3.2231 \(\int \frac{(a+b x)^{5/2} (A+B x)}{(d+e x)^{11/2}} \, dx\)

Optimal. Leaf size=95 \[ \frac{2 (a+b x)^{7/2} (-9 a B e+2 A b e+7 b B d)}{63 e (d+e x)^{7/2} (b d-a e)^2}-\frac{2 (a+b x)^{7/2} (B d-A e)}{9 e (d+e x)^{9/2} (b d-a e)} \]

[Out]

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

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Rubi [A]  time = 0.0497504, antiderivative size = 95, 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} \[ \frac{2 (a+b x)^{7/2} (-9 a B e+2 A b e+7 b B d)}{63 e (d+e x)^{7/2} (b d-a e)^2}-\frac{2 (a+b x)^{7/2} (B d-A e)}{9 e (d+e x)^{9/2} (b d-a e)} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x)^(5/2)*(A + B*x))/(d + e*x)^(11/2),x]

[Out]

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

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

Mathematica [A]  time = 0.0603764, size = 66, normalized size = 0.69 \[ \frac{2 (a+b x)^{7/2} (A (-7 a e+9 b d+2 b e x)+B (-2 a d-9 a e x+7 b d x))}{63 (d+e x)^{9/2} (b d-a e)^2} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)^(5/2)*(A + B*x))/(d + e*x)^(11/2),x]

[Out]

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

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Maple [A]  time = 0.006, size = 74, normalized size = 0.8 \begin{align*} -{\frac{-4\,Abex+18\,Baex-14\,Bbdx+14\,Aae-18\,Abd+4\,Bad}{63\,{a}^{2}{e}^{2}-126\,bead+63\,{b}^{2}{d}^{2}} \left ( bx+a \right ) ^{{\frac{7}{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)^(5/2)*(B*x+A)/(e*x+d)^(11/2),x)

[Out]

-2/63*(b*x+a)^(7/2)*(-2*A*b*e*x+9*B*a*e*x-7*B*b*d*x+7*A*a*e-9*A*b*d+2*B*a*d)/(e*x+d)^(9/2)/(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)^(5/2)*(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)^(5/2)*(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)**(5/2)*(B*x+A)/(e*x+d)**(11/2),x)

[Out]

Timed out

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Giac [B]  time = 2.91488, size = 518, normalized size = 5.45 \begin{align*} -\frac{{\left (b x + a\right )}^{\frac{7}{2}}{\left (\frac{{\left (7 \, B b^{12} d^{3}{\left | b \right |} e^{4} - 23 \, B a b^{11} d^{2}{\left | b \right |} e^{5} + 2 \, A b^{12} d^{2}{\left | b \right |} e^{5} + 25 \, B a^{2} b^{10} d{\left | b \right |} e^{6} - 4 \, A a b^{11} d{\left | b \right |} e^{6} - 9 \, B a^{3} b^{9}{\left | b \right |} e^{7} + 2 \, A a^{2} 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 (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 )}}{64512 \,{\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)^(5/2)*(B*x+A)/(e*x+d)^(11/2),x, algorithm="giac")

[Out]

-1/64512*(b*x + a)^(7/2)*((7*B*b^12*d^3*abs(b)*e^4 - 23*B*a*b^11*d^2*abs(b)*e^5 + 2*A*b^12*d^2*abs(b)*e^5 + 25
*B*a^2*b^10*d*abs(b)*e^6 - 4*A*a*b^11*d*abs(b)*e^6 - 9*B*a^3*b^9*abs(b)*e^7 + 2*A*a^2*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*(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*abs(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^2*d + (b*x + a)*b*e - a*b*e)^(9/2)