3.1304 \(\int \frac {(A+B x) (a+c x^2)^2}{(d+e x)^2} \, dx\)

Optimal. Leaf size=180 \[ \frac {\left (a e^2+c d^2\right )^2 (B d-A e)}{e^6 (d+e x)}+\frac {\left (a e^2+c d^2\right ) \log (d+e x) \left (a B e^2-4 A c d e+5 B c d^2\right )}{e^6}+\frac {c x^2 \left (2 a B e^2-2 A c d e+3 B c d^2\right )}{2 e^4}-\frac {c x \left (-2 a A e^3+4 a B d e^2-3 A c d^2 e+4 B c d^3\right )}{e^5}-\frac {c^2 x^3 (2 B d-A e)}{3 e^3}+\frac {B c^2 x^4}{4 e^2} \]

[Out]

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

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Rubi [A]  time = 0.22, antiderivative size = 180, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {772} \[ \frac {c x^2 \left (2 a B e^2-2 A c d e+3 B c d^2\right )}{2 e^4}-\frac {c x \left (-2 a A e^3+4 a B d e^2-3 A c d^2 e+4 B c d^3\right )}{e^5}+\frac {\left (a e^2+c d^2\right )^2 (B d-A e)}{e^6 (d+e x)}+\frac {\left (a e^2+c d^2\right ) \log (d+e x) \left (a B e^2-4 A c d e+5 B c d^2\right )}{e^6}-\frac {c^2 x^3 (2 B d-A e)}{3 e^3}+\frac {B c^2 x^4}{4 e^2} \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(a + c*x^2)^2)/(d + e*x)^2,x]

[Out]

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

Rule 772

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegr
and[(d + e*x)^m*(f + g*x)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && IGtQ[p, 0]

Rubi steps

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

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Mathematica [A]  time = 0.14, size = 175, normalized size = 0.97 \[ \frac {6 c e^2 x^2 \left (2 a B e^2-2 A c d e+3 B c d^2\right )+\frac {12 \left (a e^2+c d^2\right )^2 (B d-A e)}{d+e x}+12 \left (a e^2+c d^2\right ) \log (d+e x) \left (a B e^2-4 A c d e+5 B c d^2\right )+12 c e x \left (A e \left (2 a e^2+3 c d^2\right )-4 B \left (a d e^2+c d^3\right )\right )+4 c^2 e^3 x^3 (A e-2 B d)+3 B c^2 e^4 x^4}{12 e^6} \]

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(a + c*x^2)^2)/(d + e*x)^2,x]

[Out]

(12*c*e*(A*e*(3*c*d^2 + 2*a*e^2) - 4*B*(c*d^3 + a*d*e^2))*x + 6*c*e^2*(3*B*c*d^2 - 2*A*c*d*e + 2*a*B*e^2)*x^2
+ 4*c^2*e^3*(-2*B*d + A*e)*x^3 + 3*B*c^2*e^4*x^4 + (12*(B*d - A*e)*(c*d^2 + a*e^2)^2)/(d + e*x) + 12*(c*d^2 +
a*e^2)*(5*B*c*d^2 - 4*A*c*d*e + a*B*e^2)*Log[d + e*x])/(12*e^6)

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fricas [B]  time = 0.68, size = 354, normalized size = 1.97 \[ \frac {3 \, B c^{2} e^{5} x^{5} + 12 \, B c^{2} d^{5} - 12 \, A c^{2} d^{4} e + 24 \, B a c d^{3} e^{2} - 24 \, A a c d^{2} e^{3} + 12 \, B a^{2} d e^{4} - 12 \, A a^{2} e^{5} - {\left (5 \, B c^{2} d e^{4} - 4 \, A c^{2} e^{5}\right )} x^{4} + 2 \, {\left (5 \, B c^{2} d^{2} e^{3} - 4 \, A c^{2} d e^{4} + 6 \, B a c e^{5}\right )} x^{3} - 6 \, {\left (5 \, B c^{2} d^{3} e^{2} - 4 \, A c^{2} d^{2} e^{3} + 6 \, B a c d e^{4} - 4 \, A a c e^{5}\right )} x^{2} - 12 \, {\left (4 \, B c^{2} d^{4} e - 3 \, A c^{2} d^{3} e^{2} + 4 \, B a c d^{2} e^{3} - 2 \, A a c d e^{4}\right )} x + 12 \, {\left (5 \, B c^{2} d^{5} - 4 \, A c^{2} d^{4} e + 6 \, B a c d^{3} e^{2} - 4 \, A a c d^{2} e^{3} + B a^{2} d e^{4} + {\left (5 \, B c^{2} d^{4} e - 4 \, A c^{2} d^{3} e^{2} + 6 \, B a c d^{2} e^{3} - 4 \, A a c d e^{4} + B a^{2} e^{5}\right )} x\right )} \log \left (e x + d\right )}{12 \, {\left (e^{7} x + d e^{6}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+a)^2/(e*x+d)^2,x, algorithm="fricas")

[Out]

1/12*(3*B*c^2*e^5*x^5 + 12*B*c^2*d^5 - 12*A*c^2*d^4*e + 24*B*a*c*d^3*e^2 - 24*A*a*c*d^2*e^3 + 12*B*a^2*d*e^4 -
12*A*a^2*e^5 - (5*B*c^2*d*e^4 - 4*A*c^2*e^5)*x^4 + 2*(5*B*c^2*d^2*e^3 - 4*A*c^2*d*e^4 + 6*B*a*c*e^5)*x^3 - 6*
(5*B*c^2*d^3*e^2 - 4*A*c^2*d^2*e^3 + 6*B*a*c*d*e^4 - 4*A*a*c*e^5)*x^2 - 12*(4*B*c^2*d^4*e - 3*A*c^2*d^3*e^2 +
4*B*a*c*d^2*e^3 - 2*A*a*c*d*e^4)*x + 12*(5*B*c^2*d^5 - 4*A*c^2*d^4*e + 6*B*a*c*d^3*e^2 - 4*A*a*c*d^2*e^3 + B*a
^2*d*e^4 + (5*B*c^2*d^4*e - 4*A*c^2*d^3*e^2 + 6*B*a*c*d^2*e^3 - 4*A*a*c*d*e^4 + B*a^2*e^5)*x)*log(e*x + d))/(e
^7*x + d*e^6)

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giac [A]  time = 0.23, size = 317, normalized size = 1.76 \[ \frac {1}{12} \, {\left (3 \, B c^{2} - \frac {4 \, {\left (5 \, B c^{2} d e - A c^{2} e^{2}\right )} e^{\left (-1\right )}}{x e + d} + \frac {12 \, {\left (5 \, B c^{2} d^{2} e^{2} - 2 \, A c^{2} d e^{3} + B a c e^{4}\right )} e^{\left (-2\right )}}{{\left (x e + d\right )}^{2}} - \frac {24 \, {\left (5 \, B c^{2} d^{3} e^{3} - 3 \, A c^{2} d^{2} e^{4} + 3 \, B a c d e^{5} - A a c e^{6}\right )} e^{\left (-3\right )}}{{\left (x e + d\right )}^{3}}\right )} {\left (x e + d\right )}^{4} e^{\left (-6\right )} - {\left (5 \, B c^{2} d^{4} - 4 \, A c^{2} d^{3} e + 6 \, B a c d^{2} e^{2} - 4 \, A a c d e^{3} + B a^{2} e^{4}\right )} e^{\left (-6\right )} \log \left (\frac {{\left | x e + d \right |} e^{\left (-1\right )}}{{\left (x e + d\right )}^{2}}\right ) + {\left (\frac {B c^{2} d^{5} e^{4}}{x e + d} - \frac {A c^{2} d^{4} e^{5}}{x e + d} + \frac {2 \, B a c d^{3} e^{6}}{x e + d} - \frac {2 \, A a c d^{2} e^{7}}{x e + d} + \frac {B a^{2} d e^{8}}{x e + d} - \frac {A a^{2} e^{9}}{x e + d}\right )} e^{\left (-10\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+a)^2/(e*x+d)^2,x, algorithm="giac")

[Out]

1/12*(3*B*c^2 - 4*(5*B*c^2*d*e - A*c^2*e^2)*e^(-1)/(x*e + d) + 12*(5*B*c^2*d^2*e^2 - 2*A*c^2*d*e^3 + B*a*c*e^4
)*e^(-2)/(x*e + d)^2 - 24*(5*B*c^2*d^3*e^3 - 3*A*c^2*d^2*e^4 + 3*B*a*c*d*e^5 - A*a*c*e^6)*e^(-3)/(x*e + d)^3)*
(x*e + d)^4*e^(-6) - (5*B*c^2*d^4 - 4*A*c^2*d^3*e + 6*B*a*c*d^2*e^2 - 4*A*a*c*d*e^3 + B*a^2*e^4)*e^(-6)*log(ab
s(x*e + d)*e^(-1)/(x*e + d)^2) + (B*c^2*d^5*e^4/(x*e + d) - A*c^2*d^4*e^5/(x*e + d) + 2*B*a*c*d^3*e^6/(x*e + d
) - 2*A*a*c*d^2*e^7/(x*e + d) + B*a^2*d*e^8/(x*e + d) - A*a^2*e^9/(x*e + d))*e^(-10)

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maple [A]  time = 0.05, size = 309, normalized size = 1.72 \[ \frac {B \,c^{2} x^{4}}{4 e^{2}}+\frac {A \,c^{2} x^{3}}{3 e^{2}}-\frac {2 B \,c^{2} d \,x^{3}}{3 e^{3}}-\frac {A \,c^{2} d \,x^{2}}{e^{3}}+\frac {B a c \,x^{2}}{e^{2}}+\frac {3 B \,c^{2} d^{2} x^{2}}{2 e^{4}}-\frac {A \,a^{2}}{\left (e x +d \right ) e}-\frac {2 A a c \,d^{2}}{\left (e x +d \right ) e^{3}}-\frac {4 A a c d \ln \left (e x +d \right )}{e^{3}}+\frac {2 A a c x}{e^{2}}-\frac {A \,c^{2} d^{4}}{\left (e x +d \right ) e^{5}}-\frac {4 A \,c^{2} d^{3} \ln \left (e x +d \right )}{e^{5}}+\frac {3 A \,c^{2} d^{2} x}{e^{4}}+\frac {B \,a^{2} d}{\left (e x +d \right ) e^{2}}+\frac {B \,a^{2} \ln \left (e x +d \right )}{e^{2}}+\frac {2 B a c \,d^{3}}{\left (e x +d \right ) e^{4}}+\frac {6 B a c \,d^{2} \ln \left (e x +d \right )}{e^{4}}-\frac {4 B a c d x}{e^{3}}+\frac {B \,c^{2} d^{5}}{\left (e x +d \right ) e^{6}}+\frac {5 B \,c^{2} d^{4} \ln \left (e x +d \right )}{e^{6}}-\frac {4 B \,c^{2} d^{3} x}{e^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(c*x^2+a)^2/(e*x+d)^2,x)

[Out]

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

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maxima [A]  time = 0.58, size = 249, normalized size = 1.38 \[ \frac {B c^{2} d^{5} - A c^{2} d^{4} e + 2 \, B a c d^{3} e^{2} - 2 \, A a c d^{2} e^{3} + B a^{2} d e^{4} - A a^{2} e^{5}}{e^{7} x + d e^{6}} + \frac {3 \, B c^{2} e^{3} x^{4} - 4 \, {\left (2 \, B c^{2} d e^{2} - A c^{2} e^{3}\right )} x^{3} + 6 \, {\left (3 \, B c^{2} d^{2} e - 2 \, A c^{2} d e^{2} + 2 \, B a c e^{3}\right )} x^{2} - 12 \, {\left (4 \, B c^{2} d^{3} - 3 \, A c^{2} d^{2} e + 4 \, B a c d e^{2} - 2 \, A a c e^{3}\right )} x}{12 \, e^{5}} + \frac {{\left (5 \, B c^{2} d^{4} - 4 \, A c^{2} d^{3} e + 6 \, B a c d^{2} e^{2} - 4 \, A a c d e^{3} + B a^{2} e^{4}\right )} \log \left (e x + d\right )}{e^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+a)^2/(e*x+d)^2,x, algorithm="maxima")

[Out]

(B*c^2*d^5 - A*c^2*d^4*e + 2*B*a*c*d^3*e^2 - 2*A*a*c*d^2*e^3 + B*a^2*d*e^4 - A*a^2*e^5)/(e^7*x + d*e^6) + 1/12
*(3*B*c^2*e^3*x^4 - 4*(2*B*c^2*d*e^2 - A*c^2*e^3)*x^3 + 6*(3*B*c^2*d^2*e - 2*A*c^2*d*e^2 + 2*B*a*c*e^3)*x^2 -
12*(4*B*c^2*d^3 - 3*A*c^2*d^2*e + 4*B*a*c*d*e^2 - 2*A*a*c*e^3)*x)/e^5 + (5*B*c^2*d^4 - 4*A*c^2*d^3*e + 6*B*a*c
*d^2*e^2 - 4*A*a*c*d*e^3 + B*a^2*e^4)*log(e*x + d)/e^6

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mupad [B]  time = 0.09, size = 311, normalized size = 1.73 \[ x^3\,\left (\frac {A\,c^2}{3\,e^2}-\frac {2\,B\,c^2\,d}{3\,e^3}\right )-x^2\,\left (\frac {d\,\left (\frac {A\,c^2}{e^2}-\frac {2\,B\,c^2\,d}{e^3}\right )}{e}-\frac {B\,a\,c}{e^2}+\frac {B\,c^2\,d^2}{2\,e^4}\right )+x\,\left (\frac {2\,d\,\left (\frac {2\,d\,\left (\frac {A\,c^2}{e^2}-\frac {2\,B\,c^2\,d}{e^3}\right )}{e}-\frac {2\,B\,a\,c}{e^2}+\frac {B\,c^2\,d^2}{e^4}\right )}{e}-\frac {d^2\,\left (\frac {A\,c^2}{e^2}-\frac {2\,B\,c^2\,d}{e^3}\right )}{e^2}+\frac {2\,A\,a\,c}{e^2}\right )+\frac {\ln \left (d+e\,x\right )\,\left (B\,a^2\,e^4+6\,B\,a\,c\,d^2\,e^2-4\,A\,a\,c\,d\,e^3+5\,B\,c^2\,d^4-4\,A\,c^2\,d^3\,e\right )}{e^6}-\frac {-B\,a^2\,d\,e^4+A\,a^2\,e^5-2\,B\,a\,c\,d^3\,e^2+2\,A\,a\,c\,d^2\,e^3-B\,c^2\,d^5+A\,c^2\,d^4\,e}{e\,\left (x\,e^6+d\,e^5\right )}+\frac {B\,c^2\,x^4}{4\,e^2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a + c*x^2)^2*(A + B*x))/(d + e*x)^2,x)

[Out]

x^3*((A*c^2)/(3*e^2) - (2*B*c^2*d)/(3*e^3)) - x^2*((d*((A*c^2)/e^2 - (2*B*c^2*d)/e^3))/e - (B*a*c)/e^2 + (B*c^
2*d^2)/(2*e^4)) + x*((2*d*((2*d*((A*c^2)/e^2 - (2*B*c^2*d)/e^3))/e - (2*B*a*c)/e^2 + (B*c^2*d^2)/e^4))/e - (d^
2*((A*c^2)/e^2 - (2*B*c^2*d)/e^3))/e^2 + (2*A*a*c)/e^2) + (log(d + e*x)*(B*a^2*e^4 + 5*B*c^2*d^4 - 4*A*c^2*d^3
*e - 4*A*a*c*d*e^3 + 6*B*a*c*d^2*e^2))/e^6 - (A*a^2*e^5 - B*c^2*d^5 - B*a^2*d*e^4 + A*c^2*d^4*e + 2*A*a*c*d^2*
e^3 - 2*B*a*c*d^3*e^2)/(e*(d*e^5 + e^6*x)) + (B*c^2*x^4)/(4*e^2)

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sympy [A]  time = 1.19, size = 246, normalized size = 1.37 \[ \frac {B c^{2} x^{4}}{4 e^{2}} + x^{3} \left (\frac {A c^{2}}{3 e^{2}} - \frac {2 B c^{2} d}{3 e^{3}}\right ) + x^{2} \left (- \frac {A c^{2} d}{e^{3}} + \frac {B a c}{e^{2}} + \frac {3 B c^{2} d^{2}}{2 e^{4}}\right ) + x \left (\frac {2 A a c}{e^{2}} + \frac {3 A c^{2} d^{2}}{e^{4}} - \frac {4 B a c d}{e^{3}} - \frac {4 B c^{2} d^{3}}{e^{5}}\right ) + \frac {- A a^{2} e^{5} - 2 A a c d^{2} e^{3} - A c^{2} d^{4} e + B a^{2} d e^{4} + 2 B a c d^{3} e^{2} + B c^{2} d^{5}}{d e^{6} + e^{7} x} + \frac {\left (a e^{2} + c d^{2}\right ) \left (- 4 A c d e + B a e^{2} + 5 B c d^{2}\right ) \log {\left (d + e x \right )}}{e^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x**2+a)**2/(e*x+d)**2,x)

[Out]

B*c**2*x**4/(4*e**2) + x**3*(A*c**2/(3*e**2) - 2*B*c**2*d/(3*e**3)) + x**2*(-A*c**2*d/e**3 + B*a*c/e**2 + 3*B*
c**2*d**2/(2*e**4)) + x*(2*A*a*c/e**2 + 3*A*c**2*d**2/e**4 - 4*B*a*c*d/e**3 - 4*B*c**2*d**3/e**5) + (-A*a**2*e
**5 - 2*A*a*c*d**2*e**3 - A*c**2*d**4*e + B*a**2*d*e**4 + 2*B*a*c*d**3*e**2 + B*c**2*d**5)/(d*e**6 + e**7*x) +
(a*e**2 + c*d**2)*(-4*A*c*d*e + B*a*e**2 + 5*B*c*d**2)*log(d + e*x)/e**6

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