3.1468 \(\int \frac{\sqrt{c+d x}}{(a+b x)^{9/2}} \, dx\)

Optimal. Leaf size=101 \[ -\frac{16 d^2 (c+d x)^{3/2}}{105 (a+b x)^{3/2} (b c-a d)^3}+\frac{8 d (c+d x)^{3/2}}{35 (a+b x)^{5/2} (b c-a d)^2}-\frac{2 (c+d x)^{3/2}}{7 (a+b x)^{7/2} (b c-a d)} \]

[Out]

(-2*(c + d*x)^(3/2))/(7*(b*c - a*d)*(a + b*x)^(7/2)) + (8*d*(c + d*x)^(3/2))/(35*(b*c - a*d)^2*(a + b*x)^(5/2)
) - (16*d^2*(c + d*x)^(3/2))/(105*(b*c - a*d)^3*(a + b*x)^(3/2))

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Rubi [A]  time = 0.016475, antiderivative size = 101, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {45, 37} \[ -\frac{16 d^2 (c+d x)^{3/2}}{105 (a+b x)^{3/2} (b c-a d)^3}+\frac{8 d (c+d x)^{3/2}}{35 (a+b x)^{5/2} (b c-a d)^2}-\frac{2 (c+d x)^{3/2}}{7 (a+b x)^{7/2} (b c-a d)} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[c + d*x]/(a + b*x)^(9/2),x]

[Out]

(-2*(c + d*x)^(3/2))/(7*(b*c - a*d)*(a + b*x)^(7/2)) + (8*d*(c + d*x)^(3/2))/(35*(b*c - a*d)^2*(a + b*x)^(5/2)
) - (16*d^2*(c + d*x)^(3/2))/(105*(b*c - a*d)^3*(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)^{9/2}} \, dx &=-\frac{2 (c+d x)^{3/2}}{7 (b c-a d) (a+b x)^{7/2}}-\frac{(4 d) \int \frac{\sqrt{c+d x}}{(a+b x)^{7/2}} \, dx}{7 (b c-a d)}\\ &=-\frac{2 (c+d x)^{3/2}}{7 (b c-a d) (a+b x)^{7/2}}+\frac{8 d (c+d x)^{3/2}}{35 (b c-a d)^2 (a+b x)^{5/2}}+\frac{\left (8 d^2\right ) \int \frac{\sqrt{c+d x}}{(a+b x)^{5/2}} \, dx}{35 (b c-a d)^2}\\ &=-\frac{2 (c+d x)^{3/2}}{7 (b c-a d) (a+b x)^{7/2}}+\frac{8 d (c+d x)^{3/2}}{35 (b c-a d)^2 (a+b x)^{5/2}}-\frac{16 d^2 (c+d x)^{3/2}}{105 (b c-a d)^3 (a+b x)^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.0387565, size = 77, normalized size = 0.76 \[ -\frac{2 (c+d x)^{3/2} \left (35 a^2 d^2+14 a b d (2 d x-3 c)+b^2 \left (15 c^2-12 c d x+8 d^2 x^2\right )\right )}{105 (a+b x)^{7/2} (b c-a d)^3} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[c + d*x]/(a + b*x)^(9/2),x]

[Out]

(-2*(c + d*x)^(3/2)*(35*a^2*d^2 + 14*a*b*d*(-3*c + 2*d*x) + b^2*(15*c^2 - 12*c*d*x + 8*d^2*x^2)))/(105*(b*c -
a*d)^3*(a + b*x)^(7/2))

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Maple [A]  time = 0.006, size = 105, normalized size = 1. \begin{align*}{\frac{16\,{b}^{2}{d}^{2}{x}^{2}+56\,ab{d}^{2}x-24\,{b}^{2}cdx+70\,{a}^{2}{d}^{2}-84\,abcd+30\,{b}^{2}{c}^{2}}{105\,{a}^{3}{d}^{3}-315\,{a}^{2}bc{d}^{2}+315\,a{b}^{2}{c}^{2}d-105\,{b}^{3}{c}^{3}} \left ( dx+c \right ) ^{{\frac{3}{2}}} \left ( bx+a \right ) ^{-{\frac{7}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)^(1/2)/(b*x+a)^(9/2),x)

[Out]

2/105*(d*x+c)^(3/2)*(8*b^2*d^2*x^2+28*a*b*d^2*x-12*b^2*c*d*x+35*a^2*d^2-42*a*b*c*d+15*b^2*c^2)/(b*x+a)^(7/2)/(
a^3*d^3-3*a^2*b*c*d^2+3*a*b^2*c^2*d-b^3*c^3)

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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)^(9/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 10.1854, size = 682, normalized size = 6.75 \begin{align*} -\frac{2 \,{\left (8 \, b^{2} d^{3} x^{3} + 15 \, b^{2} c^{3} - 42 \, a b c^{2} d + 35 \, a^{2} c d^{2} - 4 \,{\left (b^{2} c d^{2} - 7 \, a b d^{3}\right )} x^{2} +{\left (3 \, b^{2} c^{2} d - 14 \, a b c d^{2} + 35 \, a^{2} d^{3}\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{105 \,{\left (a^{4} b^{3} c^{3} - 3 \, a^{5} b^{2} c^{2} d + 3 \, a^{6} b c d^{2} - a^{7} d^{3} +{\left (b^{7} c^{3} - 3 \, a b^{6} c^{2} d + 3 \, a^{2} b^{5} c d^{2} - a^{3} b^{4} d^{3}\right )} x^{4} + 4 \,{\left (a b^{6} c^{3} - 3 \, a^{2} b^{5} c^{2} d + 3 \, a^{3} b^{4} c d^{2} - a^{4} b^{3} d^{3}\right )} x^{3} + 6 \,{\left (a^{2} b^{5} c^{3} - 3 \, a^{3} b^{4} c^{2} d + 3 \, a^{4} b^{3} c d^{2} - a^{5} b^{2} d^{3}\right )} x^{2} + 4 \,{\left (a^{3} b^{4} c^{3} - 3 \, a^{4} b^{3} c^{2} d + 3 \, a^{5} b^{2} c d^{2} - a^{6} b d^{3}\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)^(9/2),x, algorithm="fricas")

[Out]

-2/105*(8*b^2*d^3*x^3 + 15*b^2*c^3 - 42*a*b*c^2*d + 35*a^2*c*d^2 - 4*(b^2*c*d^2 - 7*a*b*d^3)*x^2 + (3*b^2*c^2*
d - 14*a*b*c*d^2 + 35*a^2*d^3)*x)*sqrt(b*x + a)*sqrt(d*x + c)/(a^4*b^3*c^3 - 3*a^5*b^2*c^2*d + 3*a^6*b*c*d^2 -
a^7*d^3 + (b^7*c^3 - 3*a*b^6*c^2*d + 3*a^2*b^5*c*d^2 - a^3*b^4*d^3)*x^4 + 4*(a*b^6*c^3 - 3*a^2*b^5*c^2*d + 3*
a^3*b^4*c*d^2 - a^4*b^3*d^3)*x^3 + 6*(a^2*b^5*c^3 - 3*a^3*b^4*c^2*d + 3*a^4*b^3*c*d^2 - a^5*b^2*d^3)*x^2 + 4*(
a^3*b^4*c^3 - 3*a^4*b^3*c^2*d + 3*a^5*b^2*c*d^2 - a^6*b*d^3)*x)

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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)**(9/2),x)

[Out]

Timed out

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Giac [B]  time = 1.28461, size = 930, normalized size = 9.21 \begin{align*} -\frac{32 \,{\left (\sqrt{b d} b^{10} c^{4} d^{3} - 4 \, \sqrt{b d} a b^{9} c^{3} d^{4} + 6 \, \sqrt{b d} a^{2} b^{8} c^{2} d^{5} - 4 \, \sqrt{b d} a^{3} b^{7} c d^{6} + \sqrt{b d} a^{4} b^{6} d^{7} - 7 \, \sqrt{b d}{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{2} b^{8} c^{3} d^{3} + 21 \, \sqrt{b d}{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{2} a b^{7} c^{2} d^{4} - 21 \, \sqrt{b d}{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{2} a^{2} b^{6} c d^{5} + 7 \, \sqrt{b d}{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{2} a^{3} b^{5} d^{6} + 21 \, \sqrt{b d}{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{4} b^{6} c^{2} d^{3} - 42 \, \sqrt{b d}{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{4} a b^{5} c d^{4} + 21 \, \sqrt{b d}{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{4} a^{2} b^{4} d^{5} + 35 \, \sqrt{b d}{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{6} b^{4} c d^{3} - 35 \, \sqrt{b d}{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{6} a b^{3} d^{4} + 70 \, \sqrt{b d}{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{8} b^{2} d^{3}\right )}{\left | b \right |}}{105 \,{\left (b^{2} c - a b d -{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{2}\right )}^{7} b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^(1/2)/(b*x+a)^(9/2),x, algorithm="giac")

[Out]

-32/105*(sqrt(b*d)*b^10*c^4*d^3 - 4*sqrt(b*d)*a*b^9*c^3*d^4 + 6*sqrt(b*d)*a^2*b^8*c^2*d^5 - 4*sqrt(b*d)*a^3*b^
7*c*d^6 + sqrt(b*d)*a^4*b^6*d^7 - 7*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^
2*b^8*c^3*d^3 + 21*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^7*c^2*d^4 -
21*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^6*c*d^5 + 7*sqrt(b*d)*(s
qrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^3*b^5*d^6 + 21*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x
+ a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*b^6*c^2*d^3 - 42*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*
c + (b*x + a)*b*d - a*b*d))^4*a*b^5*c*d^4 + 21*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^4*d^5 + 35*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*b^4*
c*d^3 - 35*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^3*d^4 + 70*sqrt(b*d
)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^8*b^2*d^3)*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)^7*b^2)