3.5.100 \(\int \frac {A+B x}{x^{15/2} \sqrt {a+b x}} \, dx\)

Optimal. Leaf size=216 \[ -\frac {512 b^5 \sqrt {a+b x} (12 A b-13 a B)}{9009 a^7 \sqrt {x}}+\frac {256 b^4 \sqrt {a+b x} (12 A b-13 a B)}{9009 a^6 x^{3/2}}-\frac {64 b^3 \sqrt {a+b x} (12 A b-13 a B)}{3003 a^5 x^{5/2}}+\frac {160 b^2 \sqrt {a+b x} (12 A b-13 a B)}{9009 a^4 x^{7/2}}-\frac {20 b \sqrt {a+b x} (12 A b-13 a B)}{1287 a^3 x^{9/2}}+\frac {2 \sqrt {a+b x} (12 A b-13 a B)}{143 a^2 x^{11/2}}-\frac {2 A \sqrt {a+b x}}{13 a x^{13/2}} \]

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Rubi [A]  time = 0.09, antiderivative size = 216, 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^4 \sqrt {a+b x} (12 A b-13 a B)}{9009 a^6 x^{3/2}}-\frac {64 b^3 \sqrt {a+b x} (12 A b-13 a B)}{3003 a^5 x^{5/2}}+\frac {160 b^2 \sqrt {a+b x} (12 A b-13 a B)}{9009 a^4 x^{7/2}}-\frac {512 b^5 \sqrt {a+b x} (12 A b-13 a B)}{9009 a^7 \sqrt {x}}-\frac {20 b \sqrt {a+b x} (12 A b-13 a B)}{1287 a^3 x^{9/2}}+\frac {2 \sqrt {a+b x} (12 A b-13 a B)}{143 a^2 x^{11/2}}-\frac {2 A \sqrt {a+b x}}{13 a x^{13/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(A + B*x)/(x^(15/2)*Sqrt[a + b*x]),x]

[Out]

(-2*A*Sqrt[a + b*x])/(13*a*x^(13/2)) + (2*(12*A*b - 13*a*B)*Sqrt[a + b*x])/(143*a^2*x^(11/2)) - (20*b*(12*A*b
- 13*a*B)*Sqrt[a + b*x])/(1287*a^3*x^(9/2)) + (160*b^2*(12*A*b - 13*a*B)*Sqrt[a + b*x])/(9009*a^4*x^(7/2)) - (
64*b^3*(12*A*b - 13*a*B)*Sqrt[a + b*x])/(3003*a^5*x^(5/2)) + (256*b^4*(12*A*b - 13*a*B)*Sqrt[a + b*x])/(9009*a
^6*x^(3/2)) - (512*b^5*(12*A*b - 13*a*B)*Sqrt[a + b*x])/(9009*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^{15/2} \sqrt {a+b x}} \, dx &=-\frac {2 A \sqrt {a+b x}}{13 a x^{13/2}}+\frac {\left (2 \left (-6 A b+\frac {13 a B}{2}\right )\right ) \int \frac {1}{x^{13/2} \sqrt {a+b x}} \, dx}{13 a}\\ &=-\frac {2 A \sqrt {a+b x}}{13 a x^{13/2}}+\frac {2 (12 A b-13 a B) \sqrt {a+b x}}{143 a^2 x^{11/2}}+\frac {(10 b (12 A b-13 a B)) \int \frac {1}{x^{11/2} \sqrt {a+b x}} \, dx}{143 a^2}\\ &=-\frac {2 A \sqrt {a+b x}}{13 a x^{13/2}}+\frac {2 (12 A b-13 a B) \sqrt {a+b x}}{143 a^2 x^{11/2}}-\frac {20 b (12 A b-13 a B) \sqrt {a+b x}}{1287 a^3 x^{9/2}}-\frac {\left (80 b^2 (12 A b-13 a B)\right ) \int \frac {1}{x^{9/2} \sqrt {a+b x}} \, dx}{1287 a^3}\\ &=-\frac {2 A \sqrt {a+b x}}{13 a x^{13/2}}+\frac {2 (12 A b-13 a B) \sqrt {a+b x}}{143 a^2 x^{11/2}}-\frac {20 b (12 A b-13 a B) \sqrt {a+b x}}{1287 a^3 x^{9/2}}+\frac {160 b^2 (12 A b-13 a B) \sqrt {a+b x}}{9009 a^4 x^{7/2}}+\frac {\left (160 b^3 (12 A b-13 a B)\right ) \int \frac {1}{x^{7/2} \sqrt {a+b x}} \, dx}{3003 a^4}\\ &=-\frac {2 A \sqrt {a+b x}}{13 a x^{13/2}}+\frac {2 (12 A b-13 a B) \sqrt {a+b x}}{143 a^2 x^{11/2}}-\frac {20 b (12 A b-13 a B) \sqrt {a+b x}}{1287 a^3 x^{9/2}}+\frac {160 b^2 (12 A b-13 a B) \sqrt {a+b x}}{9009 a^4 x^{7/2}}-\frac {64 b^3 (12 A b-13 a B) \sqrt {a+b x}}{3003 a^5 x^{5/2}}-\frac {\left (128 b^4 (12 A b-13 a B)\right ) \int \frac {1}{x^{5/2} \sqrt {a+b x}} \, dx}{3003 a^5}\\ &=-\frac {2 A \sqrt {a+b x}}{13 a x^{13/2}}+\frac {2 (12 A b-13 a B) \sqrt {a+b x}}{143 a^2 x^{11/2}}-\frac {20 b (12 A b-13 a B) \sqrt {a+b x}}{1287 a^3 x^{9/2}}+\frac {160 b^2 (12 A b-13 a B) \sqrt {a+b x}}{9009 a^4 x^{7/2}}-\frac {64 b^3 (12 A b-13 a B) \sqrt {a+b x}}{3003 a^5 x^{5/2}}+\frac {256 b^4 (12 A b-13 a B) \sqrt {a+b x}}{9009 a^6 x^{3/2}}+\frac {\left (256 b^5 (12 A b-13 a B)\right ) \int \frac {1}{x^{3/2} \sqrt {a+b x}} \, dx}{9009 a^6}\\ &=-\frac {2 A \sqrt {a+b x}}{13 a x^{13/2}}+\frac {2 (12 A b-13 a B) \sqrt {a+b x}}{143 a^2 x^{11/2}}-\frac {20 b (12 A b-13 a B) \sqrt {a+b x}}{1287 a^3 x^{9/2}}+\frac {160 b^2 (12 A b-13 a B) \sqrt {a+b x}}{9009 a^4 x^{7/2}}-\frac {64 b^3 (12 A b-13 a B) \sqrt {a+b x}}{3003 a^5 x^{5/2}}+\frac {256 b^4 (12 A b-13 a B) \sqrt {a+b x}}{9009 a^6 x^{3/2}}-\frac {512 b^5 (12 A b-13 a B) \sqrt {a+b x}}{9009 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 \sqrt {a+b x} \left (63 a^6 (11 A+13 B x)-14 a^5 b x (54 A+65 B x)+40 a^4 b^2 x^2 (21 A+26 B x)-96 a^3 b^3 x^3 (10 A+13 B x)+128 a^2 b^4 x^4 (9 A+13 B x)-256 a b^5 x^5 (6 A+13 B x)+3072 A b^6 x^6\right )}{9009 a^7 x^{13/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x)/(x^(15/2)*Sqrt[a + b*x]),x]

[Out]

(-2*Sqrt[a + b*x]*(3072*A*b^6*x^6 - 256*a*b^5*x^5*(6*A + 13*B*x) + 128*a^2*b^4*x^4*(9*A + 13*B*x) - 96*a^3*b^3
*x^3*(10*A + 13*B*x) + 63*a^6*(11*A + 13*B*x) + 40*a^4*b^2*x^2*(21*A + 26*B*x) - 14*a^5*b*x*(54*A + 65*B*x)))/
(9009*a^7*x^(13/2))

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IntegrateAlgebraic [A]  time = 0.28, size = 154, normalized size = 0.71 \begin {gather*} \frac {2 \sqrt {a+b x} \left (-693 a^6 A-819 a^6 B x+756 a^5 A b x+910 a^5 b B x^2-840 a^4 A b^2 x^2-1040 a^4 b^2 B x^3+960 a^3 A b^3 x^3+1248 a^3 b^3 B x^4-1152 a^2 A b^4 x^4-1664 a^2 b^4 B x^5+1536 a A b^5 x^5+3328 a b^5 B x^6-3072 A b^6 x^6\right )}{9009 a^7 x^{13/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(A + B*x)/(x^(15/2)*Sqrt[a + b*x]),x]

[Out]

(2*Sqrt[a + b*x]*(-693*a^6*A + 756*a^5*A*b*x - 819*a^6*B*x - 840*a^4*A*b^2*x^2 + 910*a^5*b*B*x^2 + 960*a^3*A*b
^3*x^3 - 1040*a^4*b^2*B*x^3 - 1152*a^2*A*b^4*x^4 + 1248*a^3*b^3*B*x^4 + 1536*a*A*b^5*x^5 - 1664*a^2*b^4*B*x^5
- 3072*A*b^6*x^6 + 3328*a*b^5*B*x^6))/(9009*a^7*x^(13/2))

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fricas [A]  time = 1.69, size = 150, normalized size = 0.69 \begin {gather*} -\frac {2 \, {\left (693 \, A a^{6} - 256 \, {\left (13 \, B a b^{5} - 12 \, A b^{6}\right )} x^{6} + 128 \, {\left (13 \, B a^{2} b^{4} - 12 \, A a b^{5}\right )} x^{5} - 96 \, {\left (13 \, B a^{3} b^{3} - 12 \, A a^{2} b^{4}\right )} x^{4} + 80 \, {\left (13 \, B a^{4} b^{2} - 12 \, A a^{3} b^{3}\right )} x^{3} - 70 \, {\left (13 \, B a^{5} b - 12 \, A a^{4} b^{2}\right )} x^{2} + 63 \, {\left (13 \, B a^{6} - 12 \, A a^{5} b\right )} x\right )} \sqrt {b x + a}}{9009 \, a^{7} x^{\frac {13}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/x^(15/2)/(b*x+a)^(1/2),x, algorithm="fricas")

[Out]

-2/9009*(693*A*a^6 - 256*(13*B*a*b^5 - 12*A*b^6)*x^6 + 128*(13*B*a^2*b^4 - 12*A*a*b^5)*x^5 - 96*(13*B*a^3*b^3
- 12*A*a^2*b^4)*x^4 + 80*(13*B*a^4*b^2 - 12*A*a^3*b^3)*x^3 - 70*(13*B*a^5*b - 12*A*a^4*b^2)*x^2 + 63*(13*B*a^6
- 12*A*a^5*b)*x)*sqrt(b*x + a)/(a^7*x^(13/2))

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giac [A]  time = 1.53, size = 233, normalized size = 1.08 \begin {gather*} \frac {2 \, {\left ({\left (2 \, {\left (8 \, {\left (2 \, {\left (b x + a\right )} {\left (4 \, {\left (b x + a\right )} {\left (\frac {2 \, {\left (13 \, B a b^{12} - 12 \, A b^{13}\right )} {\left (b x + a\right )}}{a^{7}} - \frac {13 \, {\left (13 \, B a^{2} b^{12} - 12 \, A a b^{13}\right )}}{a^{7}}\right )} + \frac {143 \, {\left (13 \, B a^{3} b^{12} - 12 \, A a^{2} b^{13}\right )}}{a^{7}}\right )} - \frac {429 \, {\left (13 \, B a^{4} b^{12} - 12 \, A a^{3} b^{13}\right )}}{a^{7}}\right )} {\left (b x + a\right )} + \frac {3003 \, {\left (13 \, B a^{5} b^{12} - 12 \, A a^{4} b^{13}\right )}}{a^{7}}\right )} {\left (b x + a\right )} - \frac {3003 \, {\left (13 \, B a^{6} b^{12} - 12 \, A a^{5} b^{13}\right )}}{a^{7}}\right )} {\left (b x + a\right )} + \frac {9009 \, {\left (B a^{7} b^{12} - A a^{6} b^{13}\right )}}{a^{7}}\right )} \sqrt {b x + a} b}{9009 \, {\left ({\left (b x + a\right )} b - a b\right )}^{\frac {13}{2}} {\left | b \right |}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/x^(15/2)/(b*x+a)^(1/2),x, algorithm="giac")

[Out]

2/9009*((2*(8*(2*(b*x + a)*(4*(b*x + a)*(2*(13*B*a*b^12 - 12*A*b^13)*(b*x + a)/a^7 - 13*(13*B*a^2*b^12 - 12*A*
a*b^13)/a^7) + 143*(13*B*a^3*b^12 - 12*A*a^2*b^13)/a^7) - 429*(13*B*a^4*b^12 - 12*A*a^3*b^13)/a^7)*(b*x + a) +
3003*(13*B*a^5*b^12 - 12*A*a^4*b^13)/a^7)*(b*x + a) - 3003*(13*B*a^6*b^12 - 12*A*a^5*b^13)/a^7)*(b*x + a) + 9
009*(B*a^7*b^12 - A*a^6*b^13)/a^7)*sqrt(b*x + a)*b/(((b*x + a)*b - a*b)^(13/2)*abs(b))

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maple [A]  time = 0.01, size = 149, normalized size = 0.69 \begin {gather*} -\frac {2 \sqrt {b x +a}\, \left (3072 A \,b^{6} x^{6}-3328 B a \,b^{5} x^{6}-1536 A a \,b^{5} x^{5}+1664 B \,a^{2} b^{4} x^{5}+1152 A \,a^{2} b^{4} x^{4}-1248 B \,a^{3} b^{3} x^{4}-960 A \,a^{3} b^{3} x^{3}+1040 B \,a^{4} b^{2} x^{3}+840 A \,a^{4} b^{2} x^{2}-910 B \,a^{5} b \,x^{2}-756 A \,a^{5} b x +819 B \,a^{6} x +693 A \,a^{6}\right )}{9009 a^{7} x^{\frac {13}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)/x^(15/2)/(b*x+a)^(1/2),x)

[Out]

-2/9009*(b*x+a)^(1/2)*(3072*A*b^6*x^6-3328*B*a*b^5*x^6-1536*A*a*b^5*x^5+1664*B*a^2*b^4*x^5+1152*A*a^2*b^4*x^4-
1248*B*a^3*b^3*x^4-960*A*a^3*b^3*x^3+1040*B*a^4*b^2*x^3+840*A*a^4*b^2*x^2-910*B*a^5*b*x^2-756*A*a^5*b*x+819*B*
a^6*x+693*A*a^6)/x^(13/2)/a^7

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maxima [A]  time = 0.95, size = 290, normalized size = 1.34 \begin {gather*} \frac {512 \, \sqrt {b x^{2} + a x} B b^{5}}{693 \, a^{6} x} - \frac {2048 \, \sqrt {b x^{2} + a x} A b^{6}}{3003 \, a^{7} x} - \frac {256 \, \sqrt {b x^{2} + a x} B b^{4}}{693 \, a^{5} x^{2}} + \frac {1024 \, \sqrt {b x^{2} + a x} A b^{5}}{3003 \, a^{6} x^{2}} + \frac {64 \, \sqrt {b x^{2} + a x} B b^{3}}{231 \, a^{4} x^{3}} - \frac {256 \, \sqrt {b x^{2} + a x} A b^{4}}{1001 \, a^{5} x^{3}} - \frac {160 \, \sqrt {b x^{2} + a x} B b^{2}}{693 \, a^{3} x^{4}} + \frac {640 \, \sqrt {b x^{2} + a x} A b^{3}}{3003 \, a^{4} x^{4}} + \frac {20 \, \sqrt {b x^{2} + a x} B b}{99 \, a^{2} x^{5}} - \frac {80 \, \sqrt {b x^{2} + a x} A b^{2}}{429 \, a^{3} x^{5}} - \frac {2 \, \sqrt {b x^{2} + a x} B}{11 \, a x^{6}} + \frac {24 \, \sqrt {b x^{2} + a x} A b}{143 \, a^{2} x^{6}} - \frac {2 \, \sqrt {b x^{2} + a x} A}{13 \, a x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/x^(15/2)/(b*x+a)^(1/2),x, algorithm="maxima")

[Out]

512/693*sqrt(b*x^2 + a*x)*B*b^5/(a^6*x) - 2048/3003*sqrt(b*x^2 + a*x)*A*b^6/(a^7*x) - 256/693*sqrt(b*x^2 + a*x
)*B*b^4/(a^5*x^2) + 1024/3003*sqrt(b*x^2 + a*x)*A*b^5/(a^6*x^2) + 64/231*sqrt(b*x^2 + a*x)*B*b^3/(a^4*x^3) - 2
56/1001*sqrt(b*x^2 + a*x)*A*b^4/(a^5*x^3) - 160/693*sqrt(b*x^2 + a*x)*B*b^2/(a^3*x^4) + 640/3003*sqrt(b*x^2 +
a*x)*A*b^3/(a^4*x^4) + 20/99*sqrt(b*x^2 + a*x)*B*b/(a^2*x^5) - 80/429*sqrt(b*x^2 + a*x)*A*b^2/(a^3*x^5) - 2/11
*sqrt(b*x^2 + a*x)*B/(a*x^6) + 24/143*sqrt(b*x^2 + a*x)*A*b/(a^2*x^6) - 2/13*sqrt(b*x^2 + a*x)*A/(a*x^7)

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mupad [B]  time = 1.04, size = 137, normalized size = 0.63 \begin {gather*} -\frac {\sqrt {a+b\,x}\,\left (\frac {2\,A}{13\,a}+\frac {x\,\left (1638\,B\,a^6-1512\,A\,a^5\,b\right )}{9009\,a^7}-\frac {160\,b^2\,x^3\,\left (12\,A\,b-13\,B\,a\right )}{9009\,a^4}+\frac {64\,b^3\,x^4\,\left (12\,A\,b-13\,B\,a\right )}{3003\,a^5}-\frac {256\,b^4\,x^5\,\left (12\,A\,b-13\,B\,a\right )}{9009\,a^6}+\frac {512\,b^5\,x^6\,\left (12\,A\,b-13\,B\,a\right )}{9009\,a^7}+\frac {20\,b\,x^2\,\left (12\,A\,b-13\,B\,a\right )}{1287\,a^3}\right )}{x^{13/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*x)/(x^(15/2)*(a + b*x)^(1/2)),x)

[Out]

-((a + b*x)^(1/2)*((2*A)/(13*a) + (x*(1638*B*a^6 - 1512*A*a^5*b))/(9009*a^7) - (160*b^2*x^3*(12*A*b - 13*B*a))
/(9009*a^4) + (64*b^3*x^4*(12*A*b - 13*B*a))/(3003*a^5) - (256*b^4*x^5*(12*A*b - 13*B*a))/(9009*a^6) + (512*b^
5*x^6*(12*A*b - 13*B*a))/(9009*a^7) + (20*b*x^2*(12*A*b - 13*B*a))/(1287*a^3)))/x^(13/2)

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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**(15/2)/(b*x+a)**(1/2),x)

[Out]

Timed out

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