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

Optimal. Leaf size=80 \[ \frac{2 \sqrt{a+b x} (b c-3 a d)}{3 d \sqrt{c+d x} (b c-a d)^2}-\frac{2 c \sqrt{a+b x}}{3 d (c+d x)^{3/2} (b c-a d)} \]

[Out]

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

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Rubi [A]  time = 0.0211434, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {78, 37} \[ \frac{2 \sqrt{a+b x} (b c-3 a d)}{3 d \sqrt{c+d x} (b c-a d)^2}-\frac{2 c \sqrt{a+b x}}{3 d (c+d x)^{3/2} (b c-a d)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 78

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> -Simp[((b*e - a*f
)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(f*(p + 1)*(c*f - d*e)), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1)
+ c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f,
n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || IntegerQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || LtQ
[p, n]))))

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

Mathematica [A]  time = 0.0176622, size = 46, normalized size = 0.57 \[ \frac{2 \sqrt{a+b x} (-2 a c-3 a d x+b c x)}{3 (c+d x)^{3/2} (b c-a d)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.005, size = 55, normalized size = 0.7 \begin{align*} -{\frac{6\,adx-2\,bcx+4\,ac}{3\,{a}^{2}{d}^{2}-6\,abcd+3\,{b}^{2}{c}^{2}}\sqrt{bx+a} \left ( dx+c \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 3.18188, size = 254, normalized size = 3.18 \begin{align*} -\frac{2 \,{\left (2 \, a c -{\left (b c - 3 \, a d\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{3 \,{\left (b^{2} c^{4} - 2 \, a b c^{3} d + a^{2} c^{2} d^{2} +{\left (b^{2} c^{2} d^{2} - 2 \, a b c d^{3} + a^{2} d^{4}\right )} x^{2} + 2 \,{\left (b^{2} c^{3} d - 2 \, a b c^{2} d^{2} + a^{2} c d^{3}\right )} x\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.25741, size = 204, normalized size = 2.55 \begin{align*} -\frac{\sqrt{b x + a}{\left (\frac{{\left (b^{5} c d{\left | b \right |} - 3 \, a b^{4} d^{2}{\left | b \right |}\right )}{\left (b x + a\right )}}{b^{8} c^{2} d^{4} - 2 \, a b^{7} c d^{5} + a^{2} b^{6} d^{6}} - \frac{3 \,{\left (a b^{5} c d{\left | b \right |} - a^{2} b^{4} d^{2}{\left | b \right |}\right )}}{b^{8} c^{2} d^{4} - 2 \, a b^{7} c d^{5} + a^{2} b^{6} d^{6}}\right )}}{12 \,{\left (b^{2} c +{\left (b x + a\right )} b d - a b d\right )}^{\frac{3}{2}} b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/12*sqrt(b*x + a)*((b^5*c*d*abs(b) - 3*a*b^4*d^2*abs(b))*(b*x + a)/(b^8*c^2*d^4 - 2*a*b^7*c*d^5 + a^2*b^6*d^
6) - 3*(a*b^5*c*d*abs(b) - a^2*b^4*d^2*abs(b))/(b^8*c^2*d^4 - 2*a*b^7*c*d^5 + a^2*b^6*d^6))/((b^2*c + (b*x + a
)*b*d - a*b*d)^(3/2)*b)