3.1469 \(\int \frac{\sqrt{c+d x}}{(a+b x)^{11/2}} \, dx\)

Optimal. Leaf size=136 \[ \frac{32 d^3 (c+d x)^{3/2}}{315 (a+b x)^{3/2} (b c-a d)^4}-\frac{16 d^2 (c+d x)^{3/2}}{105 (a+b x)^{5/2} (b c-a d)^3}+\frac{4 d (c+d x)^{3/2}}{21 (a+b x)^{7/2} (b c-a d)^2}-\frac{2 (c+d x)^{3/2}}{9 (a+b x)^{9/2} (b c-a d)} \]

[Out]

(-2*(c + d*x)^(3/2))/(9*(b*c - a*d)*(a + b*x)^(9/2)) + (4*d*(c + d*x)^(3/2))/(21*(b*c - a*d)^2*(a + b*x)^(7/2)
) - (16*d^2*(c + d*x)^(3/2))/(105*(b*c - a*d)^3*(a + b*x)^(5/2)) + (32*d^3*(c + d*x)^(3/2))/(315*(b*c - a*d)^4
*(a + b*x)^(3/2))

________________________________________________________________________________________

Rubi [A]  time = 0.0285565, antiderivative size = 136, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {45, 37} \[ \frac{32 d^3 (c+d x)^{3/2}}{315 (a+b x)^{3/2} (b c-a d)^4}-\frac{16 d^2 (c+d x)^{3/2}}{105 (a+b x)^{5/2} (b c-a d)^3}+\frac{4 d (c+d x)^{3/2}}{21 (a+b x)^{7/2} (b c-a d)^2}-\frac{2 (c+d x)^{3/2}}{9 (a+b x)^{9/2} (b c-a d)} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[c + d*x]/(a + b*x)^(11/2),x]

[Out]

(-2*(c + d*x)^(3/2))/(9*(b*c - a*d)*(a + b*x)^(9/2)) + (4*d*(c + d*x)^(3/2))/(21*(b*c - a*d)^2*(a + b*x)^(7/2)
) - (16*d^2*(c + d*x)^(3/2))/(105*(b*c - a*d)^3*(a + b*x)^(5/2)) + (32*d^3*(c + d*x)^(3/2))/(315*(b*c - a*d)^4
*(a + b*x)^(3/2))

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*Simplify[m + n + 2])/((b*c - a*d)*(m + 1)), Int[(a + b*x)^Simplify[m +
1]*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[Simplify[m + n + 2], 0] &&
NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (
SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 1])

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rubi steps

\begin{align*} \int \frac{\sqrt{c+d x}}{(a+b x)^{11/2}} \, dx &=-\frac{2 (c+d x)^{3/2}}{9 (b c-a d) (a+b x)^{9/2}}-\frac{(2 d) \int \frac{\sqrt{c+d x}}{(a+b x)^{9/2}} \, dx}{3 (b c-a d)}\\ &=-\frac{2 (c+d x)^{3/2}}{9 (b c-a d) (a+b x)^{9/2}}+\frac{4 d (c+d x)^{3/2}}{21 (b c-a d)^2 (a+b x)^{7/2}}+\frac{\left (8 d^2\right ) \int \frac{\sqrt{c+d x}}{(a+b x)^{7/2}} \, dx}{21 (b c-a d)^2}\\ &=-\frac{2 (c+d x)^{3/2}}{9 (b c-a d) (a+b x)^{9/2}}+\frac{4 d (c+d x)^{3/2}}{21 (b c-a d)^2 (a+b x)^{7/2}}-\frac{16 d^2 (c+d x)^{3/2}}{105 (b c-a d)^3 (a+b x)^{5/2}}-\frac{\left (16 d^3\right ) \int \frac{\sqrt{c+d x}}{(a+b x)^{5/2}} \, dx}{105 (b c-a d)^3}\\ &=-\frac{2 (c+d x)^{3/2}}{9 (b c-a d) (a+b x)^{9/2}}+\frac{4 d (c+d x)^{3/2}}{21 (b c-a d)^2 (a+b x)^{7/2}}-\frac{16 d^2 (c+d x)^{3/2}}{105 (b c-a d)^3 (a+b x)^{5/2}}+\frac{32 d^3 (c+d x)^{3/2}}{315 (b c-a d)^4 (a+b x)^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.0528143, size = 118, normalized size = 0.87 \[ \frac{2 (c+d x)^{3/2} \left (63 a^2 b d^2 (2 d x-3 c)+105 a^3 d^3+9 a b^2 d \left (15 c^2-12 c d x+8 d^2 x^2\right )+b^3 \left (30 c^2 d x-35 c^3-24 c d^2 x^2+16 d^3 x^3\right )\right )}{315 (a+b x)^{9/2} (b c-a d)^4} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[c + d*x]/(a + b*x)^(11/2),x]

[Out]

(2*(c + d*x)^(3/2)*(105*a^3*d^3 + 63*a^2*b*d^2*(-3*c + 2*d*x) + 9*a*b^2*d*(15*c^2 - 12*c*d*x + 8*d^2*x^2) + b^
3*(-35*c^3 + 30*c^2*d*x - 24*c*d^2*x^2 + 16*d^3*x^3)))/(315*(b*c - a*d)^4*(a + b*x)^(9/2))

________________________________________________________________________________________

Maple [A]  time = 0.009, size = 171, normalized size = 1.3 \begin{align*}{\frac{32\,{b}^{3}{d}^{3}{x}^{3}+144\,a{b}^{2}{d}^{3}{x}^{2}-48\,{b}^{3}c{d}^{2}{x}^{2}+252\,{a}^{2}b{d}^{3}x-216\,a{b}^{2}c{d}^{2}x+60\,{b}^{3}{c}^{2}dx+210\,{a}^{3}{d}^{3}-378\,{a}^{2}bc{d}^{2}+270\,a{b}^{2}{c}^{2}d-70\,{b}^{3}{c}^{3}}{315\,{d}^{4}{a}^{4}-1260\,b{d}^{3}c{a}^{3}+1890\,{b}^{2}{d}^{2}{c}^{2}{a}^{2}-1260\,{b}^{3}d{c}^{3}a+315\,{b}^{4}{c}^{4}} \left ( dx+c \right ) ^{{\frac{3}{2}}} \left ( bx+a \right ) ^{-{\frac{9}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)^(1/2)/(b*x+a)^(11/2),x)

[Out]

2/315*(d*x+c)^(3/2)*(16*b^3*d^3*x^3+72*a*b^2*d^3*x^2-24*b^3*c*d^2*x^2+126*a^2*b*d^3*x-108*a*b^2*c*d^2*x+30*b^3
*c^2*d*x+105*a^3*d^3-189*a^2*b*c*d^2+135*a*b^2*c^2*d-35*b^3*c^3)/(b*x+a)^(9/2)/(a^4*d^4-4*a^3*b*c*d^3+6*a^2*b^
2*c^2*d^2-4*a*b^3*c^3*d+b^4*c^4)

________________________________________________________________________________________

Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^(1/2)/(b*x+a)^(11/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [B]  time = 42.6591, size = 1080, normalized size = 7.94 \begin{align*} \frac{2 \,{\left (16 \, b^{3} d^{4} x^{4} - 35 \, b^{3} c^{4} + 135 \, a b^{2} c^{3} d - 189 \, a^{2} b c^{2} d^{2} + 105 \, a^{3} c d^{3} - 8 \,{\left (b^{3} c d^{3} - 9 \, a b^{2} d^{4}\right )} x^{3} + 6 \,{\left (b^{3} c^{2} d^{2} - 6 \, a b^{2} c d^{3} + 21 \, a^{2} b d^{4}\right )} x^{2} -{\left (5 \, b^{3} c^{3} d - 27 \, a b^{2} c^{2} d^{2} + 63 \, a^{2} b c d^{3} - 105 \, a^{3} d^{4}\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{315 \,{\left (a^{5} b^{4} c^{4} - 4 \, a^{6} b^{3} c^{3} d + 6 \, a^{7} b^{2} c^{2} d^{2} - 4 \, a^{8} b c d^{3} + a^{9} d^{4} +{\left (b^{9} c^{4} - 4 \, a b^{8} c^{3} d + 6 \, a^{2} b^{7} c^{2} d^{2} - 4 \, a^{3} b^{6} c d^{3} + a^{4} b^{5} d^{4}\right )} x^{5} + 5 \,{\left (a b^{8} c^{4} - 4 \, a^{2} b^{7} c^{3} d + 6 \, a^{3} b^{6} c^{2} d^{2} - 4 \, a^{4} b^{5} c d^{3} + a^{5} b^{4} d^{4}\right )} x^{4} + 10 \,{\left (a^{2} b^{7} c^{4} - 4 \, a^{3} b^{6} c^{3} d + 6 \, a^{4} b^{5} c^{2} d^{2} - 4 \, a^{5} b^{4} c d^{3} + a^{6} b^{3} d^{4}\right )} x^{3} + 10 \,{\left (a^{3} b^{6} c^{4} - 4 \, a^{4} b^{5} c^{3} d + 6 \, a^{5} b^{4} c^{2} d^{2} - 4 \, a^{6} b^{3} c d^{3} + a^{7} b^{2} d^{4}\right )} x^{2} + 5 \,{\left (a^{4} b^{5} c^{4} - 4 \, a^{5} b^{4} c^{3} d + 6 \, a^{6} b^{3} c^{2} d^{2} - 4 \, a^{7} b^{2} c d^{3} + a^{8} b d^{4}\right )} x\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^(1/2)/(b*x+a)^(11/2),x, algorithm="fricas")

[Out]

2/315*(16*b^3*d^4*x^4 - 35*b^3*c^4 + 135*a*b^2*c^3*d - 189*a^2*b*c^2*d^2 + 105*a^3*c*d^3 - 8*(b^3*c*d^3 - 9*a*
b^2*d^4)*x^3 + 6*(b^3*c^2*d^2 - 6*a*b^2*c*d^3 + 21*a^2*b*d^4)*x^2 - (5*b^3*c^3*d - 27*a*b^2*c^2*d^2 + 63*a^2*b
*c*d^3 - 105*a^3*d^4)*x)*sqrt(b*x + a)*sqrt(d*x + c)/(a^5*b^4*c^4 - 4*a^6*b^3*c^3*d + 6*a^7*b^2*c^2*d^2 - 4*a^
8*b*c*d^3 + a^9*d^4 + (b^9*c^4 - 4*a*b^8*c^3*d + 6*a^2*b^7*c^2*d^2 - 4*a^3*b^6*c*d^3 + a^4*b^5*d^4)*x^5 + 5*(a
*b^8*c^4 - 4*a^2*b^7*c^3*d + 6*a^3*b^6*c^2*d^2 - 4*a^4*b^5*c*d^3 + a^5*b^4*d^4)*x^4 + 10*(a^2*b^7*c^4 - 4*a^3*
b^6*c^3*d + 6*a^4*b^5*c^2*d^2 - 4*a^5*b^4*c*d^3 + a^6*b^3*d^4)*x^3 + 10*(a^3*b^6*c^4 - 4*a^4*b^5*c^3*d + 6*a^5
*b^4*c^2*d^2 - 4*a^6*b^3*c*d^3 + a^7*b^2*d^4)*x^2 + 5*(a^4*b^5*c^4 - 4*a^5*b^4*c^3*d + 6*a^6*b^3*c^2*d^2 - 4*a
^7*b^2*c*d^3 + a^8*b*d^4)*x)

________________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)**(1/2)/(b*x+a)**(11/2),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [B]  time = 1.41645, size = 1335, normalized size = 9.82 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^(1/2)/(b*x+a)^(11/2),x, algorithm="giac")

[Out]

64/315*(sqrt(b*d)*b^13*c^5*d^4 - 5*sqrt(b*d)*a*b^12*c^4*d^5 + 10*sqrt(b*d)*a^2*b^11*c^3*d^6 - 10*sqrt(b*d)*a^3
*b^10*c^2*d^7 + 5*sqrt(b*d)*a^4*b^9*c*d^8 - sqrt(b*d)*a^5*b^8*d^9 - 9*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqr
t(b^2*c + (b*x + a)*b*d - a*b*d))^2*b^11*c^4*d^4 + 36*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x +
a)*b*d - a*b*d))^2*a*b^10*c^3*d^5 - 54*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*
d))^2*a^2*b^9*c^2*d^6 + 36*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^3*b^8
*c*d^7 - 9*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^4*b^7*d^8 + 36*sqrt(b
*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*b^9*c^3*d^4 - 108*sqrt(b*d)*(sqrt(b*d)*s
qrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*a*b^8*c^2*d^5 + 108*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) -
sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*a^2*b^7*c*d^6 - 36*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (
b*x + a)*b*d - a*b*d))^4*a^3*b^6*d^7 - 84*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*
b*d))^6*b^7*c^2*d^4 + 168*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*a*b^6*c*
d^5 - 84*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*a^2*b^5*d^6 - 189*sqrt(b*
d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^8*b^5*c*d^4 + 189*sqrt(b*d)*(sqrt(b*d)*sqrt
(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^8*a*b^4*d^5 - 315*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b
^2*c + (b*x + a)*b*d - a*b*d))^10*b^3*d^4)*abs(b)/((b^2*c - a*b*d - (sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b
*x + a)*b*d - a*b*d))^2)^9*b^2)