3.1436 \(\int \frac{(a+b x)^3}{(c+d x)^{5/2}} \, dx\)

Optimal. Leaf size=96 \[ -\frac{6 b^2 \sqrt{c+d x} (b c-a d)}{d^4}-\frac{6 b (b c-a d)^2}{d^4 \sqrt{c+d x}}+\frac{2 (b c-a d)^3}{3 d^4 (c+d x)^{3/2}}+\frac{2 b^3 (c+d x)^{3/2}}{3 d^4} \]

[Out]

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

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Rubi [A]  time = 0.0309867, antiderivative size = 96, normalized size of antiderivative = 1., 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} \[ -\frac{6 b^2 \sqrt{c+d x} (b c-a d)}{d^4}-\frac{6 b (b c-a d)^2}{d^4 \sqrt{c+d x}}+\frac{2 (b c-a d)^3}{3 d^4 (c+d x)^{3/2}}+\frac{2 b^3 (c+d x)^{3/2}}{3 d^4} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^3/(c + d*x)^(5/2),x]

[Out]

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

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)^3}{(c+d x)^{5/2}} \, dx &=\int \left (\frac{(-b c+a d)^3}{d^3 (c+d x)^{5/2}}+\frac{3 b (b c-a d)^2}{d^3 (c+d x)^{3/2}}-\frac{3 b^2 (b c-a d)}{d^3 \sqrt{c+d x}}+\frac{b^3 \sqrt{c+d x}}{d^3}\right ) \, dx\\ &=\frac{2 (b c-a d)^3}{3 d^4 (c+d x)^{3/2}}-\frac{6 b (b c-a d)^2}{d^4 \sqrt{c+d x}}-\frac{6 b^2 (b c-a d) \sqrt{c+d x}}{d^4}+\frac{2 b^3 (c+d x)^{3/2}}{3 d^4}\\ \end{align*}

Mathematica [A]  time = 0.0555299, size = 76, normalized size = 0.79 \[ \frac{2 \left (-9 b^2 (c+d x)^2 (b c-a d)-9 b (c+d x) (b c-a d)^2+(b c-a d)^3+b^3 (c+d x)^3\right )}{3 d^4 (c+d x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*((b*c - a*d)^3 - 9*b*(b*c - a*d)^2*(c + d*x) - 9*b^2*(b*c - a*d)*(c + d*x)^2 + b^3*(c + d*x)^3))/(3*d^4*(c
+ d*x)^(3/2))

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Maple [A]  time = 0.005, size = 115, normalized size = 1.2 \begin{align*} -{\frac{-2\,{b}^{3}{x}^{3}{d}^{3}-18\,a{b}^{2}{d}^{3}{x}^{2}+12\,{b}^{3}c{d}^{2}{x}^{2}+18\,{a}^{2}b{d}^{3}x-72\,a{b}^{2}c{d}^{2}x+48\,{b}^{3}{c}^{2}dx+2\,{a}^{3}{d}^{3}+12\,{a}^{2}bc{d}^{2}-48\,a{b}^{2}{c}^{2}d+32\,{b}^{3}{c}^{3}}{3\,{d}^{4}} \left ( dx+c \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3/(d*x+c)^(3/2)*(-b^3*d^3*x^3-9*a*b^2*d^3*x^2+6*b^3*c*d^2*x^2+9*a^2*b*d^3*x-36*a*b^2*c*d^2*x+24*b^3*c^2*d*x
+a^3*d^3+6*a^2*b*c*d^2-24*a*b^2*c^2*d+16*b^3*c^3)/d^4

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Maxima [A]  time = 0.961498, size = 165, normalized size = 1.72 \begin{align*} \frac{2 \,{\left (\frac{{\left (d x + c\right )}^{\frac{3}{2}} b^{3} - 9 \,{\left (b^{3} c - a b^{2} d\right )} \sqrt{d x + c}}{d^{3}} + \frac{b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3} - 9 \,{\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )}{\left (d x + c\right )}}{{\left (d x + c\right )}^{\frac{3}{2}} d^{3}}\right )}}{3 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*(((d*x + c)^(3/2)*b^3 - 9*(b^3*c - a*b^2*d)*sqrt(d*x + c))/d^3 + (b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2
- a^3*d^3 - 9*(b^3*c^2 - 2*a*b^2*c*d + a^2*b*d^2)*(d*x + c))/((d*x + c)^(3/2)*d^3))/d

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Fricas [A]  time = 1.98774, size = 281, normalized size = 2.93 \begin{align*} \frac{2 \,{\left (b^{3} d^{3} x^{3} - 16 \, b^{3} c^{3} + 24 \, a b^{2} c^{2} d - 6 \, a^{2} b c d^{2} - a^{3} d^{3} - 3 \,{\left (2 \, b^{3} c d^{2} - 3 \, a b^{2} d^{3}\right )} x^{2} - 3 \,{\left (8 \, b^{3} c^{2} d - 12 \, a b^{2} c d^{2} + 3 \, a^{2} b d^{3}\right )} x\right )} \sqrt{d x + c}}{3 \,{\left (d^{6} x^{2} + 2 \, c d^{5} x + c^{2} d^{4}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*(b^3*d^3*x^3 - 16*b^3*c^3 + 24*a*b^2*c^2*d - 6*a^2*b*c*d^2 - a^3*d^3 - 3*(2*b^3*c*d^2 - 3*a*b^2*d^3)*x^2 -
3*(8*b^3*c^2*d - 12*a*b^2*c*d^2 + 3*a^2*b*d^3)*x)*sqrt(d*x + c)/(d^6*x^2 + 2*c*d^5*x + c^2*d^4)

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Sympy [A]  time = 1.3615, size = 461, normalized size = 4.8 \begin{align*} \begin{cases} - \frac{2 a^{3} d^{3}}{3 c d^{4} \sqrt{c + d x} + 3 d^{5} x \sqrt{c + d x}} - \frac{12 a^{2} b c d^{2}}{3 c d^{4} \sqrt{c + d x} + 3 d^{5} x \sqrt{c + d x}} - \frac{18 a^{2} b d^{3} x}{3 c d^{4} \sqrt{c + d x} + 3 d^{5} x \sqrt{c + d x}} + \frac{48 a b^{2} c^{2} d}{3 c d^{4} \sqrt{c + d x} + 3 d^{5} x \sqrt{c + d x}} + \frac{72 a b^{2} c d^{2} x}{3 c d^{4} \sqrt{c + d x} + 3 d^{5} x \sqrt{c + d x}} + \frac{18 a b^{2} d^{3} x^{2}}{3 c d^{4} \sqrt{c + d x} + 3 d^{5} x \sqrt{c + d x}} - \frac{32 b^{3} c^{3}}{3 c d^{4} \sqrt{c + d x} + 3 d^{5} x \sqrt{c + d x}} - \frac{48 b^{3} c^{2} d x}{3 c d^{4} \sqrt{c + d x} + 3 d^{5} x \sqrt{c + d x}} - \frac{12 b^{3} c d^{2} x^{2}}{3 c d^{4} \sqrt{c + d x} + 3 d^{5} x \sqrt{c + d x}} + \frac{2 b^{3} d^{3} x^{3}}{3 c d^{4} \sqrt{c + d x} + 3 d^{5} x \sqrt{c + d x}} & \text{for}\: d \neq 0 \\\frac{a^{3} x + \frac{3 a^{2} b x^{2}}{2} + a b^{2} x^{3} + \frac{b^{3} x^{4}}{4}}{c^{\frac{5}{2}}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-2*a**3*d**3/(3*c*d**4*sqrt(c + d*x) + 3*d**5*x*sqrt(c + d*x)) - 12*a**2*b*c*d**2/(3*c*d**4*sqrt(c
+ d*x) + 3*d**5*x*sqrt(c + d*x)) - 18*a**2*b*d**3*x/(3*c*d**4*sqrt(c + d*x) + 3*d**5*x*sqrt(c + d*x)) + 48*a*b
**2*c**2*d/(3*c*d**4*sqrt(c + d*x) + 3*d**5*x*sqrt(c + d*x)) + 72*a*b**2*c*d**2*x/(3*c*d**4*sqrt(c + d*x) + 3*
d**5*x*sqrt(c + d*x)) + 18*a*b**2*d**3*x**2/(3*c*d**4*sqrt(c + d*x) + 3*d**5*x*sqrt(c + d*x)) - 32*b**3*c**3/(
3*c*d**4*sqrt(c + d*x) + 3*d**5*x*sqrt(c + d*x)) - 48*b**3*c**2*d*x/(3*c*d**4*sqrt(c + d*x) + 3*d**5*x*sqrt(c
+ d*x)) - 12*b**3*c*d**2*x**2/(3*c*d**4*sqrt(c + d*x) + 3*d**5*x*sqrt(c + d*x)) + 2*b**3*d**3*x**3/(3*c*d**4*s
qrt(c + d*x) + 3*d**5*x*sqrt(c + d*x)), Ne(d, 0)), ((a**3*x + 3*a**2*b*x**2/2 + a*b**2*x**3 + b**3*x**4/4)/c**
(5/2), True))

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Giac [A]  time = 1.06457, size = 190, normalized size = 1.98 \begin{align*} -\frac{2 \,{\left (9 \,{\left (d x + c\right )} b^{3} c^{2} - b^{3} c^{3} - 18 \,{\left (d x + c\right )} a b^{2} c d + 3 \, a b^{2} c^{2} d + 9 \,{\left (d x + c\right )} a^{2} b d^{2} - 3 \, a^{2} b c d^{2} + a^{3} d^{3}\right )}}{3 \,{\left (d x + c\right )}^{\frac{3}{2}} d^{4}} + \frac{2 \,{\left ({\left (d x + c\right )}^{\frac{3}{2}} b^{3} d^{8} - 9 \, \sqrt{d x + c} b^{3} c d^{8} + 9 \, \sqrt{d x + c} a b^{2} d^{9}\right )}}{3 \, d^{12}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*(9*(d*x + c)*b^3*c^2 - b^3*c^3 - 18*(d*x + c)*a*b^2*c*d + 3*a*b^2*c^2*d + 9*(d*x + c)*a^2*b*d^2 - 3*a^2*b
*c*d^2 + a^3*d^3)/((d*x + c)^(3/2)*d^4) + 2/3*((d*x + c)^(3/2)*b^3*d^8 - 9*sqrt(d*x + c)*b^3*c*d^8 + 9*sqrt(d*
x + c)*a*b^2*d^9)/d^12