3.5.88 \(\int \frac {(a+b x)^{5/2} (A+B x)}{x^{19/2}} \, dx\)

Optimal. Leaf size=183 \[ \frac {256 b^4 (a+b x)^{7/2} (10 A b-17 a B)}{765765 a^6 x^{7/2}}-\frac {128 b^3 (a+b x)^{7/2} (10 A b-17 a B)}{109395 a^5 x^{9/2}}+\frac {32 b^2 (a+b x)^{7/2} (10 A b-17 a B)}{12155 a^4 x^{11/2}}-\frac {16 b (a+b x)^{7/2} (10 A b-17 a B)}{3315 a^3 x^{13/2}}+\frac {2 (a+b x)^{7/2} (10 A b-17 a B)}{255 a^2 x^{15/2}}-\frac {2 A (a+b x)^{7/2}}{17 a x^{17/2}} \]

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Rubi [A]  time = 0.07, antiderivative size = 183, normalized size of antiderivative = 1.00, number of steps used = 6, 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 (a+b x)^{7/2} (10 A b-17 a B)}{765765 a^6 x^{7/2}}-\frac {128 b^3 (a+b x)^{7/2} (10 A b-17 a B)}{109395 a^5 x^{9/2}}+\frac {32 b^2 (a+b x)^{7/2} (10 A b-17 a B)}{12155 a^4 x^{11/2}}-\frac {16 b (a+b x)^{7/2} (10 A b-17 a B)}{3315 a^3 x^{13/2}}+\frac {2 (a+b x)^{7/2} (10 A b-17 a B)}{255 a^2 x^{15/2}}-\frac {2 A (a+b x)^{7/2}}{17 a x^{17/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((a + b*x)^(5/2)*(A + B*x))/x^(19/2),x]

[Out]

(-2*A*(a + b*x)^(7/2))/(17*a*x^(17/2)) + (2*(10*A*b - 17*a*B)*(a + b*x)^(7/2))/(255*a^2*x^(15/2)) - (16*b*(10*
A*b - 17*a*B)*(a + b*x)^(7/2))/(3315*a^3*x^(13/2)) + (32*b^2*(10*A*b - 17*a*B)*(a + b*x)^(7/2))/(12155*a^4*x^(
11/2)) - (128*b^3*(10*A*b - 17*a*B)*(a + b*x)^(7/2))/(109395*a^5*x^(9/2)) + (256*b^4*(10*A*b - 17*a*B)*(a + b*
x)^(7/2))/(765765*a^6*x^(7/2))

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)^{5/2} (A+B x)}{x^{19/2}} \, dx &=-\frac {2 A (a+b x)^{7/2}}{17 a x^{17/2}}+\frac {\left (2 \left (-5 A b+\frac {17 a B}{2}\right )\right ) \int \frac {(a+b x)^{5/2}}{x^{17/2}} \, dx}{17 a}\\ &=-\frac {2 A (a+b x)^{7/2}}{17 a x^{17/2}}+\frac {2 (10 A b-17 a B) (a+b x)^{7/2}}{255 a^2 x^{15/2}}+\frac {(8 b (10 A b-17 a B)) \int \frac {(a+b x)^{5/2}}{x^{15/2}} \, dx}{255 a^2}\\ &=-\frac {2 A (a+b x)^{7/2}}{17 a x^{17/2}}+\frac {2 (10 A b-17 a B) (a+b x)^{7/2}}{255 a^2 x^{15/2}}-\frac {16 b (10 A b-17 a B) (a+b x)^{7/2}}{3315 a^3 x^{13/2}}-\frac {\left (16 b^2 (10 A b-17 a B)\right ) \int \frac {(a+b x)^{5/2}}{x^{13/2}} \, dx}{1105 a^3}\\ &=-\frac {2 A (a+b x)^{7/2}}{17 a x^{17/2}}+\frac {2 (10 A b-17 a B) (a+b x)^{7/2}}{255 a^2 x^{15/2}}-\frac {16 b (10 A b-17 a B) (a+b x)^{7/2}}{3315 a^3 x^{13/2}}+\frac {32 b^2 (10 A b-17 a B) (a+b x)^{7/2}}{12155 a^4 x^{11/2}}+\frac {\left (64 b^3 (10 A b-17 a B)\right ) \int \frac {(a+b x)^{5/2}}{x^{11/2}} \, dx}{12155 a^4}\\ &=-\frac {2 A (a+b x)^{7/2}}{17 a x^{17/2}}+\frac {2 (10 A b-17 a B) (a+b x)^{7/2}}{255 a^2 x^{15/2}}-\frac {16 b (10 A b-17 a B) (a+b x)^{7/2}}{3315 a^3 x^{13/2}}+\frac {32 b^2 (10 A b-17 a B) (a+b x)^{7/2}}{12155 a^4 x^{11/2}}-\frac {128 b^3 (10 A b-17 a B) (a+b x)^{7/2}}{109395 a^5 x^{9/2}}-\frac {\left (128 b^4 (10 A b-17 a B)\right ) \int \frac {(a+b x)^{5/2}}{x^{9/2}} \, dx}{109395 a^5}\\ &=-\frac {2 A (a+b x)^{7/2}}{17 a x^{17/2}}+\frac {2 (10 A b-17 a B) (a+b x)^{7/2}}{255 a^2 x^{15/2}}-\frac {16 b (10 A b-17 a B) (a+b x)^{7/2}}{3315 a^3 x^{13/2}}+\frac {32 b^2 (10 A b-17 a B) (a+b x)^{7/2}}{12155 a^4 x^{11/2}}-\frac {128 b^3 (10 A b-17 a B) (a+b x)^{7/2}}{109395 a^5 x^{9/2}}+\frac {256 b^4 (10 A b-17 a B) (a+b x)^{7/2}}{765765 a^6 x^{7/2}}\\ \end {align*}

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Mathematica [A]  time = 0.06, size = 114, normalized size = 0.62 \begin {gather*} -\frac {2 (a+b x)^{7/2} \left (3003 a^5 (15 A+17 B x)-462 a^4 b x (65 A+68 B x)+336 a^3 b^2 x^2 (55 A+51 B x)-224 a^2 b^3 x^3 (45 A+34 B x)+128 a b^4 x^4 (35 A+17 B x)-1280 A b^5 x^5\right )}{765765 a^6 x^{17/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)^(5/2)*(A + B*x))/x^(19/2),x]

[Out]

(-2*(a + b*x)^(7/2)*(-1280*A*b^5*x^5 + 3003*a^5*(15*A + 17*B*x) + 128*a*b^4*x^4*(35*A + 17*B*x) - 224*a^2*b^3*
x^3*(45*A + 34*B*x) + 336*a^3*b^2*x^2*(55*A + 51*B*x) - 462*a^4*b*x*(65*A + 68*B*x)))/(765765*a^6*x^(17/2))

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IntegrateAlgebraic [A]  time = 0.12, size = 169, normalized size = 0.92 \begin {gather*} -\frac {2 (a+b x)^{7/2} \left (\frac {425425 A b^4 (a+b x)}{x}-\frac {696150 A b^3 (a+b x)^2}{x^2}+\frac {589050 A b^2 (a+b x)^3}{x^3}+\frac {45045 A (a+b x)^5}{x^5}-\frac {255255 A b (a+b x)^4}{x^4}+109395 a b^4 B-\frac {340340 a b^3 B (a+b x)}{x}+\frac {417690 a b^2 B (a+b x)^2}{x^2}+\frac {51051 a B (a+b x)^4}{x^4}-\frac {235620 a b B (a+b x)^3}{x^3}-109395 A b^5\right )}{765765 a^6 x^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((a + b*x)^(5/2)*(A + B*x))/x^(19/2),x]

[Out]

(-2*(a + b*x)^(7/2)*(-109395*A*b^5 + 109395*a*b^4*B + (425425*A*b^4*(a + b*x))/x - (340340*a*b^3*B*(a + b*x))/
x - (696150*A*b^3*(a + b*x)^2)/x^2 + (417690*a*b^2*B*(a + b*x)^2)/x^2 + (589050*A*b^2*(a + b*x)^3)/x^3 - (2356
20*a*b*B*(a + b*x)^3)/x^3 - (255255*A*b*(a + b*x)^4)/x^4 + (51051*a*B*(a + b*x)^4)/x^4 + (45045*A*(a + b*x)^5)
/x^5))/(765765*a^6*x^(7/2))

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fricas [A]  time = 1.67, size = 198, normalized size = 1.08 \begin {gather*} -\frac {2 \, {\left (45045 \, A a^{8} + 128 \, {\left (17 \, B a b^{7} - 10 \, A b^{8}\right )} x^{8} - 64 \, {\left (17 \, B a^{2} b^{6} - 10 \, A a b^{7}\right )} x^{7} + 48 \, {\left (17 \, B a^{3} b^{5} - 10 \, A a^{2} b^{6}\right )} x^{6} - 40 \, {\left (17 \, B a^{4} b^{4} - 10 \, A a^{3} b^{5}\right )} x^{5} + 35 \, {\left (17 \, B a^{5} b^{3} - 10 \, A a^{4} b^{4}\right )} x^{4} + 63 \, {\left (1207 \, B a^{6} b^{2} + 5 \, A a^{5} b^{3}\right )} x^{3} + 231 \, {\left (527 \, B a^{7} b + 275 \, A a^{6} b^{2}\right )} x^{2} + 3003 \, {\left (17 \, B a^{8} + 35 \, A a^{7} b\right )} x\right )} \sqrt {b x + a}}{765765 \, a^{6} x^{\frac {17}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(5/2)*(B*x+A)/x^(19/2),x, algorithm="fricas")

[Out]

-2/765765*(45045*A*a^8 + 128*(17*B*a*b^7 - 10*A*b^8)*x^8 - 64*(17*B*a^2*b^6 - 10*A*a*b^7)*x^7 + 48*(17*B*a^3*b
^5 - 10*A*a^2*b^6)*x^6 - 40*(17*B*a^4*b^4 - 10*A*a^3*b^5)*x^5 + 35*(17*B*a^5*b^3 - 10*A*a^4*b^4)*x^4 + 63*(120
7*B*a^6*b^2 + 5*A*a^5*b^3)*x^3 + 231*(527*B*a^7*b + 275*A*a^6*b^2)*x^2 + 3003*(17*B*a^8 + 35*A*a^7*b)*x)*sqrt(
b*x + a)/(a^6*x^(17/2))

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giac [A]  time = 1.62, size = 208, normalized size = 1.14 \begin {gather*} -\frac {2 \, {\left ({\left (8 \, {\left (2 \, {\left (b x + a\right )} {\left (4 \, {\left (b x + a\right )} {\left (\frac {2 \, {\left (17 \, B a^{3} b^{16} - 10 \, A a^{2} b^{17}\right )} {\left (b x + a\right )}}{a^{8}} - \frac {17 \, {\left (17 \, B a^{4} b^{16} - 10 \, A a^{3} b^{17}\right )}}{a^{8}}\right )} + \frac {255 \, {\left (17 \, B a^{5} b^{16} - 10 \, A a^{4} b^{17}\right )}}{a^{8}}\right )} - \frac {1105 \, {\left (17 \, B a^{6} b^{16} - 10 \, A a^{5} b^{17}\right )}}{a^{8}}\right )} {\left (b x + a\right )} + \frac {12155 \, {\left (17 \, B a^{7} b^{16} - 10 \, A a^{6} b^{17}\right )}}{a^{8}}\right )} {\left (b x + a\right )} - \frac {109395 \, {\left (B a^{8} b^{16} - A a^{7} b^{17}\right )}}{a^{8}}\right )} {\left (b x + a\right )}^{\frac {7}{2}} b}{765765 \, {\left ({\left (b x + a\right )} b - a b\right )}^{\frac {17}{2}} {\left | b \right |}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(5/2)*(B*x+A)/x^(19/2),x, algorithm="giac")

[Out]

-2/765765*((8*(2*(b*x + a)*(4*(b*x + a)*(2*(17*B*a^3*b^16 - 10*A*a^2*b^17)*(b*x + a)/a^8 - 17*(17*B*a^4*b^16 -
10*A*a^3*b^17)/a^8) + 255*(17*B*a^5*b^16 - 10*A*a^4*b^17)/a^8) - 1105*(17*B*a^6*b^16 - 10*A*a^5*b^17)/a^8)*(b
*x + a) + 12155*(17*B*a^7*b^16 - 10*A*a^6*b^17)/a^8)*(b*x + a) - 109395*(B*a^8*b^16 - A*a^7*b^17)/a^8)*(b*x +
a)^(7/2)*b/(((b*x + a)*b - a*b)^(17/2)*abs(b))

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maple [A]  time = 0.01, size = 125, normalized size = 0.68 \begin {gather*} -\frac {2 \left (b x +a \right )^{\frac {7}{2}} \left (-1280 A \,b^{5} x^{5}+2176 B a \,b^{4} x^{5}+4480 A a \,b^{4} x^{4}-7616 B \,a^{2} b^{3} x^{4}-10080 A \,a^{2} b^{3} x^{3}+17136 B \,a^{3} b^{2} x^{3}+18480 A \,a^{3} b^{2} x^{2}-31416 B \,a^{4} b \,x^{2}-30030 A \,a^{4} b x +51051 B \,a^{5} x +45045 A \,a^{5}\right )}{765765 a^{6} x^{\frac {17}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^(5/2)*(B*x+A)/x^(19/2),x)

[Out]

-2/765765*(b*x+a)^(7/2)*(-1280*A*b^5*x^5+2176*B*a*b^4*x^5+4480*A*a*b^4*x^4-7616*B*a^2*b^3*x^4-10080*A*a^2*b^3*
x^3+17136*B*a^3*b^2*x^3+18480*A*a^3*b^2*x^2-31416*B*a^4*b*x^2-30030*A*a^4*b*x+51051*B*a^5*x+45045*A*a^5)/x^(17
/2)/a^6

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maxima [B]  time = 1.04, size = 442, normalized size = 2.42 \begin {gather*} -\frac {256 \, \sqrt {b x^{2} + a x} B b^{7}}{45045 \, a^{5} x} + \frac {512 \, \sqrt {b x^{2} + a x} A b^{8}}{153153 \, a^{6} x} + \frac {128 \, \sqrt {b x^{2} + a x} B b^{6}}{45045 \, a^{4} x^{2}} - \frac {256 \, \sqrt {b x^{2} + a x} A b^{7}}{153153 \, a^{5} x^{2}} - \frac {32 \, \sqrt {b x^{2} + a x} B b^{5}}{15015 \, a^{3} x^{3}} + \frac {64 \, \sqrt {b x^{2} + a x} A b^{6}}{51051 \, a^{4} x^{3}} + \frac {16 \, \sqrt {b x^{2} + a x} B b^{4}}{9009 \, a^{2} x^{4}} - \frac {160 \, \sqrt {b x^{2} + a x} A b^{5}}{153153 \, a^{3} x^{4}} - \frac {2 \, \sqrt {b x^{2} + a x} B b^{3}}{1287 \, a x^{5}} + \frac {20 \, \sqrt {b x^{2} + a x} A b^{4}}{21879 \, a^{2} x^{5}} + \frac {\sqrt {b x^{2} + a x} B b^{2}}{715 \, x^{6}} - \frac {2 \, \sqrt {b x^{2} + a x} A b^{3}}{2431 \, a x^{6}} - \frac {\sqrt {b x^{2} + a x} B a b}{780 \, x^{7}} + \frac {\sqrt {b x^{2} + a x} A b^{2}}{1326 \, x^{7}} - \frac {\sqrt {b x^{2} + a x} B a^{2}}{60 \, x^{8}} - \frac {\sqrt {b x^{2} + a x} A a b}{1428 \, x^{8}} + \frac {{\left (b x^{2} + a x\right )}^{\frac {3}{2}} B a}{12 \, x^{9}} - \frac {5 \, \sqrt {b x^{2} + a x} A a^{2}}{476 \, x^{9}} - \frac {{\left (b x^{2} + a x\right )}^{\frac {5}{2}} B}{5 \, x^{10}} + \frac {5 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} A a}{84 \, x^{10}} - \frac {{\left (b x^{2} + a x\right )}^{\frac {5}{2}} A}{6 \, x^{11}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(5/2)*(B*x+A)/x^(19/2),x, algorithm="maxima")

[Out]

-256/45045*sqrt(b*x^2 + a*x)*B*b^7/(a^5*x) + 512/153153*sqrt(b*x^2 + a*x)*A*b^8/(a^6*x) + 128/45045*sqrt(b*x^2
+ a*x)*B*b^6/(a^4*x^2) - 256/153153*sqrt(b*x^2 + a*x)*A*b^7/(a^5*x^2) - 32/15015*sqrt(b*x^2 + a*x)*B*b^5/(a^3
*x^3) + 64/51051*sqrt(b*x^2 + a*x)*A*b^6/(a^4*x^3) + 16/9009*sqrt(b*x^2 + a*x)*B*b^4/(a^2*x^4) - 160/153153*sq
rt(b*x^2 + a*x)*A*b^5/(a^3*x^4) - 2/1287*sqrt(b*x^2 + a*x)*B*b^3/(a*x^5) + 20/21879*sqrt(b*x^2 + a*x)*A*b^4/(a
^2*x^5) + 1/715*sqrt(b*x^2 + a*x)*B*b^2/x^6 - 2/2431*sqrt(b*x^2 + a*x)*A*b^3/(a*x^6) - 1/780*sqrt(b*x^2 + a*x)
*B*a*b/x^7 + 1/1326*sqrt(b*x^2 + a*x)*A*b^2/x^7 - 1/60*sqrt(b*x^2 + a*x)*B*a^2/x^8 - 1/1428*sqrt(b*x^2 + a*x)*
A*a*b/x^8 + 1/12*(b*x^2 + a*x)^(3/2)*B*a/x^9 - 5/476*sqrt(b*x^2 + a*x)*A*a^2/x^9 - 1/5*(b*x^2 + a*x)^(5/2)*B/x
^10 + 5/84*(b*x^2 + a*x)^(3/2)*A*a/x^10 - 1/6*(b*x^2 + a*x)^(5/2)*A/x^11

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mupad [B]  time = 1.02, size = 167, normalized size = 0.91 \begin {gather*} -\frac {\sqrt {a+b\,x}\,\left (\frac {2\,A\,a^2}{17}+\frac {2\,a\,x\,\left (35\,A\,b+17\,B\,a\right )}{255}+\frac {2\,b\,x^2\,\left (275\,A\,b+527\,B\,a\right )}{3315}-\frac {2\,b^3\,x^4\,\left (10\,A\,b-17\,B\,a\right )}{21879\,a^2}+\frac {16\,b^4\,x^5\,\left (10\,A\,b-17\,B\,a\right )}{153153\,a^3}-\frac {32\,b^5\,x^6\,\left (10\,A\,b-17\,B\,a\right )}{255255\,a^4}+\frac {128\,b^6\,x^7\,\left (10\,A\,b-17\,B\,a\right )}{765765\,a^5}-\frac {256\,b^7\,x^8\,\left (10\,A\,b-17\,B\,a\right )}{765765\,a^6}+\frac {2\,b^2\,x^3\,\left (5\,A\,b+1207\,B\,a\right )}{12155\,a}\right )}{x^{17/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x)*(a + b*x)^(5/2))/x^(19/2),x)

[Out]

-((a + b*x)^(1/2)*((2*A*a^2)/17 + (2*a*x*(35*A*b + 17*B*a))/255 + (2*b*x^2*(275*A*b + 527*B*a))/3315 - (2*b^3*
x^4*(10*A*b - 17*B*a))/(21879*a^2) + (16*b^4*x^5*(10*A*b - 17*B*a))/(153153*a^3) - (32*b^5*x^6*(10*A*b - 17*B*
a))/(255255*a^4) + (128*b^6*x^7*(10*A*b - 17*B*a))/(765765*a^5) - (256*b^7*x^8*(10*A*b - 17*B*a))/(765765*a^6)
+ (2*b^2*x^3*(5*A*b + 1207*B*a))/(12155*a)))/x^(17/2)

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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)**(5/2)*(B*x+A)/x**(19/2),x)

[Out]

Timed out

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