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

Optimal. Leaf size=210 \[ \frac{256 b^2 \sqrt{a+b x} (4 A b-3 a B)}{63 a^6 x^{3/2}}-\frac{512 b^3 \sqrt{a+b x} (4 A b-3 a B)}{63 a^7 \sqrt{x}}-\frac{64 b \sqrt{a+b x} (4 A b-3 a B)}{21 a^5 x^{5/2}}+\frac{160 \sqrt{a+b x} (4 A b-3 a B)}{63 a^4 x^{7/2}}-\frac{20 (4 A b-3 a B)}{9 a^3 x^{7/2} \sqrt{a+b x}}-\frac{2 (4 A b-3 a B)}{9 a^2 x^{7/2} (a+b x)^{3/2}}-\frac{2 A}{9 a x^{9/2} (a+b x)^{3/2}} \]

[Out]

(-2*A)/(9*a*x^(9/2)*(a + b*x)^(3/2)) - (2*(4*A*b - 3*a*B))/(9*a^2*x^(7/2)*(a + b*x)^(3/2)) - (20*(4*A*b - 3*a*
B))/(9*a^3*x^(7/2)*Sqrt[a + b*x]) + (160*(4*A*b - 3*a*B)*Sqrt[a + b*x])/(63*a^4*x^(7/2)) - (64*b*(4*A*b - 3*a*
B)*Sqrt[a + b*x])/(21*a^5*x^(5/2)) + (256*b^2*(4*A*b - 3*a*B)*Sqrt[a + b*x])/(63*a^6*x^(3/2)) - (512*b^3*(4*A*
b - 3*a*B)*Sqrt[a + b*x])/(63*a^7*Sqrt[x])

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Rubi [A]  time = 0.0829912, antiderivative size = 210, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {78, 45, 37} \[ \frac{256 b^2 \sqrt{a+b x} (4 A b-3 a B)}{63 a^6 x^{3/2}}-\frac{512 b^3 \sqrt{a+b x} (4 A b-3 a B)}{63 a^7 \sqrt{x}}-\frac{64 b \sqrt{a+b x} (4 A b-3 a B)}{21 a^5 x^{5/2}}+\frac{160 \sqrt{a+b x} (4 A b-3 a B)}{63 a^4 x^{7/2}}-\frac{20 (4 A b-3 a B)}{9 a^3 x^{7/2} \sqrt{a+b x}}-\frac{2 (4 A b-3 a B)}{9 a^2 x^{7/2} (a+b x)^{3/2}}-\frac{2 A}{9 a x^{9/2} (a+b x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*A)/(9*a*x^(9/2)*(a + b*x)^(3/2)) - (2*(4*A*b - 3*a*B))/(9*a^2*x^(7/2)*(a + b*x)^(3/2)) - (20*(4*A*b - 3*a*
B))/(9*a^3*x^(7/2)*Sqrt[a + b*x]) + (160*(4*A*b - 3*a*B)*Sqrt[a + b*x])/(63*a^4*x^(7/2)) - (64*b*(4*A*b - 3*a*
B)*Sqrt[a + b*x])/(21*a^5*x^(5/2)) + (256*b^2*(4*A*b - 3*a*B)*Sqrt[a + b*x])/(63*a^6*x^(3/2)) - (512*b^3*(4*A*
b - 3*a*B)*Sqrt[a + b*x])/(63*a^7*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)^{5/2}} \, dx &=-\frac{2 A}{9 a x^{9/2} (a+b x)^{3/2}}+\frac{\left (2 \left (-6 A b+\frac{9 a B}{2}\right )\right ) \int \frac{1}{x^{9/2} (a+b x)^{5/2}} \, dx}{9 a}\\ &=-\frac{2 A}{9 a x^{9/2} (a+b x)^{3/2}}-\frac{2 (4 A b-3 a B)}{9 a^2 x^{7/2} (a+b x)^{3/2}}-\frac{(10 (4 A b-3 a B)) \int \frac{1}{x^{9/2} (a+b x)^{3/2}} \, dx}{9 a^2}\\ &=-\frac{2 A}{9 a x^{9/2} (a+b x)^{3/2}}-\frac{2 (4 A b-3 a B)}{9 a^2 x^{7/2} (a+b x)^{3/2}}-\frac{20 (4 A b-3 a B)}{9 a^3 x^{7/2} \sqrt{a+b x}}-\frac{(80 (4 A b-3 a B)) \int \frac{1}{x^{9/2} \sqrt{a+b x}} \, dx}{9 a^3}\\ &=-\frac{2 A}{9 a x^{9/2} (a+b x)^{3/2}}-\frac{2 (4 A b-3 a B)}{9 a^2 x^{7/2} (a+b x)^{3/2}}-\frac{20 (4 A b-3 a B)}{9 a^3 x^{7/2} \sqrt{a+b x}}+\frac{160 (4 A b-3 a B) \sqrt{a+b x}}{63 a^4 x^{7/2}}+\frac{(160 b (4 A b-3 a B)) \int \frac{1}{x^{7/2} \sqrt{a+b x}} \, dx}{21 a^4}\\ &=-\frac{2 A}{9 a x^{9/2} (a+b x)^{3/2}}-\frac{2 (4 A b-3 a B)}{9 a^2 x^{7/2} (a+b x)^{3/2}}-\frac{20 (4 A b-3 a B)}{9 a^3 x^{7/2} \sqrt{a+b x}}+\frac{160 (4 A b-3 a B) \sqrt{a+b x}}{63 a^4 x^{7/2}}-\frac{64 b (4 A b-3 a B) \sqrt{a+b x}}{21 a^5 x^{5/2}}-\frac{\left (128 b^2 (4 A b-3 a B)\right ) \int \frac{1}{x^{5/2} \sqrt{a+b x}} \, dx}{21 a^5}\\ &=-\frac{2 A}{9 a x^{9/2} (a+b x)^{3/2}}-\frac{2 (4 A b-3 a B)}{9 a^2 x^{7/2} (a+b x)^{3/2}}-\frac{20 (4 A b-3 a B)}{9 a^3 x^{7/2} \sqrt{a+b x}}+\frac{160 (4 A b-3 a B) \sqrt{a+b x}}{63 a^4 x^{7/2}}-\frac{64 b (4 A b-3 a B) \sqrt{a+b x}}{21 a^5 x^{5/2}}+\frac{256 b^2 (4 A b-3 a B) \sqrt{a+b x}}{63 a^6 x^{3/2}}+\frac{\left (256 b^3 (4 A b-3 a B)\right ) \int \frac{1}{x^{3/2} \sqrt{a+b x}} \, dx}{63 a^6}\\ &=-\frac{2 A}{9 a x^{9/2} (a+b x)^{3/2}}-\frac{2 (4 A b-3 a B)}{9 a^2 x^{7/2} (a+b x)^{3/2}}-\frac{20 (4 A b-3 a B)}{9 a^3 x^{7/2} \sqrt{a+b x}}+\frac{160 (4 A b-3 a B) \sqrt{a+b x}}{63 a^4 x^{7/2}}-\frac{64 b (4 A b-3 a B) \sqrt{a+b x}}{21 a^5 x^{5/2}}+\frac{256 b^2 (4 A b-3 a B) \sqrt{a+b x}}{63 a^6 x^{3/2}}-\frac{512 b^3 (4 A b-3 a B) \sqrt{a+b x}}{63 a^7 \sqrt{x}}\\ \end{align*}

Mathematica [A]  time = 0.04152, size = 127, normalized size = 0.6 \[ -\frac{2 \left (24 a^4 b^2 x^2 (A+2 B x)-32 a^3 b^3 x^3 (2 A+9 B x)+384 a^2 b^4 x^4 (A-3 B x)-6 a^5 b x (2 A+3 B x)+a^6 (7 A+9 B x)-768 a b^5 x^5 (B x-2 A)+1024 A b^6 x^6\right )}{63 a^7 x^{9/2} (a+b x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(1024*A*b^6*x^6 + 384*a^2*b^4*x^4*(A - 3*B*x) - 768*a*b^5*x^5*(-2*A + B*x) + 24*a^4*b^2*x^2*(A + 2*B*x) -
6*a^5*b*x*(2*A + 3*B*x) - 32*a^3*b^3*x^3*(2*A + 9*B*x) + a^6*(7*A + 9*B*x)))/(63*a^7*x^(9/2)*(a + b*x)^(3/2))

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Maple [A]  time = 0.006, size = 149, normalized size = 0.7 \begin{align*} -{\frac{2048\,A{b}^{6}{x}^{6}-1536\,Ba{b}^{5}{x}^{6}+3072\,Aa{b}^{5}{x}^{5}-2304\,B{a}^{2}{b}^{4}{x}^{5}+768\,A{a}^{2}{b}^{4}{x}^{4}-576\,B{a}^{3}{b}^{3}{x}^{4}-128\,A{a}^{3}{b}^{3}{x}^{3}+96\,B{a}^{4}{b}^{2}{x}^{3}+48\,A{a}^{4}{b}^{2}{x}^{2}-36\,B{a}^{5}b{x}^{2}-24\,A{a}^{5}bx+18\,B{a}^{6}x+14\,A{a}^{6}}{63\,{a}^{7}}{x}^{-{\frac{9}{2}}} \left ( bx+a \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)/x^(11/2)/(b*x+a)^(5/2),x)

[Out]

-2/63*(1024*A*b^6*x^6-768*B*a*b^5*x^6+1536*A*a*b^5*x^5-1152*B*a^2*b^4*x^5+384*A*a^2*b^4*x^4-288*B*a^3*b^3*x^4-
64*A*a^3*b^3*x^3+48*B*a^4*b^2*x^3+24*A*a^4*b^2*x^2-18*B*a^5*b*x^2-12*A*a^5*b*x+9*B*a^6*x+7*A*a^6)/x^(9/2)/(b*x
+a)^(3/2)/a^7

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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)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.7085, size = 379, normalized size = 1.8 \begin{align*} -\frac{2 \,{\left (7 \, A a^{6} - 256 \,{\left (3 \, B a b^{5} - 4 \, A b^{6}\right )} x^{6} - 384 \,{\left (3 \, B a^{2} b^{4} - 4 \, A a b^{5}\right )} x^{5} - 96 \,{\left (3 \, B a^{3} b^{3} - 4 \, A a^{2} b^{4}\right )} x^{4} + 16 \,{\left (3 \, B a^{4} b^{2} - 4 \, A a^{3} b^{3}\right )} x^{3} - 6 \,{\left (3 \, B a^{5} b - 4 \, A a^{4} b^{2}\right )} x^{2} + 3 \,{\left (3 \, B a^{6} - 4 \, A a^{5} b\right )} x\right )} \sqrt{b x + a} \sqrt{x}}{63 \,{\left (a^{7} b^{2} x^{7} + 2 \, a^{8} b x^{6} + a^{9} 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)^(5/2),x, algorithm="fricas")

[Out]

-2/63*(7*A*a^6 - 256*(3*B*a*b^5 - 4*A*b^6)*x^6 - 384*(3*B*a^2*b^4 - 4*A*a*b^5)*x^5 - 96*(3*B*a^3*b^3 - 4*A*a^2
*b^4)*x^4 + 16*(3*B*a^4*b^2 - 4*A*a^3*b^3)*x^3 - 6*(3*B*a^5*b - 4*A*a^4*b^2)*x^2 + 3*(3*B*a^6 - 4*A*a^5*b)*x)*
sqrt(b*x + a)*sqrt(x)/(a^7*b^2*x^7 + 2*a^8*b*x^6 + a^9*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)**(5/2),x)

[Out]

Timed out

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Giac [B]  time = 2.55225, size = 537, normalized size = 2.56 \begin{align*} -\frac{{\left ({\left ({\left (b x + a\right )}{\left ({\left (b x + a\right )}{\left (\frac{{\left (474 \, B a^{19} b^{13} - 667 \, A a^{18} b^{14}\right )}{\left (b x + a\right )}}{a^{5} b^{15}} - \frac{9 \,{\left (223 \, B a^{20} b^{13} - 316 \, A a^{19} b^{14}\right )}}{a^{5} b^{15}}\right )} + \frac{63 \,{\left (51 \, B a^{21} b^{13} - 73 \, A a^{20} b^{14}\right )}}{a^{5} b^{15}}\right )} - \frac{210 \,{\left (11 \, B a^{22} b^{13} - 16 \, A a^{21} b^{14}\right )}}{a^{5} b^{15}}\right )}{\left (b x + a\right )} + \frac{315 \,{\left (2 \, B a^{23} b^{13} - 3 \, A a^{22} b^{14}\right )}}{a^{5} b^{15}}\right )} \sqrt{b x + a}}{64512 \,{\left ({\left (b x + a\right )} b - a b\right )}^{\frac{9}{2}}} + \frac{4 \,{\left (12 \, B a{\left (\sqrt{b x + a} \sqrt{b} - \sqrt{{\left (b x + a\right )} b - a b}\right )}^{4} b^{\frac{11}{2}} + 30 \, B a^{2}{\left (\sqrt{b x + a} \sqrt{b} - \sqrt{{\left (b x + a\right )} b - a b}\right )}^{2} b^{\frac{13}{2}} - 15 \, A{\left (\sqrt{b x + a} \sqrt{b} - \sqrt{{\left (b x + a\right )} b - a b}\right )}^{4} b^{\frac{13}{2}} + 14 \, B a^{3} b^{\frac{15}{2}} - 36 \, A a{\left (\sqrt{b x + a} \sqrt{b} - \sqrt{{\left (b x + a\right )} b - a b}\right )}^{2} b^{\frac{15}{2}} - 17 \, A a^{2} b^{\frac{17}{2}}\right )}}{3 \,{\left ({\left (\sqrt{b x + a} \sqrt{b} - \sqrt{{\left (b x + a\right )} b - a b}\right )}^{2} + a b\right )}^{3} a^{6}{\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)^(5/2),x, algorithm="giac")

[Out]

-1/64512*(((b*x + a)*((b*x + a)*((474*B*a^19*b^13 - 667*A*a^18*b^14)*(b*x + a)/(a^5*b^15) - 9*(223*B*a^20*b^13
- 316*A*a^19*b^14)/(a^5*b^15)) + 63*(51*B*a^21*b^13 - 73*A*a^20*b^14)/(a^5*b^15)) - 210*(11*B*a^22*b^13 - 16*
A*a^21*b^14)/(a^5*b^15))*(b*x + a) + 315*(2*B*a^23*b^13 - 3*A*a^22*b^14)/(a^5*b^15))*sqrt(b*x + a)/((b*x + a)*
b - a*b)^(9/2) + 4/3*(12*B*a*(sqrt(b*x + a)*sqrt(b) - sqrt((b*x + a)*b - a*b))^4*b^(11/2) + 30*B*a^2*(sqrt(b*x
+ a)*sqrt(b) - sqrt((b*x + a)*b - a*b))^2*b^(13/2) - 15*A*(sqrt(b*x + a)*sqrt(b) - sqrt((b*x + a)*b - a*b))^4
*b^(13/2) + 14*B*a^3*b^(15/2) - 36*A*a*(sqrt(b*x + a)*sqrt(b) - sqrt((b*x + a)*b - a*b))^2*b^(15/2) - 17*A*a^2
*b^(17/2))/(((sqrt(b*x + a)*sqrt(b) - sqrt((b*x + a)*b - a*b))^2 + a*b)^3*a^6*abs(b))