3.1.42 \(\int \frac {(a+b x^2)^2 (-a d+b x (6 c+5 d x))}{(c+d x)^2} \, dx\) [42]

Optimal. Leaf size=17 \[ \frac {\left (a+b x^2\right )^3}{c+d x} \]

[Out]

(b*x^2+a)^3/(d*x+c)

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Rubi [A]
time = 0.02, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.030, Rules used = {1604} \begin {gather*} \frac {\left (a+b x^2\right )^3}{c+d x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((a + b*x^2)^2*(-(a*d) + b*x*(6*c + 5*d*x)))/(c + d*x)^2,x]

[Out]

(a + b*x^2)^3/(c + d*x)

Rule 1604

Int[(Pp_)*(Qq_)^(m_.)*(Rr_)^(n_.), x_Symbol] :> With[{p = Expon[Pp, x], q = Expon[Qq, x], r = Expon[Rr, x]}, S
imp[Coeff[Pp, x, p]*x^(p - q - r + 1)*Qq^(m + 1)*(Rr^(n + 1)/((p + m*q + n*r + 1)*Coeff[Qq, x, q]*Coeff[Rr, x,
r])), x] /; NeQ[p + m*q + n*r + 1, 0] && EqQ[(p + m*q + n*r + 1)*Coeff[Qq, x, q]*Coeff[Rr, x, r]*Pp, Coeff[Pp
, x, p]*x^(p - q - r)*((p - q - r + 1)*Qq*Rr + (m + 1)*x*Rr*D[Qq, x] + (n + 1)*x*Qq*D[Rr, x])]] /; FreeQ[{m, n
}, x] && PolyQ[Pp, x] && PolyQ[Qq, x] && PolyQ[Rr, x] && NeQ[m, -1] && NeQ[n, -1]

Rubi steps

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

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(90\) vs. \(2(17)=34\).
time = 0.02, size = 90, normalized size = 5.29 \begin {gather*} \frac {a^3 d^6+3 a^2 b d^4 \left (c^2+c d x+d^2 x^2\right )+3 a b^2 d^2 \left (c^4+c^3 d x+d^4 x^4\right )+b^3 \left (c^6+c^5 d x+d^6 x^6\right )}{d^6 (c+d x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x^2)^2*(-(a*d) + b*x*(6*c + 5*d*x)))/(c + d*x)^2,x]

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(156\) vs. \(2(17)=34\).
time = 0.07, size = 157, normalized size = 9.24

method	result	size
gosper	\(\frac {b^{3} x^{6}+3 a \,b^{2} x^{4}+3 a^{2} b \,x^{2}+a^{3}}{d x +c}\)	\(38\)
norman	\(\frac {b^{3} x^{6}+3 a \,b^{2} x^{4}+3 a^{2} b \,x^{2}-\frac {d \,a^{3} x}{c}}{d x +c}\)	\(45\)
default	\(\frac {b \left (b^{2} x^{5} d^{4}-b^{2} c \,x^{4} d^{3}+3 a b \,d^{4} x^{3}+b^{2} c^{2} d^{2} x^{3}-3 a b c \,d^{3} x^{2}-b^{2} c^{3} d \,x^{2}+3 a^{2} d^{4} x +3 a b \,c^{2} d^{2} x +b^{2} c^{4} x \right )}{d^{5}}-\frac {-a^{3} d^{6}-3 a^{2} b \,c^{2} d^{4}-3 a \,b^{2} c^{4} d^{2}-b^{3} c^{6}}{d^{6} \left (d x +c \right )}\)	\(157\)
risch	\(\frac {b^{3} x^{5}}{d}-\frac {b^{3} c \,x^{4}}{d^{2}}+\frac {3 b^{2} a \,x^{3}}{d}+\frac {b^{3} c^{2} x^{3}}{d^{3}}-\frac {3 b^{2} a c \,x^{2}}{d^{2}}-\frac {b^{3} c^{3} x^{2}}{d^{4}}+\frac {3 b \,a^{2} x}{d}+\frac {3 b^{2} a \,c^{2} x}{d^{3}}+\frac {b^{3} c^{4} x}{d^{5}}+\frac {a^{3}}{d x +c}+\frac {3 a^{2} b \,c^{2}}{d^{2} \left (d x +c \right )}+\frac {3 a \,b^{2} c^{4}}{d^{4} \left (d x +c \right )}+\frac {b^{3} c^{6}}{d^{6} \left (d x +c \right )}\)	\(176\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^2*(-a*d+b*x*(5*d*x+6*c))/(d*x+c)^2,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 160 vs. \(2 (17) = 34\).
time = 0.28, size = 160, normalized size = 9.41 \begin {gather*} \frac {b^{3} c^{6} + 3 \, a b^{2} c^{4} d^{2} + 3 \, a^{2} b c^{2} d^{4} + a^{3} d^{6}}{d^{7} x + c d^{6}} + \frac {b^{3} d^{4} x^{5} - b^{3} c d^{3} x^{4} + {\left (b^{3} c^{2} d^{2} + 3 \, a b^{2} d^{4}\right )} x^{3} - {\left (b^{3} c^{3} d + 3 \, a b^{2} c d^{3}\right )} x^{2} + {\left (b^{3} c^{4} + 3 \, a b^{2} c^{2} d^{2} + 3 \, a^{2} b d^{4}\right )} x}{d^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^2*(-a*d+b*x*(5*d*x+6*c))/(d*x+c)^2,x, algorithm="maxima")

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 120 vs. \(2 (17) = 34\).
time = 0.33, size = 120, normalized size = 7.06 \begin {gather*} \frac {b^{3} d^{6} x^{6} + 3 \, a b^{2} d^{6} x^{4} + 3 \, a^{2} b d^{6} x^{2} + b^{3} c^{6} + 3 \, a b^{2} c^{4} d^{2} + 3 \, a^{2} b c^{2} d^{4} + a^{3} d^{6} + {\left (b^{3} c^{5} d + 3 \, a b^{2} c^{3} d^{3} + 3 \, a^{2} b c d^{5}\right )} x}{d^{7} x + c d^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^2*(-a*d+b*x*(5*d*x+6*c))/(d*x+c)^2,x, algorithm="fricas")

[Out]

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

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 153 vs. \(2 (12) = 24\).
time = 0.28, size = 153, normalized size = 9.00 \begin {gather*} - \frac {b^{3} c x^{4}}{d^{2}} + \frac {b^{3} x^{5}}{d} + x^{3} \cdot \left (\frac {3 a b^{2}}{d} + \frac {b^{3} c^{2}}{d^{3}}\right ) + x^{2} \left (- \frac {3 a b^{2} c}{d^{2}} - \frac {b^{3} c^{3}}{d^{4}}\right ) + x \left (\frac {3 a^{2} b}{d} + \frac {3 a b^{2} c^{2}}{d^{3}} + \frac {b^{3} c^{4}}{d^{5}}\right ) + \frac {a^{3} d^{6} + 3 a^{2} b c^{2} d^{4} + 3 a b^{2} c^{4} d^{2} + b^{3} c^{6}}{c d^{6} + d^{7} x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**2*(-a*d+b*x*(5*d*x+6*c))/(d*x+c)**2,x)

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 216 vs. \(2 (17) = 34\).
time = 3.62, size = 216, normalized size = 12.71 \begin {gather*} \frac {{\left (b^{3} - \frac {6 \, b^{3} c}{d x + c} + \frac {15 \, b^{3} c^{2}}{{\left (d x + c\right )}^{2}} - \frac {20 \, b^{3} c^{3}}{{\left (d x + c\right )}^{3}} + \frac {15 \, b^{3} c^{4}}{{\left (d x + c\right )}^{4}} + \frac {3 \, a b^{2} d^{2}}{{\left (d x + c\right )}^{2}} - \frac {12 \, a b^{2} c d^{2}}{{\left (d x + c\right )}^{3}} + \frac {18 \, a b^{2} c^{2} d^{2}}{{\left (d x + c\right )}^{4}} + \frac {3 \, a^{2} b d^{4}}{{\left (d x + c\right )}^{4}}\right )} {\left (d x + c\right )}^{5}}{d^{6}} + \frac {\frac {b^{3} c^{6} d^{5}}{d x + c} + \frac {3 \, a b^{2} c^{4} d^{7}}{d x + c} + \frac {3 \, a^{2} b c^{2} d^{9}}{d x + c} + \frac {a^{3} d^{11}}{d x + c}}{d^{11}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^2*(-a*d+b*x*(5*d*x+6*c))/(d*x+c)^2,x, algorithm="giac")

[Out]

(b^3 - 6*b^3*c/(d*x + c) + 15*b^3*c^2/(d*x + c)^2 - 20*b^3*c^3/(d*x + c)^3 + 15*b^3*c^4/(d*x + c)^4 + 3*a*b^2*
d^2/(d*x + c)^2 - 12*a*b^2*c*d^2/(d*x + c)^3 + 18*a*b^2*c^2*d^2/(d*x + c)^4 + 3*a^2*b*d^4/(d*x + c)^4)*(d*x +
c)^5/d^6 + (b^3*c^6*d^5/(d*x + c) + 3*a*b^2*c^4*d^7/(d*x + c) + 3*a^2*b*c^2*d^9/(d*x + c) + a^3*d^11/(d*x + c)
)/d^11

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Mupad [B]
time = 0.05, size = 252, normalized size = 14.82 \begin {gather*} x^3\,\left (\frac {3\,a\,b^2}{d}+\frac {b^3\,c^2}{d^3}\right )-x\,\left (\frac {2\,c\,\left (\frac {4\,b^3\,c^3}{d^4}-\frac {2\,c\,\left (\frac {9\,a\,b^2}{d}+\frac {3\,b^3\,c^2}{d^3}\right )}{d}+\frac {12\,a\,b^2\,c}{d^2}\right )}{d}+\frac {c^2\,\left (\frac {9\,a\,b^2}{d}+\frac {3\,b^3\,c^2}{d^3}\right )}{d^2}-\frac {3\,a^2\,b}{d}\right )+x^2\,\left (\frac {2\,b^3\,c^3}{d^4}-\frac {c\,\left (\frac {9\,a\,b^2}{d}+\frac {3\,b^3\,c^2}{d^3}\right )}{d}+\frac {6\,a\,b^2\,c}{d^2}\right )+\frac {a^3\,d^6+3\,a^2\,b\,c^2\,d^4+3\,a\,b^2\,c^4\,d^2+b^3\,c^6}{d\,\left (x\,d^6+c\,d^5\right )}+\frac {b^3\,x^5}{d}-\frac {b^3\,c\,x^4}{d^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-((a*d - b*x*(6*c + 5*d*x))*(a + b*x^2)^2)/(c + d*x)^2,x)

[Out]

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

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