3.2252 \(\int \frac{A+B x}{(a+b x)^{3/2} (d+e x)^{9/2}} \, dx\)

Optimal. Leaf size=237 \[ \frac{32 b^2 \sqrt{a+b x} (7 a B e-8 A b e+b B d)}{35 \sqrt{d+e x} (b d-a e)^5}+\frac{16 b \sqrt{a+b x} (7 a B e-8 A b e+b B d)}{35 (d+e x)^{3/2} (b d-a e)^4}+\frac{12 \sqrt{a+b x} (7 a B e-8 A b e+b B d)}{35 (d+e x)^{5/2} (b d-a e)^3}+\frac{2 \sqrt{a+b x} (7 a B e-8 A b e+b B d)}{7 b (d+e x)^{7/2} (b d-a e)^2}-\frac{2 (A b-a B)}{b \sqrt{a+b x} (d+e x)^{7/2} (b d-a e)} \]

[Out]

(-2*(A*b - a*B))/(b*(b*d - a*e)*Sqrt[a + b*x]*(d + e*x)^(7/2)) + (2*(b*B*d - 8*A*b*e + 7*a*B*e)*Sqrt[a + b*x])
/(7*b*(b*d - a*e)^2*(d + e*x)^(7/2)) + (12*(b*B*d - 8*A*b*e + 7*a*B*e)*Sqrt[a + b*x])/(35*(b*d - a*e)^3*(d + e
*x)^(5/2)) + (16*b*(b*B*d - 8*A*b*e + 7*a*B*e)*Sqrt[a + b*x])/(35*(b*d - a*e)^4*(d + e*x)^(3/2)) + (32*b^2*(b*
B*d - 8*A*b*e + 7*a*B*e)*Sqrt[a + b*x])/(35*(b*d - a*e)^5*Sqrt[d + e*x])

________________________________________________________________________________________

Rubi [A]  time = 0.146842, antiderivative size = 237, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {78, 45, 37} \[ \frac{32 b^2 \sqrt{a+b x} (7 a B e-8 A b e+b B d)}{35 \sqrt{d+e x} (b d-a e)^5}+\frac{16 b \sqrt{a+b x} (7 a B e-8 A b e+b B d)}{35 (d+e x)^{3/2} (b d-a e)^4}+\frac{12 \sqrt{a+b x} (7 a B e-8 A b e+b B d)}{35 (d+e x)^{5/2} (b d-a e)^3}+\frac{2 \sqrt{a+b x} (7 a B e-8 A b e+b B d)}{7 b (d+e x)^{7/2} (b d-a e)^2}-\frac{2 (A b-a B)}{b \sqrt{a+b x} (d+e x)^{7/2} (b d-a e)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(A*b - a*B))/(b*(b*d - a*e)*Sqrt[a + b*x]*(d + e*x)^(7/2)) + (2*(b*B*d - 8*A*b*e + 7*a*B*e)*Sqrt[a + b*x])
/(7*b*(b*d - a*e)^2*(d + e*x)^(7/2)) + (12*(b*B*d - 8*A*b*e + 7*a*B*e)*Sqrt[a + b*x])/(35*(b*d - a*e)^3*(d + e
*x)^(5/2)) + (16*b*(b*B*d - 8*A*b*e + 7*a*B*e)*Sqrt[a + b*x])/(35*(b*d - a*e)^4*(d + e*x)^(3/2)) + (32*b^2*(b*
B*d - 8*A*b*e + 7*a*B*e)*Sqrt[a + b*x])/(35*(b*d - a*e)^5*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}{(a+b x)^{3/2} (d+e x)^{9/2}} \, dx &=-\frac{2 (A b-a B)}{b (b d-a e) \sqrt{a+b x} (d+e x)^{7/2}}+\frac{(b B d-8 A b e+7 a B e) \int \frac{1}{\sqrt{a+b x} (d+e x)^{9/2}} \, dx}{b (b d-a e)}\\ &=-\frac{2 (A b-a B)}{b (b d-a e) \sqrt{a+b x} (d+e x)^{7/2}}+\frac{2 (b B d-8 A b e+7 a B e) \sqrt{a+b x}}{7 b (b d-a e)^2 (d+e x)^{7/2}}+\frac{(6 (b B d-8 A b e+7 a B e)) \int \frac{1}{\sqrt{a+b x} (d+e x)^{7/2}} \, dx}{7 (b d-a e)^2}\\ &=-\frac{2 (A b-a B)}{b (b d-a e) \sqrt{a+b x} (d+e x)^{7/2}}+\frac{2 (b B d-8 A b e+7 a B e) \sqrt{a+b x}}{7 b (b d-a e)^2 (d+e x)^{7/2}}+\frac{12 (b B d-8 A b e+7 a B e) \sqrt{a+b x}}{35 (b d-a e)^3 (d+e x)^{5/2}}+\frac{(24 b (b B d-8 A b e+7 a B e)) \int \frac{1}{\sqrt{a+b x} (d+e x)^{5/2}} \, dx}{35 (b d-a e)^3}\\ &=-\frac{2 (A b-a B)}{b (b d-a e) \sqrt{a+b x} (d+e x)^{7/2}}+\frac{2 (b B d-8 A b e+7 a B e) \sqrt{a+b x}}{7 b (b d-a e)^2 (d+e x)^{7/2}}+\frac{12 (b B d-8 A b e+7 a B e) \sqrt{a+b x}}{35 (b d-a e)^3 (d+e x)^{5/2}}+\frac{16 b (b B d-8 A b e+7 a B e) \sqrt{a+b x}}{35 (b d-a e)^4 (d+e x)^{3/2}}+\frac{\left (16 b^2 (b B d-8 A b e+7 a B e)\right ) \int \frac{1}{\sqrt{a+b x} (d+e x)^{3/2}} \, dx}{35 (b d-a e)^4}\\ &=-\frac{2 (A b-a B)}{b (b d-a e) \sqrt{a+b x} (d+e x)^{7/2}}+\frac{2 (b B d-8 A b e+7 a B e) \sqrt{a+b x}}{7 b (b d-a e)^2 (d+e x)^{7/2}}+\frac{12 (b B d-8 A b e+7 a B e) \sqrt{a+b x}}{35 (b d-a e)^3 (d+e x)^{5/2}}+\frac{16 b (b B d-8 A b e+7 a B e) \sqrt{a+b x}}{35 (b d-a e)^4 (d+e x)^{3/2}}+\frac{32 b^2 (b B d-8 A b e+7 a B e) \sqrt{a+b x}}{35 (b d-a e)^5 \sqrt{d+e x}}\\ \end{align*}

Mathematica [A]  time = 0.252802, size = 135, normalized size = 0.57 \[ \frac{2 \left (-(a+b x) \left (2 b (d+e x) \left (4 b (d+e x) (-a e+3 b d+2 b e x)+3 (b d-a e)^2\right )+5 (b d-a e)^3\right ) (-7 a B e+8 A b e-b B d)-35 (A b-a B) (b d-a e)^4\right )}{35 b \sqrt{a+b x} (d+e x)^{7/2} (b d-a e)^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(-35*(A*b - a*B)*(b*d - a*e)^4 - (-(b*B*d) + 8*A*b*e - 7*a*B*e)*(a + b*x)*(5*(b*d - a*e)^3 + 2*b*(d + e*x)*
(3*(b*d - a*e)^2 + 4*b*(d + e*x)*(3*b*d - a*e + 2*b*e*x)))))/(35*b*(b*d - a*e)^5*Sqrt[a + b*x]*(d + e*x)^(7/2)
)

________________________________________________________________________________________

Maple [B]  time = 0.008, size = 505, normalized size = 2.1 \begin{align*} -{\frac{-256\,A{b}^{4}{e}^{4}{x}^{4}+224\,Ba{b}^{3}{e}^{4}{x}^{4}+32\,B{b}^{4}d{e}^{3}{x}^{4}-128\,Aa{b}^{3}{e}^{4}{x}^{3}-896\,A{b}^{4}d{e}^{3}{x}^{3}+112\,B{a}^{2}{b}^{2}{e}^{4}{x}^{3}+800\,Ba{b}^{3}d{e}^{3}{x}^{3}+112\,B{b}^{4}{d}^{2}{e}^{2}{x}^{3}+32\,A{a}^{2}{b}^{2}{e}^{4}{x}^{2}-448\,Aa{b}^{3}d{e}^{3}{x}^{2}-1120\,A{b}^{4}{d}^{2}{e}^{2}{x}^{2}-28\,B{a}^{3}b{e}^{4}{x}^{2}+388\,B{a}^{2}{b}^{2}d{e}^{3}{x}^{2}+1036\,Ba{b}^{3}{d}^{2}{e}^{2}{x}^{2}+140\,B{b}^{4}{d}^{3}e{x}^{2}-16\,A{a}^{3}b{e}^{4}x+112\,A{a}^{2}{b}^{2}d{e}^{3}x-560\,Aa{b}^{3}{d}^{2}{e}^{2}x-560\,A{b}^{4}{d}^{3}ex+14\,B{a}^{4}{e}^{4}x-96\,B{a}^{3}bd{e}^{3}x+476\,B{a}^{2}{b}^{2}{d}^{2}{e}^{2}x+560\,Ba{b}^{3}{d}^{3}ex+70\,B{b}^{4}{d}^{4}x+10\,A{a}^{4}{e}^{4}-56\,A{a}^{3}bd{e}^{3}+140\,A{a}^{2}{b}^{2}{d}^{2}{e}^{2}-280\,Aa{b}^{3}{d}^{3}e-70\,A{b}^{4}{d}^{4}+4\,B{a}^{4}d{e}^{3}-28\,B{a}^{3}b{d}^{2}{e}^{2}+140\,B{a}^{2}{b}^{2}{d}^{3}e+140\,Ba{b}^{3}{d}^{4}}{35\,{a}^{5}{e}^{5}-175\,{a}^{4}bd{e}^{4}+350\,{a}^{3}{b}^{2}{d}^{2}{e}^{3}-350\,{a}^{2}{b}^{3}{d}^{3}{e}^{2}+175\,a{b}^{4}{d}^{4}e-35\,{b}^{5}{d}^{5}}{\frac{1}{\sqrt{bx+a}}} \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)^(3/2)/(e*x+d)^(9/2),x)

[Out]

-2/35*(-128*A*b^4*e^4*x^4+112*B*a*b^3*e^4*x^4+16*B*b^4*d*e^3*x^4-64*A*a*b^3*e^4*x^3-448*A*b^4*d*e^3*x^3+56*B*a
^2*b^2*e^4*x^3+400*B*a*b^3*d*e^3*x^3+56*B*b^4*d^2*e^2*x^3+16*A*a^2*b^2*e^4*x^2-224*A*a*b^3*d*e^3*x^2-560*A*b^4
*d^2*e^2*x^2-14*B*a^3*b*e^4*x^2+194*B*a^2*b^2*d*e^3*x^2+518*B*a*b^3*d^2*e^2*x^2+70*B*b^4*d^3*e*x^2-8*A*a^3*b*e
^4*x+56*A*a^2*b^2*d*e^3*x-280*A*a*b^3*d^2*e^2*x-280*A*b^4*d^3*e*x+7*B*a^4*e^4*x-48*B*a^3*b*d*e^3*x+238*B*a^2*b
^2*d^2*e^2*x+280*B*a*b^3*d^3*e*x+35*B*b^4*d^4*x+5*A*a^4*e^4-28*A*a^3*b*d*e^3+70*A*a^2*b^2*d^2*e^2-140*A*a*b^3*
d^3*e-35*A*b^4*d^4+2*B*a^4*d*e^3-14*B*a^3*b*d^2*e^2+70*B*a^2*b^2*d^3*e+70*B*a*b^3*d^4)/(b*x+a)^(1/2)/(e*x+d)^(
7/2)/(a^5*e^5-5*a^4*b*d*e^4+10*a^3*b^2*d^2*e^3-10*a^2*b^3*d^3*e^2+5*a*b^4*d^4*e-b^5*d^5)

________________________________________________________________________________________

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)^(3/2)/(e*x+d)^(9/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

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)/(b*x+a)^(3/2)/(e*x+d)^(9/2),x, algorithm="fricas")

[Out]

Timed out

________________________________________________________________________________________

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)**(3/2)/(e*x+d)**(9/2),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [B]  time = 6.69859, size = 2512, normalized size = 10.6 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/(b*x+a)^(3/2)/(e*x+d)^(9/2),x, algorithm="giac")

[Out]

4*(B*a*b^(9/2)*e^(1/2) - A*b^(11/2)*e^(1/2))/((b^4*d^4*abs(b) - 4*a*b^3*d^3*abs(b)*e + 6*a^2*b^2*d^2*abs(b)*e^
2 - 4*a^3*b*d*abs(b)*e^3 + a^4*abs(b)*e^4)*(b^2*d - a*b*e - (sqrt(b*x + a)*sqrt(b)*e^(1/2) - sqrt(b^2*d + (b*x
+ a)*b*e - a*b*e))^2)) - 1/26880*(((b*x + a)*((16*B*b^19*d^10*abs(b)*e^6 - 67*B*a*b^18*d^9*abs(b)*e^7 - 93*A*
b^19*d^9*abs(b)*e^7 - 117*B*a^2*b^17*d^8*abs(b)*e^8 + 837*A*a*b^18*d^8*abs(b)*e^8 + 1428*B*a^3*b^16*d^7*abs(b)
*e^9 - 3348*A*a^2*b^17*d^7*abs(b)*e^9 - 4452*B*a^4*b^15*d^6*abs(b)*e^10 + 7812*A*a^3*b^16*d^6*abs(b)*e^10 + 76
86*B*a^5*b^14*d^5*abs(b)*e^11 - 11718*A*a^4*b^15*d^5*abs(b)*e^11 - 8358*B*a^6*b^13*d^4*abs(b)*e^12 + 11718*A*a
^5*b^14*d^4*abs(b)*e^12 + 5892*B*a^7*b^12*d^3*abs(b)*e^13 - 7812*A*a^6*b^13*d^3*abs(b)*e^13 - 2628*B*a^8*b^11*
d^2*abs(b)*e^14 + 3348*A*a^7*b^12*d^2*abs(b)*e^14 + 677*B*a^9*b^10*d*abs(b)*e^15 - 837*A*a^8*b^11*d*abs(b)*e^1
5 - 77*B*a^10*b^9*abs(b)*e^16 + 93*A*a^9*b^10*abs(b)*e^16)*(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) + 28*(2*B*b^20*d^11*abs(b)*e^5 - 11*B*a*b^19*d^10*abs(b)*e^
6 - 11*A*b^20*d^10*abs(b)*e^6 + 110*A*a*b^19*d^9*abs(b)*e^7 + 165*B*a^3*b^17*d^8*abs(b)*e^8 - 495*A*a^2*b^18*d
^8*abs(b)*e^8 - 660*B*a^4*b^16*d^7*abs(b)*e^9 + 1320*A*a^3*b^17*d^7*abs(b)*e^9 + 1386*B*a^5*b^15*d^6*abs(b)*e^
10 - 2310*A*a^4*b^16*d^6*abs(b)*e^10 - 1848*B*a^6*b^14*d^5*abs(b)*e^11 + 2772*A*a^5*b^15*d^5*abs(b)*e^11 + 165
0*B*a^7*b^13*d^4*abs(b)*e^12 - 2310*A*a^6*b^14*d^4*abs(b)*e^12 - 990*B*a^8*b^12*d^3*abs(b)*e^13 + 1320*A*a^7*b
^13*d^3*abs(b)*e^13 + 385*B*a^9*b^11*d^2*abs(b)*e^14 - 495*A*a^8*b^12*d^2*abs(b)*e^14 - 88*B*a^10*b^10*d*abs(b
)*e^15 + 110*A*a^9*b^11*d*abs(b)*e^15 + 9*B*a^11*b^9*abs(b)*e^16 - 11*A*a^10*b^10*abs(b)*e^16)/(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)) + 70*(B*b^21*d^12*abs(b)*e^4 - 7
*B*a*b^20*d^11*abs(b)*e^5 - 5*A*b^21*d^11*abs(b)*e^5 + 11*B*a^2*b^19*d^10*abs(b)*e^6 + 55*A*a*b^20*d^10*abs(b)
*e^6 + 55*B*a^3*b^18*d^9*abs(b)*e^7 - 275*A*a^2*b^19*d^9*abs(b)*e^7 - 330*B*a^4*b^17*d^8*abs(b)*e^8 + 825*A*a^
3*b^18*d^8*abs(b)*e^8 + 858*B*a^5*b^16*d^7*abs(b)*e^9 - 1650*A*a^4*b^17*d^7*abs(b)*e^9 - 1386*B*a^6*b^15*d^6*a
bs(b)*e^10 + 2310*A*a^5*b^16*d^6*abs(b)*e^10 + 1518*B*a^7*b^14*d^5*abs(b)*e^11 - 2310*A*a^6*b^15*d^5*abs(b)*e^
11 - 1155*B*a^8*b^13*d^4*abs(b)*e^12 + 1650*A*a^7*b^14*d^4*abs(b)*e^12 + 605*B*a^9*b^12*d^3*abs(b)*e^13 - 825*
A*a^8*b^13*d^3*abs(b)*e^13 - 209*B*a^10*b^11*d^2*abs(b)*e^14 + 275*A*a^9*b^12*d^2*abs(b)*e^14 + 43*B*a^11*b^10
*d*abs(b)*e^15 - 55*A*a^10*b^11*d*abs(b)*e^15 - 4*B*a^12*b^9*abs(b)*e^16 + 5*A*a^11*b^10*abs(b)*e^16)/(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) + 35*(B*b^22*d^
13*abs(b)*e^3 - 9*B*a*b^21*d^12*abs(b)*e^4 - 4*A*b^22*d^12*abs(b)*e^4 + 30*B*a^2*b^20*d^11*abs(b)*e^5 + 48*A*a
*b^21*d^11*abs(b)*e^5 - 22*B*a^3*b^19*d^10*abs(b)*e^6 - 264*A*a^2*b^20*d^10*abs(b)*e^6 - 165*B*a^4*b^18*d^9*ab
s(b)*e^7 + 880*A*a^3*b^19*d^9*abs(b)*e^7 + 693*B*a^5*b^17*d^8*abs(b)*e^8 - 1980*A*a^4*b^18*d^8*abs(b)*e^8 - 14
52*B*a^6*b^16*d^7*abs(b)*e^9 + 3168*A*a^5*b^17*d^7*abs(b)*e^9 + 1980*B*a^7*b^15*d^6*abs(b)*e^10 - 3696*A*a^6*b
^16*d^6*abs(b)*e^10 - 1881*B*a^8*b^14*d^5*abs(b)*e^11 + 3168*A*a^7*b^15*d^5*abs(b)*e^11 + 1265*B*a^9*b^13*d^4*
abs(b)*e^12 - 1980*A*a^8*b^14*d^4*abs(b)*e^12 - 594*B*a^10*b^12*d^3*abs(b)*e^13 + 880*A*a^9*b^13*d^3*abs(b)*e^
13 + 186*B*a^11*b^11*d^2*abs(b)*e^14 - 264*A*a^10*b^12*d^2*abs(b)*e^14 - 35*B*a^12*b^10*d*abs(b)*e^15 + 48*A*a
^11*b^11*d*abs(b)*e^15 + 3*B*a^13*b^9*abs(b)*e^16 - 4*A*a^12*b^10*abs(b)*e^16)/(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))*sqrt(b*x + a)/(b^2*d + (b*x + a)*b*e - a*b*e)^(7
/2)