3.259 \(\int \frac {x^4 (c+d x)^2}{(a+b x)^2} \, dx\)

Optimal. Leaf size=165 \[ -\frac {a^4 (b c-a d)^2}{b^7 (a+b x)}-\frac {2 a^3 (2 b c-3 a d) (b c-a d) \log (a+b x)}{b^7}+\frac {a^2 x (3 b c-5 a d) (b c-a d)}{b^6}-\frac {a x^2 (b c-2 a d) (b c-a d)}{b^5}+\frac {x^3 (b c-3 a d) (b c-a d)}{3 b^4}+\frac {d x^4 (b c-a d)}{2 b^3}+\frac {d^2 x^5}{5 b^2} \]

[Out]

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

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Rubi [A]  time = 0.20, antiderivative size = 165, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {88} \[ \frac {a^2 x (3 b c-5 a d) (b c-a d)}{b^6}-\frac {a^4 (b c-a d)^2}{b^7 (a+b x)}-\frac {2 a^3 (2 b c-3 a d) (b c-a d) \log (a+b x)}{b^7}+\frac {d x^4 (b c-a d)}{2 b^3}+\frac {x^3 (b c-3 a d) (b c-a d)}{3 b^4}-\frac {a x^2 (b c-2 a d) (b c-a d)}{b^5}+\frac {d^2 x^5}{5 b^2} \]

Antiderivative was successfully verified.

[In]

Int[(x^4*(c + d*x)^2)/(a + b*x)^2,x]

[Out]

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

Rule 88

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

Rubi steps

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

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Mathematica [A]  time = 0.13, size = 183, normalized size = 1.11 \[ \frac {-\frac {30 a^4 (b c-a d)^2}{a+b x}-30 a b^2 x^2 \left (2 a^2 d^2-3 a b c d+b^2 c^2\right )+30 a^2 b x \left (5 a^2 d^2-8 a b c d+3 b^2 c^2\right )+10 b^3 x^3 \left (3 a^2 d^2-4 a b c d+b^2 c^2\right )-60 a^3 \left (3 a^2 d^2-5 a b c d+2 b^2 c^2\right ) \log (a+b x)+15 b^4 d x^4 (b c-a d)+6 b^5 d^2 x^5}{30 b^7} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^4*(c + d*x)^2)/(a + b*x)^2,x]

[Out]

(30*a^2*b*(3*b^2*c^2 - 8*a*b*c*d + 5*a^2*d^2)*x - 30*a*b^2*(b^2*c^2 - 3*a*b*c*d + 2*a^2*d^2)*x^2 + 10*b^3*(b^2
*c^2 - 4*a*b*c*d + 3*a^2*d^2)*x^3 + 15*b^4*d*(b*c - a*d)*x^4 + 6*b^5*d^2*x^5 - (30*a^4*(b*c - a*d)^2)/(a + b*x
) - 60*a^3*(2*b^2*c^2 - 5*a*b*c*d + 3*a^2*d^2)*Log[a + b*x])/(30*b^7)

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fricas [A]  time = 0.63, size = 286, normalized size = 1.73 \[ \frac {6 \, b^{6} d^{2} x^{6} - 30 \, a^{4} b^{2} c^{2} + 60 \, a^{5} b c d - 30 \, a^{6} d^{2} + 3 \, {\left (5 \, b^{6} c d - 3 \, a b^{5} d^{2}\right )} x^{5} + 5 \, {\left (2 \, b^{6} c^{2} - 5 \, a b^{5} c d + 3 \, a^{2} b^{4} d^{2}\right )} x^{4} - 10 \, {\left (2 \, a b^{5} c^{2} - 5 \, a^{2} b^{4} c d + 3 \, a^{3} b^{3} d^{2}\right )} x^{3} + 30 \, {\left (2 \, a^{2} b^{4} c^{2} - 5 \, a^{3} b^{3} c d + 3 \, a^{4} b^{2} d^{2}\right )} x^{2} + 30 \, {\left (3 \, a^{3} b^{3} c^{2} - 8 \, a^{4} b^{2} c d + 5 \, a^{5} b d^{2}\right )} x - 60 \, {\left (2 \, a^{4} b^{2} c^{2} - 5 \, a^{5} b c d + 3 \, a^{6} d^{2} + {\left (2 \, a^{3} b^{3} c^{2} - 5 \, a^{4} b^{2} c d + 3 \, a^{5} b d^{2}\right )} x\right )} \log \left (b x + a\right )}{30 \, {\left (b^{8} x + a b^{7}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(d*x+c)^2/(b*x+a)^2,x, algorithm="fricas")

[Out]

1/30*(6*b^6*d^2*x^6 - 30*a^4*b^2*c^2 + 60*a^5*b*c*d - 30*a^6*d^2 + 3*(5*b^6*c*d - 3*a*b^5*d^2)*x^5 + 5*(2*b^6*
c^2 - 5*a*b^5*c*d + 3*a^2*b^4*d^2)*x^4 - 10*(2*a*b^5*c^2 - 5*a^2*b^4*c*d + 3*a^3*b^3*d^2)*x^3 + 30*(2*a^2*b^4*
c^2 - 5*a^3*b^3*c*d + 3*a^4*b^2*d^2)*x^2 + 30*(3*a^3*b^3*c^2 - 8*a^4*b^2*c*d + 5*a^5*b*d^2)*x - 60*(2*a^4*b^2*
c^2 - 5*a^5*b*c*d + 3*a^6*d^2 + (2*a^3*b^3*c^2 - 5*a^4*b^2*c*d + 3*a^5*b*d^2)*x)*log(b*x + a))/(b^8*x + a*b^7)

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giac [A]  time = 1.07, size = 280, normalized size = 1.70 \[ \frac {{\left (6 \, d^{2} + \frac {15 \, {\left (b^{2} c d - 3 \, a b d^{2}\right )}}{{\left (b x + a\right )} b} + \frac {10 \, {\left (b^{4} c^{2} - 10 \, a b^{3} c d + 15 \, a^{2} b^{2} d^{2}\right )}}{{\left (b x + a\right )}^{2} b^{2}} - \frac {60 \, {\left (a b^{5} c^{2} - 5 \, a^{2} b^{4} c d + 5 \, a^{3} b^{3} d^{2}\right )}}{{\left (b x + a\right )}^{3} b^{3}} + \frac {30 \, {\left (6 \, a^{2} b^{6} c^{2} - 20 \, a^{3} b^{5} c d + 15 \, a^{4} b^{4} d^{2}\right )}}{{\left (b x + a\right )}^{4} b^{4}}\right )} {\left (b x + a\right )}^{5}}{30 \, b^{7}} + \frac {2 \, {\left (2 \, a^{3} b^{2} c^{2} - 5 \, a^{4} b c d + 3 \, a^{5} d^{2}\right )} \log \left (\frac {{\left | b x + a \right |}}{{\left (b x + a\right )}^{2} {\left | b \right |}}\right )}{b^{7}} - \frac {\frac {a^{4} b^{7} c^{2}}{b x + a} - \frac {2 \, a^{5} b^{6} c d}{b x + a} + \frac {a^{6} b^{5} d^{2}}{b x + a}}{b^{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(d*x+c)^2/(b*x+a)^2,x, algorithm="giac")

[Out]

1/30*(6*d^2 + 15*(b^2*c*d - 3*a*b*d^2)/((b*x + a)*b) + 10*(b^4*c^2 - 10*a*b^3*c*d + 15*a^2*b^2*d^2)/((b*x + a)
^2*b^2) - 60*(a*b^5*c^2 - 5*a^2*b^4*c*d + 5*a^3*b^3*d^2)/((b*x + a)^3*b^3) + 30*(6*a^2*b^6*c^2 - 20*a^3*b^5*c*
d + 15*a^4*b^4*d^2)/((b*x + a)^4*b^4))*(b*x + a)^5/b^7 + 2*(2*a^3*b^2*c^2 - 5*a^4*b*c*d + 3*a^5*d^2)*log(abs(b
*x + a)/((b*x + a)^2*abs(b)))/b^7 - (a^4*b^7*c^2/(b*x + a) - 2*a^5*b^6*c*d/(b*x + a) + a^6*b^5*d^2/(b*x + a))/
b^12

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maple [A]  time = 0.01, size = 247, normalized size = 1.50 \[ \frac {d^{2} x^{5}}{5 b^{2}}-\frac {a \,d^{2} x^{4}}{2 b^{3}}+\frac {c d \,x^{4}}{2 b^{2}}+\frac {a^{2} d^{2} x^{3}}{b^{4}}-\frac {4 a c d \,x^{3}}{3 b^{3}}+\frac {c^{2} x^{3}}{3 b^{2}}-\frac {2 a^{3} d^{2} x^{2}}{b^{5}}+\frac {3 a^{2} c d \,x^{2}}{b^{4}}-\frac {a \,c^{2} x^{2}}{b^{3}}-\frac {a^{6} d^{2}}{\left (b x +a \right ) b^{7}}+\frac {2 a^{5} c d}{\left (b x +a \right ) b^{6}}-\frac {6 a^{5} d^{2} \ln \left (b x +a \right )}{b^{7}}-\frac {a^{4} c^{2}}{\left (b x +a \right ) b^{5}}+\frac {10 a^{4} c d \ln \left (b x +a \right )}{b^{6}}+\frac {5 a^{4} d^{2} x}{b^{6}}-\frac {4 a^{3} c^{2} \ln \left (b x +a \right )}{b^{5}}-\frac {8 a^{3} c d x}{b^{5}}+\frac {3 a^{2} c^{2} x}{b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(d*x+c)^2/(b*x+a)^2,x)

[Out]

1/5*d^2*x^5/b^2-1/2/b^3*x^4*a*d^2+1/2/b^2*x^4*c*d+1/b^4*x^3*a^2*d^2-4/3/b^3*x^3*a*c*d+1/3/b^2*x^3*c^2-2/b^5*x^
2*a^3*d^2+3/b^4*x^2*a^2*c*d-1/b^3*x^2*a*c^2+5/b^6*x*a^4*d^2-8/b^5*x*a^3*c*d+3/b^4*x*a^2*c^2-6*a^5/b^7*ln(b*x+a
)*d^2+10*a^4/b^6*ln(b*x+a)*c*d-4*a^3/b^5*ln(b*x+a)*c^2-a^6/b^7/(b*x+a)*d^2+2*a^5/b^6/(b*x+a)*c*d-a^4/b^5/(b*x+
a)*c^2

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maxima [A]  time = 1.02, size = 215, normalized size = 1.30 \[ -\frac {a^{4} b^{2} c^{2} - 2 \, a^{5} b c d + a^{6} d^{2}}{b^{8} x + a b^{7}} + \frac {6 \, b^{4} d^{2} x^{5} + 15 \, {\left (b^{4} c d - a b^{3} d^{2}\right )} x^{4} + 10 \, {\left (b^{4} c^{2} - 4 \, a b^{3} c d + 3 \, a^{2} b^{2} d^{2}\right )} x^{3} - 30 \, {\left (a b^{3} c^{2} - 3 \, a^{2} b^{2} c d + 2 \, a^{3} b d^{2}\right )} x^{2} + 30 \, {\left (3 \, a^{2} b^{2} c^{2} - 8 \, a^{3} b c d + 5 \, a^{4} d^{2}\right )} x}{30 \, b^{6}} - \frac {2 \, {\left (2 \, a^{3} b^{2} c^{2} - 5 \, a^{4} b c d + 3 \, a^{5} d^{2}\right )} \log \left (b x + a\right )}{b^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(d*x+c)^2/(b*x+a)^2,x, algorithm="maxima")

[Out]

-(a^4*b^2*c^2 - 2*a^5*b*c*d + a^6*d^2)/(b^8*x + a*b^7) + 1/30*(6*b^4*d^2*x^5 + 15*(b^4*c*d - a*b^3*d^2)*x^4 +
10*(b^4*c^2 - 4*a*b^3*c*d + 3*a^2*b^2*d^2)*x^3 - 30*(a*b^3*c^2 - 3*a^2*b^2*c*d + 2*a^3*b*d^2)*x^2 + 30*(3*a^2*
b^2*c^2 - 8*a^3*b*c*d + 5*a^4*d^2)*x)/b^6 - 2*(2*a^3*b^2*c^2 - 5*a^4*b*c*d + 3*a^5*d^2)*log(b*x + a)/b^7

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mupad [B]  time = 0.10, size = 374, normalized size = 2.27 \[ x^3\,\left (\frac {c^2}{3\,b^2}+\frac {2\,a\,\left (\frac {2\,a\,d^2}{b^3}-\frac {2\,c\,d}{b^2}\right )}{3\,b}-\frac {a^2\,d^2}{3\,b^4}\right )+x^2\,\left (\frac {a^2\,\left (\frac {2\,a\,d^2}{b^3}-\frac {2\,c\,d}{b^2}\right )}{2\,b^2}-\frac {a\,\left (\frac {c^2}{b^2}+\frac {2\,a\,\left (\frac {2\,a\,d^2}{b^3}-\frac {2\,c\,d}{b^2}\right )}{b}-\frac {a^2\,d^2}{b^4}\right )}{b}\right )-x^4\,\left (\frac {a\,d^2}{2\,b^3}-\frac {c\,d}{2\,b^2}\right )-x\,\left (\frac {2\,a\,\left (\frac {a^2\,\left (\frac {2\,a\,d^2}{b^3}-\frac {2\,c\,d}{b^2}\right )}{b^2}-\frac {2\,a\,\left (\frac {c^2}{b^2}+\frac {2\,a\,\left (\frac {2\,a\,d^2}{b^3}-\frac {2\,c\,d}{b^2}\right )}{b}-\frac {a^2\,d^2}{b^4}\right )}{b}\right )}{b}+\frac {a^2\,\left (\frac {c^2}{b^2}+\frac {2\,a\,\left (\frac {2\,a\,d^2}{b^3}-\frac {2\,c\,d}{b^2}\right )}{b}-\frac {a^2\,d^2}{b^4}\right )}{b^2}\right )-\frac {a^6\,d^2-2\,a^5\,b\,c\,d+a^4\,b^2\,c^2}{b\,\left (x\,b^7+a\,b^6\right )}-\frac {\ln \left (a+b\,x\right )\,\left (6\,a^5\,d^2-10\,a^4\,b\,c\,d+4\,a^3\,b^2\,c^2\right )}{b^7}+\frac {d^2\,x^5}{5\,b^2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^4*(c + d*x)^2)/(a + b*x)^2,x)

[Out]

x^3*(c^2/(3*b^2) + (2*a*((2*a*d^2)/b^3 - (2*c*d)/b^2))/(3*b) - (a^2*d^2)/(3*b^4)) + x^2*((a^2*((2*a*d^2)/b^3 -
(2*c*d)/b^2))/(2*b^2) - (a*(c^2/b^2 + (2*a*((2*a*d^2)/b^3 - (2*c*d)/b^2))/b - (a^2*d^2)/b^4))/b) - x^4*((a*d^
2)/(2*b^3) - (c*d)/(2*b^2)) - x*((2*a*((a^2*((2*a*d^2)/b^3 - (2*c*d)/b^2))/b^2 - (2*a*(c^2/b^2 + (2*a*((2*a*d^
2)/b^3 - (2*c*d)/b^2))/b - (a^2*d^2)/b^4))/b))/b + (a^2*(c^2/b^2 + (2*a*((2*a*d^2)/b^3 - (2*c*d)/b^2))/b - (a^
2*d^2)/b^4))/b^2) - (a^6*d^2 + a^4*b^2*c^2 - 2*a^5*b*c*d)/(b*(a*b^6 + b^7*x)) - (log(a + b*x)*(6*a^5*d^2 + 4*a
^3*b^2*c^2 - 10*a^4*b*c*d))/b^7 + (d^2*x^5)/(5*b^2)

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sympy [A]  time = 1.47, size = 209, normalized size = 1.27 \[ - \frac {2 a^{3} \left (a d - b c\right ) \left (3 a d - 2 b c\right ) \log {\left (a + b x \right )}}{b^{7}} + x^{4} \left (- \frac {a d^{2}}{2 b^{3}} + \frac {c d}{2 b^{2}}\right ) + x^{3} \left (\frac {a^{2} d^{2}}{b^{4}} - \frac {4 a c d}{3 b^{3}} + \frac {c^{2}}{3 b^{2}}\right ) + x^{2} \left (- \frac {2 a^{3} d^{2}}{b^{5}} + \frac {3 a^{2} c d}{b^{4}} - \frac {a c^{2}}{b^{3}}\right ) + x \left (\frac {5 a^{4} d^{2}}{b^{6}} - \frac {8 a^{3} c d}{b^{5}} + \frac {3 a^{2} c^{2}}{b^{4}}\right ) + \frac {- a^{6} d^{2} + 2 a^{5} b c d - a^{4} b^{2} c^{2}}{a b^{7} + b^{8} x} + \frac {d^{2} x^{5}}{5 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*(d*x+c)**2/(b*x+a)**2,x)

[Out]

-2*a**3*(a*d - b*c)*(3*a*d - 2*b*c)*log(a + b*x)/b**7 + x**4*(-a*d**2/(2*b**3) + c*d/(2*b**2)) + x**3*(a**2*d*
*2/b**4 - 4*a*c*d/(3*b**3) + c**2/(3*b**2)) + x**2*(-2*a**3*d**2/b**5 + 3*a**2*c*d/b**4 - a*c**2/b**3) + x*(5*
a**4*d**2/b**6 - 8*a**3*c*d/b**5 + 3*a**2*c**2/b**4) + (-a**6*d**2 + 2*a**5*b*c*d - a**4*b**2*c**2)/(a*b**7 +
b**8*x) + d**2*x**5/(5*b**2)

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