3.2250 \(\int \frac{A+B x}{(a+b x)^{3/2} (d+e x)^{5/2}} \, dx\)

Optimal. Leaf size=139 \[ -\frac{2 (A b-a B)}{b \sqrt{a+b x} (d+e x)^{3/2} (b d-a e)}+\frac{4 \sqrt{a+b x} (3 a B e-4 A b e+b B d)}{3 \sqrt{d+e x} (b d-a e)^3}+\frac{2 \sqrt{a+b x} (3 a B e-4 A b e+b B d)}{3 b (d+e x)^{3/2} (b d-a e)^2} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0858489, antiderivative size = 139, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {78, 45, 37} \[ -\frac{2 (A b-a B)}{b \sqrt{a+b x} (d+e x)^{3/2} (b d-a e)}+\frac{4 \sqrt{a+b x} (3 a B e-4 A b e+b B d)}{3 \sqrt{d+e x} (b d-a e)^3}+\frac{2 \sqrt{a+b x} (3 a B e-4 A b e+b B d)}{3 b (d+e x)^{3/2} (b d-a e)^2} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*x)/((a + b*x)^(3/2)*(d + e*x)^(5/2)),x]

[Out]

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

Mathematica [A]  time = 0.0642196, size = 133, normalized size = 0.96 \[ \frac{2 \left (a^2 A e^2+a^2 B e (2 d+3 e x)-2 a A b e (3 d+2 e x)+2 a b B \left (3 d^2+5 d e x+3 e^2 x^2\right )-A b^2 \left (3 d^2+12 d e x+8 e^2 x^2\right )+b^2 B d x (3 d+2 e x)\right )}{3 \sqrt{a+b x} (d+e x)^{3/2} (b d-a e)^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x)/((a + b*x)^(3/2)*(d + e*x)^(5/2)),x]

[Out]

(2*(a^2*A*e^2 - 2*a*A*b*e*(3*d + 2*e*x) + b^2*B*d*x*(3*d + 2*e*x) + a^2*B*e*(2*d + 3*e*x) + 2*a*b*B*(3*d^2 + 5
*d*e*x + 3*e^2*x^2) - A*b^2*(3*d^2 + 12*d*e*x + 8*e^2*x^2)))/(3*(b*d - a*e)^3*Sqrt[a + b*x]*(d + e*x)^(3/2))

________________________________________________________________________________________

Maple [A]  time = 0.008, size = 176, normalized size = 1.3 \begin{align*} -{\frac{-16\,A{b}^{2}{e}^{2}{x}^{2}+12\,Bab{e}^{2}{x}^{2}+4\,B{b}^{2}de{x}^{2}-8\,Aab{e}^{2}x-24\,A{b}^{2}dex+6\,B{a}^{2}{e}^{2}x+20\,Babdex+6\,B{b}^{2}{d}^{2}x+2\,A{a}^{2}{e}^{2}-12\,Aabde-6\,A{b}^{2}{d}^{2}+4\,B{a}^{2}de+12\,Bab{d}^{2}}{3\,{a}^{3}{e}^{3}-9\,{a}^{2}bd{e}^{2}+9\,a{b}^{2}{d}^{2}e-3\,{b}^{3}{d}^{3}}{\frac{1}{\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)/(b*x+a)^(3/2)/(e*x+d)^(5/2),x)

[Out]

-2/3*(-8*A*b^2*e^2*x^2+6*B*a*b*e^2*x^2+2*B*b^2*d*e*x^2-4*A*a*b*e^2*x-12*A*b^2*d*e*x+3*B*a^2*e^2*x+10*B*a*b*d*e
*x+3*B*b^2*d^2*x+A*a^2*e^2-6*A*a*b*d*e-3*A*b^2*d^2+2*B*a^2*d*e+6*B*a*b*d^2)/(b*x+a)^(1/2)/(e*x+d)^(3/2)/(a^3*e
^3-3*a^2*b*d*e^2+3*a*b^2*d^2*e-b^3*d^3)

________________________________________________________________________________________

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

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [B]  time = 15.8886, size = 684, normalized size = 4.92 \begin{align*} \frac{2 \,{\left (A a^{2} e^{2} + 3 \,{\left (2 \, B a b - A b^{2}\right )} d^{2} + 2 \,{\left (B a^{2} - 3 \, A a b\right )} d e + 2 \,{\left (B b^{2} d e +{\left (3 \, B a b - 4 \, A b^{2}\right )} e^{2}\right )} x^{2} +{\left (3 \, B b^{2} d^{2} + 2 \,{\left (5 \, B a b - 6 \, A b^{2}\right )} d e +{\left (3 \, B a^{2} - 4 \, A a b\right )} e^{2}\right )} x\right )} \sqrt{b x + a} \sqrt{e x + d}}{3 \,{\left (a b^{3} d^{5} - 3 \, a^{2} b^{2} d^{4} e + 3 \, a^{3} b d^{3} e^{2} - a^{4} d^{2} e^{3} +{\left (b^{4} d^{3} e^{2} - 3 \, a b^{3} d^{2} e^{3} + 3 \, a^{2} b^{2} d e^{4} - a^{3} b e^{5}\right )} x^{3} +{\left (2 \, b^{4} d^{4} e - 5 \, a b^{3} d^{3} e^{2} + 3 \, a^{2} b^{2} d^{2} e^{3} + a^{3} b d e^{4} - a^{4} e^{5}\right )} x^{2} +{\left (b^{4} d^{5} - a b^{3} d^{4} e - 3 \, a^{2} b^{2} d^{3} e^{2} + 5 \, a^{3} b d^{2} e^{3} - 2 \, a^{4} d e^{4}\right )} x\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*(A*a^2*e^2 + 3*(2*B*a*b - A*b^2)*d^2 + 2*(B*a^2 - 3*A*a*b)*d*e + 2*(B*b^2*d*e + (3*B*a*b - 4*A*b^2)*e^2)*x
^2 + (3*B*b^2*d^2 + 2*(5*B*a*b - 6*A*b^2)*d*e + (3*B*a^2 - 4*A*a*b)*e^2)*x)*sqrt(b*x + a)*sqrt(e*x + d)/(a*b^3
*d^5 - 3*a^2*b^2*d^4*e + 3*a^3*b*d^3*e^2 - a^4*d^2*e^3 + (b^4*d^3*e^2 - 3*a*b^3*d^2*e^3 + 3*a^2*b^2*d*e^4 - a^
3*b*e^5)*x^3 + (2*b^4*d^4*e - 5*a*b^3*d^3*e^2 + 3*a^2*b^2*d^2*e^3 + a^3*b*d*e^4 - a^4*e^5)*x^2 + (b^4*d^5 - a*
b^3*d^4*e - 3*a^2*b^2*d^3*e^2 + 5*a^3*b*d^2*e^3 - 2*a^4*d*e^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((B*x+A)/(b*x+a)**(3/2)/(e*x+d)**(5/2),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [B]  time = 2.78914, size = 551, normalized size = 3.96 \begin{align*} -\frac{\sqrt{b x + a}{\left (\frac{{\left (2 \, B b^{6} d^{3}{\left | b \right |} e^{2} - B a b^{5} d^{2}{\left | b \right |} e^{3} - 5 \, A b^{6} d^{2}{\left | b \right |} e^{3} - 4 \, B a^{2} b^{4} d{\left | b \right |} e^{4} + 10 \, A a b^{5} d{\left | b \right |} e^{4} + 3 \, B a^{3} b^{3}{\left | b \right |} e^{5} - 5 \, A a^{2} b^{4}{\left | b \right |} e^{5}\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 b^{7} d^{4}{\left | b \right |} e - 2 \, B a b^{6} d^{3}{\left | b \right |} e^{2} - 2 \, A b^{7} d^{3}{\left | b \right |} e^{2} + 6 \, A a b^{6} d^{2}{\left | b \right |} e^{3} + 2 \, B a^{3} b^{4} d{\left | b \right |} e^{4} - 6 \, A a^{2} b^{5} d{\left | b \right |} e^{4} - B a^{4} b^{3}{\left | b \right |} e^{5} + 2 \, A a^{3} b^{4}{\left | b \right |} e^{5}\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}}} + \frac{4 \,{\left (B a b^{\frac{5}{2}} e^{\frac{1}{2}} - A b^{\frac{7}{2}} e^{\frac{1}{2}}\right )}}{{\left (b^{2} d^{2}{\left | b \right |} - 2 \, a b d{\left | b \right |} e + a^{2}{\left | b \right |} e^{2}\right )}{\left (b^{2} d - a b e -{\left (\sqrt{b x + a} \sqrt{b} e^{\frac{1}{2}} - \sqrt{b^{2} d +{\left (b x + a\right )} b e - a b e}\right )}^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/48*sqrt(b*x + a)*((2*B*b^6*d^3*abs(b)*e^2 - B*a*b^5*d^2*abs(b)*e^3 - 5*A*b^6*d^2*abs(b)*e^3 - 4*B*a^2*b^4*d
*abs(b)*e^4 + 10*A*a*b^5*d*abs(b)*e^4 + 3*B*a^3*b^3*abs(b)*e^5 - 5*A*a^2*b^4*abs(b)*e^5)*(b*x + a)/(b^8*d^2*e^
4 - 2*a*b^7*d*e^5 + a^2*b^6*e^6) + 3*(B*b^7*d^4*abs(b)*e - 2*B*a*b^6*d^3*abs(b)*e^2 - 2*A*b^7*d^3*abs(b)*e^2 +
6*A*a*b^6*d^2*abs(b)*e^3 + 2*B*a^3*b^4*d*abs(b)*e^4 - 6*A*a^2*b^5*d*abs(b)*e^4 - B*a^4*b^3*abs(b)*e^5 + 2*A*a
^3*b^4*abs(b)*e^5)/(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) + 4*(B*a
*b^(5/2)*e^(1/2) - A*b^(7/2)*e^(1/2))/((b^2*d^2*abs(b) - 2*a*b*d*abs(b)*e + a^2*abs(b)*e^2)*(b^2*d - a*b*e - (
sqrt(b*x + a)*sqrt(b)*e^(1/2) - sqrt(b^2*d + (b*x + a)*b*e - a*b*e))^2))