3.2241 \(\int \frac{A+B x}{\sqrt{a+b x} (d+e x)^{5/2}} \, dx\)

Optimal. Leaf size=94 \[ \frac{2 \sqrt{a+b x} (-3 a B e+2 A b e+b B d)}{3 e \sqrt{d+e x} (b d-a e)^2}-\frac{2 \sqrt{a+b x} (B d-A e)}{3 e (d+e x)^{3/2} (b d-a e)} \]

[Out]

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

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Rubi [A]  time = 0.0466166, antiderivative size = 94, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {78, 37} \[ \frac{2 \sqrt{a+b x} (-3 a B e+2 A b e+b B d)}{3 e \sqrt{d+e x} (b d-a e)^2}-\frac{2 \sqrt{a+b x} (B d-A e)}{3 e (d+e x)^{3/2} (b d-a e)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(B*d - A*e)*Sqrt[a + b*x])/(3*e*(b*d - a*e)*(d + e*x)^(3/2)) + (2*(b*B*d + 2*A*b*e - 3*a*B*e)*Sqrt[a + b*x
])/(3*e*(b*d - a*e)^2*Sqrt[d + e*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{A+B x}{\sqrt{a+b x} (d+e x)^{5/2}} \, dx &=-\frac{2 (B d-A e) \sqrt{a+b x}}{3 e (b d-a e) (d+e x)^{3/2}}+\frac{(b B d+2 A b e-3 a B e) \int \frac{1}{\sqrt{a+b x} (d+e x)^{3/2}} \, dx}{3 e (b d-a e)}\\ &=-\frac{2 (B d-A e) \sqrt{a+b x}}{3 e (b d-a e) (d+e x)^{3/2}}+\frac{2 (b B d+2 A b e-3 a B e) \sqrt{a+b x}}{3 e (b d-a e)^2 \sqrt{d+e x}}\\ \end{align*}

Mathematica [A]  time = 0.0338448, size = 65, normalized size = 0.69 \[ \frac{2 \sqrt{a+b x} (A (-a e+3 b d+2 b e x)+B (-2 a d-3 a e x+b d x))}{3 (d+e x)^{3/2} (b d-a e)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.007, size = 73, normalized size = 0.8 \begin{align*} -{\frac{-4\,Abex+6\,Baex-2\,Bbdx+2\,Aae-6\,Abd+4\,Bad}{3\,{a}^{2}{e}^{2}-6\,bead+3\,{b}^{2}{d}^{2}}\sqrt{bx+a} \left ( ex+d \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)/(e*x+d)^(5/2)/(b*x+a)^(1/2),x)

[Out]

-2/3*(b*x+a)^(1/2)*(-2*A*b*e*x+3*B*a*e*x-B*b*d*x+A*a*e-3*A*b*d+2*B*a*d)/(e*x+d)^(3/2)/(a^2*e^2-2*a*b*d*e+b^2*d
^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((B*x+A)/(e*x+d)^(5/2)/(b*x+a)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 6.37383, size = 300, normalized size = 3.19 \begin{align*} -\frac{2 \,{\left (A a e +{\left (2 \, B a - 3 \, A b\right )} d -{\left (B b d -{\left (3 \, B a - 2 \, A b\right )} e\right )} x\right )} \sqrt{b x + a} \sqrt{e x + d}}{3 \,{\left (b^{2} d^{4} - 2 \, a b d^{3} e + a^{2} d^{2} e^{2} +{\left (b^{2} d^{2} e^{2} - 2 \, a b d e^{3} + a^{2} e^{4}\right )} x^{2} + 2 \,{\left (b^{2} d^{3} e - 2 \, a b d^{2} e^{2} + a^{2} d e^{3}\right )} x\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*(A*a*e + (2*B*a - 3*A*b)*d - (B*b*d - (3*B*a - 2*A*b)*e)*x)*sqrt(b*x + a)*sqrt(e*x + d)/(b^2*d^4 - 2*a*b*
d^3*e + a^2*d^2*e^2 + (b^2*d^2*e^2 - 2*a*b*d*e^3 + a^2*e^4)*x^2 + 2*(b^2*d^3*e - 2*a*b*d^2*e^2 + a^2*d*e^3)*x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/(e*x+d)**(5/2)/(b*x+a)**(1/2),x)

[Out]

Integral((A + B*x)/(sqrt(a + b*x)*(d + e*x)**(5/2)), x)

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Giac [B]  time = 2.37559, size = 242, normalized size = 2.57 \begin{align*} -\frac{\sqrt{b x + a}{\left (\frac{{\left (B b^{4} d{\left | b \right |} e - 3 \, B a b^{3}{\left | b \right |} e^{2} + 2 \, A b^{4}{\left | b \right |} e^{2}\right )}{\left (b x + a\right )}}{b^{8} d^{2} e^{4} - 2 \, a b^{7} d e^{5} + a^{2} b^{6} e^{6}} - \frac{3 \,{\left (B a b^{4} d{\left | b \right |} e - A b^{5} d{\left | b \right |} e - B a^{2} b^{3}{\left | b \right |} e^{2} + A a b^{4}{\left | b \right |} e^{2}\right )}}{b^{8} d^{2} e^{4} - 2 \, a b^{7} d e^{5} + a^{2} b^{6} e^{6}}\right )}}{48 \,{\left (b^{2} d +{\left (b x + a\right )} b e - a b e\right )}^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/48*sqrt(b*x + a)*((B*b^4*d*abs(b)*e - 3*B*a*b^3*abs(b)*e^2 + 2*A*b^4*abs(b)*e^2)*(b*x + a)/(b^8*d^2*e^4 - 2
*a*b^7*d*e^5 + a^2*b^6*e^6) - 3*(B*a*b^4*d*abs(b)*e - A*b^5*d*abs(b)*e - B*a^2*b^3*abs(b)*e^2 + A*a*b^4*abs(b)
*e^2)/(b^8*d^2*e^4 - 2*a*b^7*d*e^5 + a^2*b^6*e^6))/(b^2*d + (b*x + a)*b*e - a*b*e)^(3/2)