3.4.71 \(\int \frac {x (a+b x^2)}{(-c+d x)^{3/2} (c+d x)^{3/2}} \, dx\) [371]

Optimal. Leaf size=76 \[ -\frac {\left (\frac {a}{c^2}+\frac {b}{d^2}\right ) x^2}{\sqrt {-c+d x} \sqrt {c+d x}}+\frac {\left (2 b c^2+a d^2\right ) \sqrt {-c+d x} \sqrt {c+d x}}{c^2 d^4} \]

[Out]

-(a/c^2+b/d^2)*x^2/(d*x-c)^(1/2)/(d*x+c)^(1/2)+(a*d^2+2*b*c^2)*(d*x-c)^(1/2)*(d*x+c)^(1/2)/c^2/d^4

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Rubi [A]
time = 0.04, antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {469, 75} \begin {gather*} \frac {\sqrt {d x-c} \sqrt {c+d x} \left (a d^2+2 b c^2\right )}{c^2 d^4}-\frac {x^2 \left (\frac {a}{c^2}+\frac {b}{d^2}\right )}{\sqrt {d x-c} \sqrt {c+d x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-(((a/c^2 + b/d^2)*x^2)/(Sqrt[-c + d*x]*Sqrt[c + d*x])) + ((2*b*c^2 + a*d^2)*Sqrt[-c + d*x]*Sqrt[c + d*x])/(c^
2*d^4)

Rule 75

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[b*(c + d*x)^
(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0] &
& EqQ[a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)), 0]

Rule 469

Int[((e_.)*(x_))^(m_.)*((a1_) + (b1_.)*(x_)^(non2_.))^(p_.)*((a2_) + (b2_.)*(x_)^(non2_.))^(p_.)*((c_) + (d_.)
*(x_)^(n_)), x_Symbol] :> Simp[(-(b1*b2*c - a1*a2*d))*(e*x)^(m + 1)*(a1 + b1*x^(n/2))^(p + 1)*((a2 + b2*x^(n/2
))^(p + 1)/(a1*a2*b1*b2*e*n*(p + 1))), x] - Dist[(a1*a2*d*(m + 1) - b1*b2*c*(m + n*(p + 1) + 1))/(a1*a2*b1*b2*
n*(p + 1)), Int[(e*x)^m*(a1 + b1*x^(n/2))^(p + 1)*(a2 + b2*x^(n/2))^(p + 1), x], x] /; FreeQ[{a1, b1, a2, b2,
c, d, e, m, n}, x] && EqQ[non2, n/2] && EqQ[a2*b1 + a1*b2, 0] && LtQ[p, -1] && (( !IntegerQ[p + 1/2] && NeQ[p,
-5/4]) ||  !RationalQ[m] || (IGtQ[n, 0] && ILtQ[p + 1/2, 0] && LeQ[-1, m, (-n)*(p + 1)]))

Rubi steps

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

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Mathematica [A]
time = 0.10, size = 45, normalized size = 0.59 \begin {gather*} \frac {-2 b c^2-a d^2+b d^2 x^2}{d^4 \sqrt {-c+d x} \sqrt {c+d x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*b*c^2 - a*d^2 + b*d^2*x^2)/(d^4*Sqrt[-c + d*x]*Sqrt[c + d*x])

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Maple [A]
time = 0.29, size = 50, normalized size = 0.66

method	result	size
gosper	\(-\frac {-b \,d^{2} x^{2}+a \,d^{2}+2 b \,c^{2}}{\sqrt {d x +c}\, d^{4} \sqrt {d x -c}}\)	\(43\)
default	\(\frac {\sqrt {d x -c}\, \left (-b \,d^{2} x^{2}+a \,d^{2}+2 b \,c^{2}\right )}{\sqrt {d x +c}\, d^{4} \left (-d x +c \right )}\)	\(50\)
risch	\(-\frac {b \left (-d x +c \right ) \sqrt {d x +c}}{d^{4} \sqrt {d x -c}}-\frac {\left (a \,d^{2}+b \,c^{2}\right ) \sqrt {\left (d x -c \right ) \left (d x +c \right )}}{d^{4} \sqrt {-\left (d x +c \right ) \left (-d x +c \right )}\, \sqrt {d x -c}\, \sqrt {d x +c}}\)	\(92\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(d*x-c)^(1/2)*(-b*d^2*x^2+a*d^2+2*b*c^2)/(d*x+c)^(1/2)/d^4/(-d*x+c)

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Maxima [A]
time = 0.28, size = 69, normalized size = 0.91 \begin {gather*} \frac {b x^{2}}{\sqrt {d^{2} x^{2} - c^{2}} d^{2}} - \frac {2 \, b c^{2}}{\sqrt {d^{2} x^{2} - c^{2}} d^{4}} - \frac {a}{\sqrt {d^{2} x^{2} - c^{2}} d^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b*x^2/(sqrt(d^2*x^2 - c^2)*d^2) - 2*b*c^2/(sqrt(d^2*x^2 - c^2)*d^4) - a/(sqrt(d^2*x^2 - c^2)*d^2)

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Fricas [A]
time = 3.00, size = 56, normalized size = 0.74 \begin {gather*} \frac {{\left (b d^{2} x^{2} - 2 \, b c^{2} - a d^{2}\right )} \sqrt {d x + c} \sqrt {d x - c}}{d^{6} x^{2} - c^{2} d^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b*d^2*x^2 - 2*b*c^2 - a*d^2)*sqrt(d*x + c)*sqrt(d*x - c)/(d^6*x^2 - c^2*d^4)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 152 vs. \(2 (68) = 136\).
time = 0.55, size = 152, normalized size = 2.00 \begin {gather*} \frac {\sqrt {d x + c} {\left (\frac {2 \, {\left (d x + c\right )} b}{d^{4}} - \frac {5 \, b c^{2} d^{8} + a d^{10}}{c d^{12}}\right )}}{2 \, \sqrt {d x - c}} + \frac {2 \, {\left (b^{2} c^{4} + 2 \, a b c^{2} d^{2} + a^{2} d^{4}\right )}}{{\left (b c^{2} {\left (\sqrt {d x + c} - \sqrt {d x - c}\right )}^{2} + a d^{2} {\left (\sqrt {d x + c} - \sqrt {d x - c}\right )}^{2} + 2 \, b c^{3} + 2 \, a c d^{2}\right )} d^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*sqrt(d*x + c)*(2*(d*x + c)*b/d^4 - (5*b*c^2*d^8 + a*d^10)/(c*d^12))/sqrt(d*x - c) + 2*(b^2*c^4 + 2*a*b*c^2
*d^2 + a^2*d^4)/((b*c^2*(sqrt(d*x + c) - sqrt(d*x - c))^2 + a*d^2*(sqrt(d*x + c) - sqrt(d*x - c))^2 + 2*b*c^3
+ 2*a*c*d^2)*d^4)

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Mupad [B]
time = 2.75, size = 67, normalized size = 0.88 \begin {gather*} \frac {a\,d^2\,\sqrt {d\,x-c}+2\,b\,c^2\,\sqrt {d\,x-c}-b\,d^2\,x^2\,\sqrt {d\,x-c}}{d^4\,\sqrt {c+d\,x}\,\left (c-d\,x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(a*d^2*(d*x - c)^(1/2) + 2*b*c^2*(d*x - c)^(1/2) - b*d^2*x^2*(d*x - c)^(1/2))/(d^4*(c + d*x)^(1/2)*(c - d*x))

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