3.6.11 \(\int \frac {A+B x}{x^{13/2} (a+b x)^{3/2}} \, dx\)

Optimal. Leaf size=213 \[ \frac {512 b^4 \sqrt {a+b x} (12 A b-11 a B)}{693 a^7 \sqrt {x}}-\frac {256 b^3 \sqrt {a+b x} (12 A b-11 a B)}{693 a^6 x^{3/2}}+\frac {64 b^2 \sqrt {a+b x} (12 A b-11 a B)}{231 a^5 x^{5/2}}-\frac {160 b \sqrt {a+b x} (12 A b-11 a B)}{693 a^4 x^{7/2}}+\frac {20 \sqrt {a+b x} (12 A b-11 a B)}{99 a^3 x^{9/2}}-\frac {2 (12 A b-11 a B)}{11 a^2 x^{9/2} \sqrt {a+b x}}-\frac {2 A}{11 a x^{11/2} \sqrt {a+b x}} \]

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Rubi [A]  time = 0.09, antiderivative size = 213, 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^3 \sqrt {a+b x} (12 A b-11 a B)}{693 a^6 x^{3/2}}+\frac {64 b^2 \sqrt {a+b x} (12 A b-11 a B)}{231 a^5 x^{5/2}}+\frac {512 b^4 \sqrt {a+b x} (12 A b-11 a B)}{693 a^7 \sqrt {x}}-\frac {160 b \sqrt {a+b x} (12 A b-11 a B)}{693 a^4 x^{7/2}}+\frac {20 \sqrt {a+b x} (12 A b-11 a B)}{99 a^3 x^{9/2}}-\frac {2 (12 A b-11 a B)}{11 a^2 x^{9/2} \sqrt {a+b x}}-\frac {2 A}{11 a x^{11/2} \sqrt {a+b x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(A + B*x)/(x^(13/2)*(a + b*x)^(3/2)),x]

[Out]

(-2*A)/(11*a*x^(11/2)*Sqrt[a + b*x]) - (2*(12*A*b - 11*a*B))/(11*a^2*x^(9/2)*Sqrt[a + b*x]) + (20*(12*A*b - 11
*a*B)*Sqrt[a + b*x])/(99*a^3*x^(9/2)) - (160*b*(12*A*b - 11*a*B)*Sqrt[a + b*x])/(693*a^4*x^(7/2)) + (64*b^2*(1
2*A*b - 11*a*B)*Sqrt[a + b*x])/(231*a^5*x^(5/2)) - (256*b^3*(12*A*b - 11*a*B)*Sqrt[a + b*x])/(693*a^6*x^(3/2))
+ (512*b^4*(12*A*b - 11*a*B)*Sqrt[a + b*x])/(693*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^{13/2} (a+b x)^{3/2}} \, dx &=-\frac {2 A}{11 a x^{11/2} \sqrt {a+b x}}+\frac {\left (2 \left (-6 A b+\frac {11 a B}{2}\right )\right ) \int \frac {1}{x^{11/2} (a+b x)^{3/2}} \, dx}{11 a}\\ &=-\frac {2 A}{11 a x^{11/2} \sqrt {a+b x}}-\frac {2 (12 A b-11 a B)}{11 a^2 x^{9/2} \sqrt {a+b x}}-\frac {(10 (12 A b-11 a B)) \int \frac {1}{x^{11/2} \sqrt {a+b x}} \, dx}{11 a^2}\\ &=-\frac {2 A}{11 a x^{11/2} \sqrt {a+b x}}-\frac {2 (12 A b-11 a B)}{11 a^2 x^{9/2} \sqrt {a+b x}}+\frac {20 (12 A b-11 a B) \sqrt {a+b x}}{99 a^3 x^{9/2}}+\frac {(80 b (12 A b-11 a B)) \int \frac {1}{x^{9/2} \sqrt {a+b x}} \, dx}{99 a^3}\\ &=-\frac {2 A}{11 a x^{11/2} \sqrt {a+b x}}-\frac {2 (12 A b-11 a B)}{11 a^2 x^{9/2} \sqrt {a+b x}}+\frac {20 (12 A b-11 a B) \sqrt {a+b x}}{99 a^3 x^{9/2}}-\frac {160 b (12 A b-11 a B) \sqrt {a+b x}}{693 a^4 x^{7/2}}-\frac {\left (160 b^2 (12 A b-11 a B)\right ) \int \frac {1}{x^{7/2} \sqrt {a+b x}} \, dx}{231 a^4}\\ &=-\frac {2 A}{11 a x^{11/2} \sqrt {a+b x}}-\frac {2 (12 A b-11 a B)}{11 a^2 x^{9/2} \sqrt {a+b x}}+\frac {20 (12 A b-11 a B) \sqrt {a+b x}}{99 a^3 x^{9/2}}-\frac {160 b (12 A b-11 a B) \sqrt {a+b x}}{693 a^4 x^{7/2}}+\frac {64 b^2 (12 A b-11 a B) \sqrt {a+b x}}{231 a^5 x^{5/2}}+\frac {\left (128 b^3 (12 A b-11 a B)\right ) \int \frac {1}{x^{5/2} \sqrt {a+b x}} \, dx}{231 a^5}\\ &=-\frac {2 A}{11 a x^{11/2} \sqrt {a+b x}}-\frac {2 (12 A b-11 a B)}{11 a^2 x^{9/2} \sqrt {a+b x}}+\frac {20 (12 A b-11 a B) \sqrt {a+b x}}{99 a^3 x^{9/2}}-\frac {160 b (12 A b-11 a B) \sqrt {a+b x}}{693 a^4 x^{7/2}}+\frac {64 b^2 (12 A b-11 a B) \sqrt {a+b x}}{231 a^5 x^{5/2}}-\frac {256 b^3 (12 A b-11 a B) \sqrt {a+b x}}{693 a^6 x^{3/2}}-\frac {\left (256 b^4 (12 A b-11 a B)\right ) \int \frac {1}{x^{3/2} \sqrt {a+b x}} \, dx}{693 a^6}\\ &=-\frac {2 A}{11 a x^{11/2} \sqrt {a+b x}}-\frac {2 (12 A b-11 a B)}{11 a^2 x^{9/2} \sqrt {a+b x}}+\frac {20 (12 A b-11 a B) \sqrt {a+b x}}{99 a^3 x^{9/2}}-\frac {160 b (12 A b-11 a B) \sqrt {a+b x}}{693 a^4 x^{7/2}}+\frac {64 b^2 (12 A b-11 a B) \sqrt {a+b x}}{231 a^5 x^{5/2}}-\frac {256 b^3 (12 A b-11 a B) \sqrt {a+b x}}{693 a^6 x^{3/2}}+\frac {512 b^4 (12 A b-11 a B) \sqrt {a+b x}}{693 a^7 \sqrt {x}}\\ \end {align*}

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Mathematica [A]  time = 0.06, size = 133, normalized size = 0.62 \begin {gather*} -\frac {2 \left (7 a^6 (9 A+11 B x)-2 a^5 b x (42 A+55 B x)+8 a^4 b^2 x^2 (15 A+22 B x)-32 a^3 b^3 x^3 (6 A+11 B x)+128 a^2 b^4 x^4 (3 A+11 B x)+256 a b^5 x^5 (11 B x-6 A)-3072 A b^6 x^6\right )}{693 a^7 x^{11/2} \sqrt {a+b x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x)/(x^(13/2)*(a + b*x)^(3/2)),x]

[Out]

(-2*(-3072*A*b^6*x^6 + 256*a*b^5*x^5*(-6*A + 11*B*x) + 128*a^2*b^4*x^4*(3*A + 11*B*x) - 32*a^3*b^3*x^3*(6*A +
11*B*x) + 7*a^6*(9*A + 11*B*x) + 8*a^4*b^2*x^2*(15*A + 22*B*x) - 2*a^5*b*x*(42*A + 55*B*x)))/(693*a^7*x^(11/2)
*Sqrt[a + b*x])

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IntegrateAlgebraic [A]  time = 0.32, size = 154, normalized size = 0.72 \begin {gather*} -\frac {2 \left (63 a^6 A+77 a^6 B x-84 a^5 A b x-110 a^5 b B x^2+120 a^4 A b^2 x^2+176 a^4 b^2 B x^3-192 a^3 A b^3 x^3-352 a^3 b^3 B x^4+384 a^2 A b^4 x^4+1408 a^2 b^4 B x^5-1536 a A b^5 x^5+2816 a b^5 B x^6-3072 A b^6 x^6\right )}{693 a^7 x^{11/2} \sqrt {a+b x}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(A + B*x)/(x^(13/2)*(a + b*x)^(3/2)),x]

[Out]

(-2*(63*a^6*A - 84*a^5*A*b*x + 77*a^6*B*x + 120*a^4*A*b^2*x^2 - 110*a^5*b*B*x^2 - 192*a^3*A*b^3*x^3 + 176*a^4*
b^2*B*x^3 + 384*a^2*A*b^4*x^4 - 352*a^3*b^3*B*x^4 - 1536*a*A*b^5*x^5 + 1408*a^2*b^4*B*x^5 - 3072*A*b^6*x^6 + 2
816*a*b^5*B*x^6))/(693*a^7*x^(11/2)*Sqrt[a + b*x])

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fricas [A]  time = 1.32, size = 165, normalized size = 0.77 \begin {gather*} -\frac {2 \, {\left (63 \, A a^{6} + 256 \, {\left (11 \, B a b^{5} - 12 \, A b^{6}\right )} x^{6} + 128 \, {\left (11 \, B a^{2} b^{4} - 12 \, A a b^{5}\right )} x^{5} - 32 \, {\left (11 \, B a^{3} b^{3} - 12 \, A a^{2} b^{4}\right )} x^{4} + 16 \, {\left (11 \, B a^{4} b^{2} - 12 \, A a^{3} b^{3}\right )} x^{3} - 10 \, {\left (11 \, B a^{5} b - 12 \, A a^{4} b^{2}\right )} x^{2} + 7 \, {\left (11 \, B a^{6} - 12 \, A a^{5} b\right )} x\right )} \sqrt {b x + a} \sqrt {x}}{693 \, {\left (a^{7} b x^{7} + a^{8} x^{6}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/x^(13/2)/(b*x+a)^(3/2),x, algorithm="fricas")

[Out]

-2/693*(63*A*a^6 + 256*(11*B*a*b^5 - 12*A*b^6)*x^6 + 128*(11*B*a^2*b^4 - 12*A*a*b^5)*x^5 - 32*(11*B*a^3*b^3 -
12*A*a^2*b^4)*x^4 + 16*(11*B*a^4*b^2 - 12*A*a^3*b^3)*x^3 - 10*(11*B*a^5*b - 12*A*a^4*b^2)*x^2 + 7*(11*B*a^6 -
12*A*a^5*b)*x)*sqrt(b*x + a)*sqrt(x)/(a^7*b*x^7 + a^8*x^6)

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giac [B]  time = 1.82, size = 368, normalized size = 1.73 \begin {gather*} -\frac {2 \, {\left ({\left ({\left ({\left (b x + a\right )} {\left ({\left (b x + a\right )} {\left (\frac {{\left (2123 \, B a^{21} b^{15} {\left | b \right |} - 2379 \, A a^{20} b^{16} {\left | b \right |}\right )} {\left (b x + a\right )}}{a^{27} b^{6}} - \frac {22 \, {\left (515 \, B a^{22} b^{15} {\left | b \right |} - 579 \, A a^{21} b^{16} {\left | b \right |}\right )}}{a^{27} b^{6}}\right )} + \frac {99 \, {\left (247 \, B a^{23} b^{15} {\left | b \right |} - 279 \, A a^{22} b^{16} {\left | b \right |}\right )}}{a^{27} b^{6}}\right )} - \frac {924 \, {\left (29 \, B a^{24} b^{15} {\left | b \right |} - 33 \, A a^{23} b^{16} {\left | b \right |}\right )}}{a^{27} b^{6}}\right )} {\left (b x + a\right )} + \frac {1155 \, {\left (13 \, B a^{25} b^{15} {\left | b \right |} - 15 \, A a^{24} b^{16} {\left | b \right |}\right )}}{a^{27} b^{6}}\right )} {\left (b x + a\right )} - \frac {693 \, {\left (5 \, B a^{26} b^{15} {\left | b \right |} - 6 \, A a^{25} b^{16} {\left | b \right |}\right )}}{a^{27} b^{6}}\right )} \sqrt {b x + a}}{693 \, {\left ({\left (b x + a\right )} b - a b\right )}^{\frac {11}{2}}} - \frac {4 \, {\left (B^{2} a^{2} b^{13} - 2 \, A B a b^{14} + A^{2} b^{15}\right )}}{{\left (B a {\left (\sqrt {b x + a} \sqrt {b} - \sqrt {{\left (b x + a\right )} b - a b}\right )}^{2} b^{\frac {13}{2}} + B a^{2} b^{\frac {15}{2}} - A {\left (\sqrt {b x + a} \sqrt {b} - \sqrt {{\left (b x + a\right )} b - a b}\right )}^{2} b^{\frac {15}{2}} - A a b^{\frac {17}{2}}\right )} a^{6} {\left | b \right |}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/x^(13/2)/(b*x+a)^(3/2),x, algorithm="giac")

[Out]

-2/693*((((b*x + a)*((b*x + a)*((2123*B*a^21*b^15*abs(b) - 2379*A*a^20*b^16*abs(b))*(b*x + a)/(a^27*b^6) - 22*
(515*B*a^22*b^15*abs(b) - 579*A*a^21*b^16*abs(b))/(a^27*b^6)) + 99*(247*B*a^23*b^15*abs(b) - 279*A*a^22*b^16*a
bs(b))/(a^27*b^6)) - 924*(29*B*a^24*b^15*abs(b) - 33*A*a^23*b^16*abs(b))/(a^27*b^6))*(b*x + a) + 1155*(13*B*a^
25*b^15*abs(b) - 15*A*a^24*b^16*abs(b))/(a^27*b^6))*(b*x + a) - 693*(5*B*a^26*b^15*abs(b) - 6*A*a^25*b^16*abs(
b))/(a^27*b^6))*sqrt(b*x + a)/((b*x + a)*b - a*b)^(11/2) - 4*(B^2*a^2*b^13 - 2*A*B*a*b^14 + A^2*b^15)/((B*a*(s
qrt(b*x + a)*sqrt(b) - sqrt((b*x + a)*b - a*b))^2*b^(13/2) + B*a^2*b^(15/2) - A*(sqrt(b*x + a)*sqrt(b) - sqrt(
(b*x + a)*b - a*b))^2*b^(15/2) - A*a*b^(17/2))*a^6*abs(b))

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maple [A]  time = 0.01, size = 149, normalized size = 0.70 \begin {gather*} -\frac {2 \left (-3072 A \,b^{6} x^{6}+2816 B a \,b^{5} x^{6}-1536 A a \,b^{5} x^{5}+1408 B \,a^{2} b^{4} x^{5}+384 A \,a^{2} b^{4} x^{4}-352 B \,a^{3} b^{3} x^{4}-192 A \,a^{3} b^{3} x^{3}+176 B \,a^{4} b^{2} x^{3}+120 A \,a^{4} b^{2} x^{2}-110 B \,a^{5} b \,x^{2}-84 A \,a^{5} b x +77 B \,a^{6} x +63 A \,a^{6}\right )}{693 \sqrt {b x +a}\, a^{7} x^{\frac {11}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)/x^(13/2)/(b*x+a)^(3/2),x)

[Out]

-2/693*(-3072*A*b^6*x^6+2816*B*a*b^5*x^6-1536*A*a*b^5*x^5+1408*B*a^2*b^4*x^5+384*A*a^2*b^4*x^4-352*B*a^3*b^3*x
^4-192*A*a^3*b^3*x^3+176*B*a^4*b^2*x^3+120*A*a^4*b^2*x^2-110*B*a^5*b*x^2-84*A*a^5*b*x+77*B*a^6*x+63*A*a^6)/(b*
x+a)^(1/2)/x^(11/2)/a^7

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maxima [A]  time = 0.99, size = 280, normalized size = 1.31 \begin {gather*} -\frac {512 \, B b^{5} x}{63 \, \sqrt {b x^{2} + a x} a^{6}} + \frac {2048 \, A b^{6} x}{231 \, \sqrt {b x^{2} + a x} a^{7}} - \frac {256 \, B b^{4}}{63 \, \sqrt {b x^{2} + a x} a^{5}} + \frac {1024 \, A b^{5}}{231 \, \sqrt {b x^{2} + a x} a^{6}} + \frac {64 \, B b^{3}}{63 \, \sqrt {b x^{2} + a x} a^{4} x} - \frac {256 \, A b^{4}}{231 \, \sqrt {b x^{2} + a x} a^{5} x} - \frac {32 \, B b^{2}}{63 \, \sqrt {b x^{2} + a x} a^{3} x^{2}} + \frac {128 \, A b^{3}}{231 \, \sqrt {b x^{2} + a x} a^{4} x^{2}} + \frac {20 \, B b}{63 \, \sqrt {b x^{2} + a x} a^{2} x^{3}} - \frac {80 \, A b^{2}}{231 \, \sqrt {b x^{2} + a x} a^{3} x^{3}} - \frac {2 \, B}{9 \, \sqrt {b x^{2} + a x} a x^{4}} + \frac {8 \, A b}{33 \, \sqrt {b x^{2} + a x} a^{2} x^{4}} - \frac {2 \, A}{11 \, \sqrt {b x^{2} + a x} a x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/x^(13/2)/(b*x+a)^(3/2),x, algorithm="maxima")

[Out]

-512/63*B*b^5*x/(sqrt(b*x^2 + a*x)*a^6) + 2048/231*A*b^6*x/(sqrt(b*x^2 + a*x)*a^7) - 256/63*B*b^4/(sqrt(b*x^2
+ a*x)*a^5) + 1024/231*A*b^5/(sqrt(b*x^2 + a*x)*a^6) + 64/63*B*b^3/(sqrt(b*x^2 + a*x)*a^4*x) - 256/231*A*b^4/(
sqrt(b*x^2 + a*x)*a^5*x) - 32/63*B*b^2/(sqrt(b*x^2 + a*x)*a^3*x^2) + 128/231*A*b^3/(sqrt(b*x^2 + a*x)*a^4*x^2)
+ 20/63*B*b/(sqrt(b*x^2 + a*x)*a^2*x^3) - 80/231*A*b^2/(sqrt(b*x^2 + a*x)*a^3*x^3) - 2/9*B/(sqrt(b*x^2 + a*x)
*a*x^4) + 8/33*A*b/(sqrt(b*x^2 + a*x)*a^2*x^4) - 2/11*A/(sqrt(b*x^2 + a*x)*a*x^5)

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mupad [B]  time = 1.02, size = 151, normalized size = 0.71 \begin {gather*} -\frac {\sqrt {a+b\,x}\,\left (\frac {2\,A}{11\,a\,b}+\frac {20\,x^2\,\left (12\,A\,b-11\,B\,a\right )}{693\,a^3}+\frac {64\,b^2\,x^4\,\left (12\,A\,b-11\,B\,a\right )}{693\,a^5}-\frac {256\,b^3\,x^5\,\left (12\,A\,b-11\,B\,a\right )}{693\,a^6}-\frac {512\,b^4\,x^6\,\left (12\,A\,b-11\,B\,a\right )}{693\,a^7}-\frac {32\,b\,x^3\,\left (12\,A\,b-11\,B\,a\right )}{693\,a^4}+\frac {x\,\left (154\,B\,a^6-168\,A\,a^5\,b\right )}{693\,a^7\,b}\right )}{x^{13/2}+\frac {a\,x^{11/2}}{b}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*x)/(x^(13/2)*(a + b*x)^(3/2)),x)

[Out]

-((a + b*x)^(1/2)*((2*A)/(11*a*b) + (20*x^2*(12*A*b - 11*B*a))/(693*a^3) + (64*b^2*x^4*(12*A*b - 11*B*a))/(693
*a^5) - (256*b^3*x^5*(12*A*b - 11*B*a))/(693*a^6) - (512*b^4*x^6*(12*A*b - 11*B*a))/(693*a^7) - (32*b*x^3*(12*
A*b - 11*B*a))/(693*a^4) + (x*(154*B*a^6 - 168*A*a^5*b))/(693*a^7*b)))/(x^(13/2) + (a*x^(11/2))/b)

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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**(13/2)/(b*x+a)**(3/2),x)

[Out]

Timed out

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