3.1498 \(\int \frac{1}{(a+b x)^{5/2} \sqrt{c+d x}} \, dx\)

Optimal. Leaf size=66 \[ \frac{4 d \sqrt{c+d x}}{3 \sqrt{a+b x} (b c-a d)^2}-\frac{2 \sqrt{c+d x}}{3 (a+b x)^{3/2} (b c-a d)} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0083011, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {45, 37} \[ \frac{4 d \sqrt{c+d x}}{3 \sqrt{a+b x} (b c-a d)^2}-\frac{2 \sqrt{c+d x}}{3 (a+b x)^{3/2} (b c-a d)} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b*x)^(5/2)*Sqrt[c + d*x]),x]

[Out]

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

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{1}{(a+b x)^{5/2} \sqrt{c+d x}} \, dx &=-\frac{2 \sqrt{c+d x}}{3 (b c-a d) (a+b x)^{3/2}}-\frac{(2 d) \int \frac{1}{(a+b x)^{3/2} \sqrt{c+d x}} \, dx}{3 (b c-a d)}\\ &=-\frac{2 \sqrt{c+d x}}{3 (b c-a d) (a+b x)^{3/2}}+\frac{4 d \sqrt{c+d x}}{3 (b c-a d)^2 \sqrt{a+b x}}\\ \end{align*}

Mathematica [A]  time = 0.0147802, size = 46, normalized size = 0.7 \[ \frac{2 \sqrt{c+d x} (3 a d-b c+2 b d x)}{3 (a+b x)^{3/2} (b c-a d)^2} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b*x)^(5/2)*Sqrt[c + d*x]),x]

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.005, size = 54, normalized size = 0.8 \begin{align*}{\frac{4\,bdx+6\,ad-2\,bc}{3\,{a}^{2}{d}^{2}-6\,abcd+3\,{b}^{2}{c}^{2}}\sqrt{dx+c} \left ( bx+a \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b*x+a)^(5/2)/(d*x+c)^(1/2),x)

[Out]

2/3*(d*x+c)^(1/2)*(2*b*d*x+3*a*d-b*c)/(b*x+a)^(3/2)/(a^2*d^2-2*a*b*c*d+b^2*c^2)

________________________________________________________________________________________

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(1/(b*x+a)^(5/2)/(d*x+c)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [B]  time = 3.04411, size = 250, normalized size = 3.79 \begin{align*} \frac{2 \,{\left (2 \, b d x - b c + 3 \, a d\right )} \sqrt{b x + a} \sqrt{d x + c}}{3 \,{\left (a^{2} b^{2} c^{2} - 2 \, a^{3} b c d + a^{4} d^{2} +{\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} x^{2} + 2 \,{\left (a b^{3} c^{2} - 2 \, a^{2} b^{2} c d + a^{3} b d^{2}\right )} x\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)^(5/2)/(d*x+c)^(1/2),x, algorithm="fricas")

[Out]

2/3*(2*b*d*x - b*c + 3*a*d)*sqrt(b*x + a)*sqrt(d*x + c)/(a^2*b^2*c^2 - 2*a^3*b*c*d + a^4*d^2 + (b^4*c^2 - 2*a*
b^3*c*d + a^2*b^2*d^2)*x^2 + 2*(a*b^3*c^2 - 2*a^2*b^2*c*d + a^3*b*d^2)*x)

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (a + b x\right )^{\frac{5}{2}} \sqrt{c + d x}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)**(5/2)/(d*x+c)**(1/2),x)

[Out]

Integral(1/((a + b*x)**(5/2)*sqrt(c + d*x)), x)

________________________________________________________________________________________

Giac [B]  time = 1.09907, size = 163, normalized size = 2.47 \begin{align*} \frac{8 \,{\left (b^{2} c - a b d - 3 \,{\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 )} \sqrt{b d} b^{2} d}{3 \,{\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 )}^{3}{\left | b \right |}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)^(5/2)/(d*x+c)^(1/2),x, algorithm="giac")

[Out]

8/3*(b^2*c - a*b*d - 3*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2)*sqrt(b*d)*b^2*d/((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)^3*abs(b))