3.14.31 \(\int \frac {(a+b x)^2}{(c+d x)^{5/2}} \, dx\)

Optimal. Leaf size=67 \[ \frac {4 b (b c-a d)}{d^3 \sqrt {c+d x}}-\frac {2 (b c-a d)^2}{3 d^3 (c+d x)^{3/2}}+\frac {2 b^2 \sqrt {c+d x}}{d^3} \]

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Rubi [A]  time = 0.02, antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {43} \begin {gather*} \frac {4 b (b c-a d)}{d^3 \sqrt {c+d x}}-\frac {2 (b c-a d)^2}{3 d^3 (c+d x)^{3/2}}+\frac {2 b^2 \sqrt {c+d x}}{d^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^2/(c + d*x)^(5/2),x]

[Out]

(-2*(b*c - a*d)^2)/(3*d^3*(c + d*x)^(3/2)) + (4*b*(b*c - a*d))/(d^3*Sqrt[c + d*x]) + (2*b^2*Sqrt[c + d*x])/d^3

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin {align*} \int \frac {(a+b x)^2}{(c+d x)^{5/2}} \, dx &=\int \left (\frac {(-b c+a d)^2}{d^2 (c+d x)^{5/2}}-\frac {2 b (b c-a d)}{d^2 (c+d x)^{3/2}}+\frac {b^2}{d^2 \sqrt {c+d x}}\right ) \, dx\\ &=-\frac {2 (b c-a d)^2}{3 d^3 (c+d x)^{3/2}}+\frac {4 b (b c-a d)}{d^3 \sqrt {c+d x}}+\frac {2 b^2 \sqrt {c+d x}}{d^3}\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 62, normalized size = 0.93 \begin {gather*} \frac {-2 a^2 d^2-4 a b d (2 c+3 d x)+2 b^2 \left (8 c^2+12 c d x+3 d^2 x^2\right )}{3 d^3 (c+d x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.05, size = 72, normalized size = 1.07 \begin {gather*} \frac {2 \left (-a^2 d^2-6 a b d (c+d x)+2 a b c d+b^2 \left (-c^2\right )+3 b^2 (c+d x)^2+6 b^2 c (c+d x)\right )}{3 d^3 (c+d x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(a + b*x)^2/(c + d*x)^(5/2),x]

[Out]

(2*(-(b^2*c^2) + 2*a*b*c*d - a^2*d^2 + 6*b^2*c*(c + d*x) - 6*a*b*d*(c + d*x) + 3*b^2*(c + d*x)^2))/(3*d^3*(c +
d*x)^(3/2))

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fricas [A]  time = 1.24, size = 85, normalized size = 1.27 \begin {gather*} \frac {2 \, {\left (3 \, b^{2} d^{2} x^{2} + 8 \, b^{2} c^{2} - 4 \, a b c d - a^{2} d^{2} + 6 \, {\left (2 \, b^{2} c d - a b d^{2}\right )} x\right )} \sqrt {d x + c}}{3 \, {\left (d^{5} x^{2} + 2 \, c d^{4} x + c^{2} d^{3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*(3*b^2*d^2*x^2 + 8*b^2*c^2 - 4*a*b*c*d - a^2*d^2 + 6*(2*b^2*c*d - a*b*d^2)*x)*sqrt(d*x + c)/(d^5*x^2 + 2*c
*d^4*x + c^2*d^3)

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giac [A]  time = 1.11, size = 72, normalized size = 1.07 \begin {gather*} \frac {2 \, \sqrt {d x + c} b^{2}}{d^{3}} + \frac {2 \, {\left (6 \, {\left (d x + c\right )} b^{2} c - b^{2} c^{2} - 6 \, {\left (d x + c\right )} a b d + 2 \, a b c d - a^{2} d^{2}\right )}}{3 \, {\left (d x + c\right )}^{\frac {3}{2}} d^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(d*x + c)*b^2/d^3 + 2/3*(6*(d*x + c)*b^2*c - b^2*c^2 - 6*(d*x + c)*a*b*d + 2*a*b*c*d - a^2*d^2)/((d*x +
c)^(3/2)*d^3)

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maple [A]  time = 0.01, size = 62, normalized size = 0.93 \begin {gather*} -\frac {2 \left (-3 b^{2} x^{2} d^{2}+6 a b \,d^{2} x -12 b^{2} c d x +a^{2} d^{2}+4 a b c d -8 b^{2} c^{2}\right )}{3 \left (d x +c \right )^{\frac {3}{2}} d^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3/(d*x+c)^(3/2)*(-3*b^2*d^2*x^2+6*a*b*d^2*x-12*b^2*c*d*x+a^2*d^2+4*a*b*c*d-8*b^2*c^2)/d^3

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maxima [A]  time = 1.39, size = 72, normalized size = 1.07 \begin {gather*} \frac {2 \, {\left (\frac {3 \, \sqrt {d x + c} b^{2}}{d^{2}} - \frac {b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2} - 6 \, {\left (b^{2} c - a b d\right )} {\left (d x + c\right )}}{{\left (d x + c\right )}^{\frac {3}{2}} d^{2}}\right )}}{3 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*(3*sqrt(d*x + c)*b^2/d^2 - (b^2*c^2 - 2*a*b*c*d + a^2*d^2 - 6*(b^2*c - a*b*d)*(d*x + c))/((d*x + c)^(3/2)*
d^2))/d

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mupad [B]  time = 0.07, size = 68, normalized size = 1.01 \begin {gather*} \frac {6\,b^2\,{\left (c+d\,x\right )}^2-2\,a^2\,d^2-2\,b^2\,c^2+12\,b^2\,c\,\left (c+d\,x\right )-12\,a\,b\,d\,\left (c+d\,x\right )+4\,a\,b\,c\,d}{3\,d^3\,{\left (c+d\,x\right )}^{3/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)^2/(c + d*x)^(5/2),x)

[Out]

(6*b^2*(c + d*x)^2 - 2*a^2*d^2 - 2*b^2*c^2 + 12*b^2*c*(c + d*x) - 12*a*b*d*(c + d*x) + 4*a*b*c*d)/(3*d^3*(c +
d*x)^(3/2))

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sympy [A]  time = 1.27, size = 265, normalized size = 3.96 \begin {gather*} \begin {cases} - \frac {2 a^{2} d^{2}}{3 c d^{3} \sqrt {c + d x} + 3 d^{4} x \sqrt {c + d x}} - \frac {8 a b c d}{3 c d^{3} \sqrt {c + d x} + 3 d^{4} x \sqrt {c + d x}} - \frac {12 a b d^{2} x}{3 c d^{3} \sqrt {c + d x} + 3 d^{4} x \sqrt {c + d x}} + \frac {16 b^{2} c^{2}}{3 c d^{3} \sqrt {c + d x} + 3 d^{4} x \sqrt {c + d x}} + \frac {24 b^{2} c d x}{3 c d^{3} \sqrt {c + d x} + 3 d^{4} x \sqrt {c + d x}} + \frac {6 b^{2} d^{2} x^{2}}{3 c d^{3} \sqrt {c + d x} + 3 d^{4} x \sqrt {c + d x}} & \text {for}\: d \neq 0 \\\frac {a^{2} x + a b x^{2} + \frac {b^{2} x^{3}}{3}}{c^{\frac {5}{2}}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-2*a**2*d**2/(3*c*d**3*sqrt(c + d*x) + 3*d**4*x*sqrt(c + d*x)) - 8*a*b*c*d/(3*c*d**3*sqrt(c + d*x)
+ 3*d**4*x*sqrt(c + d*x)) - 12*a*b*d**2*x/(3*c*d**3*sqrt(c + d*x) + 3*d**4*x*sqrt(c + d*x)) + 16*b**2*c**2/(3*
c*d**3*sqrt(c + d*x) + 3*d**4*x*sqrt(c + d*x)) + 24*b**2*c*d*x/(3*c*d**3*sqrt(c + d*x) + 3*d**4*x*sqrt(c + d*x
)) + 6*b**2*d**2*x**2/(3*c*d**3*sqrt(c + d*x) + 3*d**4*x*sqrt(c + d*x)), Ne(d, 0)), ((a**2*x + a*b*x**2 + b**2
*x**3/3)/c**(5/2), True))

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