3.275 \(\int \frac {(d+e x)^7}{(b x+c x^2)^3} \, dx\)

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

[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 = 0.27, 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 = {698} \[ \frac {3 d^5 \log (x) \left (7 b^2 e^2-7 b c d e+2 c^2 d^2\right )}{b^5}-\frac {3 (c d-b e)^5 \left (2 b^2 e^2+3 b c d e+2 c^2 d^2\right ) \log (b+c x)}{b^5 c^5}+\frac {(c d-b e)^6 (4 b e+3 c d)}{b^4 c^5 (b+c x)}+\frac {(c d-b e)^7}{2 b^3 c^5 (b+c x)^2}+\frac {d^6 (3 c d-7 b e)}{b^4 x}-\frac {d^7}{2 b^3 x^2}+\frac {e^6 x (7 c d-3 b e)}{c^4}+\frac {e^7 x^2}{2 c^3} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^7/(b*x + c*x^2)^3,x]

[Out]

-d^7/(2*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 698

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 = 0.11, size = 202, normalized size = 1.00 \[ \frac {1}{2} \left (\frac {2 (c d-b e)^6 (4 b e+3 c d)}{b^4 c^5 (b+c x)}+\frac {2 d^6 (3 c d-7 b e)}{b^4 x}+\frac {(c d-b e)^7}{b^3 c^5 (b+c x)^2}-\frac {d^7}{b^3 x^2}+\frac {6 d^5 \log (x) \left (7 b^2 e^2-7 b c d e+2 c^2 d^2\right )}{b^5}+\frac {6 (b e-c d)^5 \left (2 b^2 e^2+3 b c d e+2 c^2 d^2\right ) \log (b+c x)}{b^5 c^5}+\frac {2 e^6 x (7 c d-3 b e)}{c^4}+\frac {e^7 x^2}{c^3}\right ) \]

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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fricas [B]  time = 0.93, size = 694, normalized size = 3.42 \[ \frac {b^{5} c^{4} e^{7} x^{6} - b^{4} c^{5} d^{7} + 2 \, {\left (7 \, b^{5} c^{4} d e^{6} - 2 \, b^{6} c^{3} e^{7}\right )} x^{5} + {\left (28 \, b^{6} c^{3} d e^{6} - 11 \, b^{7} c^{2} e^{7}\right )} x^{4} + 2 \, {\left (6 \, b c^{8} d^{7} - 21 \, b^{2} c^{7} d^{6} e + 21 \, b^{3} c^{6} d^{5} e^{2} - 35 \, b^{5} c^{4} d^{3} e^{4} + 42 \, b^{6} c^{3} d^{2} e^{5} - 14 \, b^{7} c^{2} d e^{6} + b^{8} c e^{7}\right )} x^{3} + {\left (18 \, b^{2} c^{7} d^{7} - 63 \, b^{3} c^{6} d^{6} e + 63 \, b^{4} c^{5} d^{5} e^{2} - 35 \, b^{5} c^{4} d^{4} e^{3} - 35 \, b^{6} c^{3} d^{3} e^{4} + 63 \, b^{7} c^{2} d^{2} e^{5} - 35 \, b^{8} c d e^{6} + 7 \, b^{9} e^{7}\right )} x^{2} + 2 \, {\left (2 \, b^{3} c^{6} d^{7} - 7 \, b^{4} c^{5} d^{6} e\right )} x - 6 \, {\left ({\left (2 \, c^{9} d^{7} - 7 \, b c^{8} d^{6} e + 7 \, b^{2} c^{7} d^{5} e^{2} - 7 \, b^{5} c^{4} d^{2} e^{5} + 7 \, b^{6} c^{3} d e^{6} - 2 \, b^{7} c^{2} e^{7}\right )} x^{4} + 2 \, {\left (2 \, b c^{8} d^{7} - 7 \, b^{2} c^{7} d^{6} e + 7 \, b^{3} c^{6} d^{5} e^{2} - 7 \, b^{6} c^{3} d^{2} e^{5} + 7 \, b^{7} c^{2} d e^{6} - 2 \, b^{8} c e^{7}\right )} x^{3} + {\left (2 \, b^{2} c^{7} d^{7} - 7 \, b^{3} c^{6} d^{6} e + 7 \, b^{4} c^{5} d^{5} e^{2} - 7 \, b^{7} c^{2} d^{2} e^{5} + 7 \, b^{8} c d e^{6} - 2 \, b^{9} e^{7}\right )} x^{2}\right )} \log \left (c x + b\right ) + 6 \, {\left ({\left (2 \, c^{9} d^{7} - 7 \, b c^{8} d^{6} e + 7 \, b^{2} c^{7} d^{5} e^{2}\right )} x^{4} + 2 \, {\left (2 \, b c^{8} d^{7} - 7 \, b^{2} c^{7} d^{6} e + 7 \, b^{3} c^{6} d^{5} e^{2}\right )} x^{3} + {\left (2 \, b^{2} c^{7} d^{7} - 7 \, b^{3} c^{6} d^{6} e + 7 \, b^{4} c^{5} d^{5} e^{2}\right )} x^{2}\right )} \log \relax (x)}{2 \, {\left (b^{5} c^{7} x^{4} + 2 \, b^{6} c^{6} x^{3} + b^{7} c^{5} x^{2}\right )}} \]

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*(b^5*c^4*e^7*x^6 - b^4*c^5*d^7 + 2*(7*b^5*c^4*d*e^6 - 2*b^6*c^3*e^7)*x^5 + (28*b^6*c^3*d*e^6 - 11*b^7*c^2*
e^7)*x^4 + 2*(6*b*c^8*d^7 - 21*b^2*c^7*d^6*e + 21*b^3*c^6*d^5*e^2 - 35*b^5*c^4*d^3*e^4 + 42*b^6*c^3*d^2*e^5 -
14*b^7*c^2*d*e^6 + b^8*c*e^7)*x^3 + (18*b^2*c^7*d^7 - 63*b^3*c^6*d^6*e + 63*b^4*c^5*d^5*e^2 - 35*b^5*c^4*d^4*e
^3 - 35*b^6*c^3*d^3*e^4 + 63*b^7*c^2*d^2*e^5 - 35*b^8*c*d*e^6 + 7*b^9*e^7)*x^2 + 2*(2*b^3*c^6*d^7 - 7*b^4*c^5*
d^6*e)*x - 6*((2*c^9*d^7 - 7*b*c^8*d^6*e + 7*b^2*c^7*d^5*e^2 - 7*b^5*c^4*d^2*e^5 + 7*b^6*c^3*d*e^6 - 2*b^7*c^2
*e^7)*x^4 + 2*(2*b*c^8*d^7 - 7*b^2*c^7*d^6*e + 7*b^3*c^6*d^5*e^2 - 7*b^6*c^3*d^2*e^5 + 7*b^7*c^2*d*e^6 - 2*b^8
*c*e^7)*x^3 + (2*b^2*c^7*d^7 - 7*b^3*c^6*d^6*e + 7*b^4*c^5*d^5*e^2 - 7*b^7*c^2*d^2*e^5 + 7*b^8*c*d*e^6 - 2*b^9
*e^7)*x^2)*log(c*x + b) + 6*((2*c^9*d^7 - 7*b*c^8*d^6*e + 7*b^2*c^7*d^5*e^2)*x^4 + 2*(2*b*c^8*d^7 - 7*b^2*c^7*
d^6*e + 7*b^3*c^6*d^5*e^2)*x^3 + (2*b^2*c^7*d^7 - 7*b^3*c^6*d^6*e + 7*b^4*c^5*d^5*e^2)*x^2)*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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giac [A]  time = 0.18, size = 383, normalized size = 1.89 \[ \frac {3 \, {\left (2 \, c^{2} d^{7} - 7 \, b c d^{6} e + 7 \, b^{2} d^{5} e^{2}\right )} \log \left ({\left | x \right |}\right )}{b^{5}} + \frac {c^{3} x^{2} e^{7} + 14 \, c^{3} d x e^{6} - 6 \, b c^{2} x e^{7}}{2 \, c^{6}} - \frac {3 \, {\left (2 \, c^{7} d^{7} - 7 \, b c^{6} d^{6} e + 7 \, b^{2} c^{5} d^{5} e^{2} - 7 \, b^{5} c^{2} d^{2} e^{5} + 7 \, b^{6} c d e^{6} - 2 \, b^{7} e^{7}\right )} \log \left ({\left | c x + b \right |}\right )}{b^{5} c^{5}} - \frac {b^{3} c^{5} d^{7} - 2 \, {\left (6 \, c^{8} d^{7} - 21 \, b c^{7} d^{6} e + 21 \, b^{2} c^{6} d^{5} e^{2} - 35 \, b^{4} c^{4} d^{3} e^{4} + 42 \, b^{5} c^{3} d^{2} e^{5} - 21 \, b^{6} c^{2} d e^{6} + 4 \, b^{7} c e^{7}\right )} x^{3} - {\left (18 \, b c^{7} d^{7} - 63 \, b^{2} c^{6} d^{6} e + 63 \, b^{3} c^{5} d^{5} e^{2} - 35 \, b^{4} c^{4} d^{4} e^{3} - 35 \, b^{5} c^{3} d^{3} e^{4} + 63 \, b^{6} c^{2} d^{2} e^{5} - 35 \, b^{7} c d e^{6} + 7 \, b^{8} e^{7}\right )} x^{2} - 2 \, {\left (2 \, b^{2} c^{6} d^{7} - 7 \, b^{3} c^{5} d^{6} e\right )} x}{2 \, {\left (c x + b\right )}^{2} b^{4} c^{5} x^{2}} \]

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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maple [B]  time = 0.07, size = 481, normalized size = 2.37 \[ -\frac {b^{4} e^{7}}{2 \left (c x +b \right )^{2} c^{5}}+\frac {7 b^{3} d \,e^{6}}{2 \left (c x +b \right )^{2} c^{4}}-\frac {21 b^{2} d^{2} e^{5}}{2 \left (c x +b \right )^{2} c^{3}}+\frac {35 b \,d^{3} e^{4}}{2 \left (c x +b \right )^{2} c^{2}}+\frac {21 d^{5} e^{2}}{2 \left (c x +b \right )^{2} b}-\frac {7 c \,d^{6} e}{2 \left (c x +b \right )^{2} b^{2}}+\frac {c^{2} d^{7}}{2 \left (c x +b \right )^{2} b^{3}}-\frac {35 d^{4} e^{3}}{2 \left (c x +b \right )^{2} c}+\frac {e^{7} x^{2}}{2 c^{3}}+\frac {4 b^{3} e^{7}}{\left (c x +b \right ) c^{5}}-\frac {21 b^{2} d \,e^{6}}{\left (c x +b \right ) c^{4}}+\frac {6 b^{2} e^{7} \ln \left (c x +b \right )}{c^{5}}+\frac {42 b \,d^{2} e^{5}}{\left (c x +b \right ) c^{3}}-\frac {21 b d \,e^{6} \ln \left (c x +b \right )}{c^{4}}-\frac {3 b \,e^{7} x}{c^{4}}+\frac {21 d^{5} e^{2}}{\left (c x +b \right ) b^{2}}-\frac {14 c \,d^{6} e}{\left (c x +b \right ) b^{3}}+\frac {21 d^{5} e^{2} \ln \relax (x )}{b^{3}}-\frac {21 d^{5} e^{2} \ln \left (c x +b \right )}{b^{3}}+\frac {3 c^{2} d^{7}}{\left (c x +b \right ) b^{4}}-\frac {21 c \,d^{6} e \ln \relax (x )}{b^{4}}+\frac {21 c \,d^{6} e \ln \left (c x +b \right )}{b^{4}}+\frac {6 c^{2} d^{7} \ln \relax (x )}{b^{5}}-\frac {6 c^{2} d^{7} \ln \left (c x +b \right )}{b^{5}}-\frac {35 d^{3} e^{4}}{\left (c x +b \right ) c^{2}}+\frac {21 d^{2} e^{5} \ln \left (c x +b \right )}{c^{3}}+\frac {7 d \,e^{6} x}{c^{3}}-\frac {7 d^{6} e}{b^{3} x}+\frac {3 c \,d^{7}}{b^{4} x}-\frac {d^{7}}{2 b^{3} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 1.48, size = 408, normalized size = 2.01 \[ -\frac {b^{3} c^{5} d^{7} - 2 \, {\left (6 \, c^{8} d^{7} - 21 \, b c^{7} d^{6} e + 21 \, b^{2} c^{6} d^{5} e^{2} - 35 \, b^{4} c^{4} d^{3} e^{4} + 42 \, b^{5} c^{3} d^{2} e^{5} - 21 \, b^{6} c^{2} d e^{6} + 4 \, b^{7} c e^{7}\right )} x^{3} - {\left (18 \, b c^{7} d^{7} - 63 \, b^{2} c^{6} d^{6} e + 63 \, b^{3} c^{5} d^{5} e^{2} - 35 \, b^{4} c^{4} d^{4} e^{3} - 35 \, b^{5} c^{3} d^{3} e^{4} + 63 \, b^{6} c^{2} d^{2} e^{5} - 35 \, b^{7} c d e^{6} + 7 \, b^{8} e^{7}\right )} x^{2} - 2 \, {\left (2 \, b^{2} c^{6} d^{7} - 7 \, b^{3} c^{5} d^{6} e\right )} x}{2 \, {\left (b^{4} c^{7} x^{4} + 2 \, b^{5} c^{6} x^{3} + b^{6} c^{5} x^{2}\right )}} + \frac {c e^{7} x^{2} + 2 \, {\left (7 \, c d e^{6} - 3 \, b e^{7}\right )} x}{2 \, c^{4}} + \frac {3 \, {\left (2 \, c^{2} d^{7} - 7 \, b c d^{6} e + 7 \, b^{2} d^{5} e^{2}\right )} \log \relax (x)}{b^{5}} - \frac {3 \, {\left (2 \, c^{7} d^{7} - 7 \, b c^{6} d^{6} e + 7 \, b^{2} c^{5} d^{5} e^{2} - 7 \, b^{5} c^{2} d^{2} e^{5} + 7 \, b^{6} c d e^{6} - 2 \, b^{7} e^{7}\right )} \log \left (c x + b\right )}{b^{5} c^{5}} \]

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

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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sympy [B]  time = 33.37, size = 687, normalized size = 3.38 \[ x \left (- \frac {3 b e^{7}}{c^{4}} + \frac {7 d e^{6}}{c^{3}}\right ) + \frac {- b^{3} c^{5} d^{7} + x^{3} \left (8 b^{7} c e^{7} - 42 b^{6} c^{2} d e^{6} + 84 b^{5} c^{3} d^{2} e^{5} - 70 b^{4} c^{4} d^{3} e^{4} + 42 b^{2} c^{6} d^{5} e^{2} - 42 b c^{7} d^{6} e + 12 c^{8} d^{7}\right ) + x^{2} \left (7 b^{8} e^{7} - 35 b^{7} c d e^{6} + 63 b^{6} c^{2} d^{2} e^{5} - 35 b^{5} c^{3} d^{3} e^{4} - 35 b^{4} c^{4} d^{4} e^{3} + 63 b^{3} c^{5} d^{5} e^{2} - 63 b^{2} c^{6} d^{6} e + 18 b c^{7} d^{7}\right ) + x \left (- 14 b^{3} c^{5} d^{6} e + 4 b^{2} c^{6} d^{7}\right )}{2 b^{6} c^{5} x^{2} + 4 b^{5} c^{6} x^{3} + 2 b^{4} c^{7} x^{4}} + \frac {e^{7} x^{2}}{2 c^{3}} + \frac {3 d^{5} \left (7 b^{2} e^{2} - 7 b c d e + 2 c^{2} d^{2}\right ) \log {\left (x + \frac {- 21 b^{3} c^{4} d^{5} e^{2} + 21 b^{2} c^{5} d^{6} e - 6 b c^{6} d^{7} + 3 b c^{4} d^{5} \left (7 b^{2} e^{2} - 7 b c d e + 2 c^{2} d^{2}\right )}{6 b^{7} e^{7} - 21 b^{6} c d e^{6} + 21 b^{5} c^{2} d^{2} e^{5} - 42 b^{2} c^{5} d^{5} e^{2} + 42 b c^{6} d^{6} e - 12 c^{7} d^{7}} \right )}}{b^{5}} + \frac {3 \left (b e - c d\right )^{5} \left (2 b^{2} e^{2} + 3 b c d e + 2 c^{2} d^{2}\right ) \log {\left (x + \frac {- 21 b^{3} c^{4} d^{5} e^{2} + 21 b^{2} c^{5} d^{6} e - 6 b c^{6} d^{7} + \frac {3 b \left (b e - c d\right )^{5} \left (2 b^{2} e^{2} + 3 b c d e + 2 c^{2} d^{2}\right )}{c}}{6 b^{7} e^{7} - 21 b^{6} c d e^{6} + 21 b^{5} c^{2} d^{2} e^{5} - 42 b^{2} c^{5} d^{5} e^{2} + 42 b c^{6} d^{6} e - 12 c^{7} d^{7}} \right )}}{b^{5} c^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*(-3*b*e**7/c**4 + 7*d*e**6/c**3) + (-b**3*c**5*d**7 + x**3*(8*b**7*c*e**7 - 42*b**6*c**2*d*e**6 + 84*b**5*c*
*3*d**2*e**5 - 70*b**4*c**4*d**3*e**4 + 42*b**2*c**6*d**5*e**2 - 42*b*c**7*d**6*e + 12*c**8*d**7) + x**2*(7*b*
*8*e**7 - 35*b**7*c*d*e**6 + 63*b**6*c**2*d**2*e**5 - 35*b**5*c**3*d**3*e**4 - 35*b**4*c**4*d**4*e**3 + 63*b**
3*c**5*d**5*e**2 - 63*b**2*c**6*d**6*e + 18*b*c**7*d**7) + x*(-14*b**3*c**5*d**6*e + 4*b**2*c**6*d**7))/(2*b**
6*c**5*x**2 + 4*b**5*c**6*x**3 + 2*b**4*c**7*x**4) + e**7*x**2/(2*c**3) + 3*d**5*(7*b**2*e**2 - 7*b*c*d*e + 2*
c**2*d**2)*log(x + (-21*b**3*c**4*d**5*e**2 + 21*b**2*c**5*d**6*e - 6*b*c**6*d**7 + 3*b*c**4*d**5*(7*b**2*e**2
- 7*b*c*d*e + 2*c**2*d**2))/(6*b**7*e**7 - 21*b**6*c*d*e**6 + 21*b**5*c**2*d**2*e**5 - 42*b**2*c**5*d**5*e**2
+ 42*b*c**6*d**6*e - 12*c**7*d**7))/b**5 + 3*(b*e - c*d)**5*(2*b**2*e**2 + 3*b*c*d*e + 2*c**2*d**2)*log(x + (
-21*b**3*c**4*d**5*e**2 + 21*b**2*c**5*d**6*e - 6*b*c**6*d**7 + 3*b*(b*e - c*d)**5*(2*b**2*e**2 + 3*b*c*d*e +
2*c**2*d**2)/c)/(6*b**7*e**7 - 21*b**6*c*d*e**6 + 21*b**5*c**2*d**2*e**5 - 42*b**2*c**5*d**5*e**2 + 42*b*c**6*
d**6*e - 12*c**7*d**7))/(b**5*c**5)

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