3.14.44 \(\int \frac {(a+b x)^4}{(c+d x)^2} \, dx\) [1344]

Optimal. Leaf size=104 \[ \frac {6 b^2 (b c-a d)^2 x}{d^4}-\frac {(b c-a d)^4}{d^5 (c+d x)}-\frac {2 b^3 (b c-a d) (c+d x)^2}{d^5}+\frac {b^4 (c+d x)^3}{3 d^5}-\frac {4 b (b c-a d)^3 \log (c+d x)}{d^5} \]

[Out]

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

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Rubi [A]
time = 0.07, antiderivative size = 104, 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^3 (c+d x)^2 (b c-a d)}{d^5}+\frac {6 b^2 x (b c-a d)^2}{d^4}-\frac {(b c-a d)^4}{d^5 (c+d x)}-\frac {4 b (b c-a d)^3 \log (c+d x)}{d^5}+\frac {b^4 (c+d x)^3}{3 d^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^4/(c + d*x)^2,x]

[Out]

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

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

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Mathematica [A]
time = 0.04, size = 165, normalized size = 1.59 \begin {gather*} \frac {12 a^3 b c d^3-3 a^4 d^4+18 a^2 b^2 d^2 \left (-c^2+c d x+d^2 x^2\right )+6 a b^3 d \left (2 c^3-4 c^2 d x-3 c d^2 x^2+d^3 x^3\right )+b^4 \left (-3 c^4+9 c^3 d x+6 c^2 d^2 x^2-2 c d^3 x^3+d^4 x^4\right )-12 b (b c-a d)^3 (c+d x) \log (c+d x)}{3 d^5 (c+d x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^4/(c + d*x)^2,x]

[Out]

(12*a^3*b*c*d^3 - 3*a^4*d^4 + 18*a^2*b^2*d^2*(-c^2 + c*d*x + d^2*x^2) + 6*a*b^3*d*(2*c^3 - 4*c^2*d*x - 3*c*d^2
*x^2 + d^3*x^3) + b^4*(-3*c^4 + 9*c^3*d*x + 6*c^2*d^2*x^2 - 2*c*d^3*x^3 + d^4*x^4) - 12*b*(b*c - a*d)^3*(c + d
*x)*Log[c + d*x])/(3*d^5*(c + d*x))

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Maple [A]
time = 0.14, size = 175, normalized size = 1.68

method	result	size
default	\(\frac {b^{2} \left (\frac {1}{3} d^{2} x^{3} b^{2}+2 a b \,d^{2} x^{2}-b^{2} c d \,x^{2}+6 a^{2} d^{2} x -8 a b c d x +3 b^{2} c^{2} x \right )}{d^{4}}-\frac {a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}}{d^{5} \left (d x +c \right )}+\frac {4 b \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \ln \left (d x +c \right )}{d^{5}}\)	\(175\)
norman	\(\frac {\frac {\left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+12 a^{2} b^{2} c^{2} d^{2}-12 a \,b^{3} c^{3} d +4 b^{4} c^{4}\right ) x}{d^{4} c}+\frac {b^{4} x^{4}}{3 d}+\frac {2 b^{2} \left (3 a^{2} d^{2}-3 a b c d +b^{2} c^{2}\right ) x^{2}}{d^{3}}+\frac {2 b^{3} \left (3 a d -b c \right ) x^{3}}{3 d^{2}}}{d x +c}+\frac {4 b \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \ln \left (d x +c \right )}{d^{5}}\)	\(181\)
risch	\(\frac {b^{4} x^{3}}{3 d^{2}}+\frac {2 b^{3} a \,x^{2}}{d^{2}}-\frac {b^{4} c \,x^{2}}{d^{3}}+\frac {6 b^{2} a^{2} x}{d^{2}}-\frac {8 b^{3} a c x}{d^{3}}+\frac {3 b^{4} c^{2} x}{d^{4}}-\frac {a^{4}}{d \left (d x +c \right )}+\frac {4 a^{3} b c}{d^{2} \left (d x +c \right )}-\frac {6 a^{2} b^{2} c^{2}}{d^{3} \left (d x +c \right )}+\frac {4 a \,b^{3} c^{3}}{d^{4} \left (d x +c \right )}-\frac {b^{4} c^{4}}{d^{5} \left (d x +c \right )}+\frac {4 b \ln \left (d x +c \right ) a^{3}}{d^{2}}-\frac {12 b^{2} \ln \left (d x +c \right ) a^{2} c}{d^{3}}+\frac {12 b^{3} \ln \left (d x +c \right ) a \,c^{2}}{d^{4}}-\frac {4 b^{4} \ln \left (d x +c \right ) c^{3}}{d^{5}}\)	\(230\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.29, size = 183, normalized size = 1.76 \begin {gather*} -\frac {b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}}{d^{6} x + c d^{5}} + \frac {b^{4} d^{2} x^{3} - 3 \, {\left (b^{4} c d - 2 \, a b^{3} d^{2}\right )} x^{2} + 3 \, {\left (3 \, b^{4} c^{2} - 8 \, a b^{3} c d + 6 \, a^{2} b^{2} d^{2}\right )} x}{3 \, d^{4}} - \frac {4 \, {\left (b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )} \log \left (d x + c\right )}{d^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 267 vs. \(2 (102) = 204\).
time = 1.27, size = 267, normalized size = 2.57 \begin {gather*} \frac {b^{4} d^{4} x^{4} - 3 \, b^{4} c^{4} + 12 \, a b^{3} c^{3} d - 18 \, a^{2} b^{2} c^{2} d^{2} + 12 \, a^{3} b c d^{3} - 3 \, a^{4} d^{4} - 2 \, {\left (b^{4} c d^{3} - 3 \, a b^{3} d^{4}\right )} x^{3} + 6 \, {\left (b^{4} c^{2} d^{2} - 3 \, a b^{3} c d^{3} + 3 \, a^{2} b^{2} d^{4}\right )} x^{2} + 3 \, {\left (3 \, b^{4} c^{3} d - 8 \, a b^{3} c^{2} d^{2} + 6 \, a^{2} b^{2} c d^{3}\right )} x - 12 \, {\left (b^{4} c^{4} - 3 \, a b^{3} c^{3} d + 3 \, a^{2} b^{2} c^{2} d^{2} - a^{3} b c d^{3} + {\left (b^{4} c^{3} d - 3 \, a b^{3} c^{2} d^{2} + 3 \, a^{2} b^{2} c d^{3} - a^{3} b d^{4}\right )} x\right )} \log \left (d x + c\right )}{3 \, {\left (d^{6} x + c d^{5}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(b^4*d^4*x^4 - 3*b^4*c^4 + 12*a*b^3*c^3*d - 18*a^2*b^2*c^2*d^2 + 12*a^3*b*c*d^3 - 3*a^4*d^4 - 2*(b^4*c*d^3
- 3*a*b^3*d^4)*x^3 + 6*(b^4*c^2*d^2 - 3*a*b^3*c*d^3 + 3*a^2*b^2*d^4)*x^2 + 3*(3*b^4*c^3*d - 8*a*b^3*c^2*d^2 +
6*a^2*b^2*c*d^3)*x - 12*(b^4*c^4 - 3*a*b^3*c^3*d + 3*a^2*b^2*c^2*d^2 - a^3*b*c*d^3 + (b^4*c^3*d - 3*a*b^3*c^2
*d^2 + 3*a^2*b^2*c*d^3 - a^3*b*d^4)*x)*log(d*x + c))/(d^6*x + c*d^5)

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Sympy [A]
time = 0.38, size = 155, normalized size = 1.49 \begin {gather*} \frac {b^{4} x^{3}}{3 d^{2}} + \frac {4 b \left (a d - b c\right )^{3} \log {\left (c + d x \right )}}{d^{5}} + x^{2} \cdot \left (\frac {2 a b^{3}}{d^{2}} - \frac {b^{4} c}{d^{3}}\right ) + x \left (\frac {6 a^{2} b^{2}}{d^{2}} - \frac {8 a b^{3} c}{d^{3}} + \frac {3 b^{4} c^{2}}{d^{4}}\right ) + \frac {- 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}}{c d^{5} + d^{6} x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 245 vs. \(2 (102) = 204\).
time = 0.63, size = 245, normalized size = 2.36 \begin {gather*} \frac {{\left (b^{4} - \frac {6 \, {\left (b^{4} c d - a b^{3} d^{2}\right )}}{{\left (d x + c\right )} d} + \frac {18 \, {\left (b^{4} c^{2} d^{2} - 2 \, a b^{3} c d^{3} + a^{2} b^{2} d^{4}\right )}}{{\left (d x + c\right )}^{2} d^{2}}\right )} {\left (d x + c\right )}^{3}}{3 \, d^{5}} + \frac {4 \, {\left (b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )} \log \left (\frac {{\left | d x + c \right |}}{{\left (d x + c\right )}^{2} {\left | d \right |}}\right )}{d^{5}} - \frac {\frac {b^{4} c^{4} d^{3}}{d x + c} - \frac {4 \, a b^{3} c^{3} d^{4}}{d x + c} + \frac {6 \, a^{2} b^{2} c^{2} d^{5}}{d x + c} - \frac {4 \, a^{3} b c d^{6}}{d x + c} + \frac {a^{4} d^{7}}{d x + c}}{d^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(b^4 - 6*(b^4*c*d - a*b^3*d^2)/((d*x + c)*d) + 18*(b^4*c^2*d^2 - 2*a*b^3*c*d^3 + a^2*b^2*d^4)/((d*x + c)^2
*d^2))*(d*x + c)^3/d^5 + 4*(b^4*c^3 - 3*a*b^3*c^2*d + 3*a^2*b^2*c*d^2 - a^3*b*d^3)*log(abs(d*x + c)/((d*x + c)
^2*abs(d)))/d^5 - (b^4*c^4*d^3/(d*x + c) - 4*a*b^3*c^3*d^4/(d*x + c) + 6*a^2*b^2*c^2*d^5/(d*x + c) - 4*a^3*b*c
*d^6/(d*x + c) + a^4*d^7/(d*x + c))/d^8

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Mupad [B]
time = 0.07, size = 203, normalized size = 1.95 \begin {gather*} x^2\,\left (\frac {2\,a\,b^3}{d^2}-\frac {b^4\,c}{d^3}\right )-x\,\left (\frac {2\,c\,\left (\frac {4\,a\,b^3}{d^2}-\frac {2\,b^4\,c}{d^3}\right )}{d}-\frac {6\,a^2\,b^2}{d^2}+\frac {b^4\,c^2}{d^4}\right )+\frac {b^4\,x^3}{3\,d^2}-\frac {\ln \left (c+d\,x\right )\,\left (-4\,a^3\,b\,d^3+12\,a^2\,b^2\,c\,d^2-12\,a\,b^3\,c^2\,d+4\,b^4\,c^3\right )}{d^5}-\frac {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}{d\,\left (x\,d^5+c\,d^4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)^4/(c + d*x)^2,x)

[Out]

x^2*((2*a*b^3)/d^2 - (b^4*c)/d^3) - x*((2*c*((4*a*b^3)/d^2 - (2*b^4*c)/d^3))/d - (6*a^2*b^2)/d^2 + (b^4*c^2)/d
^4) + (b^4*x^3)/(3*d^2) - (log(c + d*x)*(4*b^4*c^3 - 4*a^3*b*d^3 + 12*a^2*b^2*c*d^2 - 12*a*b^3*c^2*d))/d^5 - (
a^4*d^4 + b^4*c^4 + 6*a^2*b^2*c^2*d^2 - 4*a*b^3*c^3*d - 4*a^3*b*c*d^3)/(d*(c*d^4 + d^5*x))

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