3.1346 \(\int \frac{(A+B x) (d+e x)^5}{\left (a+c x^2\right )^3} \, dx\)

Optimal. Leaf size=304 \[ -\frac{e^2 x \left (A c d \left (7 a e^2+3 c d^2\right )+5 a B e \left (c d^2-3 a e^2\right )\right )}{8 a^2 c^3}-\frac{(d+e x)^2 \left (2 a e \left (2 a A e^2+5 a B d e+A c d^2\right )-x \left (A c d \left (5 a e^2+3 c d^2\right )+5 a B e \left (c d^2-a e^2\right )\right )\right )}{8 a^2 c^2 \left (a+c x^2\right )}+\frac{\tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right ) \left (A c d \left (15 a^2 e^4+10 a c d^2 e^2+3 c^2 d^4\right )+5 a B e \left (-3 a^2 e^4+6 a c d^2 e^2+c^2 d^4\right )\right )}{8 a^{5/2} c^{7/2}}+\frac{e^4 \log \left (a+c x^2\right ) (A e+5 B d)}{2 c^3}-\frac{(d+e x)^4 (a (A e+B d)-x (A c d-a B e))}{4 a c \left (a+c x^2\right )^2} \]

[Out]

-(e^2*(5*a*B*e*(c*d^2 - 3*a*e^2) + A*c*d*(3*c*d^2 + 7*a*e^2))*x)/(8*a^2*c^3) - (
(d + e*x)^4*(a*(B*d + A*e) - (A*c*d - a*B*e)*x))/(4*a*c*(a + c*x^2)^2) - ((d + e
*x)^2*(2*a*e*(A*c*d^2 + 5*a*B*d*e + 2*a*A*e^2) - (5*a*B*e*(c*d^2 - a*e^2) + A*c*
d*(3*c*d^2 + 5*a*e^2))*x))/(8*a^2*c^2*(a + c*x^2)) + ((5*a*B*e*(c^2*d^4 + 6*a*c*
d^2*e^2 - 3*a^2*e^4) + A*c*d*(3*c^2*d^4 + 10*a*c*d^2*e^2 + 15*a^2*e^4))*ArcTan[(
Sqrt[c]*x)/Sqrt[a]])/(8*a^(5/2)*c^(7/2)) + (e^4*(5*B*d + A*e)*Log[a + c*x^2])/(2
*c^3)

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Rubi [A]  time = 0.955832, antiderivative size = 304, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ -\frac{e^2 x \left (A c d \left (7 a e^2+3 c d^2\right )+5 a B e \left (c d^2-3 a e^2\right )\right )}{8 a^2 c^3}-\frac{(d+e x)^2 \left (2 a e \left (2 a A e^2+5 a B d e+A c d^2\right )-x \left (A c d \left (5 a e^2+3 c d^2\right )+5 a B e \left (c d^2-a e^2\right )\right )\right )}{8 a^2 c^2 \left (a+c x^2\right )}+\frac{\tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right ) \left (A c d \left (15 a^2 e^4+10 a c d^2 e^2+3 c^2 d^4\right )+5 a B e \left (-3 a^2 e^4+6 a c d^2 e^2+c^2 d^4\right )\right )}{8 a^{5/2} c^{7/2}}+\frac{e^4 \log \left (a+c x^2\right ) (A e+5 B d)}{2 c^3}-\frac{(d+e x)^4 (a (A e+B d)-x (A c d-a B e))}{4 a c \left (a+c x^2\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*(d + e*x)^5)/(a + c*x^2)^3,x]

[Out]

-(e^2*(5*a*B*e*(c*d^2 - 3*a*e^2) + A*c*d*(3*c*d^2 + 7*a*e^2))*x)/(8*a^2*c^3) - (
(d + e*x)^4*(a*(B*d + A*e) - (A*c*d - a*B*e)*x))/(4*a*c*(a + c*x^2)^2) - ((d + e
*x)^2*(2*a*e*(A*c*d^2 + 5*a*B*d*e + 2*a*A*e^2) - (5*a*B*e*(c*d^2 - a*e^2) + A*c*
d*(3*c*d^2 + 5*a*e^2))*x))/(8*a^2*c^2*(a + c*x^2)) + ((5*a*B*e*(c^2*d^4 + 6*a*c*
d^2*e^2 - 3*a^2*e^4) + A*c*d*(3*c^2*d^4 + 10*a*c*d^2*e^2 + 15*a^2*e^4))*ArcTan[(
Sqrt[c]*x)/Sqrt[a]])/(8*a^(5/2)*c^(7/2)) + (e^4*(5*B*d + A*e)*Log[a + c*x^2])/(2
*c^3)

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(e*x+d)**5/(c*x**2+a)**3,x)

[Out]

Timed out

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Mathematica [A]  time = 0.53375, size = 341, normalized size = 1.12 \[ \frac{\frac{\tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right ) \left (A c d \left (15 a^2 e^4+10 a c d^2 e^2+3 c^2 d^4\right )+5 a B e \left (-3 a^2 e^4+6 a c d^2 e^2+c^2 d^4\right )\right )}{a^{5/2}}+\frac{2 \sqrt{c} \left (-a^3 e^4 (A e+5 B d+B e x)+5 a^2 c d e^2 (A e (2 d+e x)+2 B d (d+e x))-a c^2 d^3 (5 A e (d+2 e x)+B d (d+5 e x))+A c^3 d^5 x\right )}{a \left (a+c x^2\right )^2}+\frac{\sqrt{c} \left (a^3 e^4 (8 A e+40 B d+9 B e x)-5 a^2 c d e^2 (A e (8 d+5 e x)+2 B d (4 d+5 e x))+5 a c^2 d^3 e x (2 A e+B d)+3 A c^3 d^5 x\right )}{a^2 \left (a+c x^2\right )}+4 \sqrt{c} e^4 \log \left (a+c x^2\right ) (A e+5 B d)+8 B \sqrt{c} e^5 x}{8 c^{7/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*(d + e*x)^5)/(a + c*x^2)^3,x]

[Out]

(8*B*Sqrt[c]*e^5*x + (2*Sqrt[c]*(A*c^3*d^5*x - a^3*e^4*(5*B*d + A*e + B*e*x) + 5
*a^2*c*d*e^2*(2*B*d*(d + e*x) + A*e*(2*d + e*x)) - a*c^2*d^3*(5*A*e*(d + 2*e*x)
+ B*d*(d + 5*e*x))))/(a*(a + c*x^2)^2) + (Sqrt[c]*(3*A*c^3*d^5*x + 5*a*c^2*d^3*e
*(B*d + 2*A*e)*x + a^3*e^4*(40*B*d + 8*A*e + 9*B*e*x) - 5*a^2*c*d*e^2*(2*B*d*(4*
d + 5*e*x) + A*e*(8*d + 5*e*x))))/(a^2*(a + c*x^2)) + ((5*a*B*e*(c^2*d^4 + 6*a*c
*d^2*e^2 - 3*a^2*e^4) + A*c*d*(3*c^2*d^4 + 10*a*c*d^2*e^2 + 15*a^2*e^4))*ArcTan[
(Sqrt[c]*x)/Sqrt[a]])/a^(5/2) + 4*Sqrt[c]*e^4*(5*B*d + A*e)*Log[a + c*x^2])/(8*c
^(7/2))

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Maple [B]  time = 0.02, size = 686, normalized size = 2.3 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(e*x+d)^5/(c*x^2+a)^3,x)

[Out]

B*e^5*x/c^3-1/4/c/(c*x^2+a)^2*B*d^5+1/2/c^3*ln(a^2*(c*x^2+a))*A*e^5-5/4/c/(c*x^2
+a)^2*A*d^4*e+5/8/(c*x^2+a)^2/a*x*A*d^5+1/c^2/(c*x^2+a)^2*A*x^2*a*e^5-5/c/(c*x^2
+a)^2*A*x^2*d^2*e^3-5/c/(c*x^2+a)^2*B*x^2*d^3*e^2-5/4/c/(c*x^2+a)^2*x*A*d^3*e^2-
25/8/c/(c*x^2+a)^2*x^3*A*d*e^4+3/8*c/(c*x^2+a)^2/a^2*x^3*A*d^5+9/8/c^2/(c*x^2+a)
^2*a*x^3*B*e^5+7/8/c^3/(c*x^2+a)^2*a^2*x*B*e^5+15/4/c^3/(c*x^2+a)^2*B*d*a^2*e^4-
5/2/c^2/(c*x^2+a)^2*A*d^2*a*e^3-5/2/c^2/(c*x^2+a)^2*a*B*d^3*e^2+5/4/(c*x^2+a)^2/
a*x^3*A*d^3*e^2+5/8/(c*x^2+a)^2/a*x^3*B*d^4*e+15/8/c^2/(a*c)^(1/2)*arctan(c*x/(a
*c)^(1/2))*A*d*e^4+15/4/c^2/(a*c)^(1/2)*arctan(c*x/(a*c)^(1/2))*B*d^2*e^3-15/8/c
^3*a/(a*c)^(1/2)*arctan(c*x/(a*c)^(1/2))*B*e^5-5/8/c/(c*x^2+a)^2*x*B*d^4*e-25/4/
c/(c*x^2+a)^2*x^3*B*d^2*e^3+5/8/c/a/(a*c)^(1/2)*arctan(c*x/(a*c)^(1/2))*B*d^4*e+
5/4/c/a/(a*c)^(1/2)*arctan(c*x/(a*c)^(1/2))*A*d^3*e^2+3/4/c^3/(c*x^2+a)^2*A*a^2*
e^5+3/8/a^2/(a*c)^(1/2)*arctan(c*x/(a*c)^(1/2))*A*d^5+5/2/c^3*ln(a^2*(c*x^2+a))*
B*d*e^4-15/8/c^2/(c*x^2+a)^2*a*x*A*d*e^4-15/4/c^2/(c*x^2+a)^2*a*x*B*d^2*e^3+5/c^
2/(c*x^2+a)^2*B*x^2*a*d*e^4

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(e*x + d)^5/(c*x^2 + a)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.2849, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(e*x + d)^5/(c*x^2 + a)^3,x, algorithm="fricas")

[Out]

[-1/16*((3*A*a^2*c^3*d^5 + 5*B*a^3*c^2*d^4*e + 10*A*a^3*c^2*d^3*e^2 + 30*B*a^4*c
*d^2*e^3 + 15*A*a^4*c*d*e^4 - 15*B*a^5*e^5 + (3*A*c^5*d^5 + 5*B*a*c^4*d^4*e + 10
*A*a*c^4*d^3*e^2 + 30*B*a^2*c^3*d^2*e^3 + 15*A*a^2*c^3*d*e^4 - 15*B*a^3*c^2*e^5)
*x^4 + 2*(3*A*a*c^4*d^5 + 5*B*a^2*c^3*d^4*e + 10*A*a^2*c^3*d^3*e^2 + 30*B*a^3*c^
2*d^2*e^3 + 15*A*a^3*c^2*d*e^4 - 15*B*a^4*c*e^5)*x^2)*log(-(2*a*c*x - (c*x^2 - a
)*sqrt(-a*c))/(c*x^2 + a)) - 2*(8*B*a^2*c^2*e^5*x^5 - 2*B*a^2*c^2*d^5 - 10*A*a^2
*c^2*d^4*e - 20*B*a^3*c*d^3*e^2 - 20*A*a^3*c*d^2*e^3 + 30*B*a^4*d*e^4 + 6*A*a^4*
e^5 + (3*A*c^4*d^5 + 5*B*a*c^3*d^4*e + 10*A*a*c^3*d^3*e^2 - 50*B*a^2*c^2*d^2*e^3
- 25*A*a^2*c^2*d*e^4 + 25*B*a^3*c*e^5)*x^3 - 8*(5*B*a^2*c^2*d^3*e^2 + 5*A*a^2*c
^2*d^2*e^3 - 5*B*a^3*c*d*e^4 - A*a^3*c*e^5)*x^2 + 5*(A*a*c^3*d^5 - B*a^2*c^2*d^4
*e - 2*A*a^2*c^2*d^3*e^2 - 6*B*a^3*c*d^2*e^3 - 3*A*a^3*c*d*e^4 + 3*B*a^4*e^5)*x
+ 4*(5*B*a^4*d*e^4 + A*a^4*e^5 + (5*B*a^2*c^2*d*e^4 + A*a^2*c^2*e^5)*x^4 + 2*(5*
B*a^3*c*d*e^4 + A*a^3*c*e^5)*x^2)*log(c*x^2 + a))*sqrt(-a*c))/((a^2*c^5*x^4 + 2*
a^3*c^4*x^2 + a^4*c^3)*sqrt(-a*c)), 1/8*((3*A*a^2*c^3*d^5 + 5*B*a^3*c^2*d^4*e +
10*A*a^3*c^2*d^3*e^2 + 30*B*a^4*c*d^2*e^3 + 15*A*a^4*c*d*e^4 - 15*B*a^5*e^5 + (3
*A*c^5*d^5 + 5*B*a*c^4*d^4*e + 10*A*a*c^4*d^3*e^2 + 30*B*a^2*c^3*d^2*e^3 + 15*A*
a^2*c^3*d*e^4 - 15*B*a^3*c^2*e^5)*x^4 + 2*(3*A*a*c^4*d^5 + 5*B*a^2*c^3*d^4*e + 1
0*A*a^2*c^3*d^3*e^2 + 30*B*a^3*c^2*d^2*e^3 + 15*A*a^3*c^2*d*e^4 - 15*B*a^4*c*e^5
)*x^2)*arctan(sqrt(a*c)*x/a) + (8*B*a^2*c^2*e^5*x^5 - 2*B*a^2*c^2*d^5 - 10*A*a^2
*c^2*d^4*e - 20*B*a^3*c*d^3*e^2 - 20*A*a^3*c*d^2*e^3 + 30*B*a^4*d*e^4 + 6*A*a^4*
e^5 + (3*A*c^4*d^5 + 5*B*a*c^3*d^4*e + 10*A*a*c^3*d^3*e^2 - 50*B*a^2*c^2*d^2*e^3
- 25*A*a^2*c^2*d*e^4 + 25*B*a^3*c*e^5)*x^3 - 8*(5*B*a^2*c^2*d^3*e^2 + 5*A*a^2*c
^2*d^2*e^3 - 5*B*a^3*c*d*e^4 - A*a^3*c*e^5)*x^2 + 5*(A*a*c^3*d^5 - B*a^2*c^2*d^4
*e - 2*A*a^2*c^2*d^3*e^2 - 6*B*a^3*c*d^2*e^3 - 3*A*a^3*c*d*e^4 + 3*B*a^4*e^5)*x
+ 4*(5*B*a^4*d*e^4 + A*a^4*e^5 + (5*B*a^2*c^2*d*e^4 + A*a^2*c^2*e^5)*x^4 + 2*(5*
B*a^3*c*d*e^4 + A*a^3*c*e^5)*x^2)*log(c*x^2 + a))*sqrt(a*c))/((a^2*c^5*x^4 + 2*a
^3*c^4*x^2 + a^4*c^3)*sqrt(a*c))]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)*(e*x+d)**5/(c*x**2+a)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.297327, size = 544, normalized size = 1.79 \[ \frac{B x e^{5}}{c^{3}} + \frac{{\left (5 \, B d e^{4} + A e^{5}\right )}{\rm ln}\left (c x^{2} + a\right )}{2 \, c^{3}} + \frac{{\left (3 \, A c^{3} d^{5} + 5 \, B a c^{2} d^{4} e + 10 \, A a c^{2} d^{3} e^{2} + 30 \, B a^{2} c d^{2} e^{3} + 15 \, A a^{2} c d e^{4} - 15 \, B a^{3} e^{5}\right )} \arctan \left (\frac{c x}{\sqrt{a c}}\right )}{8 \, \sqrt{a c} a^{2} c^{3}} - \frac{2 \, B a^{2} c^{2} d^{5} + 10 \, A a^{2} c^{2} d^{4} e + 20 \, B a^{3} c d^{3} e^{2} + 20 \, A a^{3} c d^{2} e^{3} - 30 \, B a^{4} d e^{4} - 6 \, A a^{4} e^{5} -{\left (3 \, A c^{4} d^{5} + 5 \, B a c^{3} d^{4} e + 10 \, A a c^{3} d^{3} e^{2} - 50 \, B a^{2} c^{2} d^{2} e^{3} - 25 \, A a^{2} c^{2} d e^{4} + 9 \, B a^{3} c e^{5}\right )} x^{3} + 8 \,{\left (5 \, B a^{2} c^{2} d^{3} e^{2} + 5 \, A a^{2} c^{2} d^{2} e^{3} - 5 \, B a^{3} c d e^{4} - A a^{3} c e^{5}\right )} x^{2} -{\left (5 \, A a c^{3} d^{5} - 5 \, B a^{2} c^{2} d^{4} e - 10 \, A a^{2} c^{2} d^{3} e^{2} - 30 \, B a^{3} c d^{2} e^{3} - 15 \, A a^{3} c d e^{4} + 7 \, B a^{4} e^{5}\right )} x}{8 \,{\left (c x^{2} + a\right )}^{2} a^{2} c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(e*x + d)^5/(c*x^2 + a)^3,x, algorithm="giac")

[Out]

B*x*e^5/c^3 + 1/2*(5*B*d*e^4 + A*e^5)*ln(c*x^2 + a)/c^3 + 1/8*(3*A*c^3*d^5 + 5*B
*a*c^2*d^4*e + 10*A*a*c^2*d^3*e^2 + 30*B*a^2*c*d^2*e^3 + 15*A*a^2*c*d*e^4 - 15*B
*a^3*e^5)*arctan(c*x/sqrt(a*c))/(sqrt(a*c)*a^2*c^3) - 1/8*(2*B*a^2*c^2*d^5 + 10*
A*a^2*c^2*d^4*e + 20*B*a^3*c*d^3*e^2 + 20*A*a^3*c*d^2*e^3 - 30*B*a^4*d*e^4 - 6*A
*a^4*e^5 - (3*A*c^4*d^5 + 5*B*a*c^3*d^4*e + 10*A*a*c^3*d^3*e^2 - 50*B*a^2*c^2*d^
2*e^3 - 25*A*a^2*c^2*d*e^4 + 9*B*a^3*c*e^5)*x^3 + 8*(5*B*a^2*c^2*d^3*e^2 + 5*A*a
^2*c^2*d^2*e^3 - 5*B*a^3*c*d*e^4 - A*a^3*c*e^5)*x^2 - (5*A*a*c^3*d^5 - 5*B*a^2*c
^2*d^4*e - 10*A*a^2*c^2*d^3*e^2 - 30*B*a^3*c*d^2*e^3 - 15*A*a^3*c*d*e^4 + 7*B*a^
4*e^5)*x)/((c*x^2 + a)^2*a^2*c^3)