3.116 \(\int \frac{A+B x+C x^2}{(a+c x^2)^{7/2}} \, dx\)

Optimal. Leaf size=97 \[ \frac{2 x (a C+4 A c)}{15 a^3 c \sqrt{a+c x^2}}+\frac{x (a C+4 A c)}{15 a^2 c \left (a+c x^2\right )^{3/2}}-\frac{a B-x (A c-a C)}{5 a c \left (a+c x^2\right )^{5/2}} \]

[Out]

-(a*B - (A*c - a*C)*x)/(5*a*c*(a + c*x^2)^(5/2)) + ((4*A*c + a*C)*x)/(15*a^2*c*(a + c*x^2)^(3/2)) + (2*(4*A*c
+ a*C)*x)/(15*a^3*c*Sqrt[a + c*x^2])

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Rubi [A]  time = 0.0573023, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {1814, 12, 192, 191} \[ \frac{2 x (a C+4 A c)}{15 a^3 c \sqrt{a+c x^2}}+\frac{x (a C+4 A c)}{15 a^2 c \left (a+c x^2\right )^{3/2}}-\frac{a B-x (A c-a C)}{5 a c \left (a+c x^2\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*x + C*x^2)/(a + c*x^2)^(7/2),x]

[Out]

-(a*B - (A*c - a*C)*x)/(5*a*c*(a + c*x^2)^(5/2)) + ((4*A*c + a*C)*x)/(15*a^2*c*(a + c*x^2)^(3/2)) + (2*(4*A*c
+ a*C)*x)/(15*a^3*c*Sqrt[a + c*x^2])

Rule 1814

Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq, a + b*x^2, x], f = Coeff[P
olynomialRemainder[Pq, a + b*x^2, x], x, 0], g = Coeff[PolynomialRemainder[Pq, a + b*x^2, x], x, 1]}, Simp[((a
*g - b*f*x)*(a + b*x^2)^(p + 1))/(2*a*b*(p + 1)), x] + Dist[1/(2*a*(p + 1)), Int[(a + b*x^2)^(p + 1)*ExpandToS
um[2*a*(p + 1)*Q + f*(2*p + 3), x], x], x]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && LtQ[p, -1]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 192

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

Rule 191

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

Rubi steps

\begin{align*} \int \frac{A+B x+C x^2}{\left (a+c x^2\right )^{7/2}} \, dx &=-\frac{a B-(A c-a C) x}{5 a c \left (a+c x^2\right )^{5/2}}-\frac{\int \frac{-4 A-\frac{a C}{c}}{\left (a+c x^2\right )^{5/2}} \, dx}{5 a}\\ &=-\frac{a B-(A c-a C) x}{5 a c \left (a+c x^2\right )^{5/2}}+\frac{(4 A c+a C) \int \frac{1}{\left (a+c x^2\right )^{5/2}} \, dx}{5 a c}\\ &=-\frac{a B-(A c-a C) x}{5 a c \left (a+c x^2\right )^{5/2}}+\frac{(4 A c+a C) x}{15 a^2 c \left (a+c x^2\right )^{3/2}}+\frac{(2 (4 A c+a C)) \int \frac{1}{\left (a+c x^2\right )^{3/2}} \, dx}{15 a^2 c}\\ &=-\frac{a B-(A c-a C) x}{5 a c \left (a+c x^2\right )^{5/2}}+\frac{(4 A c+a C) x}{15 a^2 c \left (a+c x^2\right )^{3/2}}+\frac{2 (4 A c+a C) x}{15 a^3 c \sqrt{a+c x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0517778, size = 71, normalized size = 0.73 \[ \frac{5 a^2 c x \left (3 A+C x^2\right )-3 a^3 B+2 a c^2 x^3 \left (10 A+C x^2\right )+8 A c^3 x^5}{15 a^3 c \left (a+c x^2\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x + C*x^2)/(a + c*x^2)^(7/2),x]

[Out]

(-3*a^3*B + 8*A*c^3*x^5 + 5*a^2*c*x*(3*A + C*x^2) + 2*a*c^2*x^3*(10*A + C*x^2))/(15*a^3*c*(a + c*x^2)^(5/2))

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Maple [A]  time = 0.048, size = 72, normalized size = 0.7 \begin{align*}{\frac{8\,A{c}^{3}{x}^{5}+2\,Ca{c}^{2}{x}^{5}+20\,Aa{c}^{2}{x}^{3}+5\,C{a}^{2}c{x}^{3}+15\,Ax{a}^{2}c-3\,B{a}^{3}}{15\,{a}^{3}c} \left ( c{x}^{2}+a \right ) ^{-{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((C*x^2+B*x+A)/(c*x^2+a)^(7/2),x)

[Out]

1/15*(8*A*c^3*x^5+2*C*a*c^2*x^5+20*A*a*c^2*x^3+5*C*a^2*c*x^3+15*A*a^2*c*x-3*B*a^3)/(c*x^2+a)^(5/2)/a^3/c

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Maxima [A]  time = 0.992292, size = 159, normalized size = 1.64 \begin{align*} \frac{8 \, A x}{15 \, \sqrt{c x^{2} + a} a^{3}} + \frac{4 \, A x}{15 \,{\left (c x^{2} + a\right )}^{\frac{3}{2}} a^{2}} + \frac{A x}{5 \,{\left (c x^{2} + a\right )}^{\frac{5}{2}} a} - \frac{C x}{5 \,{\left (c x^{2} + a\right )}^{\frac{5}{2}} c} + \frac{2 \, C x}{15 \, \sqrt{c x^{2} + a} a^{2} c} + \frac{C x}{15 \,{\left (c x^{2} + a\right )}^{\frac{3}{2}} a c} - \frac{B}{5 \,{\left (c x^{2} + a\right )}^{\frac{5}{2}} c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((C*x^2+B*x+A)/(c*x^2+a)^(7/2),x, algorithm="maxima")

[Out]

8/15*A*x/(sqrt(c*x^2 + a)*a^3) + 4/15*A*x/((c*x^2 + a)^(3/2)*a^2) + 1/5*A*x/((c*x^2 + a)^(5/2)*a) - 1/5*C*x/((
c*x^2 + a)^(5/2)*c) + 2/15*C*x/(sqrt(c*x^2 + a)*a^2*c) + 1/15*C*x/((c*x^2 + a)^(3/2)*a*c) - 1/5*B/((c*x^2 + a)
^(5/2)*c)

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Fricas [A]  time = 1.62208, size = 215, normalized size = 2.22 \begin{align*} \frac{{\left (2 \,{\left (C a c^{2} + 4 \, A c^{3}\right )} x^{5} + 15 \, A a^{2} c x - 3 \, B a^{3} + 5 \,{\left (C a^{2} c + 4 \, A a c^{2}\right )} x^{3}\right )} \sqrt{c x^{2} + a}}{15 \,{\left (a^{3} c^{4} x^{6} + 3 \, a^{4} c^{3} x^{4} + 3 \, a^{5} c^{2} x^{2} + a^{6} c\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((C*x^2+B*x+A)/(c*x^2+a)^(7/2),x, algorithm="fricas")

[Out]

1/15*(2*(C*a*c^2 + 4*A*c^3)*x^5 + 15*A*a^2*c*x - 3*B*a^3 + 5*(C*a^2*c + 4*A*a*c^2)*x^3)*sqrt(c*x^2 + a)/(a^3*c
^4*x^6 + 3*a^4*c^3*x^4 + 3*a^5*c^2*x^2 + a^6*c)

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Sympy [B]  time = 39.5587, size = 638, normalized size = 6.58 \begin{align*} A \left (\frac{15 a^{5} x}{15 a^{\frac{17}{2}} \sqrt{1 + \frac{c x^{2}}{a}} + 45 a^{\frac{15}{2}} c x^{2} \sqrt{1 + \frac{c x^{2}}{a}} + 45 a^{\frac{13}{2}} c^{2} x^{4} \sqrt{1 + \frac{c x^{2}}{a}} + 15 a^{\frac{11}{2}} c^{3} x^{6} \sqrt{1 + \frac{c x^{2}}{a}}} + \frac{35 a^{4} c x^{3}}{15 a^{\frac{17}{2}} \sqrt{1 + \frac{c x^{2}}{a}} + 45 a^{\frac{15}{2}} c x^{2} \sqrt{1 + \frac{c x^{2}}{a}} + 45 a^{\frac{13}{2}} c^{2} x^{4} \sqrt{1 + \frac{c x^{2}}{a}} + 15 a^{\frac{11}{2}} c^{3} x^{6} \sqrt{1 + \frac{c x^{2}}{a}}} + \frac{28 a^{3} c^{2} x^{5}}{15 a^{\frac{17}{2}} \sqrt{1 + \frac{c x^{2}}{a}} + 45 a^{\frac{15}{2}} c x^{2} \sqrt{1 + \frac{c x^{2}}{a}} + 45 a^{\frac{13}{2}} c^{2} x^{4} \sqrt{1 + \frac{c x^{2}}{a}} + 15 a^{\frac{11}{2}} c^{3} x^{6} \sqrt{1 + \frac{c x^{2}}{a}}} + \frac{8 a^{2} c^{3} x^{7}}{15 a^{\frac{17}{2}} \sqrt{1 + \frac{c x^{2}}{a}} + 45 a^{\frac{15}{2}} c x^{2} \sqrt{1 + \frac{c x^{2}}{a}} + 45 a^{\frac{13}{2}} c^{2} x^{4} \sqrt{1 + \frac{c x^{2}}{a}} + 15 a^{\frac{11}{2}} c^{3} x^{6} \sqrt{1 + \frac{c x^{2}}{a}}}\right ) + B \left (\begin{cases} - \frac{1}{5 a^{2} c \sqrt{a + c x^{2}} + 10 a c^{2} x^{2} \sqrt{a + c x^{2}} + 5 c^{3} x^{4} \sqrt{a + c x^{2}}} & \text{for}\: c \neq 0 \\\frac{x^{2}}{2 a^{\frac{7}{2}}} & \text{otherwise} \end{cases}\right ) + C \left (\frac{5 a x^{3}}{15 a^{\frac{9}{2}} \sqrt{1 + \frac{c x^{2}}{a}} + 30 a^{\frac{7}{2}} c x^{2} \sqrt{1 + \frac{c x^{2}}{a}} + 15 a^{\frac{5}{2}} c^{2} x^{4} \sqrt{1 + \frac{c x^{2}}{a}}} + \frac{2 c x^{5}}{15 a^{\frac{9}{2}} \sqrt{1 + \frac{c x^{2}}{a}} + 30 a^{\frac{7}{2}} c x^{2} \sqrt{1 + \frac{c x^{2}}{a}} + 15 a^{\frac{5}{2}} c^{2} x^{4} \sqrt{1 + \frac{c x^{2}}{a}}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((C*x**2+B*x+A)/(c*x**2+a)**(7/2),x)

[Out]

A*(15*a**5*x/(15*a**(17/2)*sqrt(1 + c*x**2/a) + 45*a**(15/2)*c*x**2*sqrt(1 + c*x**2/a) + 45*a**(13/2)*c**2*x**
4*sqrt(1 + c*x**2/a) + 15*a**(11/2)*c**3*x**6*sqrt(1 + c*x**2/a)) + 35*a**4*c*x**3/(15*a**(17/2)*sqrt(1 + c*x*
*2/a) + 45*a**(15/2)*c*x**2*sqrt(1 + c*x**2/a) + 45*a**(13/2)*c**2*x**4*sqrt(1 + c*x**2/a) + 15*a**(11/2)*c**3
*x**6*sqrt(1 + c*x**2/a)) + 28*a**3*c**2*x**5/(15*a**(17/2)*sqrt(1 + c*x**2/a) + 45*a**(15/2)*c*x**2*sqrt(1 +
c*x**2/a) + 45*a**(13/2)*c**2*x**4*sqrt(1 + c*x**2/a) + 15*a**(11/2)*c**3*x**6*sqrt(1 + c*x**2/a)) + 8*a**2*c*
*3*x**7/(15*a**(17/2)*sqrt(1 + c*x**2/a) + 45*a**(15/2)*c*x**2*sqrt(1 + c*x**2/a) + 45*a**(13/2)*c**2*x**4*sqr
t(1 + c*x**2/a) + 15*a**(11/2)*c**3*x**6*sqrt(1 + c*x**2/a))) + B*Piecewise((-1/(5*a**2*c*sqrt(a + c*x**2) + 1
0*a*c**2*x**2*sqrt(a + c*x**2) + 5*c**3*x**4*sqrt(a + c*x**2)), Ne(c, 0)), (x**2/(2*a**(7/2)), True)) + C*(5*a
*x**3/(15*a**(9/2)*sqrt(1 + c*x**2/a) + 30*a**(7/2)*c*x**2*sqrt(1 + c*x**2/a) + 15*a**(5/2)*c**2*x**4*sqrt(1 +
c*x**2/a)) + 2*c*x**5/(15*a**(9/2)*sqrt(1 + c*x**2/a) + 30*a**(7/2)*c*x**2*sqrt(1 + c*x**2/a) + 15*a**(5/2)*c
**2*x**4*sqrt(1 + c*x**2/a)))

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Giac [A]  time = 1.20136, size = 108, normalized size = 1.11 \begin{align*} \frac{{\left (x^{2}{\left (\frac{2 \,{\left (C a c^{3} + 4 \, A c^{4}\right )} x^{2}}{a^{3} c^{2}} + \frac{5 \,{\left (C a^{2} c^{2} + 4 \, A a c^{3}\right )}}{a^{3} c^{2}}\right )} + \frac{15 \, A}{a}\right )} x - \frac{3 \, B}{c}}{15 \,{\left (c x^{2} + a\right )}^{\frac{5}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((C*x^2+B*x+A)/(c*x^2+a)^(7/2),x, algorithm="giac")

[Out]

1/15*((x^2*(2*(C*a*c^3 + 4*A*c^4)*x^2/(a^3*c^2) + 5*(C*a^2*c^2 + 4*A*a*c^3)/(a^3*c^2)) + 15*A/a)*x - 3*B/c)/(c
*x^2 + a)^(5/2)