3.1414 \(\int \frac{(a+b x)^4}{\sqrt{c+d x}} \, dx\)

Optimal. Leaf size=127 \[ -\frac{8 b^3 (c+d x)^{7/2} (b c-a d)}{7 d^5}+\frac{12 b^2 (c+d x)^{5/2} (b c-a d)^2}{5 d^5}-\frac{8 b (c+d x)^{3/2} (b c-a d)^3}{3 d^5}+\frac{2 \sqrt{c+d x} (b c-a d)^4}{d^5}+\frac{2 b^4 (c+d x)^{9/2}}{9 d^5} \]

[Out]

(2*(b*c - a*d)^4*Sqrt[c + d*x])/d^5 - (8*b*(b*c - a*d)^3*(c + d*x)^(3/2))/(3*d^5) + (12*b^2*(b*c - a*d)^2*(c +
d*x)^(5/2))/(5*d^5) - (8*b^3*(b*c - a*d)*(c + d*x)^(7/2))/(7*d^5) + (2*b^4*(c + d*x)^(9/2))/(9*d^5)

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Rubi [A]  time = 0.0410394, antiderivative size = 127, 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} \[ -\frac{8 b^3 (c+d x)^{7/2} (b c-a d)}{7 d^5}+\frac{12 b^2 (c+d x)^{5/2} (b c-a d)^2}{5 d^5}-\frac{8 b (c+d x)^{3/2} (b c-a d)^3}{3 d^5}+\frac{2 \sqrt{c+d x} (b c-a d)^4}{d^5}+\frac{2 b^4 (c+d x)^{9/2}}{9 d^5} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^4/Sqrt[c + d*x],x]

[Out]

(2*(b*c - a*d)^4*Sqrt[c + d*x])/d^5 - (8*b*(b*c - a*d)^3*(c + d*x)^(3/2))/(3*d^5) + (12*b^2*(b*c - a*d)^2*(c +
d*x)^(5/2))/(5*d^5) - (8*b^3*(b*c - a*d)*(c + d*x)^(7/2))/(7*d^5) + (2*b^4*(c + d*x)^(9/2))/(9*d^5)

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)^4}{\sqrt{c+d x}} \, dx &=\int \left (\frac{(-b c+a d)^4}{d^4 \sqrt{c+d x}}-\frac{4 b (b c-a d)^3 \sqrt{c+d x}}{d^4}+\frac{6 b^2 (b c-a d)^2 (c+d x)^{3/2}}{d^4}-\frac{4 b^3 (b c-a d) (c+d x)^{5/2}}{d^4}+\frac{b^4 (c+d x)^{7/2}}{d^4}\right ) \, dx\\ &=\frac{2 (b c-a d)^4 \sqrt{c+d x}}{d^5}-\frac{8 b (b c-a d)^3 (c+d x)^{3/2}}{3 d^5}+\frac{12 b^2 (b c-a d)^2 (c+d x)^{5/2}}{5 d^5}-\frac{8 b^3 (b c-a d) (c+d x)^{7/2}}{7 d^5}+\frac{2 b^4 (c+d x)^{9/2}}{9 d^5}\\ \end{align*}

Mathematica [A]  time = 0.0827054, size = 101, normalized size = 0.8 \[ \frac{2 \sqrt{c+d x} \left (378 b^2 (c+d x)^2 (b c-a d)^2-180 b^3 (c+d x)^3 (b c-a d)-420 b (c+d x) (b c-a d)^3+315 (b c-a d)^4+35 b^4 (c+d x)^4\right )}{315 d^5} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^4/Sqrt[c + d*x],x]

[Out]

(2*Sqrt[c + d*x]*(315*(b*c - a*d)^4 - 420*b*(b*c - a*d)^3*(c + d*x) + 378*b^2*(b*c - a*d)^2*(c + d*x)^2 - 180*
b^3*(b*c - a*d)*(c + d*x)^3 + 35*b^4*(c + d*x)^4))/(315*d^5)

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Maple [A]  time = 0.005, size = 186, normalized size = 1.5 \begin{align*}{\frac{70\,{b}^{4}{x}^{4}{d}^{4}+360\,a{b}^{3}{d}^{4}{x}^{3}-80\,{b}^{4}c{d}^{3}{x}^{3}+756\,{a}^{2}{b}^{2}{d}^{4}{x}^{2}-432\,a{b}^{3}c{d}^{3}{x}^{2}+96\,{b}^{4}{c}^{2}{d}^{2}{x}^{2}+840\,{a}^{3}b{d}^{4}x-1008\,{a}^{2}{b}^{2}c{d}^{3}x+576\,a{b}^{3}{c}^{2}{d}^{2}x-128\,{b}^{4}{c}^{3}dx+630\,{a}^{4}{d}^{4}-1680\,{a}^{3}bc{d}^{3}+2016\,{a}^{2}{b}^{2}{c}^{2}{d}^{2}-1152\,a{b}^{3}{c}^{3}d+256\,{b}^{4}{c}^{4}}{315\,{d}^{5}}\sqrt{dx+c}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/315*(d*x+c)^(1/2)*(35*b^4*d^4*x^4+180*a*b^3*d^4*x^3-40*b^4*c*d^3*x^3+378*a^2*b^2*d^4*x^2-216*a*b^3*c*d^3*x^2
+48*b^4*c^2*d^2*x^2+420*a^3*b*d^4*x-504*a^2*b^2*c*d^3*x+288*a*b^3*c^2*d^2*x-64*b^4*c^3*d*x+315*a^4*d^4-840*a^3
*b*c*d^3+1008*a^2*b^2*c^2*d^2-576*a*b^3*c^3*d+128*b^4*c^4)/d^5

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Maxima [A]  time = 0.965529, size = 275, normalized size = 2.17 \begin{align*} \frac{2 \,{\left (315 \, \sqrt{d x + c} a^{4} + \frac{420 \,{\left ({\left (d x + c\right )}^{\frac{3}{2}} - 3 \, \sqrt{d x + c} c\right )} a^{3} b}{d} + \frac{126 \,{\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^{2} b^{2}}{d^{2}} + \frac{36 \,{\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 )} a b^{3}}{d^{3}} + \frac{{\left (35 \,{\left (d x + c\right )}^{\frac{9}{2}} - 180 \,{\left (d x + c\right )}^{\frac{7}{2}} c + 378 \,{\left (d x + c\right )}^{\frac{5}{2}} c^{2} - 420 \,{\left (d x + c\right )}^{\frac{3}{2}} c^{3} + 315 \, \sqrt{d x + c} c^{4}\right )} b^{4}}{d^{4}}\right )}}{315 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^4/(d*x+c)^(1/2),x, algorithm="maxima")

[Out]

2/315*(315*sqrt(d*x + c)*a^4 + 420*((d*x + c)^(3/2) - 3*sqrt(d*x + c)*c)*a^3*b/d + 126*(3*(d*x + c)^(5/2) - 10
*(d*x + c)^(3/2)*c + 15*sqrt(d*x + c)*c^2)*a^2*b^2/d^2 + 36*(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)*a*b^3/d^3 + (35*(d*x + c)^(9/2) - 180*(d*x + c)^(7/2)*c + 378*(d*x +
c)^(5/2)*c^2 - 420*(d*x + c)^(3/2)*c^3 + 315*sqrt(d*x + c)*c^4)*b^4/d^4)/d

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Fricas [A]  time = 1.79735, size = 405, normalized size = 3.19 \begin{align*} \frac{2 \,{\left (35 \, b^{4} d^{4} x^{4} + 128 \, b^{4} c^{4} - 576 \, a b^{3} c^{3} d + 1008 \, a^{2} b^{2} c^{2} d^{2} - 840 \, a^{3} b c d^{3} + 315 \, a^{4} d^{4} - 20 \,{\left (2 \, b^{4} c d^{3} - 9 \, a b^{3} d^{4}\right )} x^{3} + 6 \,{\left (8 \, b^{4} c^{2} d^{2} - 36 \, a b^{3} c d^{3} + 63 \, a^{2} b^{2} d^{4}\right )} x^{2} - 4 \,{\left (16 \, b^{4} c^{3} d - 72 \, a b^{3} c^{2} d^{2} + 126 \, a^{2} b^{2} c d^{3} - 105 \, a^{3} b d^{4}\right )} x\right )} \sqrt{d x + c}}{315 \, d^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^4/(d*x+c)^(1/2),x, algorithm="fricas")

[Out]

2/315*(35*b^4*d^4*x^4 + 128*b^4*c^4 - 576*a*b^3*c^3*d + 1008*a^2*b^2*c^2*d^2 - 840*a^3*b*c*d^3 + 315*a^4*d^4 -
20*(2*b^4*c*d^3 - 9*a*b^3*d^4)*x^3 + 6*(8*b^4*c^2*d^2 - 36*a*b^3*c*d^3 + 63*a^2*b^2*d^4)*x^2 - 4*(16*b^4*c^3*
d - 72*a*b^3*c^2*d^2 + 126*a^2*b^2*c*d^3 - 105*a^3*b*d^4)*x)*sqrt(d*x + c)/d^5

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Sympy [A]  time = 39.6809, size = 532, normalized size = 4.19 \begin{align*} \begin{cases} - \frac{\frac{2 a^{4} c}{\sqrt{c + d x}} + 2 a^{4} \left (- \frac{c}{\sqrt{c + d x}} - \sqrt{c + d x}\right ) + \frac{8 a^{3} b c \left (- \frac{c}{\sqrt{c + d x}} - \sqrt{c + d x}\right )}{d} + \frac{8 a^{3} 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{12 a^{2} 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{12 a^{2} 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{8 a 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{8 a 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}} + \frac{2 b^{4} c \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^{4}} + \frac{2 b^{4} \left (- \frac{c^{5}}{\sqrt{c + d x}} - 5 c^{4} \sqrt{c + d x} + \frac{10 c^{3} \left (c + d x\right )^{\frac{3}{2}}}{3} - 2 c^{2} \left (c + d x\right )^{\frac{5}{2}} + \frac{5 c \left (c + d x\right )^{\frac{7}{2}}}{7} - \frac{\left (c + d x\right )^{\frac{9}{2}}}{9}\right )}{d^{4}}}{d} & \text{for}\: d \neq 0 \\\frac{\begin{cases} a^{4} x & \text{for}\: b = 0 \\\frac{\left (a + b x\right )^{5}}{5 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)**4/(d*x+c)**(1/2),x)

[Out]

Piecewise((-(2*a**4*c/sqrt(c + d*x) + 2*a**4*(-c/sqrt(c + d*x) - sqrt(c + d*x)) + 8*a**3*b*c*(-c/sqrt(c + d*x)
- sqrt(c + d*x))/d + 8*a**3*b*(c**2/sqrt(c + d*x) + 2*c*sqrt(c + d*x) - (c + d*x)**(3/2)/3)/d + 12*a**2*b**2*
c*(c**2/sqrt(c + d*x) + 2*c*sqrt(c + d*x) - (c + d*x)**(3/2)/3)/d**2 + 12*a**2*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 + 8*a*b**3*c*(-c**3/sqrt(c + d*x) - 3*c**2*s
qrt(c + d*x) + c*(c + d*x)**(3/2) - (c + d*x)**(5/2)/5)/d**3 + 8*a*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 + 2*b**4*c*(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**4 + 2*b
**4*(-c**5/sqrt(c + d*x) - 5*c**4*sqrt(c + d*x) + 10*c**3*(c + d*x)**(3/2)/3 - 2*c**2*(c + d*x)**(5/2) + 5*c*(
c + d*x)**(7/2)/7 - (c + d*x)**(9/2)/9)/d**4)/d, Ne(d, 0)), (Piecewise((a**4*x, Eq(b, 0)), ((a + b*x)**5/(5*b)
, True))/sqrt(c), True))

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Giac [A]  time = 1.06405, size = 275, normalized size = 2.17 \begin{align*} \frac{2 \,{\left (315 \, \sqrt{d x + c} a^{4} + \frac{420 \,{\left ({\left (d x + c\right )}^{\frac{3}{2}} - 3 \, \sqrt{d x + c} c\right )} a^{3} b}{d} + \frac{126 \,{\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^{2} b^{2}}{d^{2}} + \frac{36 \,{\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 )} a b^{3}}{d^{3}} + \frac{{\left (35 \,{\left (d x + c\right )}^{\frac{9}{2}} - 180 \,{\left (d x + c\right )}^{\frac{7}{2}} c + 378 \,{\left (d x + c\right )}^{\frac{5}{2}} c^{2} - 420 \,{\left (d x + c\right )}^{\frac{3}{2}} c^{3} + 315 \, \sqrt{d x + c} c^{4}\right )} b^{4}}{d^{4}}\right )}}{315 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^4/(d*x+c)^(1/2),x, algorithm="giac")

[Out]

2/315*(315*sqrt(d*x + c)*a^4 + 420*((d*x + c)^(3/2) - 3*sqrt(d*x + c)*c)*a^3*b/d + 126*(3*(d*x + c)^(5/2) - 10
*(d*x + c)^(3/2)*c + 15*sqrt(d*x + c)*c^2)*a^2*b^2/d^2 + 36*(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)*a*b^3/d^3 + (35*(d*x + c)^(9/2) - 180*(d*x + c)^(7/2)*c + 378*(d*x +
c)^(5/2)*c^2 - 420*(d*x + c)^(3/2)*c^3 + 315*sqrt(d*x + c)*c^4)*b^4/d^4)/d