} d^{5} + 1022 \, a^{3} b^{4} c d^{6} - 278 \, a^{4} b^{3} d^{7}\right )} x^{3} - 6 \,{\left (42 \, b^{7} c^{5} d^{2} - 210 \, a b^{6} c^{4} d^{3} + 210 \, a^{2} b^{5} c^{3} d^{4} + 63 \, a^{3} b^{4} c^{2} d^{5} - 182 \, a^{4} b^{3} c d^{6} + 68 \, a^{5} b^{2} d^{7}\right )} x^{2} - 6 \,{\left (7 \, b^{7} c^{6} d + 42 \, a b^{6} c^{5} d^{2} - 315 \, a^{2} b^{5} c^{4} d^{3} + 630 \, a^{3} b^{4} c^{3} d^{4} - 567 \, a^{4} b^{3} c^{2} d^{5} + 238 \, a^{5} b^{2} c d^{6} - 37 \, a^{6} b d^{7}\right )} x + 420 \,{\left (a^{3} b^{4} c^{4} d^{3} - 4 \, a^{4} b^{3} c^{3} d^{4} + 6 \, a^{5} b^{2} c^{2} d^{5} - 4 \, a^{6} b c d^{6} + a^{7} d^{7} +{\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} + 3 \,{\left (a b^{6} c^{4} d^{3} - 4 \, a^{2} b^{5} c^{3} d^{4} + 6 \, a^{3} b^{4} c^{2} d^{5} - 4 \, a^{4} b^{3} c d^{6} + a^{5} b^{2} d^{7}\right )} x^{2} + 3 \,{\left (a^{2} b^{5} c^{4} d^{3} - 4 \, a^{3} b^{4} c^{3} d^{4} + 6 \, a^{4} b^{3} c^{2} d^{5} - 4 \, a^{5} b^{2} c d^{6} + a^{6} b d^{7}\right )} x\right )} \log \left (b x + a\right )}{12 \,{\left (b^{11} x^{3} + 3 \, a b^{10} x^{2} + 3 \, a^{2} b^{9} x + a^{3} b^{8}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^7/(b*x+a)^4,x, algorithm="fricas")

[Out]

1/12*(3*b^7*d^7*x^7 - 4*b^7*c^7 - 14*a*b^6*c^6*d - 84*a^2*b^5*c^5*d^2 + 770*a^3*b^4*c^4*d^3 - 1820*a^4*b^3*c^3
*d^4 + 1974*a^5*b^2*c^2*d^5 - 1036*a^6*b*c*d^6 + 214*a^7*d^7 + 7*(4*b^7*c*d^6 - a*b^6*d^7)*x^6 + 21*(6*b^7*c^2
*d^5 - 4*a*b^6*c*d^6 + a^2*b^5*d^7)*x^5 + 105*(4*b^7*c^3*d^4 - 6*a*b^6*c^2*d^5 + 4*a^2*b^5*c*d^6 - a^3*b^4*d^7
)*x^4 + 2*(630*a*b^6*c^3*d^4 - 1323*a^2*b^5*c^2*d^5 + 1022*a^3*b^4*c*d^6 - 278*a^4*b^3*d^7)*x^3 - 6*(42*b^7*c^
5*d^2 - 210*a*b^6*c^4*d^3 + 210*a^2*b^5*c^3*d^4 + 63*a^3*b^4*c^2*d^5 - 182*a^4*b^3*c*d^6 + 68*a^5*b^2*d^7)*x^2
- 6*(7*b^7*c^6*d + 42*a*b^6*c^5*d^2 - 315*a^2*b^5*c^4*d^3 + 630*a^3*b^4*c^3*d^4 - 567*a^4*b^3*c^2*d^5 + 238*a
^5*b^2*c*d^6 - 37*a^6*b*d^7)*x + 420*(a^3*b^4*c^4*d^3 - 4*a^4*b^3*c^3*d^4 + 6*a^5*b^2*c^2*d^5 - 4*a^6*b*c*d^6
+ a^7*d^7 + (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 + 3*(a*b^6
*c^4*d^3 - 4*a^2*b^5*c^3*d^4 + 6*a^3*b^4*c^2*d^5 - 4*a^4*b^3*c*d^6 + a^5*b^2*d^7)*x^2 + 3*(a^2*b^5*c^4*d^3 - 4
*a^3*b^4*c^3*d^4 + 6*a^4*b^3*c^2*d^5 - 4*a^5*b^2*c*d^6 + a^6*b*d^7)*x)*log(b*x + a))/(b^11*x^3 + 3*a*b^10*x^2
+ 3*a^2*b^9*x + a^3*b^8)

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Sympy [B]  time = 8.26844, size = 468, normalized size = 2.5 \begin{align*} \frac{107 a^{7} d^{7} - 518 a^{6} b c d^{6} + 987 a^{5} b^{2} c^{2} d^{5} - 910 a^{4} b^{3} c^{3} d^{4} + 385 a^{3} b^{4} c^{4} d^{3} - 42 a^{2} b^{5} c^{5} d^{2} - 7 a b^{6} c^{6} d - 2 b^{7} c^{7} + x^{2} \left (126 a^{5} b^{2} d^{7} - 630 a^{4} b^{3} c d^{6} + 1260 a^{3} b^{4} c^{2} d^{5} - 1260 a^{2} b^{5} c^{3} d^{4} + 630 a b^{6} c^{4} d^{3} - 126 b^{7} c^{5} d^{2}\right ) + x \left (231 a^{6} b d^{7} - 1134 a^{5} b^{2} c d^{6} + 2205 a^{4} b^{3} c^{2} d^{5} - 2100 a^{3} b^{4} c^{3} d^{4} + 945 a^{2} b^{5} c^{4} d^{3} - 126 a b^{6} c^{5} d^{2} - 21 b^{7} c^{6} d\right )}{6 a^{3} b^{8} + 18 a^{2} b^{9} x + 18 a b^{10} x^{2} + 6 b^{11} x^{3}} + \frac{d^{7} x^{4}}{4 b^{4}} - \frac{x^{3} \left (4 a d^{7} - 7 b c d^{6}\right )}{3 b^{5}} + \frac{x^{2} \left (10 a^{2} d^{7} - 28 a b c d^{6} + 21 b^{2} c^{2} d^{5}\right )}{2 b^{6}} - \frac{x \left (20 a^{3} d^{7} - 70 a^{2} b c d^{6} + 84 a b^{2} c^{2} d^{5} - 35 b^{3} c^{3} d^{4}\right )}{b^{7}} + \frac{35 d^{3} \left (a d - b c\right )^{4} \log{\left (a + b x \right )}}{b^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)**7/(b*x+a)**4,x)

[Out]

(107*a**7*d**7 - 518*a**6*b*c*d**6 + 987*a**5*b**2*c**2*d**5 - 910*a**4*b**3*c**3*d**4 + 385*a**3*b**4*c**4*d*
*3 - 42*a**2*b**5*c**5*d**2 - 7*a*b**6*c**6*d - 2*b**7*c**7 + x**2*(126*a**5*b**2*d**7 - 630*a**4*b**3*c*d**6
+ 1260*a**3*b**4*c**2*d**5 - 1260*a**2*b**5*c**3*d**4 + 630*a*b**6*c**4*d**3 - 126*b**7*c**5*d**2) + x*(231*a*
*6*b*d**7 - 1134*a**5*b**2*c*d**6 + 2205*a**4*b**3*c**2*d**5 - 2100*a**3*b**4*c**3*d**4 + 945*a**2*b**5*c**4*d
**3 - 126*a*b**6*c**5*d**2 - 21*b**7*c**6*d))/(6*a**3*b**8 + 18*a**2*b**9*x + 18*a*b**10*x**2 + 6*b**11*x**3)
+ d**7*x**4/(4*b**4) - x**3*(4*a*d**7 - 7*b*c*d**6)/(3*b**5) + x**2*(10*a**2*d**7 - 28*a*b*c*d**6 + 21*b**2*c*
*2*d**5)/(2*b**6) - x*(20*a**3*d**7 - 70*a**2*b*c*d**6 + 84*a*b**2*c**2*d**5 - 35*b**3*c**3*d**4)/b**7 + 35*d*
*3*(a*d - b*c)**4*log(a + b*x)/b**8

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Giac [B]  time = 1.0549, size = 635, normalized size = 3.4 \begin{align*} \frac{35 \,{\left (b^{4} c^{4} d^{3} - 4 \, a b^{3} c^{3} d^{4} + 6 \, a^{2} b^{2} c^{2} d^{5} - 4 \, a^{3} b c d^{6} + a^{4} d^{7}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{8}} - \frac{2 \, b^{7} c^{7} + 7 \, a b^{6} c^{6} d + 42 \, a^{2} b^{5} c^{5} d^{2} - 385 \, a^{3} b^{4} c^{4} d^{3} + 910 \, a^{4} b^{3} c^{3} d^{4} - 987 \, a^{5} b^{2} c^{2} d^{5} + 518 \, a^{6} b c d^{6} - 107 \, a^{7} d^{7} + 126 \,{\left (b^{7} c^{5} d^{2} - 5 \, a b^{6} c^{4} d^{3} + 10 \, a^{2} b^{5} c^{3} d^{4} - 10 \, 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} + 21 \,{\left (b^{7} c^{6} d + 6 \, a b^{6} c^{5} d^{2} - 45 \, a^{2} b^{5} c^{4} d^{3} + 100 \, a^{3} b^{4} c^{3} d^{4} - 105 \, a^{4} b^{3} c^{2} d^{5} + 54 \, a^{5} b^{2} c d^{6} - 11 \, a^{6} b d^{7}\right )} x}{6 \,{\left (b x + a\right )}^{3} b^{8}} + \frac{3 \, b^{12} d^{7} x^{4} + 28 \, b^{12} c d^{6} x^{3} - 16 \, a b^{11} d^{7} x^{3} + 126 \, b^{12} c^{2} d^{5} x^{2} - 168 \, a b^{11} c d^{6} x^{2} + 60 \, a^{2} b^{10} d^{7} x^{2} + 420 \, b^{12} c^{3} d^{4} x - 1008 \, a b^{11} c^{2} d^{5} x + 840 \, a^{2} b^{10} c d^{6} x - 240 \, a^{3} b^{9} d^{7} x}{12 \, b^{16}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^7/(b*x+a)^4,x, algorithm="giac")

[Out]

35*(b^4*c^4*d^3 - 4*a*b^3*c^3*d^4 + 6*a^2*b^2*c^2*d^5 - 4*a^3*b*c*d^6 + a^4*d^7)*log(abs(b*x + a))/b^8 - 1/6*(
2*b^7*c^7 + 7*a*b^6*c^6*d + 42*a^2*b^5*c^5*d^2 - 385*a^3*b^4*c^4*d^3 + 910*a^4*b^3*c^3*d^4 - 987*a^5*b^2*c^2*d
^5 + 518*a^6*b*c*d^6 - 107*a^7*d^7 + 126*(b^7*c^5*d^2 - 5*a*b^6*c^4*d^3 + 10*a^2*b^5*c^3*d^4 - 10*a^3*b^4*c^2*
d^5 + 5*a^4*b^3*c*d^6 - a^5*b^2*d^7)*x^2 + 21*(b^7*c^6*d + 6*a*b^6*c^5*d^2 - 45*a^2*b^5*c^4*d^3 + 100*a^3*b^4*
c^3*d^4 - 105*a^4*b^3*c^2*d^5 + 54*a^5*b^2*c*d^6 - 11*a^6*b*d^7)*x)/((b*x + a)^3*b^8) + 1/12*(3*b^12*d^7*x^4 +
28*b^12*c*d^6*x^3 - 16*a*b^11*d^7*x^3 + 126*b^12*c^2*d^5*x^2 - 168*a*b^11*c*d^6*x^2 + 60*a^2*b^10*d^7*x^2 + 4
20*b^12*c^3*d^4*x - 1008*a*b^11*c^2*d^5*x + 840*a^2*b^10*c*d^6*x - 240*a^3*b^9*d^7*x)/b^16