3.2219 \(\int \frac{(a+b x)^{3/2} (A+B x)}{(d+e x)^{11/2}} \, dx\)

Optimal. Leaf size=147 \[ \frac{4 b (a+b x)^{5/2} (-9 a B e+4 A b e+5 b B d)}{315 e (d+e x)^{5/2} (b d-a e)^3}+\frac{2 (a+b x)^{5/2} (-9 a B e+4 A b e+5 b B d)}{63 e (d+e x)^{7/2} (b d-a e)^2}-\frac{2 (a+b x)^{5/2} (B d-A e)}{9 e (d+e x)^{9/2} (b d-a e)} \]

[Out]

(-2*(B*d - A*e)*(a + b*x)^(5/2))/(9*e*(b*d - a*e)*(d + e*x)^(9/2)) + (2*(5*b*B*d + 4*A*b*e - 9*a*B*e)*(a + b*x
)^(5/2))/(63*e*(b*d - a*e)^2*(d + e*x)^(7/2)) + (4*b*(5*b*B*d + 4*A*b*e - 9*a*B*e)*(a + b*x)^(5/2))/(315*e*(b*
d - a*e)^3*(d + e*x)^(5/2))

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Rubi [A]  time = 0.0857733, 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} \[ \frac{4 b (a+b x)^{5/2} (-9 a B e+4 A b e+5 b B d)}{315 e (d+e x)^{5/2} (b d-a e)^3}+\frac{2 (a+b x)^{5/2} (-9 a B e+4 A b e+5 b B d)}{63 e (d+e x)^{7/2} (b d-a e)^2}-\frac{2 (a+b x)^{5/2} (B d-A e)}{9 e (d+e x)^{9/2} (b d-a e)} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x)^(3/2)*(A + B*x))/(d + e*x)^(11/2),x]

[Out]

(-2*(B*d - A*e)*(a + b*x)^(5/2))/(9*e*(b*d - a*e)*(d + e*x)^(9/2)) + (2*(5*b*B*d + 4*A*b*e - 9*a*B*e)*(a + b*x
)^(5/2))/(63*e*(b*d - a*e)^2*(d + e*x)^(7/2)) + (4*b*(5*b*B*d + 4*A*b*e - 9*a*B*e)*(a + b*x)^(5/2))/(315*e*(b*
d - a*e)^3*(d + e*x)^(5/2))

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{(a+b x)^{3/2} (A+B x)}{(d+e x)^{11/2}} \, dx &=-\frac{2 (B d-A e) (a+b x)^{5/2}}{9 e (b d-a e) (d+e x)^{9/2}}+\frac{(5 b B d+4 A b e-9 a B e) \int \frac{(a+b x)^{3/2}}{(d+e x)^{9/2}} \, dx}{9 e (b d-a e)}\\ &=-\frac{2 (B d-A e) (a+b x)^{5/2}}{9 e (b d-a e) (d+e x)^{9/2}}+\frac{2 (5 b B d+4 A b e-9 a B e) (a+b x)^{5/2}}{63 e (b d-a e)^2 (d+e x)^{7/2}}+\frac{(2 b (5 b B d+4 A b e-9 a B e)) \int \frac{(a+b x)^{3/2}}{(d+e x)^{7/2}} \, dx}{63 e (b d-a e)^2}\\ &=-\frac{2 (B d-A e) (a+b x)^{5/2}}{9 e (b d-a e) (d+e x)^{9/2}}+\frac{2 (5 b B d+4 A b e-9 a B e) (a+b x)^{5/2}}{63 e (b d-a e)^2 (d+e x)^{7/2}}+\frac{4 b (5 b B d+4 A b e-9 a B e) (a+b x)^{5/2}}{315 e (b d-a e)^3 (d+e x)^{5/2}}\\ \end{align*}

Mathematica [A]  time = 0.0975123, size = 135, normalized size = 0.92 \[ \frac{2 (a+b x)^{5/2} \left (A \left (35 a^2 e^2-10 a b e (9 d+2 e x)+b^2 \left (63 d^2+36 d e x+8 e^2 x^2\right )\right )+B \left (5 a^2 e (2 d+9 e x)-2 a b \left (9 d^2+53 d e x+9 e^2 x^2\right )+5 b^2 d x (9 d+2 e x)\right )\right )}{315 (d+e x)^{9/2} (b d-a e)^3} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)^(3/2)*(A + B*x))/(d + e*x)^(11/2),x]

[Out]

(2*(a + b*x)^(5/2)*(A*(35*a^2*e^2 - 10*a*b*e*(9*d + 2*e*x) + b^2*(63*d^2 + 36*d*e*x + 8*e^2*x^2)) + B*(5*b^2*d
*x*(9*d + 2*e*x) + 5*a^2*e*(2*d + 9*e*x) - 2*a*b*(9*d^2 + 53*d*e*x + 9*e^2*x^2))))/(315*(b*d - a*e)^3*(d + e*x
)^(9/2))

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Maple [A]  time = 0.007, size = 177, normalized size = 1.2 \begin{align*} -{\frac{16\,A{b}^{2}{e}^{2}{x}^{2}-36\,Bab{e}^{2}{x}^{2}+20\,B{b}^{2}de{x}^{2}-40\,Aab{e}^{2}x+72\,A{b}^{2}dex+90\,B{a}^{2}{e}^{2}x-212\,Babdex+90\,B{b}^{2}{d}^{2}x+70\,A{a}^{2}{e}^{2}-180\,Aabde+126\,A{b}^{2}{d}^{2}+20\,B{a}^{2}de-36\,Bab{d}^{2}}{315\,{a}^{3}{e}^{3}-945\,{a}^{2}bd{e}^{2}+945\,a{b}^{2}{d}^{2}e-315\,{b}^{3}{d}^{3}} \left ( bx+a \right ) ^{{\frac{5}{2}}} \left ( ex+d \right ) ^{-{\frac{9}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^(3/2)*(B*x+A)/(e*x+d)^(11/2),x)

[Out]

-2/315*(b*x+a)^(5/2)*(8*A*b^2*e^2*x^2-18*B*a*b*e^2*x^2+10*B*b^2*d*e*x^2-20*A*a*b*e^2*x+36*A*b^2*d*e*x+45*B*a^2
*e^2*x-106*B*a*b*d*e*x+45*B*b^2*d^2*x+35*A*a^2*e^2-90*A*a*b*d*e+63*A*b^2*d^2+10*B*a^2*d*e-18*B*a*b*d^2)/(e*x+d
)^(9/2)/(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)^(3/2)*(B*x+A)/(e*x+d)^(11/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [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)^(3/2)*(B*x+A)/(e*x+d)^(11/2),x, algorithm="fricas")

[Out]

Timed out

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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)**(3/2)*(B*x+A)/(e*x+d)**(11/2),x)

[Out]

Timed out

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Giac [B]  time = 2.6864, size = 709, normalized size = 4.82 \begin{align*} -\frac{{\left ({\left (b x + a\right )}{\left (\frac{2 \,{\left (5 \, B b^{11} d^{2}{\left | b \right |} e^{5} - 14 \, B a b^{10} d{\left | b \right |} e^{6} + 4 \, A b^{11} d{\left | b \right |} e^{6} + 9 \, B a^{2} b^{9}{\left | b \right |} e^{7} - 4 \, A a b^{10}{\left | b \right |} e^{7}\right )}{\left (b x + a\right )}}{b^{20} d^{5} e^{10} - 5 \, a b^{19} d^{4} e^{11} + 10 \, a^{2} b^{18} d^{3} e^{12} - 10 \, a^{3} b^{17} d^{2} e^{13} + 5 \, a^{4} b^{16} d e^{14} - a^{5} b^{15} e^{15}} + \frac{9 \,{\left (5 \, B b^{12} d^{3}{\left | b \right |} e^{4} - 19 \, B a b^{11} d^{2}{\left | b \right |} e^{5} + 4 \, A b^{12} d^{2}{\left | b \right |} e^{5} + 23 \, B a^{2} b^{10} d{\left | b \right |} e^{6} - 8 \, A a b^{11} d{\left | b \right |} e^{6} - 9 \, B a^{3} b^{9}{\left | b \right |} e^{7} + 4 \, A a^{2} b^{10}{\left | b \right |} e^{7}\right )}}{b^{20} d^{5} e^{10} - 5 \, a b^{19} d^{4} e^{11} + 10 \, a^{2} b^{18} d^{3} e^{12} - 10 \, a^{3} b^{17} d^{2} e^{13} + 5 \, a^{4} b^{16} d e^{14} - a^{5} b^{15} e^{15}}\right )} - \frac{63 \,{\left (B a b^{12} d^{3}{\left | b \right |} e^{4} - A b^{13} d^{3}{\left | b \right |} e^{4} - 3 \, B a^{2} b^{11} d^{2}{\left | b \right |} e^{5} + 3 \, A a b^{12} d^{2}{\left | b \right |} e^{5} + 3 \, B a^{3} b^{10} d{\left | b \right |} e^{6} - 3 \, A a^{2} b^{11} d{\left | b \right |} e^{6} - B a^{4} b^{9}{\left | b \right |} e^{7} + A a^{3} b^{10}{\left | b \right |} e^{7}\right )}}{b^{20} d^{5} e^{10} - 5 \, a b^{19} d^{4} e^{11} + 10 \, a^{2} b^{18} d^{3} e^{12} - 10 \, a^{3} b^{17} d^{2} e^{13} + 5 \, a^{4} b^{16} d e^{14} - a^{5} b^{15} e^{15}}\right )}{\left (b x + a\right )}^{\frac{5}{2}}}{322560 \,{\left (b^{2} d +{\left (b x + a\right )} b e - a b e\right )}^{\frac{9}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(3/2)*(B*x+A)/(e*x+d)^(11/2),x, algorithm="giac")

[Out]

-1/322560*((b*x + a)*(2*(5*B*b^11*d^2*abs(b)*e^5 - 14*B*a*b^10*d*abs(b)*e^6 + 4*A*b^11*d*abs(b)*e^6 + 9*B*a^2*
b^9*abs(b)*e^7 - 4*A*a*b^10*abs(b)*e^7)*(b*x + a)/(b^20*d^5*e^10 - 5*a*b^19*d^4*e^11 + 10*a^2*b^18*d^3*e^12 -
10*a^3*b^17*d^2*e^13 + 5*a^4*b^16*d*e^14 - a^5*b^15*e^15) + 9*(5*B*b^12*d^3*abs(b)*e^4 - 19*B*a*b^11*d^2*abs(b
)*e^5 + 4*A*b^12*d^2*abs(b)*e^5 + 23*B*a^2*b^10*d*abs(b)*e^6 - 8*A*a*b^11*d*abs(b)*e^6 - 9*B*a^3*b^9*abs(b)*e^
7 + 4*A*a^2*b^10*abs(b)*e^7)/(b^20*d^5*e^10 - 5*a*b^19*d^4*e^11 + 10*a^2*b^18*d^3*e^12 - 10*a^3*b^17*d^2*e^13
+ 5*a^4*b^16*d*e^14 - a^5*b^15*e^15)) - 63*(B*a*b^12*d^3*abs(b)*e^4 - A*b^13*d^3*abs(b)*e^4 - 3*B*a^2*b^11*d^2
*abs(b)*e^5 + 3*A*a*b^12*d^2*abs(b)*e^5 + 3*B*a^3*b^10*d*abs(b)*e^6 - 3*A*a^2*b^11*d*abs(b)*e^6 - B*a^4*b^9*ab
s(b)*e^7 + A*a^3*b^10*abs(b)*e^7)/(b^20*d^5*e^10 - 5*a*b^19*d^4*e^11 + 10*a^2*b^18*d^3*e^12 - 10*a^3*b^17*d^2*
e^13 + 5*a^4*b^16*d*e^14 - a^5*b^15*e^15))*(b*x + a)^(5/2)/(b^2*d + (b*x + a)*b*e - a*b*e)^(9/2)