3.2260 \(\int \frac{A+B x}{(a+b x)^{5/2} (d+e x)^{7/2}} \, dx\)

Optimal. Leaf size=246 \[ -\frac{2 (A b-a B)}{3 b (a+b x)^{3/2} (d+e x)^{5/2} (b d-a e)}-\frac{32 b e \sqrt{a+b x} (5 a B e-8 A b e+3 b B d)}{15 \sqrt{d+e x} (b d-a e)^5}-\frac{16 e \sqrt{a+b x} (5 a B e-8 A b e+3 b B d)}{15 (d+e x)^{3/2} (b d-a e)^4}-\frac{4 e \sqrt{a+b x} (5 a B e-8 A b e+3 b B d)}{5 b (d+e x)^{5/2} (b d-a e)^3}-\frac{2 (5 a B e-8 A b e+3 b B d)}{3 b \sqrt{a+b x} (d+e x)^{5/2} (b d-a e)^2} \]

[Out]

(-2*(A*b - a*B))/(3*b*(b*d - a*e)*(a + b*x)^(3/2)*(d + e*x)^(5/2)) - (2*(3*b*B*d - 8*A*b*e + 5*a*B*e))/(3*b*(b
*d - a*e)^2*Sqrt[a + b*x]*(d + e*x)^(5/2)) - (4*e*(3*b*B*d - 8*A*b*e + 5*a*B*e)*Sqrt[a + b*x])/(5*b*(b*d - a*e
)^3*(d + e*x)^(5/2)) - (16*e*(3*b*B*d - 8*A*b*e + 5*a*B*e)*Sqrt[a + b*x])/(15*(b*d - a*e)^4*(d + e*x)^(3/2)) -
(32*b*e*(3*b*B*d - 8*A*b*e + 5*a*B*e)*Sqrt[a + b*x])/(15*(b*d - a*e)^5*Sqrt[d + e*x])

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Rubi [A]  time = 0.158637, antiderivative size = 246, 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} \[ -\frac{2 (A b-a B)}{3 b (a+b x)^{3/2} (d+e x)^{5/2} (b d-a e)}-\frac{32 b e \sqrt{a+b x} (5 a B e-8 A b e+3 b B d)}{15 \sqrt{d+e x} (b d-a e)^5}-\frac{16 e \sqrt{a+b x} (5 a B e-8 A b e+3 b B d)}{15 (d+e x)^{3/2} (b d-a e)^4}-\frac{4 e \sqrt{a+b x} (5 a B e-8 A b e+3 b B d)}{5 b (d+e x)^{5/2} (b d-a e)^3}-\frac{2 (5 a B e-8 A b e+3 b B d)}{3 b \sqrt{a+b x} (d+e x)^{5/2} (b d-a e)^2} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*x)/((a + b*x)^(5/2)*(d + e*x)^(7/2)),x]

[Out]

(-2*(A*b - a*B))/(3*b*(b*d - a*e)*(a + b*x)^(3/2)*(d + e*x)^(5/2)) - (2*(3*b*B*d - 8*A*b*e + 5*a*B*e))/(3*b*(b
*d - a*e)^2*Sqrt[a + b*x]*(d + e*x)^(5/2)) - (4*e*(3*b*B*d - 8*A*b*e + 5*a*B*e)*Sqrt[a + b*x])/(5*b*(b*d - a*e
)^3*(d + e*x)^(5/2)) - (16*e*(3*b*B*d - 8*A*b*e + 5*a*B*e)*Sqrt[a + b*x])/(15*(b*d - a*e)^4*(d + e*x)^(3/2)) -
(32*b*e*(3*b*B*d - 8*A*b*e + 5*a*B*e)*Sqrt[a + b*x])/(15*(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)^{5/2} (d+e x)^{7/2}} \, dx &=-\frac{2 (A b-a B)}{3 b (b d-a e) (a+b x)^{3/2} (d+e x)^{5/2}}+\frac{(3 b B d-8 A b e+5 a B e) \int \frac{1}{(a+b x)^{3/2} (d+e x)^{7/2}} \, dx}{3 b (b d-a e)}\\ &=-\frac{2 (A b-a B)}{3 b (b d-a e) (a+b x)^{3/2} (d+e x)^{5/2}}-\frac{2 (3 b B d-8 A b e+5 a B e)}{3 b (b d-a e)^2 \sqrt{a+b x} (d+e x)^{5/2}}-\frac{(2 e (3 b B d-8 A b e+5 a B e)) \int \frac{1}{\sqrt{a+b x} (d+e x)^{7/2}} \, dx}{b (b d-a e)^2}\\ &=-\frac{2 (A b-a B)}{3 b (b d-a e) (a+b x)^{3/2} (d+e x)^{5/2}}-\frac{2 (3 b B d-8 A b e+5 a B e)}{3 b (b d-a e)^2 \sqrt{a+b x} (d+e x)^{5/2}}-\frac{4 e (3 b B d-8 A b e+5 a B e) \sqrt{a+b x}}{5 b (b d-a e)^3 (d+e x)^{5/2}}-\frac{(8 e (3 b B d-8 A b e+5 a B e)) \int \frac{1}{\sqrt{a+b x} (d+e x)^{5/2}} \, dx}{5 (b d-a e)^3}\\ &=-\frac{2 (A b-a B)}{3 b (b d-a e) (a+b x)^{3/2} (d+e x)^{5/2}}-\frac{2 (3 b B d-8 A b e+5 a B e)}{3 b (b d-a e)^2 \sqrt{a+b x} (d+e x)^{5/2}}-\frac{4 e (3 b B d-8 A b e+5 a B e) \sqrt{a+b x}}{5 b (b d-a e)^3 (d+e x)^{5/2}}-\frac{16 e (3 b B d-8 A b e+5 a B e) \sqrt{a+b x}}{15 (b d-a e)^4 (d+e x)^{3/2}}-\frac{(16 b e (3 b B d-8 A b e+5 a B e)) \int \frac{1}{\sqrt{a+b x} (d+e x)^{3/2}} \, dx}{15 (b d-a e)^4}\\ &=-\frac{2 (A b-a B)}{3 b (b d-a e) (a+b x)^{3/2} (d+e x)^{5/2}}-\frac{2 (3 b B d-8 A b e+5 a B e)}{3 b (b d-a e)^2 \sqrt{a+b x} (d+e x)^{5/2}}-\frac{4 e (3 b B d-8 A b e+5 a B e) \sqrt{a+b x}}{5 b (b d-a e)^3 (d+e x)^{5/2}}-\frac{16 e (3 b B d-8 A b e+5 a B e) \sqrt{a+b x}}{15 (b d-a e)^4 (d+e x)^{3/2}}-\frac{32 b e (3 b B d-8 A b e+5 a B e) \sqrt{a+b x}}{15 (b d-a e)^5 \sqrt{d+e x}}\\ \end{align*}

Mathematica [A]  time = 0.261672, size = 134, normalized size = 0.54 \[ \frac{2 \left ((a+b x) \left (2 e (a+b x) \left (4 b (d+e x) (-a e+3 b d+2 b e x)+3 (b d-a e)^2\right )+5 (b d-a e)^3\right ) (-5 a B e+8 A b e-3 b B d)-5 (A b-a B) (b d-a e)^4\right )}{15 b (a+b x)^{3/2} (d+e x)^{5/2} (b d-a e)^5} \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x)/((a + b*x)^(5/2)*(d + e*x)^(7/2)),x]

[Out]

(2*(-5*(A*b - a*B)*(b*d - a*e)^4 + (-3*b*B*d + 8*A*b*e - 5*a*B*e)*(a + b*x)*(5*(b*d - a*e)^3 + 2*e*(a + b*x)*(
3*(b*d - a*e)^2 + 4*b*(d + e*x)*(3*b*d - a*e + 2*b*e*x)))))/(15*b*(b*d - a*e)^5*(a + b*x)^(3/2)*(d + e*x)^(5/2
))

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Maple [B]  time = 0.009, size = 505, normalized size = 2.1 \begin{align*} -{\frac{256\,A{b}^{4}{e}^{4}{x}^{4}-160\,Ba{b}^{3}{e}^{4}{x}^{4}-96\,B{b}^{4}d{e}^{3}{x}^{4}+384\,Aa{b}^{3}{e}^{4}{x}^{3}+640\,A{b}^{4}d{e}^{3}{x}^{3}-240\,B{a}^{2}{b}^{2}{e}^{4}{x}^{3}-544\,Ba{b}^{3}d{e}^{3}{x}^{3}-240\,B{b}^{4}{d}^{2}{e}^{2}{x}^{3}+96\,A{a}^{2}{b}^{2}{e}^{4}{x}^{2}+960\,Aa{b}^{3}d{e}^{3}{x}^{2}+480\,A{b}^{4}{d}^{2}{e}^{2}{x}^{2}-60\,B{a}^{3}b{e}^{4}{x}^{2}-636\,B{a}^{2}{b}^{2}d{e}^{3}{x}^{2}-660\,Ba{b}^{3}{d}^{2}{e}^{2}{x}^{2}-180\,B{b}^{4}{d}^{3}e{x}^{2}-16\,A{a}^{3}b{e}^{4}x+240\,A{a}^{2}{b}^{2}d{e}^{3}x+720\,Aa{b}^{3}{d}^{2}{e}^{2}x+80\,A{b}^{4}{d}^{3}ex+10\,B{a}^{4}{e}^{4}x-144\,B{a}^{3}bd{e}^{3}x-540\,B{a}^{2}{b}^{2}{d}^{2}{e}^{2}x-320\,Ba{b}^{3}{d}^{3}ex-30\,B{b}^{4}{d}^{4}x+6\,A{a}^{4}{e}^{4}-40\,A{a}^{3}bd{e}^{3}+180\,A{a}^{2}{b}^{2}{d}^{2}{e}^{2}+120\,Aa{b}^{3}{d}^{3}e-10\,A{b}^{4}{d}^{4}+4\,B{a}^{4}d{e}^{3}-60\,B{a}^{3}b{d}^{2}{e}^{2}-180\,B{a}^{2}{b}^{2}{d}^{3}e-20\,Ba{b}^{3}{d}^{4}}{15\,{a}^{5}{e}^{5}-75\,{a}^{4}bd{e}^{4}+150\,{a}^{3}{b}^{2}{d}^{2}{e}^{3}-150\,{a}^{2}{b}^{3}{d}^{3}{e}^{2}+75\,a{b}^{4}{d}^{4}e-15\,{b}^{5}{d}^{5}} \left ( bx+a \right ) ^{-{\frac{3}{2}}} \left ( ex+d \right ) ^{-{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/15*(128*A*b^4*e^4*x^4-80*B*a*b^3*e^4*x^4-48*B*b^4*d*e^3*x^4+192*A*a*b^3*e^4*x^3+320*A*b^4*d*e^3*x^3-120*B*a
^2*b^2*e^4*x^3-272*B*a*b^3*d*e^3*x^3-120*B*b^4*d^2*e^2*x^3+48*A*a^2*b^2*e^4*x^2+480*A*a*b^3*d*e^3*x^2+240*A*b^
4*d^2*e^2*x^2-30*B*a^3*b*e^4*x^2-318*B*a^2*b^2*d*e^3*x^2-330*B*a*b^3*d^2*e^2*x^2-90*B*b^4*d^3*e*x^2-8*A*a^3*b*
e^4*x+120*A*a^2*b^2*d*e^3*x+360*A*a*b^3*d^2*e^2*x+40*A*b^4*d^3*e*x+5*B*a^4*e^4*x-72*B*a^3*b*d*e^3*x-270*B*a^2*
b^2*d^2*e^2*x-160*B*a*b^3*d^3*e*x-15*B*b^4*d^4*x+3*A*a^4*e^4-20*A*a^3*b*d*e^3+90*A*a^2*b^2*d^2*e^2+60*A*a*b^3*
d^3*e-5*A*b^4*d^4+2*B*a^4*d*e^3-30*B*a^3*b*d^2*e^2-90*B*a^2*b^2*d^3*e-10*B*a*b^3*d^4)/(b*x+a)^(3/2)/(e*x+d)^(5
/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)

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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)^(5/2)/(e*x+d)^(7/2),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)/(b*x+a)^(5/2)/(e*x+d)^(7/2),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)/(b*x+a)**(5/2)/(e*x+d)**(7/2),x)

[Out]

Timed out

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Giac [B]  time = 8.22329, size = 2039, normalized size = 8.29 \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)^(5/2)/(e*x+d)^(7/2),x, algorithm="giac")

[Out]

1/960*((b*x + a)*((33*B*b^15*d^8*e^5 - 191*B*a*b^14*d^7*e^6 - 73*A*b^15*d^7*e^6 + 413*B*a^2*b^13*d^6*e^7 + 511
*A*a*b^14*d^6*e^7 - 315*B*a^3*b^12*d^5*e^8 - 1533*A*a^2*b^13*d^5*e^8 - 245*B*a^4*b^11*d^4*e^9 + 2555*A*a^3*b^1
2*d^4*e^9 + 707*B*a^5*b^10*d^3*e^10 - 2555*A*a^4*b^11*d^3*e^10 - 609*B*a^6*b^9*d^2*e^11 + 1533*A*a^5*b^10*d^2*
e^11 + 247*B*a^7*b^8*d*e^12 - 511*A*a^6*b^9*d*e^12 - 40*B*a^8*b^7*e^13 + 73*A*a^7*b^8*e^13)*(b*x + a)/(b^12*d^
3*e^6 - 3*a*b^11*d^2*e^7 + 3*a^2*b^10*d*e^8 - a^3*b^9*e^9) + 5*(15*B*b^16*d^9*e^4 - 103*B*a*b^15*d^8*e^5 - 32*
A*b^16*d^8*e^5 + 284*B*a^2*b^14*d^7*e^6 + 256*A*a*b^15*d^7*e^6 - 364*B*a^3*b^13*d^6*e^7 - 896*A*a^2*b^14*d^6*e
^7 + 98*B*a^4*b^12*d^5*e^8 + 1792*A*a^3*b^13*d^5*e^8 + 350*B*a^5*b^11*d^4*e^9 - 2240*A*a^4*b^12*d^4*e^9 - 532*
B*a^6*b^10*d^3*e^10 + 1792*A*a^5*b^11*d^3*e^10 + 356*B*a^7*b^9*d^2*e^11 - 896*A*a^6*b^10*d^2*e^11 - 121*B*a^8*
b^8*d*e^12 + 256*A*a^7*b^9*d*e^12 + 17*B*a^9*b^7*e^13 - 32*A*a^8*b^8*e^13)/(b^12*d^3*e^6 - 3*a*b^11*d^2*e^7 +
3*a^2*b^10*d*e^8 - a^3*b^9*e^9)) + 45*(B*b^17*d^10*e^3 - 8*B*a*b^16*d^9*e^4 - 2*A*b^17*d^9*e^4 + 27*B*a^2*b^15
*d^8*e^5 + 18*A*a*b^16*d^8*e^5 - 48*B*a^3*b^14*d^7*e^6 - 72*A*a^2*b^15*d^7*e^6 + 42*B*a^4*b^13*d^6*e^7 + 168*A
*a^3*b^14*d^6*e^7 - 252*A*a^4*b^13*d^5*e^8 - 42*B*a^6*b^11*d^4*e^9 + 252*A*a^5*b^12*d^4*e^9 + 48*B*a^7*b^10*d^
3*e^10 - 168*A*a^6*b^11*d^3*e^10 - 27*B*a^8*b^9*d^2*e^11 + 72*A*a^7*b^10*d^2*e^11 + 8*B*a^9*b^8*d*e^12 - 18*A*
a^8*b^9*d*e^12 - B*a^10*b^7*e^13 + 2*A*a^9*b^8*e^13)/(b^12*d^3*e^6 - 3*a*b^11*d^2*e^7 + 3*a^2*b^10*d*e^8 - a^3
*b^9*e^9))*sqrt(b*x + a)/(b^2*d + (b*x + a)*b*e - a*b*e)^(5/2) - 4/3*(3*B*b^(17/2)*d^3*e^(1/2) + 2*B*a*b^(15/2
)*d^2*e^(3/2) - 11*A*b^(17/2)*d^2*e^(3/2) - 6*(sqrt(b*x + a)*sqrt(b)*e^(1/2) - sqrt(b^2*d + (b*x + a)*b*e - a*
b*e))^2*B*b^(13/2)*d^2*e^(1/2) - 13*B*a^2*b^(13/2)*d*e^(5/2) + 22*A*a*b^(15/2)*d*e^(5/2) - 12*(sqrt(b*x + a)*s
qrt(b)*e^(1/2) - sqrt(b^2*d + (b*x + a)*b*e - a*b*e))^2*B*a*b^(11/2)*d*e^(3/2) + 24*(sqrt(b*x + a)*sqrt(b)*e^(
1/2) - sqrt(b^2*d + (b*x + a)*b*e - a*b*e))^2*A*b^(13/2)*d*e^(3/2) + 3*(sqrt(b*x + a)*sqrt(b)*e^(1/2) - sqrt(b
^2*d + (b*x + a)*b*e - a*b*e))^4*B*b^(9/2)*d*e^(1/2) + 8*B*a^3*b^(11/2)*e^(7/2) - 11*A*a^2*b^(13/2)*e^(7/2) +
18*(sqrt(b*x + a)*sqrt(b)*e^(1/2) - sqrt(b^2*d + (b*x + a)*b*e - a*b*e))^2*B*a^2*b^(9/2)*e^(5/2) - 24*(sqrt(b*
x + a)*sqrt(b)*e^(1/2) - sqrt(b^2*d + (b*x + a)*b*e - a*b*e))^2*A*a*b^(11/2)*e^(5/2) + 6*(sqrt(b*x + a)*sqrt(b
)*e^(1/2) - sqrt(b^2*d + (b*x + a)*b*e - a*b*e))^4*B*a*b^(7/2)*e^(3/2) - 9*(sqrt(b*x + a)*sqrt(b)*e^(1/2) - sq
rt(b^2*d + (b*x + a)*b*e - a*b*e))^4*A*b^(9/2)*e^(3/2))/((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)^3)