3.14.52 \(\int \frac {(a+b x)^6}{(c+d x)^3} \, dx\) [1352]

Optimal. Leaf size=158 \[ -\frac {20 b^3 (b c-a d)^3 x}{d^6}-\frac {(b c-a d)^6}{2 d^7 (c+d x)^2}+\frac {6 b (b c-a d)^5}{d^7 (c+d x)}+\frac {15 b^4 (b c-a d)^2 (c+d x)^2}{2 d^7}-\frac {2 b^5 (b c-a d) (c+d x)^3}{d^7}+\frac {b^6 (c+d x)^4}{4 d^7}+\frac {15 b^2 (b c-a d)^4 \log (c+d x)}{d^7} \]

[Out]

-20*b^3*(-a*d+b*c)^3*x/d^6-1/2*(-a*d+b*c)^6/d^7/(d*x+c)^2+6*b*(-a*d+b*c)^5/d^7/(d*x+c)+15/2*b^4*(-a*d+b*c)^2*(
d*x+c)^2/d^7-2*b^5*(-a*d+b*c)*(d*x+c)^3/d^7+1/4*b^6*(d*x+c)^4/d^7+15*b^2*(-a*d+b*c)^4*ln(d*x+c)/d^7

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Rubi [A]
time = 0.14, antiderivative size = 158, 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 {2 b^5 (c+d x)^3 (b c-a d)}{d^7}+\frac {15 b^4 (c+d x)^2 (b c-a d)^2}{2 d^7}-\frac {20 b^3 x (b c-a d)^3}{d^6}+\frac {15 b^2 (b c-a d)^4 \log (c+d x)}{d^7}+\frac {6 b (b c-a d)^5}{d^7 (c+d x)}-\frac {(b c-a d)^6}{2 d^7 (c+d x)^2}+\frac {b^6 (c+d x)^4}{4 d^7} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^6/(c + d*x)^3,x]

[Out]

(-20*b^3*(b*c - a*d)^3*x)/d^6 - (b*c - a*d)^6/(2*d^7*(c + d*x)^2) + (6*b*(b*c - a*d)^5)/(d^7*(c + d*x)) + (15*
b^4*(b*c - a*d)^2*(c + d*x)^2)/(2*d^7) - (2*b^5*(b*c - a*d)*(c + d*x)^3)/d^7 + (b^6*(c + d*x)^4)/(4*d^7) + (15
*b^2*(b*c - a*d)^4*Log[c + d*x])/d^7

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 {(a+b x)^6}{(c+d x)^3} \, dx &=\int \left (-\frac {20 b^3 (b c-a d)^3}{d^6}+\frac {(-b c+a d)^6}{d^6 (c+d x)^3}-\frac {6 b (b c-a d)^5}{d^6 (c+d x)^2}+\frac {15 b^2 (b c-a d)^4}{d^6 (c+d x)}+\frac {15 b^4 (b c-a d)^2 (c+d x)}{d^6}-\frac {6 b^5 (b c-a d) (c+d x)^2}{d^6}+\frac {b^6 (c+d x)^3}{d^6}\right ) \, dx\\ &=-\frac {20 b^3 (b c-a d)^3 x}{d^6}-\frac {(b c-a d)^6}{2 d^7 (c+d x)^2}+\frac {6 b (b c-a d)^5}{d^7 (c+d x)}+\frac {15 b^4 (b c-a d)^2 (c+d x)^2}{2 d^7}-\frac {2 b^5 (b c-a d) (c+d x)^3}{d^7}+\frac {b^6 (c+d x)^4}{4 d^7}+\frac {15 b^2 (b c-a d)^4 \log (c+d x)}{d^7}\\ \end {align*}

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Mathematica [A]
time = 0.07, size = 303, normalized size = 1.92 \begin {gather*} \frac {-2 a^6 d^6-12 a^5 b d^5 (c+2 d x)+30 a^4 b^2 c d^4 (3 c+4 d x)+40 a^3 b^3 d^3 \left (-5 c^3-4 c^2 d x+4 c d^2 x^2+2 d^3 x^3\right )+30 a^2 b^4 d^2 \left (7 c^4+2 c^3 d x-11 c^2 d^2 x^2-4 c d^3 x^3+d^4 x^4\right )+4 a b^5 d \left (-27 c^5+6 c^4 d x+63 c^3 d^2 x^2+20 c^2 d^3 x^3-5 c d^4 x^4+2 d^5 x^5\right )+b^6 \left (22 c^6-16 c^5 d x-68 c^4 d^2 x^2-20 c^3 d^3 x^3+5 c^2 d^4 x^4-2 c d^5 x^5+d^6 x^6\right )+60 b^2 (b c-a d)^4 (c+d x)^2 \log (c+d x)}{4 d^7 (c+d x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^6/(c + d*x)^3,x]

[Out]

(-2*a^6*d^6 - 12*a^5*b*d^5*(c + 2*d*x) + 30*a^4*b^2*c*d^4*(3*c + 4*d*x) + 40*a^3*b^3*d^3*(-5*c^3 - 4*c^2*d*x +
4*c*d^2*x^2 + 2*d^3*x^3) + 30*a^2*b^4*d^2*(7*c^4 + 2*c^3*d*x - 11*c^2*d^2*x^2 - 4*c*d^3*x^3 + d^4*x^4) + 4*a*
b^5*d*(-27*c^5 + 6*c^4*d*x + 63*c^3*d^2*x^2 + 20*c^2*d^3*x^3 - 5*c*d^4*x^4 + 2*d^5*x^5) + b^6*(22*c^6 - 16*c^5
*d*x - 68*c^4*d^2*x^2 - 20*c^3*d^3*x^3 + 5*c^2*d^4*x^4 - 2*c*d^5*x^5 + d^6*x^6) + 60*b^2*(b*c - a*d)^4*(c + d*
x)^2*Log[c + d*x])/(4*d^7*(c + d*x)^2)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(350\) vs. \(2(152)=304\).
time = 0.14, size = 351, normalized size = 2.22

method	result	size
norman	\(\frac {-\frac {a^{6} d^{6}+6 a^{5} b c \,d^{5}-45 a^{4} b^{2} c^{2} d^{4}+180 a^{3} b^{3} c^{3} d^{3}-270 a^{2} b^{4} c^{4} d^{2}+180 a \,b^{5} c^{5} d -45 b^{6} c^{6}}{2 d^{7}}+\frac {b^{6} x^{6}}{4 d}-\frac {2 \left (3 a^{5} b \,d^{5}-15 a^{4} b^{2} c \,d^{4}+60 a^{3} b^{3} c^{2} d^{3}-90 a^{2} b^{4} c^{3} d^{2}+60 a \,b^{5} c^{4} d -15 b^{6} c^{5}\right ) x}{d^{6}}+\frac {5 b^{3} \left (4 a^{3} d^{3}-6 a^{2} b c \,d^{2}+4 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) x^{3}}{d^{4}}+\frac {5 b^{4} \left (6 a^{2} d^{2}-4 a b c d +b^{2} c^{2}\right ) x^{4}}{4 d^{3}}+\frac {b^{5} \left (4 a d -b c \right ) x^{5}}{2 d^{2}}}{\left (d x +c \right )^{2}}+\frac {15 b^{2} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right ) \ln \left (d x +c \right )}{d^{7}}\)	\(343\)
default	\(\frac {b^{3} \left (\frac {1}{4} d^{3} x^{4} b^{3}+2 a \,b^{2} d^{3} x^{3}-b^{3} c \,d^{2} x^{3}+\frac {15}{2} a^{2} b \,d^{3} x^{2}-9 a \,b^{2} c \,d^{2} x^{2}+3 b^{3} c^{2} d \,x^{2}+20 a^{3} d^{3} x -45 a^{2} b c \,d^{2} x +36 a \,b^{2} c^{2} d x -10 b^{3} c^{3} x \right )}{d^{6}}-\frac {6 b \left (a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -b^{5} c^{5}\right )}{d^{7} \left (d x +c \right )}+\frac {15 b^{2} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right ) \ln \left (d x +c \right )}{d^{7}}-\frac {a^{6} d^{6}-6 a^{5} b c \,d^{5}+15 a^{4} b^{2} c^{2} d^{4}-20 a^{3} b^{3} c^{3} d^{3}+15 a^{2} b^{4} c^{4} d^{2}-6 a \,b^{5} c^{5} d +b^{6} c^{6}}{2 d^{7} \left (d x +c \right )^{2}}\)	\(351\)
risch	\(\frac {b^{6} x^{4}}{4 d^{3}}+\frac {2 b^{5} a \,x^{3}}{d^{3}}-\frac {b^{6} c \,x^{3}}{d^{4}}+\frac {15 b^{4} a^{2} x^{2}}{2 d^{3}}-\frac {9 b^{5} a c \,x^{2}}{d^{4}}+\frac {3 b^{6} c^{2} x^{2}}{d^{5}}+\frac {20 b^{3} a^{3} x}{d^{3}}-\frac {45 b^{4} a^{2} c x}{d^{4}}+\frac {36 b^{5} a \,c^{2} x}{d^{5}}-\frac {10 b^{6} c^{3} x}{d^{6}}+\frac {\left (-6 a^{5} b \,d^{5}+30 a^{4} b^{2} c \,d^{4}-60 a^{3} b^{3} c^{2} d^{3}+60 a^{2} b^{4} c^{3} d^{2}-30 a \,b^{5} c^{4} d +6 b^{6} c^{5}\right ) x -\frac {a^{6} d^{6}+6 a^{5} b c \,d^{5}-45 a^{4} b^{2} c^{2} d^{4}+100 a^{3} b^{3} c^{3} d^{3}-105 a^{2} b^{4} c^{4} d^{2}+54 a \,b^{5} c^{5} d -11 b^{6} c^{6}}{2 d}}{d^{6} \left (d x +c \right )^{2}}+\frac {15 b^{2} \ln \left (d x +c \right ) a^{4}}{d^{3}}-\frac {60 b^{3} \ln \left (d x +c \right ) a^{3} c}{d^{4}}+\frac {90 b^{4} \ln \left (d x +c \right ) a^{2} c^{2}}{d^{5}}-\frac {60 b^{5} \ln \left (d x +c \right ) a \,c^{3}}{d^{6}}+\frac {15 b^{6} \ln \left (d x +c \right ) c^{4}}{d^{7}}\)	\(383\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^6/(d*x+c)^3,x,method=_RETURNVERBOSE)

[Out]

b^3/d^6*(1/4*d^3*x^4*b^3+2*a*b^2*d^3*x^3-b^3*c*d^2*x^3+15/2*a^2*b*d^3*x^2-9*a*b^2*c*d^2*x^2+3*b^3*c^2*d*x^2+20
*a^3*d^3*x-45*a^2*b*c*d^2*x+36*a*b^2*c^2*d*x-10*b^3*c^3*x)-6*b/d^7*(a^5*d^5-5*a^4*b*c*d^4+10*a^3*b^2*c^2*d^3-1
0*a^2*b^3*c^3*d^2+5*a*b^4*c^4*d-b^5*c^5)/(d*x+c)+15*b^2/d^7*(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)*ln(d*x+c)-1/2/d^7*(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)/(d*x+c)^2

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 364 vs. \(2 (152) = 304\).
time = 0.33, size = 364, normalized size = 2.30 \begin {gather*} \frac {11 \, b^{6} c^{6} - 54 \, a b^{5} c^{5} d + 105 \, a^{2} b^{4} c^{4} d^{2} - 100 \, a^{3} b^{3} c^{3} d^{3} + 45 \, a^{4} b^{2} c^{2} d^{4} - 6 \, a^{5} b c d^{5} - a^{6} d^{6} + 12 \, {\left (b^{6} c^{5} d - 5 \, a b^{5} c^{4} d^{2} + 10 \, a^{2} b^{4} c^{3} d^{3} - 10 \, a^{3} b^{3} c^{2} d^{4} + 5 \, a^{4} b^{2} c d^{5} - a^{5} b d^{6}\right )} x}{2 \, {\left (d^{9} x^{2} + 2 \, c d^{8} x + c^{2} d^{7}\right )}} + \frac {b^{6} d^{3} x^{4} - 4 \, {\left (b^{6} c d^{2} - 2 \, a b^{5} d^{3}\right )} x^{3} + 6 \, {\left (2 \, b^{6} c^{2} d - 6 \, a b^{5} c d^{2} + 5 \, a^{2} b^{4} d^{3}\right )} x^{2} - 4 \, {\left (10 \, b^{6} c^{3} - 36 \, a b^{5} c^{2} d + 45 \, a^{2} b^{4} c d^{2} - 20 \, a^{3} b^{3} d^{3}\right )} x}{4 \, d^{6}} + \frac {15 \, {\left (b^{6} c^{4} - 4 \, a b^{5} c^{3} d + 6 \, a^{2} b^{4} c^{2} d^{2} - 4 \, a^{3} b^{3} c d^{3} + a^{4} b^{2} d^{4}\right )} \log \left (d x + c\right )}{d^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^6/(d*x+c)^3,x, algorithm="maxima")

[Out]

1/2*(11*b^6*c^6 - 54*a*b^5*c^5*d + 105*a^2*b^4*c^4*d^2 - 100*a^3*b^3*c^3*d^3 + 45*a^4*b^2*c^2*d^4 - 6*a^5*b*c*
d^5 - a^6*d^6 + 12*(b^6*c^5*d - 5*a*b^5*c^4*d^2 + 10*a^2*b^4*c^3*d^3 - 10*a^3*b^3*c^2*d^4 + 5*a^4*b^2*c*d^5 -
a^5*b*d^6)*x)/(d^9*x^2 + 2*c*d^8*x + c^2*d^7) + 1/4*(b^6*d^3*x^4 - 4*(b^6*c*d^2 - 2*a*b^5*d^3)*x^3 + 6*(2*b^6*
c^2*d - 6*a*b^5*c*d^2 + 5*a^2*b^4*d^3)*x^2 - 4*(10*b^6*c^3 - 36*a*b^5*c^2*d + 45*a^2*b^4*c*d^2 - 20*a^3*b^3*d^
3)*x)/d^6 + 15*(b^6*c^4 - 4*a*b^5*c^3*d + 6*a^2*b^4*c^2*d^2 - 4*a^3*b^3*c*d^3 + a^4*b^2*d^4)*log(d*x + c)/d^7

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 548 vs. \(2 (152) = 304\).
time = 0.68, size = 548, normalized size = 3.47 \begin {gather*} \frac {b^{6} d^{6} x^{6} + 22 \, b^{6} c^{6} - 108 \, a b^{5} c^{5} d + 210 \, a^{2} b^{4} c^{4} d^{2} - 200 \, a^{3} b^{3} c^{3} d^{3} + 90 \, a^{4} b^{2} c^{2} d^{4} - 12 \, a^{5} b c d^{5} - 2 \, a^{6} d^{6} - 2 \, {\left (b^{6} c d^{5} - 4 \, a b^{5} d^{6}\right )} x^{5} + 5 \, {\left (b^{6} c^{2} d^{4} - 4 \, a b^{5} c d^{5} + 6 \, a^{2} b^{4} d^{6}\right )} x^{4} - 20 \, {\left (b^{6} c^{3} d^{3} - 4 \, a b^{5} c^{2} d^{4} + 6 \, a^{2} b^{4} c d^{5} - 4 \, a^{3} b^{3} d^{6}\right )} x^{3} - 2 \, {\left (34 \, b^{6} c^{4} d^{2} - 126 \, a b^{5} c^{3} d^{3} + 165 \, a^{2} b^{4} c^{2} d^{4} - 80 \, a^{3} b^{3} c d^{5}\right )} x^{2} - 4 \, {\left (4 \, b^{6} c^{5} d - 6 \, a b^{5} c^{4} d^{2} - 15 \, a^{2} b^{4} c^{3} d^{3} + 40 \, a^{3} b^{3} c^{2} d^{4} - 30 \, a^{4} b^{2} c d^{5} + 6 \, a^{5} b d^{6}\right )} x + 60 \, {\left (b^{6} c^{6} - 4 \, a b^{5} c^{5} d + 6 \, a^{2} b^{4} c^{4} d^{2} - 4 \, a^{3} b^{3} c^{3} d^{3} + a^{4} b^{2} c^{2} d^{4} + {\left (b^{6} c^{4} d^{2} - 4 \, a b^{5} c^{3} d^{3} + 6 \, a^{2} b^{4} c^{2} d^{4} - 4 \, a^{3} b^{3} c d^{5} + a^{4} b^{2} d^{6}\right )} x^{2} + 2 \, {\left (b^{6} c^{5} d - 4 \, a b^{5} c^{4} d^{2} + 6 \, a^{2} b^{4} c^{3} d^{3} - 4 \, a^{3} b^{3} c^{2} d^{4} + a^{4} b^{2} c d^{5}\right )} x\right )} \log \left (d x + c\right )}{4 \, {\left (d^{9} x^{2} + 2 \, c d^{8} x + c^{2} d^{7}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^6/(d*x+c)^3,x, algorithm="fricas")

[Out]

1/4*(b^6*d^6*x^6 + 22*b^6*c^6 - 108*a*b^5*c^5*d + 210*a^2*b^4*c^4*d^2 - 200*a^3*b^3*c^3*d^3 + 90*a^4*b^2*c^2*d
^4 - 12*a^5*b*c*d^5 - 2*a^6*d^6 - 2*(b^6*c*d^5 - 4*a*b^5*d^6)*x^5 + 5*(b^6*c^2*d^4 - 4*a*b^5*c*d^5 + 6*a^2*b^4
*d^6)*x^4 - 20*(b^6*c^3*d^3 - 4*a*b^5*c^2*d^4 + 6*a^2*b^4*c*d^5 - 4*a^3*b^3*d^6)*x^3 - 2*(34*b^6*c^4*d^2 - 126
*a*b^5*c^3*d^3 + 165*a^2*b^4*c^2*d^4 - 80*a^3*b^3*c*d^5)*x^2 - 4*(4*b^6*c^5*d - 6*a*b^5*c^4*d^2 - 15*a^2*b^4*c
^3*d^3 + 40*a^3*b^3*c^2*d^4 - 30*a^4*b^2*c*d^5 + 6*a^5*b*d^6)*x + 60*(b^6*c^6 - 4*a*b^5*c^5*d + 6*a^2*b^4*c^4*
d^2 - 4*a^3*b^3*c^3*d^3 + a^4*b^2*c^2*d^4 + (b^6*c^4*d^2 - 4*a*b^5*c^3*d^3 + 6*a^2*b^4*c^2*d^4 - 4*a^3*b^3*c*d
^5 + a^4*b^2*d^6)*x^2 + 2*(b^6*c^5*d - 4*a*b^5*c^4*d^2 + 6*a^2*b^4*c^3*d^3 - 4*a^3*b^3*c^2*d^4 + a^4*b^2*c*d^5
)*x)*log(d*x + c))/(d^9*x^2 + 2*c*d^8*x + c^2*d^7)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 340 vs. \(2 (144) = 288\).
time = 1.23, size = 340, normalized size = 2.15 \begin {gather*} \frac {b^{6} x^{4}}{4 d^{3}} + \frac {15 b^{2} \left (a d - b c\right )^{4} \log {\left (c + d x \right )}}{d^{7}} + x^{3} \cdot \left (\frac {2 a b^{5}}{d^{3}} - \frac {b^{6} c}{d^{4}}\right ) + x^{2} \cdot \left (\frac {15 a^{2} b^{4}}{2 d^{3}} - \frac {9 a b^{5} c}{d^{4}} + \frac {3 b^{6} c^{2}}{d^{5}}\right ) + x \left (\frac {20 a^{3} b^{3}}{d^{3}} - \frac {45 a^{2} b^{4} c}{d^{4}} + \frac {36 a b^{5} c^{2}}{d^{5}} - \frac {10 b^{6} c^{3}}{d^{6}}\right ) + \frac {- a^{6} d^{6} - 6 a^{5} b c d^{5} + 45 a^{4} b^{2} c^{2} d^{4} - 100 a^{3} b^{3} c^{3} d^{3} + 105 a^{2} b^{4} c^{4} d^{2} - 54 a b^{5} c^{5} d + 11 b^{6} c^{6} + x \left (- 12 a^{5} b d^{6} + 60 a^{4} b^{2} c d^{5} - 120 a^{3} b^{3} c^{2} d^{4} + 120 a^{2} b^{4} c^{3} d^{3} - 60 a b^{5} c^{4} d^{2} + 12 b^{6} c^{5} d\right )}{2 c^{2} d^{7} + 4 c d^{8} x + 2 d^{9} x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**6/(d*x+c)**3,x)

[Out]

b**6*x**4/(4*d**3) + 15*b**2*(a*d - b*c)**4*log(c + d*x)/d**7 + x**3*(2*a*b**5/d**3 - b**6*c/d**4) + x**2*(15*
a**2*b**4/(2*d**3) - 9*a*b**5*c/d**4 + 3*b**6*c**2/d**5) + x*(20*a**3*b**3/d**3 - 45*a**2*b**4*c/d**4 + 36*a*b
**5*c**2/d**5 - 10*b**6*c**3/d**6) + (-a**6*d**6 - 6*a**5*b*c*d**5 + 45*a**4*b**2*c**2*d**4 - 100*a**3*b**3*c*
*3*d**3 + 105*a**2*b**4*c**4*d**2 - 54*a*b**5*c**5*d + 11*b**6*c**6 + x*(-12*a**5*b*d**6 + 60*a**4*b**2*c*d**5
- 120*a**3*b**3*c**2*d**4 + 120*a**2*b**4*c**3*d**3 - 60*a*b**5*c**4*d**2 + 12*b**6*c**5*d))/(2*c**2*d**7 + 4
*c*d**8*x + 2*d**9*x**2)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 362 vs. \(2 (152) = 304\).
time = 1.41, size = 362, normalized size = 2.29 \begin {gather*} \frac {15 \, {\left (b^{6} c^{4} - 4 \, a b^{5} c^{3} d + 6 \, a^{2} b^{4} c^{2} d^{2} - 4 \, a^{3} b^{3} c d^{3} + a^{4} b^{2} d^{4}\right )} \log \left ({\left | d x + c \right |}\right )}{d^{7}} + \frac {11 \, b^{6} c^{6} - 54 \, a b^{5} c^{5} d + 105 \, a^{2} b^{4} c^{4} d^{2} - 100 \, a^{3} b^{3} c^{3} d^{3} + 45 \, a^{4} b^{2} c^{2} d^{4} - 6 \, a^{5} b c d^{5} - a^{6} d^{6} + 12 \, {\left (b^{6} c^{5} d - 5 \, a b^{5} c^{4} d^{2} + 10 \, a^{2} b^{4} c^{3} d^{3} - 10 \, a^{3} b^{3} c^{2} d^{4} + 5 \, a^{4} b^{2} c d^{5} - a^{5} b d^{6}\right )} x}{2 \, {\left (d x + c\right )}^{2} d^{7}} + \frac {b^{6} d^{9} x^{4} - 4 \, b^{6} c d^{8} x^{3} + 8 \, a b^{5} d^{9} x^{3} + 12 \, b^{6} c^{2} d^{7} x^{2} - 36 \, a b^{5} c d^{8} x^{2} + 30 \, a^{2} b^{4} d^{9} x^{2} - 40 \, b^{6} c^{3} d^{6} x + 144 \, a b^{5} c^{2} d^{7} x - 180 \, a^{2} b^{4} c d^{8} x + 80 \, a^{3} b^{3} d^{9} x}{4 \, d^{12}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^6/(d*x+c)^3,x, algorithm="giac")

[Out]

15*(b^6*c^4 - 4*a*b^5*c^3*d + 6*a^2*b^4*c^2*d^2 - 4*a^3*b^3*c*d^3 + a^4*b^2*d^4)*log(abs(d*x + c))/d^7 + 1/2*(
11*b^6*c^6 - 54*a*b^5*c^5*d + 105*a^2*b^4*c^4*d^2 - 100*a^3*b^3*c^3*d^3 + 45*a^4*b^2*c^2*d^4 - 6*a^5*b*c*d^5 -
a^6*d^6 + 12*(b^6*c^5*d - 5*a*b^5*c^4*d^2 + 10*a^2*b^4*c^3*d^3 - 10*a^3*b^3*c^2*d^4 + 5*a^4*b^2*c*d^5 - a^5*b
*d^6)*x)/((d*x + c)^2*d^7) + 1/4*(b^6*d^9*x^4 - 4*b^6*c*d^8*x^3 + 8*a*b^5*d^9*x^3 + 12*b^6*c^2*d^7*x^2 - 36*a*
b^5*c*d^8*x^2 + 30*a^2*b^4*d^9*x^2 - 40*b^6*c^3*d^6*x + 144*a*b^5*c^2*d^7*x - 180*a^2*b^4*c*d^8*x + 80*a^3*b^3
*d^9*x)/d^12

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Mupad [B]
time = 0.27, size = 441, normalized size = 2.79 \begin {gather*} x^3\,\left (\frac {2\,a\,b^5}{d^3}-\frac {b^6\,c}{d^4}\right )-\frac {\frac {a^6\,d^6+6\,a^5\,b\,c\,d^5-45\,a^4\,b^2\,c^2\,d^4+100\,a^3\,b^3\,c^3\,d^3-105\,a^2\,b^4\,c^4\,d^2+54\,a\,b^5\,c^5\,d-11\,b^6\,c^6}{2\,d}-x\,\left (-6\,a^5\,b\,d^5+30\,a^4\,b^2\,c\,d^4-60\,a^3\,b^3\,c^2\,d^3+60\,a^2\,b^4\,c^3\,d^2-30\,a\,b^5\,c^4\,d+6\,b^6\,c^5\right )}{c^2\,d^6+2\,c\,d^7\,x+d^8\,x^2}-x^2\,\left (\frac {3\,c\,\left (\frac {6\,a\,b^5}{d^3}-\frac {3\,b^6\,c}{d^4}\right )}{2\,d}-\frac {15\,a^2\,b^4}{2\,d^3}+\frac {3\,b^6\,c^2}{2\,d^5}\right )+x\,\left (\frac {3\,c\,\left (\frac {3\,c\,\left (\frac {6\,a\,b^5}{d^3}-\frac {3\,b^6\,c}{d^4}\right )}{d}-\frac {15\,a^2\,b^4}{d^3}+\frac {3\,b^6\,c^2}{d^5}\right )}{d}+\frac {20\,a^3\,b^3}{d^3}-\frac {b^6\,c^3}{d^6}-\frac {3\,c^2\,\left (\frac {6\,a\,b^5}{d^3}-\frac {3\,b^6\,c}{d^4}\right )}{d^2}\right )+\frac {\ln \left (c+d\,x\right )\,\left (15\,a^4\,b^2\,d^4-60\,a^3\,b^3\,c\,d^3+90\,a^2\,b^4\,c^2\,d^2-60\,a\,b^5\,c^3\,d+15\,b^6\,c^4\right )}{d^7}+\frac {b^6\,x^4}{4\,d^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)^6/(c + d*x)^3,x)

[Out]

x^3*((2*a*b^5)/d^3 - (b^6*c)/d^4) - ((a^6*d^6 - 11*b^6*c^6 - 105*a^2*b^4*c^4*d^2 + 100*a^3*b^3*c^3*d^3 - 45*a^
4*b^2*c^2*d^4 + 54*a*b^5*c^5*d + 6*a^5*b*c*d^5)/(2*d) - x*(6*b^6*c^5 - 6*a^5*b*d^5 + 30*a^4*b^2*c*d^4 + 60*a^2
*b^4*c^3*d^2 - 60*a^3*b^3*c^2*d^3 - 30*a*b^5*c^4*d))/(c^2*d^6 + d^8*x^2 + 2*c*d^7*x) - x^2*((3*c*((6*a*b^5)/d^
3 - (3*b^6*c)/d^4))/(2*d) - (15*a^2*b^4)/(2*d^3) + (3*b^6*c^2)/(2*d^5)) + x*((3*c*((3*c*((6*a*b^5)/d^3 - (3*b^
6*c)/d^4))/d - (15*a^2*b^4)/d^3 + (3*b^6*c^2)/d^5))/d + (20*a^3*b^3)/d^3 - (b^6*c^3)/d^6 - (3*c^2*((6*a*b^5)/d
^3 - (3*b^6*c)/d^4))/d^2) + (log(c + d*x)*(15*b^6*c^4 + 15*a^4*b^2*d^4 - 60*a^3*b^3*c*d^3 + 90*a^2*b^4*c^2*d^2
- 60*a*b^5*c^3*d))/d^7 + (b^6*x^4)/(4*d^3)

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