3.1.19 \(\int \frac {(a+b x^2) (c+d x^2)^3}{e+f x^2} \, dx\) [19]

Optimal. Leaf size=227 \[ \frac {\left (7 a d f \left (15 d^2 e^2-40 c d e f+33 c^2 f^2\right )-b \left (105 d^3 e^3-280 c d^2 e^2 f+231 c^2 d e f^2-48 c^3 f^3\right )\right ) x}{105 f^4}-\frac {\left (7 a d f (5 d e-9 c f)-b \left (35 d^2 e^2-63 c d e f+24 c^2 f^2\right )\right ) x \left (c+d x^2\right )}{105 f^3}-\frac {(7 b d e-6 b c f-7 a d f) x \left (c+d x^2\right )^2}{35 f^2}+\frac {b x \left (c+d x^2\right )^3}{7 f}+\frac {(b e-a f) (d e-c f)^3 \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {e} f^{9/2}} \]

[Out]

1/105*(7*a*d*f*(33*c^2*f^2-40*c*d*e*f+15*d^2*e^2)-b*(-48*c^3*f^3+231*c^2*d*e*f^2-280*c*d^2*e^2*f+105*d^3*e^3))
*x/f^4-1/105*(7*a*d*f*(-9*c*f+5*d*e)-b*(24*c^2*f^2-63*c*d*e*f+35*d^2*e^2))*x*(d*x^2+c)/f^3-1/35*(-7*a*d*f-6*b*
c*f+7*b*d*e)*x*(d*x^2+c)^2/f^2+1/7*b*x*(d*x^2+c)^3/f+(-a*f+b*e)*(-c*f+d*e)^3*arctan(x*f^(1/2)/e^(1/2))/f^(9/2)
/e^(1/2)

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Rubi [A]
time = 0.25, antiderivative size = 227, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {542, 396, 211} \begin {gather*} \frac {(b e-a f) \text {ArcTan}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) (d e-c f)^3}{\sqrt {e} f^{9/2}}-\frac {x \left (c+d x^2\right ) \left (7 a d f (5 d e-9 c f)-b \left (24 c^2 f^2-63 c d e f+35 d^2 e^2\right )\right )}{105 f^3}+\frac {x \left (7 a d f \left (33 c^2 f^2-40 c d e f+15 d^2 e^2\right )-b \left (-48 c^3 f^3+231 c^2 d e f^2-280 c d^2 e^2 f+105 d^3 e^3\right )\right )}{105 f^4}-\frac {x \left (c+d x^2\right )^2 (-7 a d f-6 b c f+7 b d e)}{35 f^2}+\frac {b x \left (c+d x^2\right )^3}{7 f} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((a + b*x^2)*(c + d*x^2)^3)/(e + f*x^2),x]

[Out]

((7*a*d*f*(15*d^2*e^2 - 40*c*d*e*f + 33*c^2*f^2) - b*(105*d^3*e^3 - 280*c*d^2*e^2*f + 231*c^2*d*e*f^2 - 48*c^3
*f^3))*x)/(105*f^4) - ((7*a*d*f*(5*d*e - 9*c*f) - b*(35*d^2*e^2 - 63*c*d*e*f + 24*c^2*f^2))*x*(c + d*x^2))/(10
5*f^3) - ((7*b*d*e - 6*b*c*f - 7*a*d*f)*x*(c + d*x^2)^2)/(35*f^2) + (b*x*(c + d*x^2)^3)/(7*f) + ((b*e - a*f)*(
d*e - c*f)^3*ArcTan[(Sqrt[f]*x)/Sqrt[e]])/(Sqrt[e]*f^(9/2))

Rule 211

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/Rt[a/b, 2]], x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 396

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[d*x*((a + b*x^n)^(p + 1)/(b*(n*(
p + 1) + 1))), x] - Dist[(a*d - b*c*(n*(p + 1) + 1))/(b*(n*(p + 1) + 1)), Int[(a + b*x^n)^p, x], x] /; FreeQ[{
a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && NeQ[n*(p + 1) + 1, 0]

Rule 542

Int[((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> Simp[
f*x*(a + b*x^n)^(p + 1)*((c + d*x^n)^q/(b*(n*(p + q + 1) + 1))), x] + Dist[1/(b*(n*(p + q + 1) + 1)), Int[(a +
b*x^n)^p*(c + d*x^n)^(q - 1)*Simp[c*(b*e - a*f + b*e*n*(p + q + 1)) + (d*(b*e - a*f) + f*n*q*(b*c - a*d) + b*
d*e*n*(p + q + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && GtQ[q, 0] && NeQ[n*(p + q + 1) + 1
, 0]

Rubi steps

\begin {align*} \int \frac {\left (a+b x^2\right ) \left (c+d x^2\right )^3}{e+f x^2} \, dx &=\frac {b x \left (c+d x^2\right )^3}{7 f}+\frac {\int \frac {\left (c+d x^2\right )^2 \left (-c (b e-7 a f)+(-7 b d e+6 b c f+7 a d f) x^2\right )}{e+f x^2} \, dx}{7 f}\\ &=-\frac {(7 b d e-6 b c f-7 a d f) x \left (c+d x^2\right )^2}{35 f^2}+\frac {b x \left (c+d x^2\right )^3}{7 f}+\frac {\int \frac {\left (c+d x^2\right ) \left (c (b e (7 d e-11 c f)-7 a f (d e-5 c f))+\left (-7 a d f (5 d e-9 c f)+b \left (35 d^2 e^2-63 c d e f+24 c^2 f^2\right )\right ) x^2\right )}{e+f x^2} \, dx}{35 f^2}\\ &=-\frac {\left (7 a d f (5 d e-9 c f)-b \left (35 d^2 e^2-63 c d e f+24 c^2 f^2\right )\right ) x \left (c+d x^2\right )}{105 f^3}-\frac {(7 b d e-6 b c f-7 a d f) x \left (c+d x^2\right )^2}{35 f^2}+\frac {b x \left (c+d x^2\right )^3}{7 f}+\frac {\int \frac {c \left (7 a f \left (5 d^2 e^2-12 c d e f+15 c^2 f^2\right )-b e \left (35 d^2 e^2-84 c d e f+57 c^2 f^2\right )\right )+\left (7 a d f \left (15 d^2 e^2-40 c d e f+33 c^2 f^2\right )-b \left (105 d^3 e^3-280 c d^2 e^2 f+231 c^2 d e f^2-48 c^3 f^3\right )\right ) x^2}{e+f x^2} \, dx}{105 f^3}\\ &=\frac {\left (7 a d f \left (15 d^2 e^2-40 c d e f+33 c^2 f^2\right )-b \left (105 d^3 e^3-280 c d^2 e^2 f+231 c^2 d e f^2-48 c^3 f^3\right )\right ) x}{105 f^4}-\frac {\left (7 a d f (5 d e-9 c f)-b \left (35 d^2 e^2-63 c d e f+24 c^2 f^2\right )\right ) x \left (c+d x^2\right )}{105 f^3}-\frac {(7 b d e-6 b c f-7 a d f) x \left (c+d x^2\right )^2}{35 f^2}+\frac {b x \left (c+d x^2\right )^3}{7 f}+\frac {\left ((b e-a f) (d e-c f)^3\right ) \int \frac {1}{e+f x^2} \, dx}{f^4}\\ &=\frac {\left (7 a d f \left (15 d^2 e^2-40 c d e f+33 c^2 f^2\right )-b \left (105 d^3 e^3-280 c d^2 e^2 f+231 c^2 d e f^2-48 c^3 f^3\right )\right ) x}{105 f^4}-\frac {\left (7 a d f (5 d e-9 c f)-b \left (35 d^2 e^2-63 c d e f+24 c^2 f^2\right )\right ) x \left (c+d x^2\right )}{105 f^3}-\frac {(7 b d e-6 b c f-7 a d f) x \left (c+d x^2\right )^2}{35 f^2}+\frac {b x \left (c+d x^2\right )^3}{7 f}+\frac {(b e-a f) (d e-c f)^3 \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {e} f^{9/2}}\\ \end {align*}

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Mathematica [A]
time = 0.07, size = 179, normalized size = 0.79 \begin {gather*} \frac {\left (-b (d e-c f)^3+a d f \left (d^2 e^2-3 c d e f+3 c^2 f^2\right )\right ) x}{f^4}+\frac {d \left (a d f (-d e+3 c f)+b \left (d^2 e^2-3 c d e f+3 c^2 f^2\right )\right ) x^3}{3 f^3}+\frac {d^2 (-b d e+3 b c f+a d f) x^5}{5 f^2}+\frac {b d^3 x^7}{7 f}+\frac {(b e-a f) (d e-c f)^3 \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {e} f^{9/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x^2)*(c + d*x^2)^3)/(e + f*x^2),x]

[Out]

((-(b*(d*e - c*f)^3) + a*d*f*(d^2*e^2 - 3*c*d*e*f + 3*c^2*f^2))*x)/f^4 + (d*(a*d*f*(-(d*e) + 3*c*f) + b*(d^2*e
^2 - 3*c*d*e*f + 3*c^2*f^2))*x^3)/(3*f^3) + (d^2*(-(b*d*e) + 3*b*c*f + a*d*f)*x^5)/(5*f^2) + (b*d^3*x^7)/(7*f)
+ ((b*e - a*f)*(d*e - c*f)^3*ArcTan[(Sqrt[f]*x)/Sqrt[e]])/(Sqrt[e]*f^(9/2))

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Maple [A]
time = 0.14, size = 300, normalized size = 1.32

method	result	size
default	\(\frac {\frac {1}{7} b \,d^{3} x^{7} f^{3}+\frac {1}{5} a \,d^{3} f^{3} x^{5}+\frac {3}{5} b c \,d^{2} f^{3} x^{5}-\frac {1}{5} b \,d^{3} e \,f^{2} x^{5}+a c \,d^{2} f^{3} x^{3}-\frac {1}{3} a \,d^{3} e \,f^{2} x^{3}+b \,c^{2} d \,f^{3} x^{3}-b c \,d^{2} e \,f^{2} x^{3}+\frac {1}{3} b \,d^{3} e^{2} f \,x^{3}+3 a \,c^{2} d \,f^{3} x -3 a c \,d^{2} e \,f^{2} x +a \,d^{3} e^{2} f x +b \,c^{3} f^{3} x -3 b \,c^{2} d e \,f^{2} x +3 b c \,d^{2} e^{2} f x -b \,d^{3} e^{3} x}{f^{4}}+\frac {\left (a \,c^{3} f^{4}-3 a \,c^{2} d e \,f^{3}+3 a c \,d^{2} e^{2} f^{2}-a \,d^{3} e^{3} f -b \,c^{3} e \,f^{3}+3 b \,c^{2} d \,e^{2} f^{2}-3 b c \,d^{2} e^{3} f +b \,d^{3} e^{4}\right ) \arctan \left (\frac {f x}{\sqrt {f e}}\right )}{f^{4} \sqrt {f e}}\)	\(300\)
risch	\(\frac {\ln \left (f x +\sqrt {-f e}\right ) a \,d^{3} e^{3}}{2 f^{3} \sqrt {-f e}}+\frac {\ln \left (f x +\sqrt {-f e}\right ) b \,c^{3} e}{2 f \sqrt {-f e}}-\frac {\ln \left (f x +\sqrt {-f e}\right ) b \,d^{3} e^{4}}{2 f^{4} \sqrt {-f e}}-\frac {\ln \left (-f x +\sqrt {-f e}\right ) a \,d^{3} e^{3}}{2 f^{3} \sqrt {-f e}}-\frac {\ln \left (-f x +\sqrt {-f e}\right ) b \,c^{3} e}{2 f \sqrt {-f e}}+\frac {\ln \left (-f x +\sqrt {-f e}\right ) b \,d^{3} e^{4}}{2 f^{4} \sqrt {-f e}}+\frac {3 b c \,d^{2} x^{5}}{5 f}-\frac {b \,d^{3} e \,x^{5}}{5 f^{2}}-\frac {3 a c \,d^{2} e x}{f^{2}}-\frac {3 b \,c^{2} d e x}{f^{2}}+\frac {3 b c \,d^{2} e^{2} x}{f^{3}}+\frac {3 \ln \left (-f x +\sqrt {-f e}\right ) b \,c^{2} d \,e^{2}}{2 f^{2} \sqrt {-f e}}-\frac {3 \ln \left (-f x +\sqrt {-f e}\right ) b c \,d^{2} e^{3}}{2 f^{3} \sqrt {-f e}}+\frac {a c \,d^{2} x^{3}}{f}-\frac {a \,d^{3} e \,x^{3}}{3 f^{2}}+\frac {b \,c^{2} d \,x^{3}}{f}+\frac {b \,d^{3} e^{2} x^{3}}{3 f^{3}}+\frac {3 a \,c^{2} d x}{f}+\frac {a \,d^{3} e^{2} x}{f^{3}}-\frac {b \,d^{3} e^{3} x}{f^{4}}+\frac {b \,d^{3} x^{7}}{7 f}+\frac {a \,d^{3} x^{5}}{5 f}+\frac {b \,c^{3} x}{f}-\frac {\ln \left (f x +\sqrt {-f e}\right ) a \,c^{3}}{2 \sqrt {-f e}}+\frac {\ln \left (-f x +\sqrt {-f e}\right ) a \,c^{3}}{2 \sqrt {-f e}}-\frac {b c \,d^{2} e \,x^{3}}{f^{2}}+\frac {3 \ln \left (f x +\sqrt {-f e}\right ) a \,c^{2} d e}{2 f \sqrt {-f e}}-\frac {3 \ln \left (f x +\sqrt {-f e}\right ) a c \,d^{2} e^{2}}{2 f^{2} \sqrt {-f e}}-\frac {3 \ln \left (f x +\sqrt {-f e}\right ) b \,c^{2} d \,e^{2}}{2 f^{2} \sqrt {-f e}}+\frac {3 \ln \left (f x +\sqrt {-f e}\right ) b c \,d^{2} e^{3}}{2 f^{3} \sqrt {-f e}}-\frac {3 \ln \left (-f x +\sqrt {-f e}\right ) a \,c^{2} d e}{2 f \sqrt {-f e}}+\frac {3 \ln \left (-f x +\sqrt {-f e}\right ) a c \,d^{2} e^{2}}{2 f^{2} \sqrt {-f e}}\)	\(661\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)*(d*x^2+c)^3/(f*x^2+e),x,method=_RETURNVERBOSE)

[Out]

1/f^4*(1/7*b*d^3*x^7*f^3+1/5*a*d^3*f^3*x^5+3/5*b*c*d^2*f^3*x^5-1/5*b*d^3*e*f^2*x^5+a*c*d^2*f^3*x^3-1/3*a*d^3*e
*f^2*x^3+b*c^2*d*f^3*x^3-b*c*d^2*e*f^2*x^3+1/3*b*d^3*e^2*f*x^3+3*a*c^2*d*f^3*x-3*a*c*d^2*e*f^2*x+a*d^3*e^2*f*x
+b*c^3*f^3*x-3*b*c^2*d*e*f^2*x+3*b*c*d^2*e^2*f*x-b*d^3*e^3*x)+(a*c^3*f^4-3*a*c^2*d*e*f^3+3*a*c*d^2*e^2*f^2-a*d
^3*e^3*f-b*c^3*e*f^3+3*b*c^2*d*e^2*f^2-3*b*c*d^2*e^3*f+b*d^3*e^4)/f^4/(f*e)^(1/2)*arctan(f*x/(f*e)^(1/2))

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Maxima [A]
time = 0.50, size = 272, normalized size = 1.20 \begin {gather*} \frac {{\left (a c^{3} f^{4} + b d^{3} e^{4} - {\left (b c^{3} e + 3 \, a c^{2} d e\right )} f^{3} + 3 \, {\left (b c^{2} d e^{2} + a c d^{2} e^{2}\right )} f^{2} - {\left (3 \, b c d^{2} e^{3} + a d^{3} e^{3}\right )} f\right )} \arctan \left (\sqrt {f} x e^{\left (-\frac {1}{2}\right )}\right ) e^{\left (-\frac {1}{2}\right )}}{f^{\frac {9}{2}}} + \frac {15 \, b d^{3} f^{3} x^{7} - 21 \, {\left (b d^{3} f^{2} e - {\left (3 \, b c d^{2} + a d^{3}\right )} f^{3}\right )} x^{5} + 35 \, {\left (b d^{3} f e^{2} + 3 \, {\left (b c^{2} d + a c d^{2}\right )} f^{3} - {\left (3 \, b c d^{2} e + a d^{3} e\right )} f^{2}\right )} x^{3} - 105 \, {\left (b d^{3} e^{3} - {\left (b c^{3} + 3 \, a c^{2} d\right )} f^{3} + 3 \, {\left (b c^{2} d e + a c d^{2} e\right )} f^{2} - {\left (3 \, b c d^{2} e^{2} + a d^{3} e^{2}\right )} f\right )} x}{105 \, f^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)*(d*x^2+c)^3/(f*x^2+e),x, algorithm="maxima")

[Out]

(a*c^3*f^4 + b*d^3*e^4 - (b*c^3*e + 3*a*c^2*d*e)*f^3 + 3*(b*c^2*d*e^2 + a*c*d^2*e^2)*f^2 - (3*b*c*d^2*e^3 + a*
d^3*e^3)*f)*arctan(sqrt(f)*x*e^(-1/2))*e^(-1/2)/f^(9/2) + 1/105*(15*b*d^3*f^3*x^7 - 21*(b*d^3*f^2*e - (3*b*c*d
^2 + a*d^3)*f^3)*x^5 + 35*(b*d^3*f*e^2 + 3*(b*c^2*d + a*c*d^2)*f^3 - (3*b*c*d^2*e + a*d^3*e)*f^2)*x^3 - 105*(b
*d^3*e^3 - (b*c^3 + 3*a*c^2*d)*f^3 + 3*(b*c^2*d*e + a*c*d^2*e)*f^2 - (3*b*c*d^2*e^2 + a*d^3*e^2)*f)*x)/f^4

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Fricas [A]
time = 1.87, size = 581, normalized size = 2.56 \begin {gather*} \left [-\frac {{\left (210 \, b d^{3} f x e^{4} + 105 \, {\left (a c^{3} f^{4} + b d^{3} e^{4} - {\left (b c^{3} + 3 \, a c^{2} d\right )} f^{3} e + 3 \, {\left (b c^{2} d + a c d^{2}\right )} f^{2} e^{2} - {\left (3 \, b c d^{2} + a d^{3}\right )} f e^{3}\right )} \sqrt {-f e} \log \left (\frac {f x^{2} - 2 \, \sqrt {-f e} x - e}{f x^{2} + e}\right ) - 70 \, {\left (b d^{3} f^{2} x^{3} + 3 \, {\left (3 \, b c d^{2} + a d^{3}\right )} f^{2} x\right )} e^{3} + 14 \, {\left (3 \, b d^{3} f^{3} x^{5} + 5 \, {\left (3 \, b c d^{2} + a d^{3}\right )} f^{3} x^{3} + 45 \, {\left (b c^{2} d + a c d^{2}\right )} f^{3} x\right )} e^{2} - 6 \, {\left (5 \, b d^{3} f^{4} x^{7} + 7 \, {\left (3 \, b c d^{2} + a d^{3}\right )} f^{4} x^{5} + 35 \, {\left (b c^{2} d + a c d^{2}\right )} f^{4} x^{3} + 35 \, {\left (b c^{3} + 3 \, a c^{2} d\right )} f^{4} x\right )} e\right )} e^{\left (-1\right )}}{210 \, f^{5}}, -\frac {{\left (105 \, b d^{3} f x e^{4} - 105 \, {\left (a c^{3} f^{4} + b d^{3} e^{4} - {\left (b c^{3} + 3 \, a c^{2} d\right )} f^{3} e + 3 \, {\left (b c^{2} d + a c d^{2}\right )} f^{2} e^{2} - {\left (3 \, b c d^{2} + a d^{3}\right )} f e^{3}\right )} \sqrt {f} \arctan \left (\sqrt {f} x e^{\left (-\frac {1}{2}\right )}\right ) e^{\frac {1}{2}} - 35 \, {\left (b d^{3} f^{2} x^{3} + 3 \, {\left (3 \, b c d^{2} + a d^{3}\right )} f^{2} x\right )} e^{3} + 7 \, {\left (3 \, b d^{3} f^{3} x^{5} + 5 \, {\left (3 \, b c d^{2} + a d^{3}\right )} f^{3} x^{3} + 45 \, {\left (b c^{2} d + a c d^{2}\right )} f^{3} x\right )} e^{2} - 3 \, {\left (5 \, b d^{3} f^{4} x^{7} + 7 \, {\left (3 \, b c d^{2} + a d^{3}\right )} f^{4} x^{5} + 35 \, {\left (b c^{2} d + a c d^{2}\right )} f^{4} x^{3} + 35 \, {\left (b c^{3} + 3 \, a c^{2} d\right )} f^{4} x\right )} e\right )} e^{\left (-1\right )}}{105 \, f^{5}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)*(d*x^2+c)^3/(f*x^2+e),x, algorithm="fricas")

[Out]

[-1/210*(210*b*d^3*f*x*e^4 + 105*(a*c^3*f^4 + b*d^3*e^4 - (b*c^3 + 3*a*c^2*d)*f^3*e + 3*(b*c^2*d + a*c*d^2)*f^
2*e^2 - (3*b*c*d^2 + a*d^3)*f*e^3)*sqrt(-f*e)*log((f*x^2 - 2*sqrt(-f*e)*x - e)/(f*x^2 + e)) - 70*(b*d^3*f^2*x^
3 + 3*(3*b*c*d^2 + a*d^3)*f^2*x)*e^3 + 14*(3*b*d^3*f^3*x^5 + 5*(3*b*c*d^2 + a*d^3)*f^3*x^3 + 45*(b*c^2*d + a*c
*d^2)*f^3*x)*e^2 - 6*(5*b*d^3*f^4*x^7 + 7*(3*b*c*d^2 + a*d^3)*f^4*x^5 + 35*(b*c^2*d + a*c*d^2)*f^4*x^3 + 35*(b
*c^3 + 3*a*c^2*d)*f^4*x)*e)*e^(-1)/f^5, -1/105*(105*b*d^3*f*x*e^4 - 105*(a*c^3*f^4 + b*d^3*e^4 - (b*c^3 + 3*a*
c^2*d)*f^3*e + 3*(b*c^2*d + a*c*d^2)*f^2*e^2 - (3*b*c*d^2 + a*d^3)*f*e^3)*sqrt(f)*arctan(sqrt(f)*x*e^(-1/2))*e
^(1/2) - 35*(b*d^3*f^2*x^3 + 3*(3*b*c*d^2 + a*d^3)*f^2*x)*e^3 + 7*(3*b*d^3*f^3*x^5 + 5*(3*b*c*d^2 + a*d^3)*f^3
*x^3 + 45*(b*c^2*d + a*c*d^2)*f^3*x)*e^2 - 3*(5*b*d^3*f^4*x^7 + 7*(3*b*c*d^2 + a*d^3)*f^4*x^5 + 35*(b*c^2*d +
a*c*d^2)*f^4*x^3 + 35*(b*c^3 + 3*a*c^2*d)*f^4*x)*e)*e^(-1)/f^5]

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 508 vs. \(2 (228) = 456\).
time = 0.92, size = 508, normalized size = 2.24 \begin {gather*} \frac {b d^{3} x^{7}}{7 f} + x^{5} \left (\frac {a d^{3}}{5 f} + \frac {3 b c d^{2}}{5 f} - \frac {b d^{3} e}{5 f^{2}}\right ) + x^{3} \left (\frac {a c d^{2}}{f} - \frac {a d^{3} e}{3 f^{2}} + \frac {b c^{2} d}{f} - \frac {b c d^{2} e}{f^{2}} + \frac {b d^{3} e^{2}}{3 f^{3}}\right ) + x \left (\frac {3 a c^{2} d}{f} - \frac {3 a c d^{2} e}{f^{2}} + \frac {a d^{3} e^{2}}{f^{3}} + \frac {b c^{3}}{f} - \frac {3 b c^{2} d e}{f^{2}} + \frac {3 b c d^{2} e^{2}}{f^{3}} - \frac {b d^{3} e^{3}}{f^{4}}\right ) - \frac {\sqrt {- \frac {1}{e f^{9}}} \left (a f - b e\right ) \left (c f - d e\right )^{3} \log {\left (- \frac {e f^{4} \sqrt {- \frac {1}{e f^{9}}} \left (a f - b e\right ) \left (c f - d e\right )^{3}}{a c^{3} f^{4} - 3 a c^{2} d e f^{3} + 3 a c d^{2} e^{2} f^{2} - a d^{3} e^{3} f - b c^{3} e f^{3} + 3 b c^{2} d e^{2} f^{2} - 3 b c d^{2} e^{3} f + b d^{3} e^{4}} + x \right )}}{2} + \frac {\sqrt {- \frac {1}{e f^{9}}} \left (a f - b e\right ) \left (c f - d e\right )^{3} \log {\left (\frac {e f^{4} \sqrt {- \frac {1}{e f^{9}}} \left (a f - b e\right ) \left (c f - d e\right )^{3}}{a c^{3} f^{4} - 3 a c^{2} d e f^{3} + 3 a c d^{2} e^{2} f^{2} - a d^{3} e^{3} f - b c^{3} e f^{3} + 3 b c^{2} d e^{2} f^{2} - 3 b c d^{2} e^{3} f + b d^{3} e^{4}} + x \right )}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b*d**3*x**7/(7*f) + x**5*(a*d**3/(5*f) + 3*b*c*d**2/(5*f) - b*d**3*e/(5*f**2)) + x**3*(a*c*d**2/f - a*d**3*e/(
3*f**2) + b*c**2*d/f - b*c*d**2*e/f**2 + b*d**3*e**2/(3*f**3)) + x*(3*a*c**2*d/f - 3*a*c*d**2*e/f**2 + a*d**3*
e**2/f**3 + b*c**3/f - 3*b*c**2*d*e/f**2 + 3*b*c*d**2*e**2/f**3 - b*d**3*e**3/f**4) - sqrt(-1/(e*f**9))*(a*f -
b*e)*(c*f - d*e)**3*log(-e*f**4*sqrt(-1/(e*f**9))*(a*f - b*e)*(c*f - d*e)**3/(a*c**3*f**4 - 3*a*c**2*d*e*f**3
+ 3*a*c*d**2*e**2*f**2 - a*d**3*e**3*f - b*c**3*e*f**3 + 3*b*c**2*d*e**2*f**2 - 3*b*c*d**2*e**3*f + b*d**3*e*
*4) + x)/2 + sqrt(-1/(e*f**9))*(a*f - b*e)*(c*f - d*e)**3*log(e*f**4*sqrt(-1/(e*f**9))*(a*f - b*e)*(c*f - d*e)
**3/(a*c**3*f**4 - 3*a*c**2*d*e*f**3 + 3*a*c*d**2*e**2*f**2 - a*d**3*e**3*f - b*c**3*e*f**3 + 3*b*c**2*d*e**2*
f**2 - 3*b*c*d**2*e**3*f + b*d**3*e**4) + x)/2

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Giac [A]
time = 1.00, size = 307, normalized size = 1.35 \begin {gather*} \frac {{\left (a c^{3} f^{4} - b c^{3} f^{3} e - 3 \, a c^{2} d f^{3} e + 3 \, b c^{2} d f^{2} e^{2} + 3 \, a c d^{2} f^{2} e^{2} - 3 \, b c d^{2} f e^{3} - a d^{3} f e^{3} + b d^{3} e^{4}\right )} \arctan \left (\sqrt {f} x e^{\left (-\frac {1}{2}\right )}\right ) e^{\left (-\frac {1}{2}\right )}}{f^{\frac {9}{2}}} + \frac {15 \, b d^{3} f^{6} x^{7} + 63 \, b c d^{2} f^{6} x^{5} + 21 \, a d^{3} f^{6} x^{5} - 21 \, b d^{3} f^{5} x^{5} e + 105 \, b c^{2} d f^{6} x^{3} + 105 \, a c d^{2} f^{6} x^{3} - 105 \, b c d^{2} f^{5} x^{3} e - 35 \, a d^{3} f^{5} x^{3} e + 35 \, b d^{3} f^{4} x^{3} e^{2} + 105 \, b c^{3} f^{6} x + 315 \, a c^{2} d f^{6} x - 315 \, b c^{2} d f^{5} x e - 315 \, a c d^{2} f^{5} x e + 315 \, b c d^{2} f^{4} x e^{2} + 105 \, a d^{3} f^{4} x e^{2} - 105 \, b d^{3} f^{3} x e^{3}}{105 \, f^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)*(d*x^2+c)^3/(f*x^2+e),x, algorithm="giac")

[Out]

(a*c^3*f^4 - b*c^3*f^3*e - 3*a*c^2*d*f^3*e + 3*b*c^2*d*f^2*e^2 + 3*a*c*d^2*f^2*e^2 - 3*b*c*d^2*f*e^3 - a*d^3*f
*e^3 + b*d^3*e^4)*arctan(sqrt(f)*x*e^(-1/2))*e^(-1/2)/f^(9/2) + 1/105*(15*b*d^3*f^6*x^7 + 63*b*c*d^2*f^6*x^5 +
21*a*d^3*f^6*x^5 - 21*b*d^3*f^5*x^5*e + 105*b*c^2*d*f^6*x^3 + 105*a*c*d^2*f^6*x^3 - 105*b*c*d^2*f^5*x^3*e - 3
5*a*d^3*f^5*x^3*e + 35*b*d^3*f^4*x^3*e^2 + 105*b*c^3*f^6*x + 315*a*c^2*d*f^6*x - 315*b*c^2*d*f^5*x*e - 315*a*c
*d^2*f^5*x*e + 315*b*c*d^2*f^4*x*e^2 + 105*a*d^3*f^4*x*e^2 - 105*b*d^3*f^3*x*e^3)/f^7

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Mupad [B]
time = 0.85, size = 312, normalized size = 1.37 \begin {gather*} x\,\left (\frac {b\,c^3+3\,a\,d\,c^2}{f}+\frac {e\,\left (\frac {e\,\left (\frac {a\,d^3+3\,b\,c\,d^2}{f}-\frac {b\,d^3\,e}{f^2}\right )}{f}-\frac {3\,c\,d\,\left (a\,d+b\,c\right )}{f}\right )}{f}\right )+x^5\,\left (\frac {a\,d^3+3\,b\,c\,d^2}{5\,f}-\frac {b\,d^3\,e}{5\,f^2}\right )-x^3\,\left (\frac {e\,\left (\frac {a\,d^3+3\,b\,c\,d^2}{f}-\frac {b\,d^3\,e}{f^2}\right )}{3\,f}-\frac {c\,d\,\left (a\,d+b\,c\right )}{f}\right )+\frac {b\,d^3\,x^7}{7\,f}+\frac {\mathrm {atan}\left (\frac {\sqrt {f}\,x\,\left (a\,f-b\,e\right )\,{\left (c\,f-d\,e\right )}^3}{\sqrt {e}\,\left (-b\,c^3\,e\,f^3+a\,c^3\,f^4+3\,b\,c^2\,d\,e^2\,f^2-3\,a\,c^2\,d\,e\,f^3-3\,b\,c\,d^2\,e^3\,f+3\,a\,c\,d^2\,e^2\,f^2+b\,d^3\,e^4-a\,d^3\,e^3\,f\right )}\right )\,\left (a\,f-b\,e\right )\,{\left (c\,f-d\,e\right )}^3}{\sqrt {e}\,f^{9/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*((b*c^3 + 3*a*c^2*d)/f + (e*((e*((a*d^3 + 3*b*c*d^2)/f - (b*d^3*e)/f^2))/f - (3*c*d*(a*d + b*c))/f))/f) + x^
5*((a*d^3 + 3*b*c*d^2)/(5*f) - (b*d^3*e)/(5*f^2)) - x^3*((e*((a*d^3 + 3*b*c*d^2)/f - (b*d^3*e)/f^2))/(3*f) - (
c*d*(a*d + b*c))/f) + (b*d^3*x^7)/(7*f) + (atan((f^(1/2)*x*(a*f - b*e)*(c*f - d*e)^3)/(e^(1/2)*(a*c^3*f^4 + b*
d^3*e^4 - a*d^3*e^3*f - b*c^3*e*f^3 - 3*a*c^2*d*e*f^3 - 3*b*c*d^2*e^3*f + 3*a*c*d^2*e^2*f^2 + 3*b*c^2*d*e^2*f^
2)))*(a*f - b*e)*(c*f - d*e)^3)/(e^(1/2)*f^(9/2))

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