3.5.87 \(\int \frac {(a+b x)^{5/2} (A+B x)}{x^{17/2}} \, dx\)

Optimal. Leaf size=150 \[ -\frac {32 b^3 (a+b x)^{7/2} (8 A b-15 a B)}{45045 a^5 x^{7/2}}+\frac {16 b^2 (a+b x)^{7/2} (8 A b-15 a B)}{6435 a^4 x^{9/2}}-\frac {4 b (a+b x)^{7/2} (8 A b-15 a B)}{715 a^3 x^{11/2}}+\frac {2 (a+b x)^{7/2} (8 A b-15 a B)}{195 a^2 x^{13/2}}-\frac {2 A (a+b x)^{7/2}}{15 a x^{15/2}} \]

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Rubi [A]  time = 0.05, antiderivative size = 150, normalized size of antiderivative = 1.00, number of steps used = 5, 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 {32 b^3 (a+b x)^{7/2} (8 A b-15 a B)}{45045 a^5 x^{7/2}}+\frac {16 b^2 (a+b x)^{7/2} (8 A b-15 a B)}{6435 a^4 x^{9/2}}-\frac {4 b (a+b x)^{7/2} (8 A b-15 a B)}{715 a^3 x^{11/2}}+\frac {2 (a+b x)^{7/2} (8 A b-15 a B)}{195 a^2 x^{13/2}}-\frac {2 A (a+b x)^{7/2}}{15 a x^{15/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((a + b*x)^(5/2)*(A + B*x))/x^(17/2),x]

[Out]

(-2*A*(a + b*x)^(7/2))/(15*a*x^(15/2)) + (2*(8*A*b - 15*a*B)*(a + b*x)^(7/2))/(195*a^2*x^(13/2)) - (4*b*(8*A*b
- 15*a*B)*(a + b*x)^(7/2))/(715*a^3*x^(11/2)) + (16*b^2*(8*A*b - 15*a*B)*(a + b*x)^(7/2))/(6435*a^4*x^(9/2))
- (32*b^3*(8*A*b - 15*a*B)*(a + b*x)^(7/2))/(45045*a^5*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^{17/2}} \, dx &=-\frac {2 A (a+b x)^{7/2}}{15 a x^{15/2}}+\frac {\left (2 \left (-4 A b+\frac {15 a B}{2}\right )\right ) \int \frac {(a+b x)^{5/2}}{x^{15/2}} \, dx}{15 a}\\ &=-\frac {2 A (a+b x)^{7/2}}{15 a x^{15/2}}+\frac {2 (8 A b-15 a B) (a+b x)^{7/2}}{195 a^2 x^{13/2}}+\frac {(2 b (8 A b-15 a B)) \int \frac {(a+b x)^{5/2}}{x^{13/2}} \, dx}{65 a^2}\\ &=-\frac {2 A (a+b x)^{7/2}}{15 a x^{15/2}}+\frac {2 (8 A b-15 a B) (a+b x)^{7/2}}{195 a^2 x^{13/2}}-\frac {4 b (8 A b-15 a B) (a+b x)^{7/2}}{715 a^3 x^{11/2}}-\frac {\left (8 b^2 (8 A b-15 a B)\right ) \int \frac {(a+b x)^{5/2}}{x^{11/2}} \, dx}{715 a^3}\\ &=-\frac {2 A (a+b x)^{7/2}}{15 a x^{15/2}}+\frac {2 (8 A b-15 a B) (a+b x)^{7/2}}{195 a^2 x^{13/2}}-\frac {4 b (8 A b-15 a B) (a+b x)^{7/2}}{715 a^3 x^{11/2}}+\frac {16 b^2 (8 A b-15 a B) (a+b x)^{7/2}}{6435 a^4 x^{9/2}}+\frac {\left (16 b^3 (8 A b-15 a B)\right ) \int \frac {(a+b x)^{5/2}}{x^{9/2}} \, dx}{6435 a^4}\\ &=-\frac {2 A (a+b x)^{7/2}}{15 a x^{15/2}}+\frac {2 (8 A b-15 a B) (a+b x)^{7/2}}{195 a^2 x^{13/2}}-\frac {4 b (8 A b-15 a B) (a+b x)^{7/2}}{715 a^3 x^{11/2}}+\frac {16 b^2 (8 A b-15 a B) (a+b x)^{7/2}}{6435 a^4 x^{9/2}}-\frac {32 b^3 (8 A b-15 a B) (a+b x)^{7/2}}{45045 a^5 x^{7/2}}\\ \end {align*}

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Mathematica [A]  time = 0.06, size = 95, normalized size = 0.63 \begin {gather*} -\frac {2 (a+b x)^{7/2} \left (231 a^4 (13 A+15 B x)-42 a^3 b x (44 A+45 B x)+168 a^2 b^2 x^2 (6 A+5 B x)-16 a b^3 x^3 (28 A+15 B x)+128 A b^4 x^4\right )}{45045 a^5 x^{15/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)^(5/2)*(A + B*x))/x^(17/2),x]

[Out]

(-2*(a + b*x)^(7/2)*(128*A*b^4*x^4 + 168*a^2*b^2*x^2*(6*A + 5*B*x) + 231*a^4*(13*A + 15*B*x) - 16*a*b^3*x^3*(2
8*A + 15*B*x) - 42*a^3*b*x*(44*A + 45*B*x)))/(45045*a^5*x^(15/2))

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IntegrateAlgebraic [A]  time = 0.12, size = 136, normalized size = 0.91 \begin {gather*} -\frac {2 (a+b x)^{7/2} \left (-\frac {20020 A b^3 (a+b x)}{x}+\frac {24570 A b^2 (a+b x)^2}{x^2}+\frac {3003 A (a+b x)^4}{x^4}-\frac {13860 A b (a+b x)^3}{x^3}-6435 a b^3 B+\frac {15015 a b^2 B (a+b x)}{x}+\frac {3465 a B (a+b x)^3}{x^3}-\frac {12285 a b B (a+b x)^2}{x^2}+6435 A b^4\right )}{45045 a^5 x^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((a + b*x)^(5/2)*(A + B*x))/x^(17/2),x]

[Out]

(-2*(a + b*x)^(7/2)*(6435*A*b^4 - 6435*a*b^3*B - (20020*A*b^3*(a + b*x))/x + (15015*a*b^2*B*(a + b*x))/x + (24
570*A*b^2*(a + b*x)^2)/x^2 - (12285*a*b*B*(a + b*x)^2)/x^2 - (13860*A*b*(a + b*x)^3)/x^3 + (3465*a*B*(a + b*x)
^3)/x^3 + (3003*A*(a + b*x)^4)/x^4))/(45045*a^5*x^(7/2))

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fricas [A]  time = 1.60, size = 173, normalized size = 1.15 \begin {gather*} -\frac {2 \, {\left (3003 \, A a^{7} - 16 \, {\left (15 \, B a b^{6} - 8 \, A b^{7}\right )} x^{7} + 8 \, {\left (15 \, B a^{2} b^{5} - 8 \, A a b^{6}\right )} x^{6} - 6 \, {\left (15 \, B a^{3} b^{4} - 8 \, A a^{2} b^{5}\right )} x^{5} + 5 \, {\left (15 \, B a^{4} b^{3} - 8 \, A a^{3} b^{4}\right )} x^{4} + 35 \, {\left (159 \, B a^{5} b^{2} + A a^{4} b^{3}\right )} x^{3} + 63 \, {\left (135 \, B a^{6} b + 71 \, A a^{5} b^{2}\right )} x^{2} + 231 \, {\left (15 \, B a^{7} + 31 \, A a^{6} b\right )} x\right )} \sqrt {b x + a}}{45045 \, a^{5} x^{\frac {15}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(5/2)*(B*x+A)/x^(17/2),x, algorithm="fricas")

[Out]

-2/45045*(3003*A*a^7 - 16*(15*B*a*b^6 - 8*A*b^7)*x^7 + 8*(15*B*a^2*b^5 - 8*A*a*b^6)*x^6 - 6*(15*B*a^3*b^4 - 8*
A*a^2*b^5)*x^5 + 5*(15*B*a^4*b^3 - 8*A*a^3*b^4)*x^4 + 35*(159*B*a^5*b^2 + A*a^4*b^3)*x^3 + 63*(135*B*a^6*b + 7
1*A*a^5*b^2)*x^2 + 231*(15*B*a^7 + 31*A*a^6*b)*x)*sqrt(b*x + a)/(a^5*x^(15/2))

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giac [A]  time = 1.64, size = 176, normalized size = 1.17 \begin {gather*} \frac {2 \, {\left ({\left (2 \, {\left (b x + a\right )} {\left (4 \, {\left (b x + a\right )} {\left (\frac {2 \, {\left (15 \, B a^{3} b^{14} - 8 \, A a^{2} b^{15}\right )} {\left (b x + a\right )}}{a^{7}} - \frac {15 \, {\left (15 \, B a^{4} b^{14} - 8 \, A a^{3} b^{15}\right )}}{a^{7}}\right )} + \frac {195 \, {\left (15 \, B a^{5} b^{14} - 8 \, A a^{4} b^{15}\right )}}{a^{7}}\right )} - \frac {715 \, {\left (15 \, B a^{6} b^{14} - 8 \, A a^{5} b^{15}\right )}}{a^{7}}\right )} {\left (b x + a\right )} + \frac {6435 \, {\left (B a^{7} b^{14} - A a^{6} b^{15}\right )}}{a^{7}}\right )} {\left (b x + a\right )}^{\frac {7}{2}} b}{45045 \, {\left ({\left (b x + a\right )} b - a b\right )}^{\frac {15}{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^(17/2),x, algorithm="giac")

[Out]

2/45045*((2*(b*x + a)*(4*(b*x + a)*(2*(15*B*a^3*b^14 - 8*A*a^2*b^15)*(b*x + a)/a^7 - 15*(15*B*a^4*b^14 - 8*A*a
^3*b^15)/a^7) + 195*(15*B*a^5*b^14 - 8*A*a^4*b^15)/a^7) - 715*(15*B*a^6*b^14 - 8*A*a^5*b^15)/a^7)*(b*x + a) +
6435*(B*a^7*b^14 - A*a^6*b^15)/a^7)*(b*x + a)^(7/2)*b/(((b*x + a)*b - a*b)^(15/2)*abs(b))

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maple [A]  time = 0.01, size = 101, normalized size = 0.67 \begin {gather*} -\frac {2 \left (b x +a \right )^{\frac {7}{2}} \left (128 A \,b^{4} x^{4}-240 B a \,b^{3} x^{4}-448 A a \,b^{3} x^{3}+840 B \,a^{2} b^{2} x^{3}+1008 A \,a^{2} b^{2} x^{2}-1890 B \,a^{3} b \,x^{2}-1848 A \,a^{3} b x +3465 B \,a^{4} x +3003 A \,a^{4}\right )}{45045 a^{5} x^{\frac {15}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^(5/2)*(B*x+A)/x^(17/2),x)

[Out]

-2/45045*(b*x+a)^(7/2)*(128*A*b^4*x^4-240*B*a*b^3*x^4-448*A*a*b^3*x^3+840*B*a^2*b^2*x^3+1008*A*a^2*b^2*x^2-189
0*B*a^3*b*x^2-1848*A*a^3*b*x+3465*B*a^4*x+3003*A*a^4)/x^(15/2)/a^5

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maxima [B]  time = 1.03, size = 396, normalized size = 2.64 \begin {gather*} \frac {32 \, \sqrt {b x^{2} + a x} B b^{6}}{3003 \, a^{4} x} - \frac {256 \, \sqrt {b x^{2} + a x} A b^{7}}{45045 \, a^{5} x} - \frac {16 \, \sqrt {b x^{2} + a x} B b^{5}}{3003 \, a^{3} x^{2}} + \frac {128 \, \sqrt {b x^{2} + a x} A b^{6}}{45045 \, a^{4} x^{2}} + \frac {4 \, \sqrt {b x^{2} + a x} B b^{4}}{1001 \, a^{2} x^{3}} - \frac {32 \, \sqrt {b x^{2} + a x} A b^{5}}{15015 \, a^{3} x^{3}} - \frac {10 \, \sqrt {b x^{2} + a x} B b^{3}}{3003 \, a x^{4}} + \frac {16 \, \sqrt {b x^{2} + a x} A b^{4}}{9009 \, a^{2} x^{4}} + \frac {5 \, \sqrt {b x^{2} + a x} B b^{2}}{1716 \, x^{5}} - \frac {2 \, \sqrt {b x^{2} + a x} A b^{3}}{1287 \, a x^{5}} - \frac {3 \, \sqrt {b x^{2} + a x} B a b}{1144 \, x^{6}} + \frac {\sqrt {b x^{2} + a x} A b^{2}}{715 \, x^{6}} - \frac {3 \, \sqrt {b x^{2} + a x} B a^{2}}{104 \, x^{7}} - \frac {\sqrt {b x^{2} + a x} A a b}{780 \, x^{7}} + \frac {{\left (b x^{2} + a x\right )}^{\frac {3}{2}} B a}{8 \, x^{8}} - \frac {\sqrt {b x^{2} + a x} A a^{2}}{60 \, x^{8}} - \frac {{\left (b x^{2} + a x\right )}^{\frac {5}{2}} B}{4 \, x^{9}} + \frac {{\left (b x^{2} + a x\right )}^{\frac {3}{2}} A a}{12 \, x^{9}} - \frac {{\left (b x^{2} + a x\right )}^{\frac {5}{2}} A}{5 \, x^{10}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(5/2)*(B*x+A)/x^(17/2),x, algorithm="maxima")

[Out]

32/3003*sqrt(b*x^2 + a*x)*B*b^6/(a^4*x) - 256/45045*sqrt(b*x^2 + a*x)*A*b^7/(a^5*x) - 16/3003*sqrt(b*x^2 + a*x
)*B*b^5/(a^3*x^2) + 128/45045*sqrt(b*x^2 + a*x)*A*b^6/(a^4*x^2) + 4/1001*sqrt(b*x^2 + a*x)*B*b^4/(a^2*x^3) - 3
2/15015*sqrt(b*x^2 + a*x)*A*b^5/(a^3*x^3) - 10/3003*sqrt(b*x^2 + a*x)*B*b^3/(a*x^4) + 16/9009*sqrt(b*x^2 + a*x
)*A*b^4/(a^2*x^4) + 5/1716*sqrt(b*x^2 + a*x)*B*b^2/x^5 - 2/1287*sqrt(b*x^2 + a*x)*A*b^3/(a*x^5) - 3/1144*sqrt(
b*x^2 + a*x)*B*a*b/x^6 + 1/715*sqrt(b*x^2 + a*x)*A*b^2/x^6 - 3/104*sqrt(b*x^2 + a*x)*B*a^2/x^7 - 1/780*sqrt(b*
x^2 + a*x)*A*a*b/x^7 + 1/8*(b*x^2 + a*x)^(3/2)*B*a/x^8 - 1/60*sqrt(b*x^2 + a*x)*A*a^2/x^8 - 1/4*(b*x^2 + a*x)^
(5/2)*B/x^9 + 1/12*(b*x^2 + a*x)^(3/2)*A*a/x^9 - 1/5*(b*x^2 + a*x)^(5/2)*A/x^10

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mupad [B]  time = 1.00, size = 148, normalized size = 0.99 \begin {gather*} -\frac {\sqrt {a+b\,x}\,\left (\frac {2\,A\,a^2}{15}+\frac {x^7\,\left (256\,A\,b^7-480\,B\,a\,b^6\right )}{45045\,a^5}+\frac {2\,a\,x\,\left (31\,A\,b+15\,B\,a\right )}{195}+\frac {2\,b\,x^2\,\left (71\,A\,b+135\,B\,a\right )}{715}-\frac {2\,b^3\,x^4\,\left (8\,A\,b-15\,B\,a\right )}{9009\,a^2}+\frac {4\,b^4\,x^5\,\left (8\,A\,b-15\,B\,a\right )}{15015\,a^3}-\frac {16\,b^5\,x^6\,\left (8\,A\,b-15\,B\,a\right )}{45045\,a^4}+\frac {2\,b^2\,x^3\,\left (A\,b+159\,B\,a\right )}{1287\,a}\right )}{x^{15/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x)*(a + b*x)^(5/2))/x^(17/2),x)

[Out]

-((a + b*x)^(1/2)*((2*A*a^2)/15 + (x^7*(256*A*b^7 - 480*B*a*b^6))/(45045*a^5) + (2*a*x*(31*A*b + 15*B*a))/195
+ (2*b*x^2*(71*A*b + 135*B*a))/715 - (2*b^3*x^4*(8*A*b - 15*B*a))/(9009*a^2) + (4*b^4*x^5*(8*A*b - 15*B*a))/(1
5015*a^3) - (16*b^5*x^6*(8*A*b - 15*B*a))/(45045*a^4) + (2*b^2*x^3*(A*b + 159*B*a))/(1287*a)))/x^(15/2)

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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**(17/2),x)

[Out]

Timed out

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