3.1415 \(\int \frac{(a+b x)^3}{\sqrt{c+d x}} \, dx\)

Optimal. Leaf size=96 \[ -\frac{6 b^2 (c+d x)^{5/2} (b c-a d)}{5 d^4}+\frac{2 b (c+d x)^{3/2} (b c-a d)^2}{d^4}-\frac{2 \sqrt{c+d x} (b c-a d)^3}{d^4}+\frac{2 b^3 (c+d x)^{7/2}}{7 d^4} \]

[Out]

(-2*(b*c - a*d)^3*Sqrt[c + d*x])/d^4 + (2*b*(b*c - a*d)^2*(c + d*x)^(3/2))/d^4 - (6*b^2*(b*c - a*d)*(c + d*x)^
(5/2))/(5*d^4) + (2*b^3*(c + d*x)^(7/2))/(7*d^4)

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Rubi [A]  time = 0.0308993, 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} \[ -\frac{6 b^2 (c+d x)^{5/2} (b c-a d)}{5 d^4}+\frac{2 b (c+d x)^{3/2} (b c-a d)^2}{d^4}-\frac{2 \sqrt{c+d x} (b c-a d)^3}{d^4}+\frac{2 b^3 (c+d x)^{7/2}}{7 d^4} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^3/Sqrt[c + d*x],x]

[Out]

(-2*(b*c - a*d)^3*Sqrt[c + d*x])/d^4 + (2*b*(b*c - a*d)^2*(c + d*x)^(3/2))/d^4 - (6*b^2*(b*c - a*d)*(c + d*x)^
(5/2))/(5*d^4) + (2*b^3*(c + d*x)^(7/2))/(7*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}{\sqrt{c+d x}} \, dx &=\int \left (\frac{(-b c+a d)^3}{d^3 \sqrt{c+d x}}+\frac{3 b (b c-a d)^2 \sqrt{c+d x}}{d^3}-\frac{3 b^2 (b c-a d) (c+d x)^{3/2}}{d^3}+\frac{b^3 (c+d x)^{5/2}}{d^3}\right ) \, dx\\ &=-\frac{2 (b c-a d)^3 \sqrt{c+d x}}{d^4}+\frac{2 b (b c-a d)^2 (c+d x)^{3/2}}{d^4}-\frac{6 b^2 (b c-a d) (c+d x)^{5/2}}{5 d^4}+\frac{2 b^3 (c+d x)^{7/2}}{7 d^4}\\ \end{align*}

Mathematica [A]  time = 0.0667105, size = 79, normalized size = 0.82 \[ \frac{2 \sqrt{c+d x} \left (-21 b^2 (c+d x)^2 (b c-a d)+35 b (c+d x) (b c-a d)^2-35 (b c-a d)^3+5 b^3 (c+d x)^3\right )}{35 d^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[c + d*x]*(-35*(b*c - a*d)^3 + 35*b*(b*c - a*d)^2*(c + d*x) - 21*b^2*(b*c - a*d)*(c + d*x)^2 + 5*b^3*(c
+ d*x)^3))/(35*d^4)

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Maple [A]  time = 0.005, size = 116, normalized size = 1.2 \begin{align*}{\frac{10\,{b}^{3}{x}^{3}{d}^{3}+42\,a{b}^{2}{d}^{3}{x}^{2}-12\,{b}^{3}c{d}^{2}{x}^{2}+70\,{a}^{2}b{d}^{3}x-56\,a{b}^{2}c{d}^{2}x+16\,{b}^{3}{c}^{2}dx+70\,{a}^{3}{d}^{3}-140\,{a}^{2}bc{d}^{2}+112\,a{b}^{2}{c}^{2}d-32\,{b}^{3}{c}^{3}}{35\,{d}^{4}}\sqrt{dx+c}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^3/(d*x+c)^(1/2),x)

[Out]

2/35*(d*x+c)^(1/2)*(5*b^3*d^3*x^3+21*a*b^2*d^3*x^2-6*b^3*c*d^2*x^2+35*a^2*b*d^3*x-28*a*b^2*c*d^2*x+8*b^3*c^2*d
*x+35*a^3*d^3-70*a^2*b*c*d^2+56*a*b^2*c^2*d-16*b^3*c^3)/d^4

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Maxima [A]  time = 0.982013, size = 185, normalized size = 1.93 \begin{align*} \frac{2 \,{\left (35 \, \sqrt{d x + c} a^{3} + \frac{35 \,{\left ({\left (d x + c\right )}^{\frac{3}{2}} - 3 \, \sqrt{d x + c} c\right )} a^{2} b}{d} + \frac{7 \,{\left (3 \,{\left (d x + c\right )}^{\frac{5}{2}} - 10 \,{\left (d x + c\right )}^{\frac{3}{2}} c + 15 \, \sqrt{d x + c} c^{2}\right )} a b^{2}}{d^{2}} + \frac{{\left (5 \,{\left (d x + c\right )}^{\frac{7}{2}} - 21 \,{\left (d x + c\right )}^{\frac{5}{2}} c + 35 \,{\left (d x + c\right )}^{\frac{3}{2}} c^{2} - 35 \, \sqrt{d x + c} c^{3}\right )} b^{3}}{d^{3}}\right )}}{35 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3/(d*x+c)^(1/2),x, algorithm="maxima")

[Out]

2/35*(35*sqrt(d*x + c)*a^3 + 35*((d*x + c)^(3/2) - 3*sqrt(d*x + c)*c)*a^2*b/d + 7*(3*(d*x + c)^(5/2) - 10*(d*x
+ c)^(3/2)*c + 15*sqrt(d*x + c)*c^2)*a*b^2/d^2 + (5*(d*x + c)^(7/2) - 21*(d*x + c)^(5/2)*c + 35*(d*x + c)^(3/
2)*c^2 - 35*sqrt(d*x + c)*c^3)*b^3/d^3)/d

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Fricas [A]  time = 1.78684, size = 251, normalized size = 2.61 \begin{align*} \frac{2 \,{\left (5 \, b^{3} d^{3} x^{3} - 16 \, b^{3} c^{3} + 56 \, a b^{2} c^{2} d - 70 \, a^{2} b c d^{2} + 35 \, a^{3} d^{3} - 3 \,{\left (2 \, b^{3} c d^{2} - 7 \, a b^{2} d^{3}\right )} x^{2} +{\left (8 \, b^{3} c^{2} d - 28 \, a b^{2} c d^{2} + 35 \, a^{2} b d^{3}\right )} x\right )} \sqrt{d x + c}}{35 \, d^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3/(d*x+c)^(1/2),x, algorithm="fricas")

[Out]

2/35*(5*b^3*d^3*x^3 - 16*b^3*c^3 + 56*a*b^2*c^2*d - 70*a^2*b*c*d^2 + 35*a^3*d^3 - 3*(2*b^3*c*d^2 - 7*a*b^2*d^3
)*x^2 + (8*b^3*c^2*d - 28*a*b^2*c*d^2 + 35*a^2*b*d^3)*x)*sqrt(d*x + c)/d^4

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Sympy [A]  time = 25.7393, size = 366, normalized size = 3.81 \begin{align*} \begin{cases} - \frac{\frac{2 a^{3} c}{\sqrt{c + d x}} + 2 a^{3} \left (- \frac{c}{\sqrt{c + d x}} - \sqrt{c + d x}\right ) + \frac{6 a^{2} b c \left (- \frac{c}{\sqrt{c + d x}} - \sqrt{c + d x}\right )}{d} + \frac{6 a^{2} b \left (\frac{c^{2}}{\sqrt{c + d x}} + 2 c \sqrt{c + d x} - \frac{\left (c + d x\right )^{\frac{3}{2}}}{3}\right )}{d} + \frac{6 a b^{2} c \left (\frac{c^{2}}{\sqrt{c + d x}} + 2 c \sqrt{c + d x} - \frac{\left (c + d x\right )^{\frac{3}{2}}}{3}\right )}{d^{2}} + \frac{6 a b^{2} \left (- \frac{c^{3}}{\sqrt{c + d x}} - 3 c^{2} \sqrt{c + d x} + c \left (c + d x\right )^{\frac{3}{2}} - \frac{\left (c + d x\right )^{\frac{5}{2}}}{5}\right )}{d^{2}} + \frac{2 b^{3} c \left (- \frac{c^{3}}{\sqrt{c + d x}} - 3 c^{2} \sqrt{c + d x} + c \left (c + d x\right )^{\frac{3}{2}} - \frac{\left (c + d x\right )^{\frac{5}{2}}}{5}\right )}{d^{3}} + \frac{2 b^{3} \left (\frac{c^{4}}{\sqrt{c + d x}} + 4 c^{3} \sqrt{c + d x} - 2 c^{2} \left (c + d x\right )^{\frac{3}{2}} + \frac{4 c \left (c + d x\right )^{\frac{5}{2}}}{5} - \frac{\left (c + d x\right )^{\frac{7}{2}}}{7}\right )}{d^{3}}}{d} & \text{for}\: d \neq 0 \\\frac{\begin{cases} a^{3} x & \text{for}\: b = 0 \\\frac{\left (a + b x\right )^{4}}{4 b} & \text{otherwise} \end{cases}}{\sqrt{c}} & \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)**(1/2),x)

[Out]

Piecewise((-(2*a**3*c/sqrt(c + d*x) + 2*a**3*(-c/sqrt(c + d*x) - sqrt(c + d*x)) + 6*a**2*b*c*(-c/sqrt(c + d*x)
- sqrt(c + d*x))/d + 6*a**2*b*(c**2/sqrt(c + d*x) + 2*c*sqrt(c + d*x) - (c + d*x)**(3/2)/3)/d + 6*a*b**2*c*(c
**2/sqrt(c + d*x) + 2*c*sqrt(c + d*x) - (c + d*x)**(3/2)/3)/d**2 + 6*a*b**2*(-c**3/sqrt(c + d*x) - 3*c**2*sqrt
(c + d*x) + c*(c + d*x)**(3/2) - (c + d*x)**(5/2)/5)/d**2 + 2*b**3*c*(-c**3/sqrt(c + d*x) - 3*c**2*sqrt(c + d*
x) + c*(c + d*x)**(3/2) - (c + d*x)**(5/2)/5)/d**3 + 2*b**3*(c**4/sqrt(c + d*x) + 4*c**3*sqrt(c + d*x) - 2*c**
2*(c + d*x)**(3/2) + 4*c*(c + d*x)**(5/2)/5 - (c + d*x)**(7/2)/7)/d**3)/d, Ne(d, 0)), (Piecewise((a**3*x, Eq(b
, 0)), ((a + b*x)**4/(4*b), True))/sqrt(c), True))

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Giac [A]  time = 1.07964, size = 185, normalized size = 1.93 \begin{align*} \frac{2 \,{\left (35 \, \sqrt{d x + c} a^{3} + \frac{35 \,{\left ({\left (d x + c\right )}^{\frac{3}{2}} - 3 \, \sqrt{d x + c} c\right )} a^{2} b}{d} + \frac{7 \,{\left (3 \,{\left (d x + c\right )}^{\frac{5}{2}} - 10 \,{\left (d x + c\right )}^{\frac{3}{2}} c + 15 \, \sqrt{d x + c} c^{2}\right )} a b^{2}}{d^{2}} + \frac{{\left (5 \,{\left (d x + c\right )}^{\frac{7}{2}} - 21 \,{\left (d x + c\right )}^{\frac{5}{2}} c + 35 \,{\left (d x + c\right )}^{\frac{3}{2}} c^{2} - 35 \, \sqrt{d x + c} c^{3}\right )} b^{3}}{d^{3}}\right )}}{35 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3/(d*x+c)^(1/2),x, algorithm="giac")

[Out]

2/35*(35*sqrt(d*x + c)*a^3 + 35*((d*x + c)^(3/2) - 3*sqrt(d*x + c)*c)*a^2*b/d + 7*(3*(d*x + c)^(5/2) - 10*(d*x
+ c)^(3/2)*c + 15*sqrt(d*x + c)*c^2)*a*b^2/d^2 + (5*(d*x + c)^(7/2) - 21*(d*x + c)^(5/2)*c + 35*(d*x + c)^(3/
2)*c^2 - 35*sqrt(d*x + c)*c^3)*b^3/d^3)/d