3.579 \(\int \frac {(d+e x)^3}{(a+c x^2)^{5/2}} \, dx\)

Optimal. Leaf size=79 \[ -\frac {2 \left (a e^2+c d^2\right ) (a e-c d x)}{3 a^2 c^2 \sqrt {a+c x^2}}-\frac {(d+e x)^2 (a e-c d x)}{3 a c \left (a+c x^2\right )^{3/2}} \]

[Out]

-1/3*(-c*d*x+a*e)*(e*x+d)^2/a/c/(c*x^2+a)^(3/2)-2/3*(a*e^2+c*d^2)*(-c*d*x+a*e)/a^2/c^2/(c*x^2+a)^(1/2)

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Rubi [A]  time = 0.03, antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {723, 637} \[ -\frac {2 \left (a e^2+c d^2\right ) (a e-c d x)}{3 a^2 c^2 \sqrt {a+c x^2}}-\frac {(d+e x)^2 (a e-c d x)}{3 a c \left (a+c x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^3/(a + c*x^2)^(5/2),x]

[Out]

-((a*e - c*d*x)*(d + e*x)^2)/(3*a*c*(a + c*x^2)^(3/2)) - (2*(c*d^2 + a*e^2)*(a*e - c*d*x))/(3*a^2*c^2*Sqrt[a +
c*x^2])

Rule 637

Int[((d_) + (e_.)*(x_))/((a_) + (c_.)*(x_)^2)^(3/2), x_Symbol] :> Simp[(-(a*e) + c*d*x)/(a*c*Sqrt[a + c*x^2]),
x] /; FreeQ[{a, c, d, e}, x]

Rule 723

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((d + e*x)^(m - 1)*(a*e - c*d*x)*(a
+ c*x^2)^(p + 1))/(2*a*c*(p + 1)), x] + Dist[((2*p + 3)*(c*d^2 + a*e^2))/(2*a*c*(p + 1)), Int[(d + e*x)^(m -
2)*(a + c*x^2)^(p + 1), x], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && EqQ[m + 2*p + 2, 0] && Lt
Q[p, -1]

Rubi steps

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

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Mathematica [A]  time = 0.10, size = 78, normalized size = 0.99 \[ \frac {-2 a^3 e^3-3 a^2 c e \left (d^2+e^2 x^2\right )+3 a c^2 d x \left (d^2+e^2 x^2\right )+2 c^3 d^3 x^3}{3 a^2 c^2 \left (a+c x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 1.19, size = 107, normalized size = 1.35 \[ -\frac {{\left (3 \, a^{2} c e^{3} x^{2} - 3 \, a c^{2} d^{3} x + 3 \, a^{2} c d^{2} e + 2 \, a^{3} e^{3} - {\left (2 \, c^{3} d^{3} + 3 \, a c^{2} d e^{2}\right )} x^{3}\right )} \sqrt {c x^{2} + a}}{3 \, {\left (a^{2} c^{4} x^{4} + 2 \, a^{3} c^{3} x^{2} + a^{4} c^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(3*a^2*c*e^3*x^2 - 3*a*c^2*d^3*x + 3*a^2*c*d^2*e + 2*a^3*e^3 - (2*c^3*d^3 + 3*a*c^2*d*e^2)*x^3)*sqrt(c*x^
2 + a)/(a^2*c^4*x^4 + 2*a^3*c^3*x^2 + a^4*c^2)

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giac [A]  time = 0.25, size = 88, normalized size = 1.11 \[ \frac {{\left (\frac {3 \, d^{3}}{a} - x {\left (\frac {3 \, e^{3}}{c} - \frac {{\left (2 \, c^{3} d^{3} + 3 \, a c^{2} d e^{2}\right )} x}{a^{2} c^{2}}\right )}\right )} x - \frac {3 \, a^{2} c d^{2} e + 2 \, a^{3} e^{3}}{a^{2} c^{2}}}{3 \, {\left (c x^{2} + a\right )}^{\frac {3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*((3*d^3/a - x*(3*e^3/c - (2*c^3*d^3 + 3*a*c^2*d*e^2)*x/(a^2*c^2)))*x - (3*a^2*c*d^2*e + 2*a^3*e^3)/(a^2*c^
2))/(c*x^2 + a)^(3/2)

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maple [A]  time = 0.05, size = 83, normalized size = 1.05 \[ -\frac {-3 a \,c^{2} d \,e^{2} x^{3}-2 c^{3} d^{3} x^{3}+3 e^{3} x^{2} a^{2} c -3 d^{3} x a \,c^{2}+2 e^{3} a^{3}+3 d^{2} e \,a^{2} c}{3 \left (c \,x^{2}+a \right )^{\frac {3}{2}} a^{2} c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^3/(c*x^2+a)^(5/2),x)

[Out]

-1/3*(-3*a*c^2*d*e^2*x^3-2*c^3*d^3*x^3+3*a^2*c*e^3*x^2-3*a*c^2*d^3*x+2*a^3*e^3+3*a^2*c*d^2*e)/(c*x^2+a)^(3/2)/
a^2/c^2

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maxima [A]  time = 1.36, size = 133, normalized size = 1.68 \[ -\frac {e^{3} x^{2}}{{\left (c x^{2} + a\right )}^{\frac {3}{2}} c} + \frac {2 \, d^{3} x}{3 \, \sqrt {c x^{2} + a} a^{2}} + \frac {d^{3} x}{3 \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} a} - \frac {d e^{2} x}{{\left (c x^{2} + a\right )}^{\frac {3}{2}} c} + \frac {d e^{2} x}{\sqrt {c x^{2} + a} a c} - \frac {d^{2} e}{{\left (c x^{2} + a\right )}^{\frac {3}{2}} c} - \frac {2 \, a e^{3}}{3 \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-e^3*x^2/((c*x^2 + a)^(3/2)*c) + 2/3*d^3*x/(sqrt(c*x^2 + a)*a^2) + 1/3*d^3*x/((c*x^2 + a)^(3/2)*a) - d*e^2*x/(
(c*x^2 + a)^(3/2)*c) + d*e^2*x/(sqrt(c*x^2 + a)*a*c) - d^2*e/((c*x^2 + a)^(3/2)*c) - 2/3*a*e^3/((c*x^2 + a)^(3
/2)*c^2)

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mupad [B]  time = 0.51, size = 82, normalized size = 1.04 \[ -\frac {2\,a^3\,e^3+3\,a^2\,c\,d^2\,e+3\,a^2\,c\,e^3\,x^2-3\,a\,c^2\,d^3\,x-3\,a\,c^2\,d\,e^2\,x^3-2\,c^3\,d^3\,x^3}{3\,a^2\,c^2\,{\left (c\,x^2+a\right )}^{3/2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)^3/(a + c*x^2)^(5/2),x)

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (d + e x\right )^{3}}{\left (a + c x^{2}\right )^{\frac {5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**3/(c*x**2+a)**(5/2),x)

[Out]

Integral((d + e*x)**3/(a + c*x**2)**(5/2), x)

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