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

Optimal. Leaf size=117 \[ \frac {16 b^2 (a+b x)^{5/2} (6 A b-11 a B)}{3465 a^4 x^{5/2}}-\frac {8 b (a+b x)^{5/2} (6 A b-11 a B)}{693 a^3 x^{7/2}}+\frac {2 (a+b x)^{5/2} (6 A b-11 a B)}{99 a^2 x^{9/2}}-\frac {2 A (a+b x)^{5/2}}{11 a x^{11/2}} \]

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Rubi [A]  time = 0.04, antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps used = 4, 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 {16 b^2 (a+b x)^{5/2} (6 A b-11 a B)}{3465 a^4 x^{5/2}}-\frac {8 b (a+b x)^{5/2} (6 A b-11 a B)}{693 a^3 x^{7/2}}+\frac {2 (a+b x)^{5/2} (6 A b-11 a B)}{99 a^2 x^{9/2}}-\frac {2 A (a+b x)^{5/2}}{11 a x^{11/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*A*(a + b*x)^(5/2))/(11*a*x^(11/2)) + (2*(6*A*b - 11*a*B)*(a + b*x)^(5/2))/(99*a^2*x^(9/2)) - (8*b*(6*A*b -
11*a*B)*(a + b*x)^(5/2))/(693*a^3*x^(7/2)) + (16*b^2*(6*A*b - 11*a*B)*(a + b*x)^(5/2))/(3465*a^4*x^(5/2))

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)^{3/2} (A+B x)}{x^{13/2}} \, dx &=-\frac {2 A (a+b x)^{5/2}}{11 a x^{11/2}}+\frac {\left (2 \left (-3 A b+\frac {11 a B}{2}\right )\right ) \int \frac {(a+b x)^{3/2}}{x^{11/2}} \, dx}{11 a}\\ &=-\frac {2 A (a+b x)^{5/2}}{11 a x^{11/2}}+\frac {2 (6 A b-11 a B) (a+b x)^{5/2}}{99 a^2 x^{9/2}}+\frac {(4 b (6 A b-11 a B)) \int \frac {(a+b x)^{3/2}}{x^{9/2}} \, dx}{99 a^2}\\ &=-\frac {2 A (a+b x)^{5/2}}{11 a x^{11/2}}+\frac {2 (6 A b-11 a B) (a+b x)^{5/2}}{99 a^2 x^{9/2}}-\frac {8 b (6 A b-11 a B) (a+b x)^{5/2}}{693 a^3 x^{7/2}}-\frac {\left (8 b^2 (6 A b-11 a B)\right ) \int \frac {(a+b x)^{3/2}}{x^{7/2}} \, dx}{693 a^3}\\ &=-\frac {2 A (a+b x)^{5/2}}{11 a x^{11/2}}+\frac {2 (6 A b-11 a B) (a+b x)^{5/2}}{99 a^2 x^{9/2}}-\frac {8 b (6 A b-11 a B) (a+b x)^{5/2}}{693 a^3 x^{7/2}}+\frac {16 b^2 (6 A b-11 a B) (a+b x)^{5/2}}{3465 a^4 x^{5/2}}\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 76, normalized size = 0.65 \begin {gather*} -\frac {2 (a+b x)^{5/2} \left (35 a^3 (9 A+11 B x)-10 a^2 b x (21 A+22 B x)+8 a b^2 x^2 (15 A+11 B x)-48 A b^3 x^3\right )}{3465 a^4 x^{11/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(a + b*x)^(5/2)*(-48*A*b^3*x^3 + 35*a^3*(9*A + 11*B*x) + 8*a*b^2*x^2*(15*A + 11*B*x) - 10*a^2*b*x*(21*A +
22*B*x)))/(3465*a^4*x^(11/2))

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IntegrateAlgebraic [A]  time = 0.36, size = 130, normalized size = 1.11 \begin {gather*} -\frac {2 \sqrt {a+b x} \left (315 a^5 A+385 a^5 B x+420 a^4 A b x+550 a^4 b B x^2+15 a^3 A b^2 x^2+33 a^3 b^2 B x^3-18 a^2 A b^3 x^3-44 a^2 b^3 B x^4+24 a A b^4 x^4+88 a b^4 B x^5-48 A b^5 x^5\right )}{3465 a^4 x^{11/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[a + b*x]*(315*a^5*A + 420*a^4*A*b*x + 385*a^5*B*x + 15*a^3*A*b^2*x^2 + 550*a^4*b*B*x^2 - 18*a^2*A*b^3
*x^3 + 33*a^3*b^2*B*x^3 + 24*a*A*b^4*x^4 - 44*a^2*b^3*B*x^4 - 48*A*b^5*x^5 + 88*a*b^4*B*x^5))/(3465*a^4*x^(11/
2))

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fricas [A]  time = 1.44, size = 126, normalized size = 1.08 \begin {gather*} -\frac {2 \, {\left (315 \, A a^{5} + 8 \, {\left (11 \, B a b^{4} - 6 \, A b^{5}\right )} x^{5} - 4 \, {\left (11 \, B a^{2} b^{3} - 6 \, A a b^{4}\right )} x^{4} + 3 \, {\left (11 \, B a^{3} b^{2} - 6 \, A a^{2} b^{3}\right )} x^{3} + 5 \, {\left (110 \, B a^{4} b + 3 \, A a^{3} b^{2}\right )} x^{2} + 35 \, {\left (11 \, B a^{5} + 12 \, A a^{4} b\right )} x\right )} \sqrt {b x + a}}{3465 \, a^{4} x^{\frac {11}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3465*(315*A*a^5 + 8*(11*B*a*b^4 - 6*A*b^5)*x^5 - 4*(11*B*a^2*b^3 - 6*A*a*b^4)*x^4 + 3*(11*B*a^3*b^2 - 6*A*a
^2*b^3)*x^3 + 5*(110*B*a^4*b + 3*A*a^3*b^2)*x^2 + 35*(11*B*a^5 + 12*A*a^4*b)*x)*sqrt(b*x + a)/(a^4*x^(11/2))

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giac [A]  time = 2.07, size = 142, normalized size = 1.21 \begin {gather*} -\frac {2 \, {\left ({\left (b x + a\right )} {\left (4 \, {\left (b x + a\right )} {\left (\frac {2 \, {\left (11 \, B a^{2} b^{10} - 6 \, A a b^{11}\right )} {\left (b x + a\right )}}{a^{5}} - \frac {11 \, {\left (11 \, B a^{3} b^{10} - 6 \, A a^{2} b^{11}\right )}}{a^{5}}\right )} + \frac {99 \, {\left (11 \, B a^{4} b^{10} - 6 \, A a^{3} b^{11}\right )}}{a^{5}}\right )} - \frac {693 \, {\left (B a^{5} b^{10} - A a^{4} b^{11}\right )}}{a^{5}}\right )} {\left (b x + a\right )}^{\frac {5}{2}} b}{3465 \, {\left ({\left (b x + a\right )} b - a b\right )}^{\frac {11}{2}} {\left | b \right |}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3465*((b*x + a)*(4*(b*x + a)*(2*(11*B*a^2*b^10 - 6*A*a*b^11)*(b*x + a)/a^5 - 11*(11*B*a^3*b^10 - 6*A*a^2*b^
11)/a^5) + 99*(11*B*a^4*b^10 - 6*A*a^3*b^11)/a^5) - 693*(B*a^5*b^10 - A*a^4*b^11)/a^5)*(b*x + a)^(5/2)*b/(((b*
x + a)*b - a*b)^(11/2)*abs(b))

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maple [A]  time = 0.01, size = 77, normalized size = 0.66 \begin {gather*} -\frac {2 \left (b x +a \right )^{\frac {5}{2}} \left (-48 A \,b^{3} x^{3}+88 B a \,b^{2} x^{3}+120 A a \,b^{2} x^{2}-220 B \,a^{2} b \,x^{2}-210 A \,a^{2} b x +385 B \,a^{3} x +315 A \,a^{3}\right )}{3465 a^{4} x^{\frac {11}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3465*(b*x+a)^(5/2)*(-48*A*b^3*x^3+88*B*a*b^2*x^3+120*A*a*b^2*x^2-220*B*a^2*b*x^2-210*A*a^2*b*x+385*B*a^3*x+
315*A*a^3)/x^(11/2)/a^4

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maxima [B]  time = 0.97, size = 268, normalized size = 2.29 \begin {gather*} -\frac {16 \, \sqrt {b x^{2} + a x} B b^{4}}{315 \, a^{3} x} + \frac {32 \, \sqrt {b x^{2} + a x} A b^{5}}{1155 \, a^{4} x} + \frac {8 \, \sqrt {b x^{2} + a x} B b^{3}}{315 \, a^{2} x^{2}} - \frac {16 \, \sqrt {b x^{2} + a x} A b^{4}}{1155 \, a^{3} x^{2}} - \frac {2 \, \sqrt {b x^{2} + a x} B b^{2}}{105 \, a x^{3}} + \frac {4 \, \sqrt {b x^{2} + a x} A b^{3}}{385 \, a^{2} x^{3}} + \frac {\sqrt {b x^{2} + a x} B b}{63 \, x^{4}} - \frac {2 \, \sqrt {b x^{2} + a x} A b^{2}}{231 \, a x^{4}} + \frac {\sqrt {b x^{2} + a x} B a}{9 \, x^{5}} + \frac {\sqrt {b x^{2} + a x} A b}{132 \, x^{5}} - \frac {{\left (b x^{2} + a x\right )}^{\frac {3}{2}} B}{3 \, x^{6}} + \frac {3 \, \sqrt {b x^{2} + a x} A a}{44 \, x^{6}} - \frac {{\left (b x^{2} + a x\right )}^{\frac {3}{2}} A}{4 \, x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-16/315*sqrt(b*x^2 + a*x)*B*b^4/(a^3*x) + 32/1155*sqrt(b*x^2 + a*x)*A*b^5/(a^4*x) + 8/315*sqrt(b*x^2 + a*x)*B*
b^3/(a^2*x^2) - 16/1155*sqrt(b*x^2 + a*x)*A*b^4/(a^3*x^2) - 2/105*sqrt(b*x^2 + a*x)*B*b^2/(a*x^3) + 4/385*sqrt
(b*x^2 + a*x)*A*b^3/(a^2*x^3) + 1/63*sqrt(b*x^2 + a*x)*B*b/x^4 - 2/231*sqrt(b*x^2 + a*x)*A*b^2/(a*x^4) + 1/9*s
qrt(b*x^2 + a*x)*B*a/x^5 + 1/132*sqrt(b*x^2 + a*x)*A*b/x^5 - 1/3*(b*x^2 + a*x)^(3/2)*B/x^6 + 3/44*sqrt(b*x^2 +
a*x)*A*a/x^6 - 1/4*(b*x^2 + a*x)^(3/2)*A/x^7

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mupad [B]  time = 0.85, size = 117, normalized size = 1.00 \begin {gather*} -\frac {\sqrt {a+b\,x}\,\left (\frac {2\,A\,a}{11}+\frac {x\,\left (770\,B\,a^5+840\,A\,b\,a^4\right )}{3465\,a^4}-\frac {x^5\,\left (96\,A\,b^5-176\,B\,a\,b^4\right )}{3465\,a^4}-\frac {2\,b^2\,x^3\,\left (6\,A\,b-11\,B\,a\right )}{1155\,a^2}+\frac {8\,b^3\,x^4\,\left (6\,A\,b-11\,B\,a\right )}{3465\,a^3}+\frac {2\,b\,x^2\,\left (3\,A\,b+110\,B\,a\right )}{693\,a}\right )}{x^{11/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-((a + b*x)^(1/2)*((2*A*a)/11 + (x*(770*B*a^5 + 840*A*a^4*b))/(3465*a^4) - (x^5*(96*A*b^5 - 176*B*a*b^4))/(346
5*a^4) - (2*b^2*x^3*(6*A*b - 11*B*a))/(1155*a^2) + (8*b^3*x^4*(6*A*b - 11*B*a))/(3465*a^3) + (2*b*x^2*(3*A*b +
110*B*a))/(693*a)))/x^(11/2)

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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)**(3/2)*(B*x+A)/x**(13/2),x)

[Out]

Timed out

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