3.3.75 \(\int \frac {(d+e x)^7}{(b x+c x^2)^3} \, dx\) [275]

Optimal. Leaf size=203 \[ -\frac {d^7}{2 b^3 x^2}+\frac {d^6 (3 c d-7 b e)}{b^4 x}+\frac {e^6 (7 c d-3 b e) x}{c^4}+\frac {e^7 x^2}{2 c^3}+\frac {(c d-b e)^7}{2 b^3 c^5 (b+c x)^2}+\frac {(c d-b e)^6 (3 c d+4 b e)}{b^4 c^5 (b+c x)}+\frac {3 d^5 \left (2 c^2 d^2-7 b c d e+7 b^2 e^2\right ) \log (x)}{b^5}-\frac {3 (c d-b e)^5 \left (2 c^2 d^2+3 b c d e+2 b^2 e^2\right ) \log (b+c x)}{b^5 c^5} \]

[Out]

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

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Rubi [A]
time = 0.18, antiderivative size = 203, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {712} \begin {gather*} \frac {(c d-b e)^6 (4 b e+3 c d)}{b^4 c^5 (b+c x)}+\frac {d^6 (3 c d-7 b e)}{b^4 x}+\frac {(c d-b e)^7}{2 b^3 c^5 (b+c x)^2}-\frac {d^7}{2 b^3 x^2}+\frac {3 d^5 \log (x) \left (7 b^2 e^2-7 b c d e+2 c^2 d^2\right )}{b^5}-\frac {3 (c d-b e)^5 \left (2 b^2 e^2+3 b c d e+2 c^2 d^2\right ) \log (b+c x)}{b^5 c^5}+\frac {e^6 x (7 c d-3 b e)}{c^4}+\frac {e^7 x^2}{2 c^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^7/(b*x + c*x^2)^3,x]

[Out]

-1/2*d^7/(b^3*x^2) + (d^6*(3*c*d - 7*b*e))/(b^4*x) + (e^6*(7*c*d - 3*b*e)*x)/c^4 + (e^7*x^2)/(2*c^3) + (c*d -
b*e)^7/(2*b^3*c^5*(b + c*x)^2) + ((c*d - b*e)^6*(3*c*d + 4*b*e))/(b^4*c^5*(b + c*x)) + (3*d^5*(2*c^2*d^2 - 7*b
*c*d*e + 7*b^2*e^2)*Log[x])/b^5 - (3*(c*d - b*e)^5*(2*c^2*d^2 + 3*b*c*d*e + 2*b^2*e^2)*Log[b + c*x])/(b^5*c^5)

Rule 712

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d +
e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*
e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

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

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Mathematica [A]
time = 0.08, size = 202, normalized size = 1.00 \begin {gather*} \frac {1}{2} \left (-\frac {d^7}{b^3 x^2}+\frac {2 d^6 (3 c d-7 b e)}{b^4 x}+\frac {2 e^6 (7 c d-3 b e) x}{c^4}+\frac {e^7 x^2}{c^3}+\frac {(c d-b e)^7}{b^3 c^5 (b+c x)^2}+\frac {2 (c d-b e)^6 (3 c d+4 b e)}{b^4 c^5 (b+c x)}+\frac {6 d^5 \left (2 c^2 d^2-7 b c d e+7 b^2 e^2\right ) \log (x)}{b^5}+\frac {6 (-c d+b e)^5 \left (2 c^2 d^2+3 b c d e+2 b^2 e^2\right ) \log (b+c x)}{b^5 c^5}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^7/(b*x + c*x^2)^3,x]

[Out]

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

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Maple [A]
time = 0.49, size = 371, normalized size = 1.83

method	result	size
default	\(-\frac {e^{6} \left (-\frac {1}{2} c e \,x^{2}+3 b e x -7 c d x \right )}{c^{4}}+\frac {\left (6 b^{7} e^{7}-21 b^{6} d \,e^{6} c +21 d^{2} e^{5} b^{5} c^{2}-21 c^{5} d^{5} e^{2} b^{2}+21 c^{6} d^{6} e b -6 c^{7} d^{7}\right ) \ln \left (c x +b \right )}{b^{5} c^{5}}-\frac {-4 b^{7} e^{7}+21 b^{6} d \,e^{6} c -42 d^{2} e^{5} b^{5} c^{2}+35 d^{3} e^{4} b^{4} c^{3}-21 c^{5} d^{5} e^{2} b^{2}+14 c^{6} d^{6} e b -3 c^{7} d^{7}}{b^{4} c^{5} \left (c x +b \right )}-\frac {b^{7} e^{7}-7 b^{6} d \,e^{6} c +21 d^{2} e^{5} b^{5} c^{2}-35 d^{3} e^{4} b^{4} c^{3}+35 c^{4} d^{4} e^{3} b^{3}-21 c^{5} d^{5} e^{2} b^{2}+7 c^{6} d^{6} e b -c^{7} d^{7}}{2 c^{5} b^{3} \left (c x +b \right )^{2}}-\frac {d^{7}}{2 b^{3} x^{2}}-\frac {d^{6} \left (7 b e -3 c d \right )}{b^{4} x}+\frac {3 d^{5} \left (7 b^{2} e^{2}-7 b c d e +2 d^{2} c^{2}\right ) \ln \left (x \right )}{b^{5}}\)	\(371\)
norman	\(\frac {\frac {\left (12 b^{7} e^{7}-42 b^{6} d \,e^{6} c +42 d^{2} e^{5} b^{5} c^{2}-35 d^{3} e^{4} b^{4} c^{3}+21 c^{5} d^{5} e^{2} b^{2}-21 c^{6} d^{6} e b +6 c^{7} d^{7}\right ) x^{3}}{b^{4} c^{4}}-\frac {d^{7}}{2 b}+\frac {e^{7} x^{6}}{2 c}-\frac {d^{6} \left (7 b e -2 c d \right ) x}{b^{2}}-\frac {e^{6} \left (2 b e -7 c d \right ) x^{5}}{c^{2}}+\frac {\left (18 b^{7} e^{7}-63 b^{6} d \,e^{6} c +63 d^{2} e^{5} b^{5} c^{2}-35 d^{3} e^{4} b^{4} c^{3}-35 c^{4} d^{4} e^{3} b^{3}+63 c^{5} d^{5} e^{2} b^{2}-63 c^{6} d^{6} e b +18 c^{7} d^{7}\right ) x^{2}}{2 b^{3} c^{5}}}{x^{2} \left (c x +b \right )^{2}}+\frac {3 d^{5} \left (7 b^{2} e^{2}-7 b c d e +2 d^{2} c^{2}\right ) \ln \left (x \right )}{b^{5}}+\frac {3 \left (2 b^{7} e^{7}-7 b^{6} d \,e^{6} c +7 d^{2} e^{5} b^{5} c^{2}-7 c^{5} d^{5} e^{2} b^{2}+7 c^{6} d^{6} e b -2 c^{7} d^{7}\right ) \ln \left (c x +b \right )}{b^{5} c^{5}}\)	\(376\)
risch	\(\frac {e^{7} x^{2}}{2 c^{3}}-\frac {3 e^{7} b x}{c^{4}}+\frac {7 e^{6} d x}{c^{3}}+\frac {\frac {\left (4 b^{7} e^{7}-21 b^{6} d \,e^{6} c +42 d^{2} e^{5} b^{5} c^{2}-35 d^{3} e^{4} b^{4} c^{3}+21 c^{5} d^{5} e^{2} b^{2}-21 c^{6} d^{6} e b +6 c^{7} d^{7}\right ) x^{3}}{b^{4}}+\frac {\left (7 b^{7} e^{7}-35 b^{6} d \,e^{6} c +63 d^{2} e^{5} b^{5} c^{2}-35 d^{3} e^{4} b^{4} c^{3}-35 c^{4} d^{4} e^{3} b^{3}+63 c^{5} d^{5} e^{2} b^{2}-63 c^{6} d^{6} e b +18 c^{7} d^{7}\right ) x^{2}}{2 b^{3} c}-\frac {c^{4} d^{6} \left (7 b e -2 c d \right ) x}{b^{2}}-\frac {c^{4} d^{7}}{2 b}}{c^{4} x^{2} \left (c x +b \right )^{2}}+\frac {6 b^{2} \ln \left (-c x -b \right ) e^{7}}{c^{5}}-\frac {21 b \ln \left (-c x -b \right ) d \,e^{6}}{c^{4}}+\frac {21 \ln \left (-c x -b \right ) d^{2} e^{5}}{c^{3}}-\frac {21 \ln \left (-c x -b \right ) d^{5} e^{2}}{b^{3}}+\frac {21 c \ln \left (-c x -b \right ) d^{6} e}{b^{4}}-\frac {6 c^{2} \ln \left (-c x -b \right ) d^{7}}{b^{5}}+\frac {21 d^{5} \ln \left (x \right ) e^{2}}{b^{3}}-\frac {21 d^{6} \ln \left (x \right ) c e}{b^{4}}+\frac {6 d^{7} \ln \left (x \right ) c^{2}}{b^{5}}\)	\(426\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^7/(c*x^2+b*x)^3,x,method=_RETURNVERBOSE)

[Out]

-e^6/c^4*(-1/2*c*e*x^2+3*b*e*x-7*c*d*x)+(6*b^7*e^7-21*b^6*c*d*e^6+21*b^5*c^2*d^2*e^5-21*b^2*c^5*d^5*e^2+21*b*c
^6*d^6*e-6*c^7*d^7)/b^5/c^5*ln(c*x+b)-(-4*b^7*e^7+21*b^6*c*d*e^6-42*b^5*c^2*d^2*e^5+35*b^4*c^3*d^3*e^4-21*b^2*
c^5*d^5*e^2+14*b*c^6*d^6*e-3*c^7*d^7)/b^4/c^5/(c*x+b)-1/2/c^5*(b^7*e^7-7*b^6*c*d*e^6+21*b^5*c^2*d^2*e^5-35*b^4
*c^3*d^3*e^4+35*b^3*c^4*d^4*e^3-21*b^2*c^5*d^5*e^2+7*b*c^6*d^6*e-c^7*d^7)/b^3/(c*x+b)^2-1/2*d^7/b^3/x^2-d^6*(7
*b*e-3*c*d)/b^4/x+3*d^5*(7*b^2*e^2-7*b*c*d*e+2*c^2*d^2)*ln(x)/b^5

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Maxima [A]
time = 0.30, size = 394, normalized size = 1.94 \begin {gather*} -\frac {b^{3} c^{5} d^{7} - 2 \, {\left (6 \, c^{8} d^{7} - 21 \, b c^{7} d^{6} e + 21 \, b^{2} c^{6} d^{5} e^{2} - 35 \, b^{4} c^{4} d^{3} e^{4} + 42 \, b^{5} c^{3} d^{2} e^{5} - 21 \, b^{6} c^{2} d e^{6} + 4 \, b^{7} c e^{7}\right )} x^{3} - {\left (18 \, b c^{7} d^{7} - 63 \, b^{2} c^{6} d^{6} e + 63 \, b^{3} c^{5} d^{5} e^{2} - 35 \, b^{4} c^{4} d^{4} e^{3} - 35 \, b^{5} c^{3} d^{3} e^{4} + 63 \, b^{6} c^{2} d^{2} e^{5} - 35 \, b^{7} c d e^{6} + 7 \, b^{8} e^{7}\right )} x^{2} - 2 \, {\left (2 \, b^{2} c^{6} d^{7} - 7 \, b^{3} c^{5} d^{6} e\right )} x}{2 \, {\left (b^{4} c^{7} x^{4} + 2 \, b^{5} c^{6} x^{3} + b^{6} c^{5} x^{2}\right )}} + \frac {c x^{2} e^{7} + 2 \, {\left (7 \, c d e^{6} - 3 \, b e^{7}\right )} x}{2 \, c^{4}} + \frac {3 \, {\left (2 \, c^{2} d^{7} - 7 \, b c d^{6} e + 7 \, b^{2} d^{5} e^{2}\right )} \log \left (x\right )}{b^{5}} - \frac {3 \, {\left (2 \, c^{7} d^{7} - 7 \, b c^{6} d^{6} e + 7 \, b^{2} c^{5} d^{5} e^{2} - 7 \, b^{5} c^{2} d^{2} e^{5} + 7 \, b^{6} c d e^{6} - 2 \, b^{7} e^{7}\right )} \log \left (c x + b\right )}{b^{5} c^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^7/(c*x^2+b*x)^3,x, algorithm="maxima")

[Out]

-1/2*(b^3*c^5*d^7 - 2*(6*c^8*d^7 - 21*b*c^7*d^6*e + 21*b^2*c^6*d^5*e^2 - 35*b^4*c^4*d^3*e^4 + 42*b^5*c^3*d^2*e
^5 - 21*b^6*c^2*d*e^6 + 4*b^7*c*e^7)*x^3 - (18*b*c^7*d^7 - 63*b^2*c^6*d^6*e + 63*b^3*c^5*d^5*e^2 - 35*b^4*c^4*
d^4*e^3 - 35*b^5*c^3*d^3*e^4 + 63*b^6*c^2*d^2*e^5 - 35*b^7*c*d*e^6 + 7*b^8*e^7)*x^2 - 2*(2*b^2*c^6*d^7 - 7*b^3
*c^5*d^6*e)*x)/(b^4*c^7*x^4 + 2*b^5*c^6*x^3 + b^6*c^5*x^2) + 1/2*(c*x^2*e^7 + 2*(7*c*d*e^6 - 3*b*e^7)*x)/c^4 +
3*(2*c^2*d^7 - 7*b*c*d^6*e + 7*b^2*d^5*e^2)*log(x)/b^5 - 3*(2*c^7*d^7 - 7*b*c^6*d^6*e + 7*b^2*c^5*d^5*e^2 - 7
*b^5*c^2*d^2*e^5 + 7*b^6*c*d*e^6 - 2*b^7*e^7)*log(c*x + b)/(b^5*c^5)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 726 vs. \(2 (201) = 402\).
time = 2.50, size = 726, normalized size = 3.58 \begin {gather*} \frac {12 \, b c^{8} d^{7} x^{3} + 18 \, b^{2} c^{7} d^{7} x^{2} + 4 \, b^{3} c^{6} d^{7} x - b^{4} c^{5} d^{7} - 35 \, b^{5} c^{4} d^{4} x^{2} e^{3} + {\left (b^{5} c^{4} x^{6} - 4 \, b^{6} c^{3} x^{5} - 11 \, b^{7} c^{2} x^{4} + 2 \, b^{8} c x^{3} + 7 \, b^{9} x^{2}\right )} e^{7} + 7 \, {\left (2 \, b^{5} c^{4} d x^{5} + 4 \, b^{6} c^{3} d x^{4} - 4 \, b^{7} c^{2} d x^{3} - 5 \, b^{8} c d x^{2}\right )} e^{6} + 21 \, {\left (4 \, b^{6} c^{3} d^{2} x^{3} + 3 \, b^{7} c^{2} d^{2} x^{2}\right )} e^{5} - 35 \, {\left (2 \, b^{5} c^{4} d^{3} x^{3} + b^{6} c^{3} d^{3} x^{2}\right )} e^{4} + 21 \, {\left (2 \, b^{3} c^{6} d^{5} x^{3} + 3 \, b^{4} c^{5} d^{5} x^{2}\right )} e^{2} - 7 \, {\left (6 \, b^{2} c^{7} d^{6} x^{3} + 9 \, b^{3} c^{6} d^{6} x^{2} + 2 \, b^{4} c^{5} d^{6} x\right )} e - 6 \, {\left (2 \, c^{9} d^{7} x^{4} + 4 \, b c^{8} d^{7} x^{3} + 2 \, b^{2} c^{7} d^{7} x^{2} - 2 \, {\left (b^{7} c^{2} x^{4} + 2 \, b^{8} c x^{3} + b^{9} x^{2}\right )} e^{7} + 7 \, {\left (b^{6} c^{3} d x^{4} + 2 \, b^{7} c^{2} d x^{3} + b^{8} c d x^{2}\right )} e^{6} - 7 \, {\left (b^{5} c^{4} d^{2} x^{4} + 2 \, b^{6} c^{3} d^{2} x^{3} + b^{7} c^{2} d^{2} x^{2}\right )} e^{5} + 7 \, {\left (b^{2} c^{7} d^{5} x^{4} + 2 \, b^{3} c^{6} d^{5} x^{3} + b^{4} c^{5} d^{5} x^{2}\right )} e^{2} - 7 \, {\left (b c^{8} d^{6} x^{4} + 2 \, b^{2} c^{7} d^{6} x^{3} + b^{3} c^{6} d^{6} x^{2}\right )} e\right )} \log \left (c x + b\right ) + 6 \, {\left (2 \, c^{9} d^{7} x^{4} + 4 \, b c^{8} d^{7} x^{3} + 2 \, b^{2} c^{7} d^{7} x^{2} + 7 \, {\left (b^{2} c^{7} d^{5} x^{4} + 2 \, b^{3} c^{6} d^{5} x^{3} + b^{4} c^{5} d^{5} x^{2}\right )} e^{2} - 7 \, {\left (b c^{8} d^{6} x^{4} + 2 \, b^{2} c^{7} d^{6} x^{3} + b^{3} c^{6} d^{6} x^{2}\right )} e\right )} \log \left (x\right )}{2 \, {\left (b^{5} c^{7} x^{4} + 2 \, b^{6} c^{6} x^{3} + b^{7} c^{5} x^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^7/(c*x^2+b*x)^3,x, algorithm="fricas")

[Out]

1/2*(12*b*c^8*d^7*x^3 + 18*b^2*c^7*d^7*x^2 + 4*b^3*c^6*d^7*x - b^4*c^5*d^7 - 35*b^5*c^4*d^4*x^2*e^3 + (b^5*c^4
*x^6 - 4*b^6*c^3*x^5 - 11*b^7*c^2*x^4 + 2*b^8*c*x^3 + 7*b^9*x^2)*e^7 + 7*(2*b^5*c^4*d*x^5 + 4*b^6*c^3*d*x^4 -
4*b^7*c^2*d*x^3 - 5*b^8*c*d*x^2)*e^6 + 21*(4*b^6*c^3*d^2*x^3 + 3*b^7*c^2*d^2*x^2)*e^5 - 35*(2*b^5*c^4*d^3*x^3
+ b^6*c^3*d^3*x^2)*e^4 + 21*(2*b^3*c^6*d^5*x^3 + 3*b^4*c^5*d^5*x^2)*e^2 - 7*(6*b^2*c^7*d^6*x^3 + 9*b^3*c^6*d^6
*x^2 + 2*b^4*c^5*d^6*x)*e - 6*(2*c^9*d^7*x^4 + 4*b*c^8*d^7*x^3 + 2*b^2*c^7*d^7*x^2 - 2*(b^7*c^2*x^4 + 2*b^8*c*
x^3 + b^9*x^2)*e^7 + 7*(b^6*c^3*d*x^4 + 2*b^7*c^2*d*x^3 + b^8*c*d*x^2)*e^6 - 7*(b^5*c^4*d^2*x^4 + 2*b^6*c^3*d^
2*x^3 + b^7*c^2*d^2*x^2)*e^5 + 7*(b^2*c^7*d^5*x^4 + 2*b^3*c^6*d^5*x^3 + b^4*c^5*d^5*x^2)*e^2 - 7*(b*c^8*d^6*x^
4 + 2*b^2*c^7*d^6*x^3 + b^3*c^6*d^6*x^2)*e)*log(c*x + b) + 6*(2*c^9*d^7*x^4 + 4*b*c^8*d^7*x^3 + 2*b^2*c^7*d^7*
x^2 + 7*(b^2*c^7*d^5*x^4 + 2*b^3*c^6*d^5*x^3 + b^4*c^5*d^5*x^2)*e^2 - 7*(b*c^8*d^6*x^4 + 2*b^2*c^7*d^6*x^3 + b
^3*c^6*d^6*x^2)*e)*log(x))/(b^5*c^7*x^4 + 2*b^6*c^6*x^3 + b^7*c^5*x^2)

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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((e*x+d)**7/(c*x**2+b*x)**3,x)

[Out]

Timed out

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Giac [A]
time = 1.59, size = 383, normalized size = 1.89 \begin {gather*} \frac {3 \, {\left (2 \, c^{2} d^{7} - 7 \, b c d^{6} e + 7 \, b^{2} d^{5} e^{2}\right )} \log \left ({\left | x \right |}\right )}{b^{5}} + \frac {c^{3} x^{2} e^{7} + 14 \, c^{3} d x e^{6} - 6 \, b c^{2} x e^{7}}{2 \, c^{6}} - \frac {3 \, {\left (2 \, c^{7} d^{7} - 7 \, b c^{6} d^{6} e + 7 \, b^{2} c^{5} d^{5} e^{2} - 7 \, b^{5} c^{2} d^{2} e^{5} + 7 \, b^{6} c d e^{6} - 2 \, b^{7} e^{7}\right )} \log \left ({\left | c x + b \right |}\right )}{b^{5} c^{5}} - \frac {b^{3} c^{5} d^{7} - 2 \, {\left (6 \, c^{8} d^{7} - 21 \, b c^{7} d^{6} e + 21 \, b^{2} c^{6} d^{5} e^{2} - 35 \, b^{4} c^{4} d^{3} e^{4} + 42 \, b^{5} c^{3} d^{2} e^{5} - 21 \, b^{6} c^{2} d e^{6} + 4 \, b^{7} c e^{7}\right )} x^{3} - {\left (18 \, b c^{7} d^{7} - 63 \, b^{2} c^{6} d^{6} e + 63 \, b^{3} c^{5} d^{5} e^{2} - 35 \, b^{4} c^{4} d^{4} e^{3} - 35 \, b^{5} c^{3} d^{3} e^{4} + 63 \, b^{6} c^{2} d^{2} e^{5} - 35 \, b^{7} c d e^{6} + 7 \, b^{8} e^{7}\right )} x^{2} - 2 \, {\left (2 \, b^{2} c^{6} d^{7} - 7 \, b^{3} c^{5} d^{6} e\right )} x}{2 \, {\left (c x + b\right )}^{2} b^{4} c^{5} x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^7/(c*x^2+b*x)^3,x, algorithm="giac")

[Out]

3*(2*c^2*d^7 - 7*b*c*d^6*e + 7*b^2*d^5*e^2)*log(abs(x))/b^5 + 1/2*(c^3*x^2*e^7 + 14*c^3*d*x*e^6 - 6*b*c^2*x*e^
7)/c^6 - 3*(2*c^7*d^7 - 7*b*c^6*d^6*e + 7*b^2*c^5*d^5*e^2 - 7*b^5*c^2*d^2*e^5 + 7*b^6*c*d*e^6 - 2*b^7*e^7)*log
(abs(c*x + b))/(b^5*c^5) - 1/2*(b^3*c^5*d^7 - 2*(6*c^8*d^7 - 21*b*c^7*d^6*e + 21*b^2*c^6*d^5*e^2 - 35*b^4*c^4*
d^3*e^4 + 42*b^5*c^3*d^2*e^5 - 21*b^6*c^2*d*e^6 + 4*b^7*c*e^7)*x^3 - (18*b*c^7*d^7 - 63*b^2*c^6*d^6*e + 63*b^3
*c^5*d^5*e^2 - 35*b^4*c^4*d^4*e^3 - 35*b^5*c^3*d^3*e^4 + 63*b^6*c^2*d^2*e^5 - 35*b^7*c*d*e^6 + 7*b^8*e^7)*x^2
- 2*(2*b^2*c^6*d^7 - 7*b^3*c^5*d^6*e)*x)/((c*x + b)^2*b^4*c^5*x^2)

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Mupad [B]
time = 0.49, size = 399, normalized size = 1.97 \begin {gather*} \frac {e^7\,x^2}{2\,c^3}-x\,\left (\frac {3\,b\,e^7}{c^4}-\frac {7\,d\,e^6}{c^3}\right )-\frac {\frac {c^4\,d^7}{2\,b}-\frac {x^3\,\left (4\,b^7\,e^7-21\,b^6\,c\,d\,e^6+42\,b^5\,c^2\,d^2\,e^5-35\,b^4\,c^3\,d^3\,e^4+21\,b^2\,c^5\,d^5\,e^2-21\,b\,c^6\,d^6\,e+6\,c^7\,d^7\right )}{b^4}-\frac {x^2\,\left (7\,b^7\,e^7-35\,b^6\,c\,d\,e^6+63\,b^5\,c^2\,d^2\,e^5-35\,b^4\,c^3\,d^3\,e^4-35\,b^3\,c^4\,d^4\,e^3+63\,b^2\,c^5\,d^5\,e^2-63\,b\,c^6\,d^6\,e+18\,c^7\,d^7\right )}{2\,b^3\,c}+\frac {c^4\,d^6\,x\,\left (7\,b\,e-2\,c\,d\right )}{b^2}}{b^2\,c^4\,x^2+2\,b\,c^5\,x^3+c^6\,x^4}+\frac {\ln \left (b+c\,x\right )\,\left (6\,b^7\,e^7-21\,b^6\,c\,d\,e^6+21\,b^5\,c^2\,d^2\,e^5-21\,b^2\,c^5\,d^5\,e^2+21\,b\,c^6\,d^6\,e-6\,c^7\,d^7\right )}{b^5\,c^5}+\frac {3\,d^5\,\ln \left (x\right )\,\left (7\,b^2\,e^2-7\,b\,c\,d\,e+2\,c^2\,d^2\right )}{b^5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)^7/(b*x + c*x^2)^3,x)

[Out]

(e^7*x^2)/(2*c^3) - x*((3*b*e^7)/c^4 - (7*d*e^6)/c^3) - ((c^4*d^7)/(2*b) - (x^3*(4*b^7*e^7 + 6*c^7*d^7 + 21*b^
2*c^5*d^5*e^2 - 35*b^4*c^3*d^3*e^4 + 42*b^5*c^2*d^2*e^5 - 21*b*c^6*d^6*e - 21*b^6*c*d*e^6))/b^4 - (x^2*(7*b^7*
e^7 + 18*c^7*d^7 + 63*b^2*c^5*d^5*e^2 - 35*b^3*c^4*d^4*e^3 - 35*b^4*c^3*d^3*e^4 + 63*b^5*c^2*d^2*e^5 - 63*b*c^
6*d^6*e - 35*b^6*c*d*e^6))/(2*b^3*c) + (c^4*d^6*x*(7*b*e - 2*c*d))/b^2)/(c^6*x^4 + 2*b*c^5*x^3 + b^2*c^4*x^2)
+ (log(b + c*x)*(6*b^7*e^7 - 6*c^7*d^7 - 21*b^2*c^5*d^5*e^2 + 21*b^5*c^2*d^2*e^5 + 21*b*c^6*d^6*e - 21*b^6*c*d
*e^6))/(b^5*c^5) + (3*d^5*log(x)*(7*b^2*e^2 + 2*c^2*d^2 - 7*b*c*d*e))/b^5

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