3.2218 \(\int \frac{(a+b x)^{3/2} (A+B x)}{(d+e x)^{9/2}} \, dx\)

Optimal. Leaf size=95 \[ \frac{2 (a+b x)^{5/2} (-7 a B e+2 A b e+5 b B d)}{35 e (d+e x)^{5/2} (b d-a e)^2}-\frac{2 (a+b x)^{5/2} (B d-A e)}{7 e (d+e x)^{7/2} (b d-a e)} \]

[Out]

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

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Rubi [A]  time = 0.051143, 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} \[ \frac{2 (a+b x)^{5/2} (-7 a B e+2 A b e+5 b B d)}{35 e (d+e x)^{5/2} (b d-a e)^2}-\frac{2 (a+b x)^{5/2} (B d-A e)}{7 e (d+e x)^{7/2} (b d-a e)} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x)^(3/2)*(A + B*x))/(d + e*x)^(9/2),x]

[Out]

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

Mathematica [A]  time = 0.0530805, size = 66, normalized size = 0.69 \[ \frac{2 (a+b x)^{5/2} (A (-5 a e+7 b d+2 b e x)+B (-2 a d-7 a e x+5 b d x))}{35 (d+e x)^{7/2} (b d-a e)^2} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)^(3/2)*(A + B*x))/(d + e*x)^(9/2),x]

[Out]

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

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Maple [A]  time = 0.006, size = 74, normalized size = 0.8 \begin{align*} -{\frac{-4\,Abex+14\,Baex-10\,Bbdx+10\,Aae-14\,Abd+4\,Bad}{35\,{a}^{2}{e}^{2}-70\,bead+35\,{b}^{2}{d}^{2}} \left ( bx+a \right ) ^{{\frac{5}{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)^(3/2)*(B*x+A)/(e*x+d)^(9/2),x)

[Out]

-2/35*(b*x+a)^(5/2)*(-2*A*b*e*x+7*B*a*e*x-5*B*b*d*x+5*A*a*e-7*A*b*d+2*B*a*d)/(e*x+d)^(7/2)/(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)^(3/2)*(B*x+A)/(e*x+d)^(9/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 56.1361, size = 628, normalized size = 6.61 \begin{align*} -\frac{2 \,{\left (5 \, A a^{3} e -{\left (5 \, B b^{3} d -{\left (7 \, B a b^{2} - 2 \, A b^{3}\right )} e\right )} x^{3} -{\left ({\left (8 \, B a b^{2} + 7 \, A b^{3}\right )} d -{\left (14 \, B a^{2} b + A a b^{2}\right )} e\right )} x^{2} +{\left (2 \, B a^{3} - 7 \, A a^{2} b\right )} d -{\left ({\left (B a^{2} b + 14 \, A a b^{2}\right )} d -{\left (7 \, B a^{3} + 8 \, A a^{2} b\right )} e\right )} x\right )} \sqrt{b x + a} \sqrt{e x + d}}{35 \,{\left (b^{2} d^{6} - 2 \, a b d^{5} e + a^{2} d^{4} e^{2} +{\left (b^{2} d^{2} e^{4} - 2 \, a b d e^{5} + a^{2} e^{6}\right )} x^{4} + 4 \,{\left (b^{2} d^{3} e^{3} - 2 \, a b d^{2} e^{4} + a^{2} d e^{5}\right )} x^{3} + 6 \,{\left (b^{2} d^{4} e^{2} - 2 \, a b d^{3} e^{3} + a^{2} d^{2} e^{4}\right )} x^{2} + 4 \,{\left (b^{2} d^{5} e - 2 \, a b d^{4} e^{2} + a^{2} d^{3} e^{3}\right )} x\right )}} \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)^(9/2),x, algorithm="fricas")

[Out]

-2/35*(5*A*a^3*e - (5*B*b^3*d - (7*B*a*b^2 - 2*A*b^3)*e)*x^3 - ((8*B*a*b^2 + 7*A*b^3)*d - (14*B*a^2*b + A*a*b^
2)*e)*x^2 + (2*B*a^3 - 7*A*a^2*b)*d - ((B*a^2*b + 14*A*a*b^2)*d - (7*B*a^3 + 8*A*a^2*b)*e)*x)*sqrt(b*x + a)*sq
rt(e*x + d)/(b^2*d^6 - 2*a*b*d^5*e + a^2*d^4*e^2 + (b^2*d^2*e^4 - 2*a*b*d*e^5 + a^2*e^6)*x^4 + 4*(b^2*d^3*e^3
- 2*a*b*d^2*e^4 + a^2*d*e^5)*x^3 + 6*(b^2*d^4*e^2 - 2*a*b*d^3*e^3 + a^2*d^2*e^4)*x^2 + 4*(b^2*d^5*e - 2*a*b*d^
4*e^2 + a^2*d^3*e^3)*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)**(3/2)*(B*x+A)/(e*x+d)**(9/2),x)

[Out]

Timed out

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Giac [B]  time = 2.37289, size = 394, normalized size = 4.15 \begin{align*} -\frac{{\left (b x + a\right )}^{\frac{5}{2}}{\left (\frac{{\left (5 \, B b^{9} d^{2}{\left | b \right |} e^{3} - 12 \, B a b^{8} d{\left | b \right |} e^{4} + 2 \, A b^{9} d{\left | b \right |} e^{4} + 7 \, B a^{2} b^{7}{\left | b \right |} e^{5} - 2 \, A 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 (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 )}}{26880 \,{\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)^(3/2)*(B*x+A)/(e*x+d)^(9/2),x, algorithm="giac")

[Out]

-1/26880*(b*x + a)^(5/2)*((5*B*b^9*d^2*abs(b)*e^3 - 12*B*a*b^8*d*abs(b)*e^4 + 2*A*b^9*d*abs(b)*e^4 + 7*B*a^2*b
^7*abs(b)*e^5 - 2*A*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*(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^2*d + (b*x + a)*b*e - a*b*e)^(7/2)