3.2242 \(\int \frac{A+B x}{\sqrt{a+b x} (d+e x)^{7/2}} \, dx\)

Optimal. Leaf size=145 \[ -\frac{2 \sqrt{a+b x} (B d-A e)}{5 e (d+e x)^{5/2} (b d-a e)}+\frac{4 b \sqrt{a+b x} (-5 a B e+4 A b e+b B d)}{15 e \sqrt{d+e x} (b d-a e)^3}+\frac{2 \sqrt{a+b x} (-5 a B e+4 A b e+b B d)}{15 e (d+e x)^{3/2} (b d-a e)^2} \]

[Out]

(-2*(B*d - A*e)*Sqrt[a + b*x])/(5*e*(b*d - a*e)*(d + e*x)^(5/2)) + (2*(b*B*d + 4*A*b*e - 5*a*B*e)*Sqrt[a + b*x
])/(15*e*(b*d - a*e)^2*(d + e*x)^(3/2)) + (4*b*(b*B*d + 4*A*b*e - 5*a*B*e)*Sqrt[a + b*x])/(15*e*(b*d - a*e)^3*
Sqrt[d + e*x])

________________________________________________________________________________________

Rubi [A]  time = 0.0852252, antiderivative size = 145, 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} \[ -\frac{2 \sqrt{a+b x} (B d-A e)}{5 e (d+e x)^{5/2} (b d-a e)}+\frac{4 b \sqrt{a+b x} (-5 a B e+4 A b e+b B d)}{15 e \sqrt{d+e x} (b d-a e)^3}+\frac{2 \sqrt{a+b x} (-5 a B e+4 A b e+b B d)}{15 e (d+e x)^{3/2} (b d-a e)^2} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*x)/(Sqrt[a + b*x]*(d + e*x)^(7/2)),x]

[Out]

(-2*(B*d - A*e)*Sqrt[a + b*x])/(5*e*(b*d - a*e)*(d + e*x)^(5/2)) + (2*(b*B*d + 4*A*b*e - 5*a*B*e)*Sqrt[a + b*x
])/(15*e*(b*d - a*e)^2*(d + e*x)^(3/2)) + (4*b*(b*B*d + 4*A*b*e - 5*a*B*e)*Sqrt[a + b*x])/(15*e*(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}{\sqrt{a+b x} (d+e x)^{7/2}} \, dx &=-\frac{2 (B d-A e) \sqrt{a+b x}}{5 e (b d-a e) (d+e x)^{5/2}}+\frac{(b B d+4 A b e-5 a B e) \int \frac{1}{\sqrt{a+b x} (d+e x)^{5/2}} \, dx}{5 e (b d-a e)}\\ &=-\frac{2 (B d-A e) \sqrt{a+b x}}{5 e (b d-a e) (d+e x)^{5/2}}+\frac{2 (b B d+4 A b e-5 a B e) \sqrt{a+b x}}{15 e (b d-a e)^2 (d+e x)^{3/2}}+\frac{(2 b (b B d+4 A b e-5 a B e)) \int \frac{1}{\sqrt{a+b x} (d+e x)^{3/2}} \, dx}{15 e (b d-a e)^2}\\ &=-\frac{2 (B d-A e) \sqrt{a+b x}}{5 e (b d-a e) (d+e x)^{5/2}}+\frac{2 (b B d+4 A b e-5 a B e) \sqrt{a+b x}}{15 e (b d-a e)^2 (d+e x)^{3/2}}+\frac{4 b (b B d+4 A b e-5 a B e) \sqrt{a+b x}}{15 e (b d-a e)^3 \sqrt{d+e x}}\\ \end{align*}

Mathematica [A]  time = 0.0748377, size = 133, normalized size = 0.92 \[ \frac{2 \sqrt{a+b x} \left (A \left (3 a^2 e^2-2 a b e (5 d+2 e x)+b^2 \left (15 d^2+20 d e x+8 e^2 x^2\right )\right )+B \left (a^2 e (2 d+5 e x)-2 a b \left (5 d^2+13 d e x+5 e^2 x^2\right )+b^2 d x (5 d+2 e x)\right )\right )}{15 (d+e x)^{5/2} (b d-a e)^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x)/(Sqrt[a + b*x]*(d + e*x)^(7/2)),x]

[Out]

(2*Sqrt[a + b*x]*(B*(b^2*d*x*(5*d + 2*e*x) + a^2*e*(2*d + 5*e*x) - 2*a*b*(5*d^2 + 13*d*e*x + 5*e^2*x^2)) + A*(
3*a^2*e^2 - 2*a*b*e*(5*d + 2*e*x) + b^2*(15*d^2 + 20*d*e*x + 8*e^2*x^2))))/(15*(b*d - a*e)^3*(d + e*x)^(5/2))

________________________________________________________________________________________

Maple [A]  time = 0.004, size = 177, normalized size = 1.2 \begin{align*} -{\frac{16\,A{b}^{2}{e}^{2}{x}^{2}-20\,Bab{e}^{2}{x}^{2}+4\,B{b}^{2}de{x}^{2}-8\,Aab{e}^{2}x+40\,A{b}^{2}dex+10\,B{a}^{2}{e}^{2}x-52\,Babdex+10\,B{b}^{2}{d}^{2}x+6\,A{a}^{2}{e}^{2}-20\,Aabde+30\,A{b}^{2}{d}^{2}+4\,B{a}^{2}de-20\,Bab{d}^{2}}{15\,{a}^{3}{e}^{3}-45\,{a}^{2}bd{e}^{2}+45\,a{b}^{2}{d}^{2}e-15\,{b}^{3}{d}^{3}}\sqrt{bx+a} \left ( ex+d \right ) ^{-{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)/(e*x+d)^(7/2)/(b*x+a)^(1/2),x)

[Out]

-2/15*(b*x+a)^(1/2)*(8*A*b^2*e^2*x^2-10*B*a*b*e^2*x^2+2*B*b^2*d*e*x^2-4*A*a*b*e^2*x+20*A*b^2*d*e*x+5*B*a^2*e^2
*x-26*B*a*b*d*e*x+5*B*b^2*d^2*x+3*A*a^2*e^2-10*A*a*b*d*e+15*A*b^2*d^2+2*B*a^2*d*e-10*B*a*b*d^2)/(e*x+d)^(5/2)/
(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)/(e*x+d)^(7/2)/(b*x+a)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [B]  time = 19.7202, size = 651, normalized size = 4.49 \begin{align*} \frac{2 \,{\left (3 \, A a^{2} e^{2} - 5 \,{\left (2 \, B a b - 3 \, A b^{2}\right )} d^{2} + 2 \,{\left (B a^{2} - 5 \, A a b\right )} d e + 2 \,{\left (B b^{2} d e -{\left (5 \, B a b - 4 \, A b^{2}\right )} e^{2}\right )} x^{2} +{\left (5 \, B b^{2} d^{2} - 2 \,{\left (13 \, B a b - 10 \, A b^{2}\right )} d e +{\left (5 \, B a^{2} - 4 \, A a b\right )} e^{2}\right )} x\right )} \sqrt{b x + a} \sqrt{e x + d}}{15 \,{\left (b^{3} d^{6} - 3 \, a b^{2} d^{5} e + 3 \, a^{2} b d^{4} e^{2} - a^{3} d^{3} e^{3} +{\left (b^{3} d^{3} e^{3} - 3 \, a b^{2} d^{2} e^{4} + 3 \, a^{2} b d e^{5} - a^{3} e^{6}\right )} x^{3} + 3 \,{\left (b^{3} d^{4} e^{2} - 3 \, a b^{2} d^{3} e^{3} + 3 \, a^{2} b d^{2} e^{4} - a^{3} d e^{5}\right )} x^{2} + 3 \,{\left (b^{3} d^{5} e - 3 \, a b^{2} d^{4} e^{2} + 3 \, a^{2} b d^{3} e^{3} - a^{3} d^{2} e^{4}\right )} x\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/(e*x+d)^(7/2)/(b*x+a)^(1/2),x, algorithm="fricas")

[Out]

2/15*(3*A*a^2*e^2 - 5*(2*B*a*b - 3*A*b^2)*d^2 + 2*(B*a^2 - 5*A*a*b)*d*e + 2*(B*b^2*d*e - (5*B*a*b - 4*A*b^2)*e
^2)*x^2 + (5*B*b^2*d^2 - 2*(13*B*a*b - 10*A*b^2)*d*e + (5*B*a^2 - 4*A*a*b)*e^2)*x)*sqrt(b*x + a)*sqrt(e*x + d)
/(b^3*d^6 - 3*a*b^2*d^5*e + 3*a^2*b*d^4*e^2 - a^3*d^3*e^3 + (b^3*d^3*e^3 - 3*a*b^2*d^2*e^4 + 3*a^2*b*d*e^5 - a
^3*e^6)*x^3 + 3*(b^3*d^4*e^2 - 3*a*b^2*d^3*e^3 + 3*a^2*b*d^2*e^4 - a^3*d*e^5)*x^2 + 3*(b^3*d^5*e - 3*a*b^2*d^4
*e^2 + 3*a^2*b*d^3*e^3 - a^3*d^2*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)/(e*x+d)**(7/2)/(b*x+a)**(1/2),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [B]  time = 3.02829, size = 477, normalized size = 3.29 \begin{align*} -\frac{{\left ({\left (b x + a\right )}{\left (\frac{2 \,{\left (B b^{6} d{\left | b \right |} e^{3} - 5 \, B a b^{5}{\left | b \right |} e^{4} + 4 \, A b^{6}{\left | b \right |} e^{4}\right )}{\left (b x + a\right )}}{b^{12} d^{3} e^{6} - 3 \, a b^{11} d^{2} e^{7} + 3 \, a^{2} b^{10} d e^{8} - a^{3} b^{9} e^{9}} + \frac{5 \,{\left (B b^{7} d^{2}{\left | b \right |} e^{2} - 6 \, B a b^{6} d{\left | b \right |} e^{3} + 4 \, A b^{7} d{\left | b \right |} e^{3} + 5 \, B a^{2} b^{5}{\left | b \right |} e^{4} - 4 \, A a b^{6}{\left | b \right |} e^{4}\right )}}{b^{12} d^{3} e^{6} - 3 \, a b^{11} d^{2} e^{7} + 3 \, a^{2} b^{10} d e^{8} - a^{3} b^{9} e^{9}}\right )} - \frac{15 \,{\left (B a b^{7} d^{2}{\left | b \right |} e^{2} - A b^{8} d^{2}{\left | b \right |} e^{2} - 2 \, B a^{2} b^{6} d{\left | b \right |} e^{3} + 2 \, A a b^{7} d{\left | b \right |} e^{3} + B a^{3} b^{5}{\left | b \right |} e^{4} - A a^{2} b^{6}{\left | b \right |} e^{4}\right )}}{b^{12} d^{3} e^{6} - 3 \, a b^{11} d^{2} e^{7} + 3 \, a^{2} b^{10} d e^{8} - a^{3} b^{9} e^{9}}\right )} \sqrt{b x + a}}{960 \,{\left (b^{2} d +{\left (b x + a\right )} b e - a b e\right )}^{\frac{5}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/(e*x+d)^(7/2)/(b*x+a)^(1/2),x, algorithm="giac")

[Out]

-1/960*((b*x + a)*(2*(B*b^6*d*abs(b)*e^3 - 5*B*a*b^5*abs(b)*e^4 + 4*A*b^6*abs(b)*e^4)*(b*x + a)/(b^12*d^3*e^6
- 3*a*b^11*d^2*e^7 + 3*a^2*b^10*d*e^8 - a^3*b^9*e^9) + 5*(B*b^7*d^2*abs(b)*e^2 - 6*B*a*b^6*d*abs(b)*e^3 + 4*A*
b^7*d*abs(b)*e^3 + 5*B*a^2*b^5*abs(b)*e^4 - 4*A*a*b^6*abs(b)*e^4)/(b^12*d^3*e^6 - 3*a*b^11*d^2*e^7 + 3*a^2*b^1
0*d*e^8 - a^3*b^9*e^9)) - 15*(B*a*b^7*d^2*abs(b)*e^2 - A*b^8*d^2*abs(b)*e^2 - 2*B*a^2*b^6*d*abs(b)*e^3 + 2*A*a
*b^7*d*abs(b)*e^3 + B*a^3*b^5*abs(b)*e^4 - A*a^2*b^6*abs(b)*e^4)/(b^12*d^3*e^6 - 3*a*b^11*d^2*e^7 + 3*a^2*b^10
*d*e^8 - a^3*b^9*e^9))*sqrt(b*x + a)/(b^2*d + (b*x + a)*b*e - a*b*e)^(5/2)