3.1425 \(\int \frac{(a+b x)^4}{(c+d x)^{3/2}} \, dx\)

Optimal. Leaf size=123 \[ -\frac{8 b^3 (c+d x)^{5/2} (b c-a d)}{5 d^5}+\frac{4 b^2 (c+d x)^{3/2} (b c-a d)^2}{d^5}-\frac{8 b \sqrt{c+d x} (b c-a d)^3}{d^5}-\frac{2 (b c-a d)^4}{d^5 \sqrt{c+d x}}+\frac{2 b^4 (c+d x)^{7/2}}{7 d^5} \]

[Out]

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

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Rubi [A]  time = 0.0371394, antiderivative size = 123, 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} \[ -\frac{8 b^3 (c+d x)^{5/2} (b c-a d)}{5 d^5}+\frac{4 b^2 (c+d x)^{3/2} (b c-a d)^2}{d^5}-\frac{8 b \sqrt{c+d x} (b c-a d)^3}{d^5}-\frac{2 (b c-a d)^4}{d^5 \sqrt{c+d x}}+\frac{2 b^4 (c+d x)^{7/2}}{7 d^5} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^4/(c + d*x)^(3/2),x]

[Out]

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

Mathematica [A]  time = 0.0763483, size = 101, normalized size = 0.82 \[ \frac{2 \left (70 b^2 (c+d x)^2 (b c-a d)^2-28 b^3 (c+d x)^3 (b c-a d)-140 b (c+d x) (b c-a d)^3-35 (b c-a d)^4+5 b^4 (c+d x)^4\right )}{35 d^5 \sqrt{c+d x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(-35*(b*c - a*d)^4 - 140*b*(b*c - a*d)^3*(c + d*x) + 70*b^2*(b*c - a*d)^2*(c + d*x)^2 - 28*b^3*(b*c - a*d)*
(c + d*x)^3 + 5*b^4*(c + d*x)^4))/(35*d^5*Sqrt[c + d*x])

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Maple [A]  time = 0.006, size = 186, normalized size = 1.5 \begin{align*} -{\frac{-10\,{b}^{4}{x}^{4}{d}^{4}-56\,a{b}^{3}{d}^{4}{x}^{3}+16\,{b}^{4}c{d}^{3}{x}^{3}-140\,{a}^{2}{b}^{2}{d}^{4}{x}^{2}+112\,a{b}^{3}c{d}^{3}{x}^{2}-32\,{b}^{4}{c}^{2}{d}^{2}{x}^{2}-280\,{a}^{3}b{d}^{4}x+560\,{a}^{2}{b}^{2}c{d}^{3}x-448\,a{b}^{3}{c}^{2}{d}^{2}x+128\,{b}^{4}{c}^{3}dx+70\,{a}^{4}{d}^{4}-560\,{a}^{3}bc{d}^{3}+1120\,{a}^{2}{b}^{2}{c}^{2}{d}^{2}-896\,a{b}^{3}{c}^{3}d+256\,{b}^{4}{c}^{4}}{35\,{d}^{5}}{\frac{1}{\sqrt{dx+c}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^4/(d*x+c)^(3/2),x)

[Out]

-2/35/(d*x+c)^(1/2)*(-5*b^4*d^4*x^4-28*a*b^3*d^4*x^3+8*b^4*c*d^3*x^3-70*a^2*b^2*d^4*x^2+56*a*b^3*c*d^3*x^2-16*
b^4*c^2*d^2*x^2-140*a^3*b*d^4*x+280*a^2*b^2*c*d^3*x-224*a*b^3*c^2*d^2*x+64*b^4*c^3*d*x+35*a^4*d^4-280*a^3*b*c*
d^3+560*a^2*b^2*c^2*d^2-448*a*b^3*c^3*d+128*b^4*c^4)/d^5

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Maxima [A]  time = 0.952235, size = 255, normalized size = 2.07 \begin{align*} \frac{2 \,{\left (\frac{5 \,{\left (d x + c\right )}^{\frac{7}{2}} b^{4} - 28 \,{\left (b^{4} c - a b^{3} d\right )}{\left (d x + c\right )}^{\frac{5}{2}} + 70 \,{\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )}{\left (d x + c\right )}^{\frac{3}{2}} - 140 \,{\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 )} \sqrt{d x + c}}{d^{4}} - \frac{35 \,{\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}\right )}}{\sqrt{d x + c} d^{4}}\right )}}{35 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^4/(d*x+c)^(3/2),x, algorithm="maxima")

[Out]

2/35*((5*(d*x + c)^(7/2)*b^4 - 28*(b^4*c - a*b^3*d)*(d*x + c)^(5/2) + 70*(b^4*c^2 - 2*a*b^3*c*d + a^2*b^2*d^2)
*(d*x + c)^(3/2) - 140*(b^4*c^3 - 3*a*b^3*c^2*d + 3*a^2*b^2*c*d^2 - a^3*b*d^3)*sqrt(d*x + c))/d^4 - 35*(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)/(sqrt(d*x + c)*d^4))/d

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Fricas [A]  time = 2.01547, size = 412, normalized size = 3.35 \begin{align*} \frac{2 \,{\left (5 \, b^{4} d^{4} x^{4} - 128 \, b^{4} c^{4} + 448 \, a b^{3} c^{3} d - 560 \, a^{2} b^{2} c^{2} d^{2} + 280 \, a^{3} b c d^{3} - 35 \, a^{4} d^{4} - 4 \,{\left (2 \, b^{4} c d^{3} - 7 \, a b^{3} d^{4}\right )} x^{3} + 2 \,{\left (8 \, b^{4} c^{2} d^{2} - 28 \, a b^{3} c d^{3} + 35 \, a^{2} b^{2} d^{4}\right )} x^{2} - 4 \,{\left (16 \, b^{4} c^{3} d - 56 \, a b^{3} c^{2} d^{2} + 70 \, a^{2} b^{2} c d^{3} - 35 \, a^{3} b d^{4}\right )} x\right )} \sqrt{d x + c}}{35 \,{\left (d^{6} x + c d^{5}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^4/(d*x+c)^(3/2),x, algorithm="fricas")

[Out]

2/35*(5*b^4*d^4*x^4 - 128*b^4*c^4 + 448*a*b^3*c^3*d - 560*a^2*b^2*c^2*d^2 + 280*a^3*b*c*d^3 - 35*a^4*d^4 - 4*(
2*b^4*c*d^3 - 7*a*b^3*d^4)*x^3 + 2*(8*b^4*c^2*d^2 - 28*a*b^3*c*d^3 + 35*a^2*b^2*d^4)*x^2 - 4*(16*b^4*c^3*d - 5
6*a*b^3*c^2*d^2 + 70*a^2*b^2*c*d^3 - 35*a^3*b*d^4)*x)*sqrt(d*x + c)/(d^6*x + c*d^5)

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Sympy [A]  time = 20.2228, size = 168, normalized size = 1.37 \begin{align*} \frac{2 b^{4} \left (c + d x\right )^{\frac{7}{2}}}{7 d^{5}} + \frac{\left (c + d x\right )^{\frac{5}{2}} \left (8 a b^{3} d - 8 b^{4} c\right )}{5 d^{5}} + \frac{\left (c + d x\right )^{\frac{3}{2}} \left (12 a^{2} b^{2} d^{2} - 24 a b^{3} c d + 12 b^{4} c^{2}\right )}{3 d^{5}} + \frac{\sqrt{c + d x} \left (8 a^{3} b d^{3} - 24 a^{2} b^{2} c d^{2} + 24 a b^{3} c^{2} d - 8 b^{4} c^{3}\right )}{d^{5}} - \frac{2 \left (a d - b c\right )^{4}}{d^{5} \sqrt{c + d x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**4/(d*x+c)**(3/2),x)

[Out]

2*b**4*(c + d*x)**(7/2)/(7*d**5) + (c + d*x)**(5/2)*(8*a*b**3*d - 8*b**4*c)/(5*d**5) + (c + d*x)**(3/2)*(12*a*
*2*b**2*d**2 - 24*a*b**3*c*d + 12*b**4*c**2)/(3*d**5) + sqrt(c + d*x)*(8*a**3*b*d**3 - 24*a**2*b**2*c*d**2 + 2
4*a*b**3*c**2*d - 8*b**4*c**3)/d**5 - 2*(a*d - b*c)**4/(d**5*sqrt(c + d*x))

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Giac [B]  time = 1.08478, size = 324, normalized size = 2.63 \begin{align*} -\frac{2 \,{\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}\right )}}{\sqrt{d x + c} d^{5}} + \frac{2 \,{\left (5 \,{\left (d x + c\right )}^{\frac{7}{2}} b^{4} d^{30} - 28 \,{\left (d x + c\right )}^{\frac{5}{2}} b^{4} c d^{30} + 70 \,{\left (d x + c\right )}^{\frac{3}{2}} b^{4} c^{2} d^{30} - 140 \, \sqrt{d x + c} b^{4} c^{3} d^{30} + 28 \,{\left (d x + c\right )}^{\frac{5}{2}} a b^{3} d^{31} - 140 \,{\left (d x + c\right )}^{\frac{3}{2}} a b^{3} c d^{31} + 420 \, \sqrt{d x + c} a b^{3} c^{2} d^{31} + 70 \,{\left (d x + c\right )}^{\frac{3}{2}} a^{2} b^{2} d^{32} - 420 \, \sqrt{d x + c} a^{2} b^{2} c d^{32} + 140 \, \sqrt{d x + c} a^{3} b d^{33}\right )}}{35 \, d^{35}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^4/(d*x+c)^(3/2),x, algorithm="giac")

[Out]

-2*(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)/(sqrt(d*x + c)*d^5) + 2/35*(5*(d*x
+ c)^(7/2)*b^4*d^30 - 28*(d*x + c)^(5/2)*b^4*c*d^30 + 70*(d*x + c)^(3/2)*b^4*c^2*d^30 - 140*sqrt(d*x + c)*b^4*
c^3*d^30 + 28*(d*x + c)^(5/2)*a*b^3*d^31 - 140*(d*x + c)^(3/2)*a*b^3*c*d^31 + 420*sqrt(d*x + c)*a*b^3*c^2*d^31
+ 70*(d*x + c)^(3/2)*a^2*b^2*d^32 - 420*sqrt(d*x + c)*a^2*b^2*c*d^32 + 140*sqrt(d*x + c)*a^3*b*d^33)/d^35