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

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

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Rubi [A]  time = 0.08, antiderivative size = 210, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {78, 45, 37} \begin {gather*} \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}} \end {gather*}

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 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]

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 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]))))

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*}

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Mathematica [A]  time = 0.06, size = 127, normalized size = 0.60 \begin {gather*} -\frac {2 \left (a^6 (7 A+9 B x)-6 a^5 b x (2 A+3 B x)+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)-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}} \end {gather*}

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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IntegrateAlgebraic [A]  time = 0.34, size = 154, normalized size = 0.73 \begin {gather*} \frac {2 \left (-7 a^6 A-9 a^6 B x+12 a^5 A b x+18 a^5 b B x^2-24 a^4 A b^2 x^2-48 a^4 b^2 B x^3+64 a^3 A b^3 x^3+288 a^3 b^3 B x^4-384 a^2 A b^4 x^4+1152 a^2 b^4 B x^5-1536 a A b^5 x^5+768 a b^5 B x^6-1024 A b^6 x^6\right )}{63 a^7 x^{9/2} (a+b x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.70, size = 176, normalized size = 0.84 \begin {gather*} -\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 {gather*}

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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giac [B]  time = 2.80, size = 418, normalized size = 1.99 \begin {gather*} \frac {2 \, {\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^{25} b^{4} {\left | b \right |}} - \frac {9 \, {\left (223 \, B a^{20} b^{13} - 316 \, A a^{19} b^{14}\right )}}{a^{25} b^{4} {\left | b \right |}}\right )} + \frac {63 \, {\left (51 \, B a^{21} b^{13} - 73 \, A a^{20} b^{14}\right )}}{a^{25} b^{4} {\left | b \right |}}\right )} - \frac {210 \, {\left (11 \, B a^{22} b^{13} - 16 \, A a^{21} b^{14}\right )}}{a^{25} b^{4} {\left | b \right |}}\right )} {\left (b x + a\right )} + \frac {315 \, {\left (2 \, B a^{23} b^{13} - 3 \, A a^{22} b^{14}\right )}}{a^{25} b^{4} {\left | b \right |}}\right )} \sqrt {b x + a}}{63 \, {\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 {gather*}

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]

2/63*(((b*x + a)*((b*x + a)*((474*B*a^19*b^13 - 667*A*a^18*b^14)*(b*x + a)/(a^25*b^4*abs(b)) - 9*(223*B*a^20*b
^13 - 316*A*a^19*b^14)/(a^25*b^4*abs(b))) + 63*(51*B*a^21*b^13 - 73*A*a^20*b^14)/(a^25*b^4*abs(b))) - 210*(11*
B*a^22*b^13 - 16*A*a^21*b^14)/(a^25*b^4*abs(b)))*(b*x + a) + 315*(2*B*a^23*b^13 - 3*A*a^22*b^14)/(a^25*b^4*abs
(b)))*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*a
bs(b))

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maple [A]  time = 0.01, size = 149, normalized size = 0.71 \begin {gather*} -\frac {2 \left (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}\right )}{63 \left (b x +a \right )^{\frac {3}{2}} a^{7} x^{\frac {9}{2}}} \end {gather*}

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)/(b*x+a)^(3/2
)/x^(9/2)/a^7

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maxima [A]  time = 0.96, size = 270, normalized size = 1.29 \begin {gather*} -\frac {64 \, B b^{3} x}{21 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} a^{4}} + \frac {512 \, B b^{4} x}{21 \, \sqrt {b x^{2} + a x} a^{6}} + \frac {256 \, A b^{4} x}{63 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} a^{5}} - \frac {2048 \, A b^{5} x}{63 \, \sqrt {b x^{2} + a x} a^{7}} - \frac {32 \, B b^{2}}{21 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} a^{3}} + \frac {256 \, B b^{3}}{21 \, \sqrt {b x^{2} + a x} a^{5}} + \frac {128 \, A b^{3}}{63 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} a^{4}} - \frac {1024 \, A b^{4}}{63 \, \sqrt {b x^{2} + a x} a^{6}} + \frac {4 \, B b}{7 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} a^{2} x} - \frac {16 \, A b^{2}}{21 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} a^{3} x} - \frac {2 \, B}{7 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} a x^{2}} + \frac {8 \, A b}{21 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} a^{2} x^{2}} - \frac {2 \, A}{9 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} a x^{3}} \end {gather*}

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]

-64/21*B*b^3*x/((b*x^2 + a*x)^(3/2)*a^4) + 512/21*B*b^4*x/(sqrt(b*x^2 + a*x)*a^6) + 256/63*A*b^4*x/((b*x^2 + a
*x)^(3/2)*a^5) - 2048/63*A*b^5*x/(sqrt(b*x^2 + a*x)*a^7) - 32/21*B*b^2/((b*x^2 + a*x)^(3/2)*a^3) + 256/21*B*b^
3/(sqrt(b*x^2 + a*x)*a^5) + 128/63*A*b^3/((b*x^2 + a*x)^(3/2)*a^4) - 1024/63*A*b^4/(sqrt(b*x^2 + a*x)*a^6) + 4
/7*B*b/((b*x^2 + a*x)^(3/2)*a^2*x) - 16/21*A*b^2/((b*x^2 + a*x)^(3/2)*a^3*x) - 2/7*B/((b*x^2 + a*x)^(3/2)*a*x^
2) + 8/21*A*b/((b*x^2 + a*x)^(3/2)*a^2*x^2) - 2/9*A/((b*x^2 + a*x)^(3/2)*a*x^3)

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mupad [B]  time = 1.08, size = 162, normalized size = 0.77 \begin {gather*} -\frac {\sqrt {a+b\,x}\,\left (\frac {2\,A}{9\,a\,b^2}-\frac {32\,x^3\,\left (4\,A\,b-3\,B\,a\right )}{63\,a^4}+\frac {4\,x^2\,\left (4\,A\,b-3\,B\,a\right )}{21\,a^3\,b}+\frac {256\,b^2\,x^5\,\left (4\,A\,b-3\,B\,a\right )}{21\,a^6}+\frac {512\,b^3\,x^6\,\left (4\,A\,b-3\,B\,a\right )}{63\,a^7}+\frac {64\,b\,x^4\,\left (4\,A\,b-3\,B\,a\right )}{21\,a^5}+\frac {x\,\left (18\,B\,a^6-24\,A\,a^5\,b\right )}{63\,a^7\,b^2}\right )}{x^{13/2}+\frac {2\,a\,x^{11/2}}{b}+\frac {a^2\,x^{9/2}}{b^2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-((a + b*x)^(1/2)*((2*A)/(9*a*b^2) - (32*x^3*(4*A*b - 3*B*a))/(63*a^4) + (4*x^2*(4*A*b - 3*B*a))/(21*a^3*b) +
(256*b^2*x^5*(4*A*b - 3*B*a))/(21*a^6) + (512*b^3*x^6*(4*A*b - 3*B*a))/(63*a^7) + (64*b*x^4*(4*A*b - 3*B*a))/(
21*a^5) + (x*(18*B*a^6 - 24*A*a^5*b))/(63*a^7*b^2)))/(x^(13/2) + (2*a*x^(11/2))/b + (a^2*x^(9/2))/b^2)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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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