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

Optimal. Leaf size=180 \[ \frac{128 b^2 \sqrt{a+b x} (10 A b-9 a B)}{315 a^5 x^{3/2}}-\frac{256 b^3 \sqrt{a+b x} (10 A b-9 a B)}{315 a^6 \sqrt{x}}-\frac{32 b \sqrt{a+b x} (10 A b-9 a B)}{105 a^4 x^{5/2}}+\frac{16 \sqrt{a+b x} (10 A b-9 a B)}{63 a^3 x^{7/2}}-\frac{2 (10 A b-9 a B)}{9 a^2 x^{7/2} \sqrt{a+b x}}-\frac{2 A}{9 a x^{9/2} \sqrt{a+b x}} \]

[Out]

(-2*A)/(9*a*x^(9/2)*Sqrt[a + b*x]) - (2*(10*A*b - 9*a*B))/(9*a^2*x^(7/2)*Sqrt[a + b*x]) + (16*(10*A*b - 9*a*B)
*Sqrt[a + b*x])/(63*a^3*x^(7/2)) - (32*b*(10*A*b - 9*a*B)*Sqrt[a + b*x])/(105*a^4*x^(5/2)) + (128*b^2*(10*A*b
- 9*a*B)*Sqrt[a + b*x])/(315*a^5*x^(3/2)) - (256*b^3*(10*A*b - 9*a*B)*Sqrt[a + b*x])/(315*a^6*Sqrt[x])

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Rubi [A]  time = 0.0706605, antiderivative size = 180, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {78, 45, 37} \[ \frac{128 b^2 \sqrt{a+b x} (10 A b-9 a B)}{315 a^5 x^{3/2}}-\frac{256 b^3 \sqrt{a+b x} (10 A b-9 a B)}{315 a^6 \sqrt{x}}-\frac{32 b \sqrt{a+b x} (10 A b-9 a B)}{105 a^4 x^{5/2}}+\frac{16 \sqrt{a+b x} (10 A b-9 a B)}{63 a^3 x^{7/2}}-\frac{2 (10 A b-9 a B)}{9 a^2 x^{7/2} \sqrt{a+b x}}-\frac{2 A}{9 a x^{9/2} \sqrt{a+b x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*A)/(9*a*x^(9/2)*Sqrt[a + b*x]) - (2*(10*A*b - 9*a*B))/(9*a^2*x^(7/2)*Sqrt[a + b*x]) + (16*(10*A*b - 9*a*B)
*Sqrt[a + b*x])/(63*a^3*x^(7/2)) - (32*b*(10*A*b - 9*a*B)*Sqrt[a + b*x])/(105*a^4*x^(5/2)) + (128*b^2*(10*A*b
- 9*a*B)*Sqrt[a + b*x])/(315*a^5*x^(3/2)) - (256*b^3*(10*A*b - 9*a*B)*Sqrt[a + b*x])/(315*a^6*Sqrt[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}{x^{11/2} (a+b x)^{3/2}} \, dx &=-\frac{2 A}{9 a x^{9/2} \sqrt{a+b x}}+\frac{\left (2 \left (-5 A b+\frac{9 a B}{2}\right )\right ) \int \frac{1}{x^{9/2} (a+b x)^{3/2}} \, dx}{9 a}\\ &=-\frac{2 A}{9 a x^{9/2} \sqrt{a+b x}}-\frac{2 (10 A b-9 a B)}{9 a^2 x^{7/2} \sqrt{a+b x}}-\frac{(8 (10 A b-9 a B)) \int \frac{1}{x^{9/2} \sqrt{a+b x}} \, dx}{9 a^2}\\ &=-\frac{2 A}{9 a x^{9/2} \sqrt{a+b x}}-\frac{2 (10 A b-9 a B)}{9 a^2 x^{7/2} \sqrt{a+b x}}+\frac{16 (10 A b-9 a B) \sqrt{a+b x}}{63 a^3 x^{7/2}}+\frac{(16 b (10 A b-9 a B)) \int \frac{1}{x^{7/2} \sqrt{a+b x}} \, dx}{21 a^3}\\ &=-\frac{2 A}{9 a x^{9/2} \sqrt{a+b x}}-\frac{2 (10 A b-9 a B)}{9 a^2 x^{7/2} \sqrt{a+b x}}+\frac{16 (10 A b-9 a B) \sqrt{a+b x}}{63 a^3 x^{7/2}}-\frac{32 b (10 A b-9 a B) \sqrt{a+b x}}{105 a^4 x^{5/2}}-\frac{\left (64 b^2 (10 A b-9 a B)\right ) \int \frac{1}{x^{5/2} \sqrt{a+b x}} \, dx}{105 a^4}\\ &=-\frac{2 A}{9 a x^{9/2} \sqrt{a+b x}}-\frac{2 (10 A b-9 a B)}{9 a^2 x^{7/2} \sqrt{a+b x}}+\frac{16 (10 A b-9 a B) \sqrt{a+b x}}{63 a^3 x^{7/2}}-\frac{32 b (10 A b-9 a B) \sqrt{a+b x}}{105 a^4 x^{5/2}}+\frac{128 b^2 (10 A b-9 a B) \sqrt{a+b x}}{315 a^5 x^{3/2}}+\frac{\left (128 b^3 (10 A b-9 a B)\right ) \int \frac{1}{x^{3/2} \sqrt{a+b x}} \, dx}{315 a^5}\\ &=-\frac{2 A}{9 a x^{9/2} \sqrt{a+b x}}-\frac{2 (10 A b-9 a B)}{9 a^2 x^{7/2} \sqrt{a+b x}}+\frac{16 (10 A b-9 a B) \sqrt{a+b x}}{63 a^3 x^{7/2}}-\frac{32 b (10 A b-9 a B) \sqrt{a+b x}}{105 a^4 x^{5/2}}+\frac{128 b^2 (10 A b-9 a B) \sqrt{a+b x}}{315 a^5 x^{3/2}}-\frac{256 b^3 (10 A b-9 a B) \sqrt{a+b x}}{315 a^6 \sqrt{x}}\\ \end{align*}

Mathematica [A]  time = 0.0318225, size = 114, normalized size = 0.63 \[ -\frac{2 \left (16 a^3 b^2 x^2 (5 A+9 B x)-32 a^2 b^3 x^3 (5 A+18 B x)-2 a^4 b x (25 A+36 B x)+5 a^5 (7 A+9 B x)+128 a b^4 x^4 (5 A-9 B x)+1280 A b^5 x^5\right )}{315 a^6 x^{9/2} \sqrt{a+b x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(1280*A*b^5*x^5 + 128*a*b^4*x^4*(5*A - 9*B*x) + 16*a^3*b^2*x^2*(5*A + 9*B*x) + 5*a^5*(7*A + 9*B*x) - 32*a^
2*b^3*x^3*(5*A + 18*B*x) - 2*a^4*b*x*(25*A + 36*B*x)))/(315*a^6*x^(9/2)*Sqrt[a + b*x])

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Maple [A]  time = 0.005, size = 125, normalized size = 0.7 \begin{align*} -{\frac{2560\,A{b}^{5}{x}^{5}-2304\,B{x}^{5}a{b}^{4}+1280\,aA{b}^{4}{x}^{4}-1152\,B{x}^{4}{a}^{2}{b}^{3}-320\,{a}^{2}A{b}^{3}{x}^{3}+288\,B{x}^{3}{a}^{3}{b}^{2}+160\,{a}^{3}A{b}^{2}{x}^{2}-144\,B{x}^{2}{a}^{4}b-100\,{a}^{4}Abx+90\,{a}^{5}Bx+70\,A{a}^{5}}{315\,{a}^{6}}{x}^{-{\frac{9}{2}}}{\frac{1}{\sqrt{bx+a}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/315*(1280*A*b^5*x^5-1152*B*a*b^4*x^5+640*A*a*b^4*x^4-576*B*a^2*b^3*x^4-160*A*a^2*b^3*x^3+144*B*a^3*b^2*x^3+
80*A*a^3*b^2*x^2-72*B*a^4*b*x^2-50*A*a^4*b*x+45*B*a^5*x+35*A*a^5)/x^(9/2)/(b*x+a)^(1/2)/a^6

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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)/x^(11/2)/(b*x+a)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.69928, size = 316, normalized size = 1.76 \begin{align*} -\frac{2 \,{\left (35 \, A a^{5} - 128 \,{\left (9 \, B a b^{4} - 10 \, A b^{5}\right )} x^{5} - 64 \,{\left (9 \, B a^{2} b^{3} - 10 \, A a b^{4}\right )} x^{4} + 16 \,{\left (9 \, B a^{3} b^{2} - 10 \, A a^{2} b^{3}\right )} x^{3} - 8 \,{\left (9 \, B a^{4} b - 10 \, A a^{3} b^{2}\right )} x^{2} + 5 \,{\left (9 \, B a^{5} - 10 \, A a^{4} b\right )} x\right )} \sqrt{b x + a} \sqrt{x}}{315 \,{\left (a^{6} b x^{6} + a^{7} x^{5}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/315*(35*A*a^5 - 128*(9*B*a*b^4 - 10*A*b^5)*x^5 - 64*(9*B*a^2*b^3 - 10*A*a*b^4)*x^4 + 16*(9*B*a^3*b^2 - 10*A
*a^2*b^3)*x^3 - 8*(9*B*a^4*b - 10*A*a^3*b^2)*x^2 + 5*(9*B*a^5 - 10*A*a^4*b)*x)*sqrt(b*x + a)*sqrt(x)/(a^6*b*x^
6 + a^7*x^5)

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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)/x**(11/2)/(b*x+a)**(3/2),x)

[Out]

Timed out

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Giac [A]  time = 1.39282, size = 328, normalized size = 1.82 \begin{align*} -\frac{{\left ({\left ({\left (b x + a\right )}{\left ({\left (b x + a\right )}{\left (\frac{{\left (837 \, B a^{15} b^{13} - 965 \, A a^{14} b^{14}\right )}{\left (b x + a\right )}}{a^{5} b^{15}} - \frac{9 \,{\left (401 \, B a^{16} b^{13} - 465 \, A a^{15} b^{14}\right )}}{a^{5} b^{15}}\right )} + \frac{126 \,{\left (47 \, B a^{17} b^{13} - 55 \, A a^{16} b^{14}\right )}}{a^{5} b^{15}}\right )} - \frac{210 \,{\left (21 \, B a^{18} b^{13} - 25 \, A a^{17} b^{14}\right )}}{a^{5} b^{15}}\right )}{\left (b x + a\right )} + \frac{315 \,{\left (4 \, B a^{19} b^{13} - 5 \, A a^{18} b^{14}\right )}}{a^{5} b^{15}}\right )} \sqrt{b x + a}}{322560 \,{\left ({\left (b x + a\right )} b - a b\right )}^{\frac{9}{2}}} + \frac{4 \,{\left (B a b^{\frac{11}{2}} - A b^{\frac{13}{2}}\right )}}{{\left ({\left (\sqrt{b x + a} \sqrt{b} - \sqrt{{\left (b x + a\right )} b - a b}\right )}^{2} + a b\right )} a^{5}{\left | b \right |}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/322560*(((b*x + a)*((b*x + a)*((837*B*a^15*b^13 - 965*A*a^14*b^14)*(b*x + a)/(a^5*b^15) - 9*(401*B*a^16*b^1
3 - 465*A*a^15*b^14)/(a^5*b^15)) + 126*(47*B*a^17*b^13 - 55*A*a^16*b^14)/(a^5*b^15)) - 210*(21*B*a^18*b^13 - 2
5*A*a^17*b^14)/(a^5*b^15))*(b*x + a) + 315*(4*B*a^19*b^13 - 5*A*a^18*b^14)/(a^5*b^15))*sqrt(b*x + a)/((b*x + a
)*b - a*b)^(9/2) + 4*(B*a*b^(11/2) - A*b^(13/2))/(((sqrt(b*x + a)*sqrt(b) - sqrt((b*x + a)*b - a*b))^2 + a*b)*
a^5*abs(b))