3.2207 \(\int \frac{\sqrt{a+b x} (A+B x)}{(d+e x)^{9/2}} \, dx\)

Optimal. Leaf size=147 \[ \frac{4 b (a+b x)^{3/2} (-7 a B e+4 A b e+3 b B d)}{105 e (d+e x)^{3/2} (b d-a e)^3}+\frac{2 (a+b x)^{3/2} (-7 a B e+4 A b e+3 b B d)}{35 e (d+e x)^{5/2} (b d-a e)^2}-\frac{2 (a+b x)^{3/2} (B d-A e)}{7 e (d+e x)^{7/2} (b d-a e)} \]

[Out]

(-2*(B*d - A*e)*(a + b*x)^(3/2))/(7*e*(b*d - a*e)*(d + e*x)^(7/2)) + (2*(3*b*B*d + 4*A*b*e - 7*a*B*e)*(a + b*x
)^(3/2))/(35*e*(b*d - a*e)^2*(d + e*x)^(5/2)) + (4*b*(3*b*B*d + 4*A*b*e - 7*a*B*e)*(a + b*x)^(3/2))/(105*e*(b*
d - a*e)^3*(d + e*x)^(3/2))

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Rubi [A]  time = 0.0896367, antiderivative size = 147, 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} \[ \frac{4 b (a+b x)^{3/2} (-7 a B e+4 A b e+3 b B d)}{105 e (d+e x)^{3/2} (b d-a e)^3}+\frac{2 (a+b x)^{3/2} (-7 a B e+4 A b e+3 b B d)}{35 e (d+e x)^{5/2} (b d-a e)^2}-\frac{2 (a+b x)^{3/2} (B d-A e)}{7 e (d+e x)^{7/2} (b d-a e)} \]

Antiderivative was successfully verified.

[In]

Int[(Sqrt[a + b*x]*(A + B*x))/(d + e*x)^(9/2),x]

[Out]

(-2*(B*d - A*e)*(a + b*x)^(3/2))/(7*e*(b*d - a*e)*(d + e*x)^(7/2)) + (2*(3*b*B*d + 4*A*b*e - 7*a*B*e)*(a + b*x
)^(3/2))/(35*e*(b*d - a*e)^2*(d + e*x)^(5/2)) + (4*b*(3*b*B*d + 4*A*b*e - 7*a*B*e)*(a + b*x)^(3/2))/(105*e*(b*
d - a*e)^3*(d + e*x)^(3/2))

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{\sqrt{a+b x} (A+B x)}{(d+e x)^{9/2}} \, dx &=-\frac{2 (B d-A e) (a+b x)^{3/2}}{7 e (b d-a e) (d+e x)^{7/2}}+\frac{(3 b B d+4 A b e-7 a B e) \int \frac{\sqrt{a+b x}}{(d+e x)^{7/2}} \, dx}{7 e (b d-a e)}\\ &=-\frac{2 (B d-A e) (a+b x)^{3/2}}{7 e (b d-a e) (d+e x)^{7/2}}+\frac{2 (3 b B d+4 A b e-7 a B e) (a+b x)^{3/2}}{35 e (b d-a e)^2 (d+e x)^{5/2}}+\frac{(2 b (3 b B d+4 A b e-7 a B e)) \int \frac{\sqrt{a+b x}}{(d+e x)^{5/2}} \, dx}{35 e (b d-a e)^2}\\ &=-\frac{2 (B d-A e) (a+b x)^{3/2}}{7 e (b d-a e) (d+e x)^{7/2}}+\frac{2 (3 b B d+4 A b e-7 a B e) (a+b x)^{3/2}}{35 e (b d-a e)^2 (d+e x)^{5/2}}+\frac{4 b (3 b B d+4 A b e-7 a B e) (a+b x)^{3/2}}{105 e (b d-a e)^3 (d+e x)^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.0877609, size = 135, normalized size = 0.92 \[ \frac{2 (a+b x)^{3/2} \left (A \left (15 a^2 e^2-6 a b e (7 d+2 e x)+b^2 \left (35 d^2+28 d e x+8 e^2 x^2\right )\right )+B \left (3 a^2 e (2 d+7 e x)-2 a b \left (7 d^2+29 d e x+7 e^2 x^2\right )+3 b^2 d x (7 d+2 e x)\right )\right )}{105 (d+e x)^{7/2} (b d-a e)^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(Sqrt[a + b*x]*(A + B*x))/(d + e*x)^(9/2),x]

[Out]

(2*(a + b*x)^(3/2)*(B*(3*b^2*d*x*(7*d + 2*e*x) + 3*a^2*e*(2*d + 7*e*x) - 2*a*b*(7*d^2 + 29*d*e*x + 7*e^2*x^2))
+ A*(15*a^2*e^2 - 6*a*b*e*(7*d + 2*e*x) + b^2*(35*d^2 + 28*d*e*x + 8*e^2*x^2))))/(105*(b*d - a*e)^3*(d + e*x)
^(7/2))

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Maple [A]  time = 0.006, size = 177, normalized size = 1.2 \begin{align*} -{\frac{16\,A{b}^{2}{e}^{2}{x}^{2}-28\,Bab{e}^{2}{x}^{2}+12\,B{b}^{2}de{x}^{2}-24\,Aab{e}^{2}x+56\,A{b}^{2}dex+42\,B{a}^{2}{e}^{2}x-116\,Babdex+42\,B{b}^{2}{d}^{2}x+30\,A{a}^{2}{e}^{2}-84\,Aabde+70\,A{b}^{2}{d}^{2}+12\,B{a}^{2}de-28\,Bab{d}^{2}}{105\,{a}^{3}{e}^{3}-315\,{a}^{2}bd{e}^{2}+315\,a{b}^{2}{d}^{2}e-105\,{b}^{3}{d}^{3}} \left ( bx+a \right ) ^{{\frac{3}{2}}} \left ( ex+d \right ) ^{-{\frac{7}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(b*x+a)^(1/2)/(e*x+d)^(9/2),x)

[Out]

-2/105*(b*x+a)^(3/2)*(8*A*b^2*e^2*x^2-14*B*a*b*e^2*x^2+6*B*b^2*d*e*x^2-12*A*a*b*e^2*x+28*A*b^2*d*e*x+21*B*a^2*
e^2*x-58*B*a*b*d*e*x+21*B*b^2*d^2*x+15*A*a^2*e^2-42*A*a*b*d*e+35*A*b^2*d^2+6*B*a^2*d*e-14*B*a*b*d^2)/(e*x+d)^(
7/2)/(a^3*e^3-3*a^2*b*d*e^2+3*a*b^2*d^2*e-b^3*d^3)

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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)*(b*x+a)^(1/2)/(e*x+d)^(9/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 54.2922, size = 900, normalized size = 6.12 \begin{align*} \frac{2 \,{\left (15 \, A a^{3} e^{2} + 2 \,{\left (3 \, B b^{3} d e -{\left (7 \, B a b^{2} - 4 \, A b^{3}\right )} e^{2}\right )} x^{3} - 7 \,{\left (2 \, B a^{2} b - 5 \, A a b^{2}\right )} d^{2} + 6 \,{\left (B a^{3} - 7 \, A a^{2} b\right )} d e +{\left (21 \, B b^{3} d^{2} - 4 \,{\left (13 \, B a b^{2} - 7 \, A b^{3}\right )} d e +{\left (7 \, B a^{2} b - 4 \, A a b^{2}\right )} e^{2}\right )} x^{2} +{\left (7 \,{\left (B a b^{2} + 5 \, A b^{3}\right )} d^{2} - 2 \,{\left (26 \, B a^{2} b + 7 \, A a b^{2}\right )} d e + 3 \,{\left (7 \, B a^{3} + A a^{2} b\right )} e^{2}\right )} x\right )} \sqrt{b x + a} \sqrt{e x + d}}{105 \,{\left (b^{3} d^{7} - 3 \, a b^{2} d^{6} e + 3 \, a^{2} b d^{5} e^{2} - a^{3} d^{4} e^{3} +{\left (b^{3} d^{3} e^{4} - 3 \, a b^{2} d^{2} e^{5} + 3 \, a^{2} b d e^{6} - a^{3} e^{7}\right )} x^{4} + 4 \,{\left (b^{3} d^{4} e^{3} - 3 \, a b^{2} d^{3} e^{4} + 3 \, a^{2} b d^{2} e^{5} - a^{3} d e^{6}\right )} x^{3} + 6 \,{\left (b^{3} d^{5} e^{2} - 3 \, a b^{2} d^{4} e^{3} + 3 \, a^{2} b d^{3} e^{4} - a^{3} d^{2} e^{5}\right )} x^{2} + 4 \,{\left (b^{3} d^{6} e - 3 \, a b^{2} d^{5} e^{2} + 3 \, a^{2} b d^{4} e^{3} - a^{3} d^{3} 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)^(1/2)/(e*x+d)^(9/2),x, algorithm="fricas")

[Out]

2/105*(15*A*a^3*e^2 + 2*(3*B*b^3*d*e - (7*B*a*b^2 - 4*A*b^3)*e^2)*x^3 - 7*(2*B*a^2*b - 5*A*a*b^2)*d^2 + 6*(B*a
^3 - 7*A*a^2*b)*d*e + (21*B*b^3*d^2 - 4*(13*B*a*b^2 - 7*A*b^3)*d*e + (7*B*a^2*b - 4*A*a*b^2)*e^2)*x^2 + (7*(B*
a*b^2 + 5*A*b^3)*d^2 - 2*(26*B*a^2*b + 7*A*a*b^2)*d*e + 3*(7*B*a^3 + A*a^2*b)*e^2)*x)*sqrt(b*x + a)*sqrt(e*x +
d)/(b^3*d^7 - 3*a*b^2*d^6*e + 3*a^2*b*d^5*e^2 - a^3*d^4*e^3 + (b^3*d^3*e^4 - 3*a*b^2*d^2*e^5 + 3*a^2*b*d*e^6
- a^3*e^7)*x^4 + 4*(b^3*d^4*e^3 - 3*a*b^2*d^3*e^4 + 3*a^2*b*d^2*e^5 - a^3*d*e^6)*x^3 + 6*(b^3*d^5*e^2 - 3*a*b^
2*d^4*e^3 + 3*a^2*b*d^3*e^4 - a^3*d^2*e^5)*x^2 + 4*(b^3*d^6*e - 3*a*b^2*d^5*e^2 + 3*a^2*b*d^4*e^3 - a^3*d^3*e^
4)*x)

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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)**(1/2)/(e*x+d)**(9/2),x)

[Out]

Timed out

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Giac [B]  time = 2.85944, size = 528, normalized size = 3.59 \begin{align*} -\frac{{\left ({\left (b x + a\right )}{\left (\frac{2 \,{\left (3 \, B b^{8} d{\left | b \right |} e^{4} - 7 \, B a b^{7}{\left | b \right |} e^{5} + 4 \, A b^{8}{\left | b \right |} e^{5}\right )}{\left (b x + a\right )}}{b^{16} d^{4} e^{8} - 4 \, a b^{15} d^{3} e^{9} + 6 \, a^{2} b^{14} d^{2} e^{10} - 4 \, a^{3} b^{13} d e^{11} + a^{4} b^{12} e^{12}} + \frac{7 \,{\left (3 \, B b^{9} d^{2}{\left | b \right |} e^{3} - 10 \, B a b^{8} d{\left | b \right |} e^{4} + 4 \, A b^{9} d{\left | b \right |} e^{4} + 7 \, B a^{2} b^{7}{\left | b \right |} e^{5} - 4 \, A a b^{8}{\left | b \right |} e^{5}\right )}}{b^{16} d^{4} e^{8} - 4 \, a b^{15} d^{3} e^{9} + 6 \, a^{2} b^{14} d^{2} e^{10} - 4 \, a^{3} b^{13} d e^{11} + a^{4} b^{12} e^{12}}\right )} - \frac{35 \,{\left (B a b^{9} d^{2}{\left | b \right |} e^{3} - A b^{10} d^{2}{\left | b \right |} e^{3} - 2 \, B a^{2} b^{8} d{\left | b \right |} e^{4} + 2 \, A a b^{9} d{\left | b \right |} e^{4} + B a^{3} b^{7}{\left | b \right |} e^{5} - A a^{2} b^{8}{\left | b \right |} e^{5}\right )}}{b^{16} d^{4} e^{8} - 4 \, a b^{15} d^{3} e^{9} + 6 \, a^{2} b^{14} d^{2} e^{10} - 4 \, a^{3} b^{13} d e^{11} + a^{4} b^{12} e^{12}}\right )}{\left (b x + a\right )}^{\frac{3}{2}}}{80640 \,{\left (b^{2} d +{\left (b x + a\right )} b e - a b e\right )}^{\frac{7}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(b*x+a)^(1/2)/(e*x+d)^(9/2),x, algorithm="giac")

[Out]

-1/80640*((b*x + a)*(2*(3*B*b^8*d*abs(b)*e^4 - 7*B*a*b^7*abs(b)*e^5 + 4*A*b^8*abs(b)*e^5)*(b*x + a)/(b^16*d^4*
e^8 - 4*a*b^15*d^3*e^9 + 6*a^2*b^14*d^2*e^10 - 4*a^3*b^13*d*e^11 + a^4*b^12*e^12) + 7*(3*B*b^9*d^2*abs(b)*e^3
- 10*B*a*b^8*d*abs(b)*e^4 + 4*A*b^9*d*abs(b)*e^4 + 7*B*a^2*b^7*abs(b)*e^5 - 4*A*a*b^8*abs(b)*e^5)/(b^16*d^4*e^
8 - 4*a*b^15*d^3*e^9 + 6*a^2*b^14*d^2*e^10 - 4*a^3*b^13*d*e^11 + a^4*b^12*e^12)) - 35*(B*a*b^9*d^2*abs(b)*e^3
- A*b^10*d^2*abs(b)*e^3 - 2*B*a^2*b^8*d*abs(b)*e^4 + 2*A*a*b^9*d*abs(b)*e^4 + B*a^3*b^7*abs(b)*e^5 - A*a^2*b^8
*abs(b)*e^5)/(b^16*d^4*e^8 - 4*a*b^15*d^3*e^9 + 6*a^2*b^14*d^2*e^10 - 4*a^3*b^13*d*e^11 + a^4*b^12*e^12))*(b*x
+ a)^(3/2)/(b^2*d + (b*x + a)*b*e - a*b*e)^(7/2)