3.1434 \(\int \frac{(a+b x)^5}{(c+d x)^{5/2}} \, dx\)

Optimal. Leaf size=152 \[ -\frac{2 b^4 (c+d x)^{5/2} (b c-a d)}{d^6}+\frac{20 b^3 (c+d x)^{3/2} (b c-a d)^2}{3 d^6}-\frac{20 b^2 \sqrt{c+d x} (b c-a d)^3}{d^6}-\frac{10 b (b c-a d)^4}{d^6 \sqrt{c+d x}}+\frac{2 (b c-a d)^5}{3 d^6 (c+d x)^{3/2}}+\frac{2 b^5 (c+d x)^{7/2}}{7 d^6} \]

[Out]

(2*(b*c - a*d)^5)/(3*d^6*(c + d*x)^(3/2)) - (10*b*(b*c - a*d)^4)/(d^6*Sqrt[c + d*x]) - (20*b^2*(b*c - a*d)^3*S
qrt[c + d*x])/d^6 + (20*b^3*(b*c - a*d)^2*(c + d*x)^(3/2))/(3*d^6) - (2*b^4*(b*c - a*d)*(c + d*x)^(5/2))/d^6 +
(2*b^5*(c + d*x)^(7/2))/(7*d^6)

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Rubi [A]  time = 0.0483056, antiderivative size = 152, 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} \[ -\frac{2 b^4 (c+d x)^{5/2} (b c-a d)}{d^6}+\frac{20 b^3 (c+d x)^{3/2} (b c-a d)^2}{3 d^6}-\frac{20 b^2 \sqrt{c+d x} (b c-a d)^3}{d^6}-\frac{10 b (b c-a d)^4}{d^6 \sqrt{c+d x}}+\frac{2 (b c-a d)^5}{3 d^6 (c+d x)^{3/2}}+\frac{2 b^5 (c+d x)^{7/2}}{7 d^6} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^5/(c + d*x)^(5/2),x]

[Out]

(2*(b*c - a*d)^5)/(3*d^6*(c + d*x)^(3/2)) - (10*b*(b*c - a*d)^4)/(d^6*Sqrt[c + d*x]) - (20*b^2*(b*c - a*d)^3*S
qrt[c + d*x])/d^6 + (20*b^3*(b*c - a*d)^2*(c + d*x)^(3/2))/(3*d^6) - (2*b^4*(b*c - a*d)*(c + d*x)^(5/2))/d^6 +
(2*b^5*(c + d*x)^(7/2))/(7*d^6)

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)^5}{(c+d x)^{5/2}} \, dx &=\int \left (\frac{(-b c+a d)^5}{d^5 (c+d x)^{5/2}}+\frac{5 b (b c-a d)^4}{d^5 (c+d x)^{3/2}}-\frac{10 b^2 (b c-a d)^3}{d^5 \sqrt{c+d x}}+\frac{10 b^3 (b c-a d)^2 \sqrt{c+d x}}{d^5}-\frac{5 b^4 (b c-a d) (c+d x)^{3/2}}{d^5}+\frac{b^5 (c+d x)^{5/2}}{d^5}\right ) \, dx\\ &=\frac{2 (b c-a d)^5}{3 d^6 (c+d x)^{3/2}}-\frac{10 b (b c-a d)^4}{d^6 \sqrt{c+d x}}-\frac{20 b^2 (b c-a d)^3 \sqrt{c+d x}}{d^6}+\frac{20 b^3 (b c-a d)^2 (c+d x)^{3/2}}{3 d^6}-\frac{2 b^4 (b c-a d) (c+d x)^{5/2}}{d^6}+\frac{2 b^5 (c+d x)^{7/2}}{7 d^6}\\ \end{align*}

Mathematica [A]  time = 0.112737, size = 123, normalized size = 0.81 \[ \frac{2 \left (-210 b^2 (c+d x)^2 (b c-a d)^3+70 b^3 (c+d x)^3 (b c-a d)^2-21 b^4 (c+d x)^4 (b c-a d)-105 b (c+d x) (b c-a d)^4+7 (b c-a d)^5+3 b^5 (c+d x)^5\right )}{21 d^6 (c+d x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(7*(b*c - a*d)^5 - 105*b*(b*c - a*d)^4*(c + d*x) - 210*b^2*(b*c - a*d)^3*(c + d*x)^2 + 70*b^3*(b*c - a*d)^2
*(c + d*x)^3 - 21*b^4*(b*c - a*d)*(c + d*x)^4 + 3*b^5*(c + d*x)^5))/(21*d^6*(c + d*x)^(3/2))

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Maple [B]  time = 0.006, size = 273, normalized size = 1.8 \begin{align*} -{\frac{-6\,{b}^{5}{x}^{5}{d}^{5}-42\,a{b}^{4}{d}^{5}{x}^{4}+12\,{b}^{5}c{d}^{4}{x}^{4}-140\,{a}^{2}{b}^{3}{d}^{5}{x}^{3}+112\,a{b}^{4}c{d}^{4}{x}^{3}-32\,{b}^{5}{c}^{2}{d}^{3}{x}^{3}-420\,{a}^{3}{b}^{2}{d}^{5}{x}^{2}+840\,{a}^{2}{b}^{3}c{d}^{4}{x}^{2}-672\,a{b}^{4}{c}^{2}{d}^{3}{x}^{2}+192\,{b}^{5}{c}^{3}{d}^{2}{x}^{2}+210\,{a}^{4}b{d}^{5}x-1680\,{a}^{3}{b}^{2}c{d}^{4}x+3360\,{a}^{2}{b}^{3}{c}^{2}{d}^{3}x-2688\,a{b}^{4}{c}^{3}{d}^{2}x+768\,{b}^{5}{c}^{4}dx+14\,{a}^{5}{d}^{5}+140\,{a}^{4}bc{d}^{4}-1120\,{a}^{3}{b}^{2}{c}^{2}{d}^{3}+2240\,{a}^{2}{b}^{3}{c}^{3}{d}^{2}-1792\,a{b}^{4}{c}^{4}d+512\,{b}^{5}{c}^{5}}{21\,{d}^{6}} \left ( dx+c \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/21/(d*x+c)^(3/2)*(-3*b^5*d^5*x^5-21*a*b^4*d^5*x^4+6*b^5*c*d^4*x^4-70*a^2*b^3*d^5*x^3+56*a*b^4*c*d^4*x^3-16*
b^5*c^2*d^3*x^3-210*a^3*b^2*d^5*x^2+420*a^2*b^3*c*d^4*x^2-336*a*b^4*c^2*d^3*x^2+96*b^5*c^3*d^2*x^2+105*a^4*b*d
^5*x-840*a^3*b^2*c*d^4*x+1680*a^2*b^3*c^2*d^3*x-1344*a*b^4*c^3*d^2*x+384*b^5*c^4*d*x+7*a^5*d^5+70*a^4*b*c*d^4-
560*a^3*b^2*c^2*d^3+1120*a^2*b^3*c^3*d^2-896*a*b^4*c^4*d+256*b^5*c^5)/d^6

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Maxima [A]  time = 0.961379, size = 358, normalized size = 2.36 \begin{align*} \frac{2 \,{\left (\frac{3 \,{\left (d x + c\right )}^{\frac{7}{2}} b^{5} - 21 \,{\left (b^{5} c - a b^{4} d\right )}{\left (d x + c\right )}^{\frac{5}{2}} + 70 \,{\left (b^{5} c^{2} - 2 \, a b^{4} c d + a^{2} b^{3} d^{2}\right )}{\left (d x + c\right )}^{\frac{3}{2}} - 210 \,{\left (b^{5} c^{3} - 3 \, a b^{4} c^{2} d + 3 \, a^{2} b^{3} c d^{2} - a^{3} b^{2} d^{3}\right )} \sqrt{d x + c}}{d^{5}} + \frac{7 \,{\left (b^{5} c^{5} - 5 \, a b^{4} c^{4} d + 10 \, a^{2} b^{3} c^{3} d^{2} - 10 \, a^{3} b^{2} c^{2} d^{3} + 5 \, a^{4} b c d^{4} - a^{5} d^{5} - 15 \,{\left (b^{5} c^{4} - 4 \, a b^{4} c^{3} d + 6 \, a^{2} b^{3} c^{2} d^{2} - 4 \, a^{3} b^{2} c d^{3} + a^{4} b d^{4}\right )}{\left (d x + c\right )}\right )}}{{\left (d x + c\right )}^{\frac{3}{2}} d^{5}}\right )}}{21 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/21*((3*(d*x + c)^(7/2)*b^5 - 21*(b^5*c - a*b^4*d)*(d*x + c)^(5/2) + 70*(b^5*c^2 - 2*a*b^4*c*d + a^2*b^3*d^2)
*(d*x + c)^(3/2) - 210*(b^5*c^3 - 3*a*b^4*c^2*d + 3*a^2*b^3*c*d^2 - a^3*b^2*d^3)*sqrt(d*x + c))/d^5 + 7*(b^5*c
^5 - 5*a*b^4*c^4*d + 10*a^2*b^3*c^3*d^2 - 10*a^3*b^2*c^2*d^3 + 5*a^4*b*c*d^4 - a^5*d^5 - 15*(b^5*c^4 - 4*a*b^4
*c^3*d + 6*a^2*b^3*c^2*d^2 - 4*a^3*b^2*c*d^3 + a^4*b*d^4)*(d*x + c))/((d*x + c)^(3/2)*d^5))/d

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Fricas [B]  time = 2.08844, size = 603, normalized size = 3.97 \begin{align*} \frac{2 \,{\left (3 \, b^{5} d^{5} x^{5} - 256 \, b^{5} c^{5} + 896 \, a b^{4} c^{4} d - 1120 \, a^{2} b^{3} c^{3} d^{2} + 560 \, a^{3} b^{2} c^{2} d^{3} - 70 \, a^{4} b c d^{4} - 7 \, a^{5} d^{5} - 3 \,{\left (2 \, b^{5} c d^{4} - 7 \, a b^{4} d^{5}\right )} x^{4} + 2 \,{\left (8 \, b^{5} c^{2} d^{3} - 28 \, a b^{4} c d^{4} + 35 \, a^{2} b^{3} d^{5}\right )} x^{3} - 6 \,{\left (16 \, b^{5} c^{3} d^{2} - 56 \, a b^{4} c^{2} d^{3} + 70 \, a^{2} b^{3} c d^{4} - 35 \, a^{3} b^{2} d^{5}\right )} x^{2} - 3 \,{\left (128 \, b^{5} c^{4} d - 448 \, a b^{4} c^{3} d^{2} + 560 \, a^{2} b^{3} c^{2} d^{3} - 280 \, a^{3} b^{2} c d^{4} + 35 \, a^{4} b d^{5}\right )} x\right )} \sqrt{d x + c}}{21 \,{\left (d^{8} x^{2} + 2 \, c d^{7} x + c^{2} d^{6}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/21*(3*b^5*d^5*x^5 - 256*b^5*c^5 + 896*a*b^4*c^4*d - 1120*a^2*b^3*c^3*d^2 + 560*a^3*b^2*c^2*d^3 - 70*a^4*b*c*
d^4 - 7*a^5*d^5 - 3*(2*b^5*c*d^4 - 7*a*b^4*d^5)*x^4 + 2*(8*b^5*c^2*d^3 - 28*a*b^4*c*d^4 + 35*a^2*b^3*d^5)*x^3
- 6*(16*b^5*c^3*d^2 - 56*a*b^4*c^2*d^3 + 70*a^2*b^3*c*d^4 - 35*a^3*b^2*d^5)*x^2 - 3*(128*b^5*c^4*d - 448*a*b^4
*c^3*d^2 + 560*a^2*b^3*c^2*d^3 - 280*a^3*b^2*c*d^4 + 35*a^4*b*d^5)*x)*sqrt(d*x + c)/(d^8*x^2 + 2*c*d^7*x + c^2
*d^6)

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Sympy [A]  time = 38.8678, size = 196, normalized size = 1.29 \begin{align*} \frac{2 b^{5} \left (c + d x\right )^{\frac{7}{2}}}{7 d^{6}} - \frac{10 b \left (a d - b c\right )^{4}}{d^{6} \sqrt{c + d x}} + \frac{\left (c + d x\right )^{\frac{5}{2}} \left (10 a b^{4} d - 10 b^{5} c\right )}{5 d^{6}} + \frac{\left (c + d x\right )^{\frac{3}{2}} \left (20 a^{2} b^{3} d^{2} - 40 a b^{4} c d + 20 b^{5} c^{2}\right )}{3 d^{6}} + \frac{\sqrt{c + d x} \left (20 a^{3} b^{2} d^{3} - 60 a^{2} b^{3} c d^{2} + 60 a b^{4} c^{2} d - 20 b^{5} c^{3}\right )}{d^{6}} - \frac{2 \left (a d - b c\right )^{5}}{3 d^{6} \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)**5/(d*x+c)**(5/2),x)

[Out]

2*b**5*(c + d*x)**(7/2)/(7*d**6) - 10*b*(a*d - b*c)**4/(d**6*sqrt(c + d*x)) + (c + d*x)**(5/2)*(10*a*b**4*d -
10*b**5*c)/(5*d**6) + (c + d*x)**(3/2)*(20*a**2*b**3*d**2 - 40*a*b**4*c*d + 20*b**5*c**2)/(3*d**6) + sqrt(c +
d*x)*(20*a**3*b**2*d**3 - 60*a**2*b**3*c*d**2 + 60*a*b**4*c**2*d - 20*b**5*c**3)/d**6 - 2*(a*d - b*c)**5/(3*d*
*6*(c + d*x)**(3/2))

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Giac [B]  time = 1.07182, size = 452, normalized size = 2.97 \begin{align*} -\frac{2 \,{\left (15 \,{\left (d x + c\right )} b^{5} c^{4} - b^{5} c^{5} - 60 \,{\left (d x + c\right )} a b^{4} c^{3} d + 5 \, a b^{4} c^{4} d + 90 \,{\left (d x + c\right )} a^{2} b^{3} c^{2} d^{2} - 10 \, a^{2} b^{3} c^{3} d^{2} - 60 \,{\left (d x + c\right )} a^{3} b^{2} c d^{3} + 10 \, a^{3} b^{2} c^{2} d^{3} + 15 \,{\left (d x + c\right )} a^{4} b d^{4} - 5 \, a^{4} b c d^{4} + a^{5} d^{5}\right )}}{3 \,{\left (d x + c\right )}^{\frac{3}{2}} d^{6}} + \frac{2 \,{\left (3 \,{\left (d x + c\right )}^{\frac{7}{2}} b^{5} d^{36} - 21 \,{\left (d x + c\right )}^{\frac{5}{2}} b^{5} c d^{36} + 70 \,{\left (d x + c\right )}^{\frac{3}{2}} b^{5} c^{2} d^{36} - 210 \, \sqrt{d x + c} b^{5} c^{3} d^{36} + 21 \,{\left (d x + c\right )}^{\frac{5}{2}} a b^{4} d^{37} - 140 \,{\left (d x + c\right )}^{\frac{3}{2}} a b^{4} c d^{37} + 630 \, \sqrt{d x + c} a b^{4} c^{2} d^{37} + 70 \,{\left (d x + c\right )}^{\frac{3}{2}} a^{2} b^{3} d^{38} - 630 \, \sqrt{d x + c} a^{2} b^{3} c d^{38} + 210 \, \sqrt{d x + c} a^{3} b^{2} d^{39}\right )}}{21 \, d^{42}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*(15*(d*x + c)*b^5*c^4 - b^5*c^5 - 60*(d*x + c)*a*b^4*c^3*d + 5*a*b^4*c^4*d + 90*(d*x + c)*a^2*b^3*c^2*d^2
- 10*a^2*b^3*c^3*d^2 - 60*(d*x + c)*a^3*b^2*c*d^3 + 10*a^3*b^2*c^2*d^3 + 15*(d*x + c)*a^4*b*d^4 - 5*a^4*b*c*d
^4 + a^5*d^5)/((d*x + c)^(3/2)*d^6) + 2/21*(3*(d*x + c)^(7/2)*b^5*d^36 - 21*(d*x + c)^(5/2)*b^5*c*d^36 + 70*(d
*x + c)^(3/2)*b^5*c^2*d^36 - 210*sqrt(d*x + c)*b^5*c^3*d^36 + 21*(d*x + c)^(5/2)*a*b^4*d^37 - 140*(d*x + c)^(3
/2)*a*b^4*c*d^37 + 630*sqrt(d*x + c)*a*b^4*c^2*d^37 + 70*(d*x + c)^(3/2)*a^2*b^3*d^38 - 630*sqrt(d*x + c)*a^2*
b^3*c*d^38 + 210*sqrt(d*x + c)*a^3*b^2*d^39)/d^42