3.1435 \(\int \frac{(a+b x)^4}{(c+d x)^{5/2}} \, dx\)

Optimal. Leaf size=125 \[ -\frac{8 b^3 (c+d x)^{3/2} (b c-a d)}{3 d^5}+\frac{12 b^2 \sqrt{c+d x} (b c-a d)^2}{d^5}+\frac{8 b (b c-a d)^3}{d^5 \sqrt{c+d x}}-\frac{2 (b c-a d)^4}{3 d^5 (c+d x)^{3/2}}+\frac{2 b^4 (c+d x)^{5/2}}{5 d^5} \]

[Out]

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

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Rubi [A]  time = 0.0385926, antiderivative size = 125, 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} \[ -\frac{8 b^3 (c+d x)^{3/2} (b c-a d)}{3 d^5}+\frac{12 b^2 \sqrt{c+d x} (b c-a d)^2}{d^5}+\frac{8 b (b c-a d)^3}{d^5 \sqrt{c+d x}}-\frac{2 (b c-a d)^4}{3 d^5 (c+d x)^{3/2}}+\frac{2 b^4 (c+d x)^{5/2}}{5 d^5} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^4/(c + d*x)^(5/2),x]

[Out]

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

Mathematica [A]  time = 0.082142, size = 101, normalized size = 0.81 \[ \frac{2 \left (90 b^2 (c+d x)^2 (b c-a d)^2-20 b^3 (c+d x)^3 (b c-a d)+60 b (c+d x) (b c-a d)^3-5 (b c-a d)^4+3 b^4 (c+d x)^4\right )}{15 d^5 (c+d x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^4/(c + d*x)^(5/2),x]

[Out]

(2*(-5*(b*c - a*d)^4 + 60*b*(b*c - a*d)^3*(c + d*x) + 90*b^2*(b*c - a*d)^2*(c + d*x)^2 - 20*b^3*(b*c - a*d)*(c
+ d*x)^3 + 3*b^4*(c + d*x)^4))/(15*d^5*(c + d*x)^(3/2))

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Maple [A]  time = 0.005, size = 186, normalized size = 1.5 \begin{align*} -{\frac{-6\,{b}^{4}{x}^{4}{d}^{4}-40\,a{b}^{3}{d}^{4}{x}^{3}+16\,{b}^{4}c{d}^{3}{x}^{3}-180\,{a}^{2}{b}^{2}{d}^{4}{x}^{2}+240\,a{b}^{3}c{d}^{3}{x}^{2}-96\,{b}^{4}{c}^{2}{d}^{2}{x}^{2}+120\,{a}^{3}b{d}^{4}x-720\,{a}^{2}{b}^{2}c{d}^{3}x+960\,a{b}^{3}{c}^{2}{d}^{2}x-384\,{b}^{4}{c}^{3}dx+10\,{a}^{4}{d}^{4}+80\,{a}^{3}bc{d}^{3}-480\,{a}^{2}{b}^{2}{c}^{2}{d}^{2}+640\,a{b}^{3}{c}^{3}d-256\,{b}^{4}{c}^{4}}{15\,{d}^{5}} \left ( dx+c \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^4/(d*x+c)^(5/2),x)

[Out]

-2/15/(d*x+c)^(3/2)*(-3*b^4*d^4*x^4-20*a*b^3*d^4*x^3+8*b^4*c*d^3*x^3-90*a^2*b^2*d^4*x^2+120*a*b^3*c*d^3*x^2-48
*b^4*c^2*d^2*x^2+60*a^3*b*d^4*x-360*a^2*b^2*c*d^3*x+480*a*b^3*c^2*d^2*x-192*b^4*c^3*d*x+5*a^4*d^4+40*a^3*b*c*d
^3-240*a^2*b^2*c^2*d^2+320*a*b^3*c^3*d-128*b^4*c^4)/d^5

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Maxima [A]  time = 0.973523, size = 252, normalized size = 2.02 \begin{align*} \frac{2 \,{\left (\frac{3 \,{\left (d x + c\right )}^{\frac{5}{2}} b^{4} - 20 \,{\left (b^{4} c - a b^{3} d\right )}{\left (d x + c\right )}^{\frac{3}{2}} + 90 \,{\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} \sqrt{d x + c}}{d^{4}} - \frac{5 \,{\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4} - 12 \,{\left (b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )}{\left (d x + c\right )}\right )}}{{\left (d x + c\right )}^{\frac{3}{2}} d^{4}}\right )}}{15 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^4/(d*x+c)^(5/2),x, algorithm="maxima")

[Out]

2/15*((3*(d*x + c)^(5/2)*b^4 - 20*(b^4*c - a*b^3*d)*(d*x + c)^(3/2) + 90*(b^4*c^2 - 2*a*b^3*c*d + a^2*b^2*d^2)
*sqrt(d*x + c))/d^4 - 5*(b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4 - 12*(b^4*c^3 -
3*a*b^3*c^2*d + 3*a^2*b^2*c*d^2 - a^3*b*d^3)*(d*x + c))/((d*x + c)^(3/2)*d^4))/d

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Fricas [A]  time = 2.08512, size = 431, normalized size = 3.45 \begin{align*} \frac{2 \,{\left (3 \, b^{4} d^{4} x^{4} + 128 \, b^{4} c^{4} - 320 \, a b^{3} c^{3} d + 240 \, a^{2} b^{2} c^{2} d^{2} - 40 \, a^{3} b c d^{3} - 5 \, a^{4} d^{4} - 4 \,{\left (2 \, b^{4} c d^{3} - 5 \, a b^{3} d^{4}\right )} x^{3} + 6 \,{\left (8 \, b^{4} c^{2} d^{2} - 20 \, a b^{3} c d^{3} + 15 \, a^{2} b^{2} d^{4}\right )} x^{2} + 12 \,{\left (16 \, b^{4} c^{3} d - 40 \, a b^{3} c^{2} d^{2} + 30 \, a^{2} b^{2} c d^{3} - 5 \, a^{3} b d^{4}\right )} x\right )} \sqrt{d x + c}}{15 \,{\left (d^{7} x^{2} + 2 \, c d^{6} x + c^{2} d^{5}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^4/(d*x+c)^(5/2),x, algorithm="fricas")

[Out]

2/15*(3*b^4*d^4*x^4 + 128*b^4*c^4 - 320*a*b^3*c^3*d + 240*a^2*b^2*c^2*d^2 - 40*a^3*b*c*d^3 - 5*a^4*d^4 - 4*(2*
b^4*c*d^3 - 5*a*b^3*d^4)*x^3 + 6*(8*b^4*c^2*d^2 - 20*a*b^3*c*d^3 + 15*a^2*b^2*d^4)*x^2 + 12*(16*b^4*c^3*d - 40
*a*b^3*c^2*d^2 + 30*a^2*b^2*c*d^3 - 5*a^3*b*d^4)*x)*sqrt(d*x + c)/(d^7*x^2 + 2*c*d^6*x + c^2*d^5)

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Sympy [A]  time = 28.054, size = 136, normalized size = 1.09 \begin{align*} \frac{2 b^{4} \left (c + d x\right )^{\frac{5}{2}}}{5 d^{5}} - \frac{8 b \left (a d - b c\right )^{3}}{d^{5} \sqrt{c + d x}} + \frac{\left (c + d x\right )^{\frac{3}{2}} \left (8 a b^{3} d - 8 b^{4} c\right )}{3 d^{5}} + \frac{\sqrt{c + d x} \left (12 a^{2} b^{2} d^{2} - 24 a b^{3} c d + 12 b^{4} c^{2}\right )}{d^{5}} - \frac{2 \left (a d - b c\right )^{4}}{3 d^{5} \left (c + d x\right )^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**4/(d*x+c)**(5/2),x)

[Out]

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

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Giac [B]  time = 1.07748, size = 309, normalized size = 2.47 \begin{align*} \frac{2 \,{\left (12 \,{\left (d x + c\right )} b^{4} c^{3} - b^{4} c^{4} - 36 \,{\left (d x + c\right )} a b^{3} c^{2} d + 4 \, a b^{3} c^{3} d + 36 \,{\left (d x + c\right )} a^{2} b^{2} c d^{2} - 6 \, a^{2} b^{2} c^{2} d^{2} - 12 \,{\left (d x + c\right )} a^{3} b d^{3} + 4 \, a^{3} b c d^{3} - a^{4} d^{4}\right )}}{3 \,{\left (d x + c\right )}^{\frac{3}{2}} d^{5}} + \frac{2 \,{\left (3 \,{\left (d x + c\right )}^{\frac{5}{2}} b^{4} d^{20} - 20 \,{\left (d x + c\right )}^{\frac{3}{2}} b^{4} c d^{20} + 90 \, \sqrt{d x + c} b^{4} c^{2} d^{20} + 20 \,{\left (d x + c\right )}^{\frac{3}{2}} a b^{3} d^{21} - 180 \, \sqrt{d x + c} a b^{3} c d^{21} + 90 \, \sqrt{d x + c} a^{2} b^{2} d^{22}\right )}}{15 \, d^{25}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^4/(d*x+c)^(5/2),x, algorithm="giac")

[Out]

2/3*(12*(d*x + c)*b^4*c^3 - b^4*c^4 - 36*(d*x + c)*a*b^3*c^2*d + 4*a*b^3*c^3*d + 36*(d*x + c)*a^2*b^2*c*d^2 -
6*a^2*b^2*c^2*d^2 - 12*(d*x + c)*a^3*b*d^3 + 4*a^3*b*c*d^3 - a^4*d^4)/((d*x + c)^(3/2)*d^5) + 2/15*(3*(d*x + c
)^(5/2)*b^4*d^20 - 20*(d*x + c)^(3/2)*b^4*c*d^20 + 90*sqrt(d*x + c)*b^4*c^2*d^20 + 20*(d*x + c)^(3/2)*a*b^3*d^
21 - 180*sqrt(d*x + c)*a*b^3*c*d^21 + 90*sqrt(d*x + c)*a^2*b^2*d^22)/d^25