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

Optimal. Leaf size=183 \[ \frac {256 b^4 \sqrt {a+b x} (10 A b-11 a B)}{3465 a^6 \sqrt {x}}-\frac {128 b^3 \sqrt {a+b x} (10 A b-11 a B)}{3465 a^5 x^{3/2}}+\frac {32 b^2 \sqrt {a+b x} (10 A b-11 a B)}{1155 a^4 x^{5/2}}-\frac {16 b \sqrt {a+b x} (10 A b-11 a B)}{693 a^3 x^{7/2}}+\frac {2 \sqrt {a+b x} (10 A b-11 a B)}{99 a^2 x^{9/2}}-\frac {2 A \sqrt {a+b x}}{11 a x^{11/2}} \]

[Out]

-2/11*A*(b*x+a)^(1/2)/a/x^(11/2)+2/99*(10*A*b-11*B*a)*(b*x+a)^(1/2)/a^2/x^(9/2)-16/693*b*(10*A*b-11*B*a)*(b*x+
a)^(1/2)/a^3/x^(7/2)+32/1155*b^2*(10*A*b-11*B*a)*(b*x+a)^(1/2)/a^4/x^(5/2)-128/3465*b^3*(10*A*b-11*B*a)*(b*x+a
)^(1/2)/a^5/x^(3/2)+256/3465*b^4*(10*A*b-11*B*a)*(b*x+a)^(1/2)/a^6/x^(1/2)

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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} \[ -\frac {128 b^3 \sqrt {a+b x} (10 A b-11 a B)}{3465 a^5 x^{3/2}}+\frac {32 b^2 \sqrt {a+b x} (10 A b-11 a B)}{1155 a^4 x^{5/2}}+\frac {256 b^4 \sqrt {a+b x} (10 A b-11 a B)}{3465 a^6 \sqrt {x}}-\frac {16 b \sqrt {a+b x} (10 A b-11 a B)}{693 a^3 x^{7/2}}+\frac {2 \sqrt {a+b x} (10 A b-11 a B)}{99 a^2 x^{9/2}}-\frac {2 A \sqrt {a+b x}}{11 a x^{11/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*A*Sqrt[a + b*x])/(11*a*x^(11/2)) + (2*(10*A*b - 11*a*B)*Sqrt[a + b*x])/(99*a^2*x^(9/2)) - (16*b*(10*A*b -
11*a*B)*Sqrt[a + b*x])/(693*a^3*x^(7/2)) + (32*b^2*(10*A*b - 11*a*B)*Sqrt[a + b*x])/(1155*a^4*x^(5/2)) - (128*
b^3*(10*A*b - 11*a*B)*Sqrt[a + b*x])/(3465*a^5*x^(3/2)) + (256*b^4*(10*A*b - 11*a*B)*Sqrt[a + b*x])/(3465*a^6*
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} \sqrt {a+b x}} \, dx &=-\frac {2 A \sqrt {a+b x}}{11 a x^{11/2}}+\frac {\left (2 \left (-5 A b+\frac {11 a B}{2}\right )\right ) \int \frac {1}{x^{11/2} \sqrt {a+b x}} \, dx}{11 a}\\ &=-\frac {2 A \sqrt {a+b x}}{11 a x^{11/2}}+\frac {2 (10 A b-11 a B) \sqrt {a+b x}}{99 a^2 x^{9/2}}+\frac {(8 b (10 A b-11 a B)) \int \frac {1}{x^{9/2} \sqrt {a+b x}} \, dx}{99 a^2}\\ &=-\frac {2 A \sqrt {a+b x}}{11 a x^{11/2}}+\frac {2 (10 A b-11 a B) \sqrt {a+b x}}{99 a^2 x^{9/2}}-\frac {16 b (10 A b-11 a B) \sqrt {a+b x}}{693 a^3 x^{7/2}}-\frac {\left (16 b^2 (10 A b-11 a B)\right ) \int \frac {1}{x^{7/2} \sqrt {a+b x}} \, dx}{231 a^3}\\ &=-\frac {2 A \sqrt {a+b x}}{11 a x^{11/2}}+\frac {2 (10 A b-11 a B) \sqrt {a+b x}}{99 a^2 x^{9/2}}-\frac {16 b (10 A b-11 a B) \sqrt {a+b x}}{693 a^3 x^{7/2}}+\frac {32 b^2 (10 A b-11 a B) \sqrt {a+b x}}{1155 a^4 x^{5/2}}+\frac {\left (64 b^3 (10 A b-11 a B)\right ) \int \frac {1}{x^{5/2} \sqrt {a+b x}} \, dx}{1155 a^4}\\ &=-\frac {2 A \sqrt {a+b x}}{11 a x^{11/2}}+\frac {2 (10 A b-11 a B) \sqrt {a+b x}}{99 a^2 x^{9/2}}-\frac {16 b (10 A b-11 a B) \sqrt {a+b x}}{693 a^3 x^{7/2}}+\frac {32 b^2 (10 A b-11 a B) \sqrt {a+b x}}{1155 a^4 x^{5/2}}-\frac {128 b^3 (10 A b-11 a B) \sqrt {a+b x}}{3465 a^5 x^{3/2}}-\frac {\left (128 b^4 (10 A b-11 a B)\right ) \int \frac {1}{x^{3/2} \sqrt {a+b x}} \, dx}{3465 a^5}\\ &=-\frac {2 A \sqrt {a+b x}}{11 a x^{11/2}}+\frac {2 (10 A b-11 a B) \sqrt {a+b x}}{99 a^2 x^{9/2}}-\frac {16 b (10 A b-11 a B) \sqrt {a+b x}}{693 a^3 x^{7/2}}+\frac {32 b^2 (10 A b-11 a B) \sqrt {a+b x}}{1155 a^4 x^{5/2}}-\frac {128 b^3 (10 A b-11 a B) \sqrt {a+b x}}{3465 a^5 x^{3/2}}+\frac {256 b^4 (10 A b-11 a B) \sqrt {a+b x}}{3465 a^6 \sqrt {x}}\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 114, normalized size = 0.62 \[ -\frac {2 \sqrt {a+b x} \left (35 a^5 (9 A+11 B x)-10 a^4 b x (35 A+44 B x)+16 a^3 b^2 x^2 (25 A+33 B x)-32 a^2 b^3 x^3 (15 A+22 B x)+128 a b^4 x^4 (5 A+11 B x)-1280 A b^5 x^5\right )}{3465 a^6 x^{11/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[a + b*x]*(-1280*A*b^5*x^5 + 128*a*b^4*x^4*(5*A + 11*B*x) + 35*a^5*(9*A + 11*B*x) - 32*a^2*b^3*x^3*(15
*A + 22*B*x) + 16*a^3*b^2*x^2*(25*A + 33*B*x) - 10*a^4*b*x*(35*A + 44*B*x)))/(3465*a^6*x^(11/2))

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fricas [A]  time = 0.67, size = 126, normalized size = 0.69 \[ -\frac {2 \, {\left (315 \, A a^{5} + 128 \, {\left (11 \, B a b^{4} - 10 \, A b^{5}\right )} x^{5} - 64 \, {\left (11 \, B a^{2} b^{3} - 10 \, A a b^{4}\right )} x^{4} + 48 \, {\left (11 \, B a^{3} b^{2} - 10 \, A a^{2} b^{3}\right )} x^{3} - 40 \, {\left (11 \, B a^{4} b - 10 \, A a^{3} b^{2}\right )} x^{2} + 35 \, {\left (11 \, B a^{5} - 10 \, A a^{4} b\right )} x\right )} \sqrt {b x + a}}{3465 \, a^{6} x^{\frac {11}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/x^(13/2)/(b*x+a)^(1/2),x, algorithm="fricas")

[Out]

-2/3465*(315*A*a^5 + 128*(11*B*a*b^4 - 10*A*b^5)*x^5 - 64*(11*B*a^2*b^3 - 10*A*a*b^4)*x^4 + 48*(11*B*a^3*b^2 -
10*A*a^2*b^3)*x^3 - 40*(11*B*a^4*b - 10*A*a^3*b^2)*x^2 + 35*(11*B*a^5 - 10*A*a^4*b)*x)*sqrt(b*x + a)/(a^6*x^(
11/2))

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giac [A]  time = 1.53, size = 201, normalized size = 1.10 \[ -\frac {2 \, {\left ({\left (8 \, {\left (2 \, {\left (b x + a\right )} {\left (4 \, {\left (b x + a\right )} {\left (\frac {2 \, {\left (11 \, B a b^{10} - 10 \, A b^{11}\right )} {\left (b x + a\right )}}{a^{6}} - \frac {11 \, {\left (11 \, B a^{2} b^{10} - 10 \, A a b^{11}\right )}}{a^{6}}\right )} + \frac {99 \, {\left (11 \, B a^{3} b^{10} - 10 \, A a^{2} b^{11}\right )}}{a^{6}}\right )} - \frac {231 \, {\left (11 \, B a^{4} b^{10} - 10 \, A a^{3} b^{11}\right )}}{a^{6}}\right )} {\left (b x + a\right )} + \frac {1155 \, {\left (11 \, B a^{5} b^{10} - 10 \, A a^{4} b^{11}\right )}}{a^{6}}\right )} {\left (b x + a\right )} - \frac {3465 \, {\left (B a^{6} b^{10} - A a^{5} b^{11}\right )}}{a^{6}}\right )} \sqrt {b x + a} b}{3465 \, {\left ({\left (b x + a\right )} b - a b\right )}^{\frac {11}{2}} {\left | b \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/x^(13/2)/(b*x+a)^(1/2),x, algorithm="giac")

[Out]

-2/3465*((8*(2*(b*x + a)*(4*(b*x + a)*(2*(11*B*a*b^10 - 10*A*b^11)*(b*x + a)/a^6 - 11*(11*B*a^2*b^10 - 10*A*a*
b^11)/a^6) + 99*(11*B*a^3*b^10 - 10*A*a^2*b^11)/a^6) - 231*(11*B*a^4*b^10 - 10*A*a^3*b^11)/a^6)*(b*x + a) + 11
55*(11*B*a^5*b^10 - 10*A*a^4*b^11)/a^6)*(b*x + a) - 3465*(B*a^6*b^10 - A*a^5*b^11)/a^6)*sqrt(b*x + a)*b/(((b*x
+ a)*b - a*b)^(11/2)*abs(b))

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maple [A]  time = 0.01, size = 125, normalized size = 0.68 \[ -\frac {2 \sqrt {b x +a}\, \left (-1280 A \,b^{5} x^{5}+1408 B a \,b^{4} x^{5}+640 A a \,b^{4} x^{4}-704 B \,a^{2} b^{3} x^{4}-480 A \,a^{2} b^{3} x^{3}+528 B \,a^{3} b^{2} x^{3}+400 A \,a^{3} b^{2} x^{2}-440 B \,a^{4} b \,x^{2}-350 A \,a^{4} b x +385 B \,a^{5} x +315 A \,a^{5}\right )}{3465 a^{6} x^{\frac {11}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)/x^(13/2)/(b*x+a)^(1/2),x)

[Out]

-2/3465*(b*x+a)^(1/2)*(-1280*A*b^5*x^5+1408*B*a*b^4*x^5+640*A*a*b^4*x^4-704*B*a^2*b^3*x^4-480*A*a^2*b^3*x^3+52
8*B*a^3*b^2*x^3+400*A*a^3*b^2*x^2-440*B*a^4*b*x^2-350*A*a^4*b*x+385*B*a^5*x+315*A*a^5)/x^(11/2)/a^6

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maxima [A]  time = 0.83, size = 244, normalized size = 1.33 \[ -\frac {256 \, \sqrt {b x^{2} + a x} B b^{4}}{315 \, a^{5} x} + \frac {512 \, \sqrt {b x^{2} + a x} A b^{5}}{693 \, a^{6} x} + \frac {128 \, \sqrt {b x^{2} + a x} B b^{3}}{315 \, a^{4} x^{2}} - \frac {256 \, \sqrt {b x^{2} + a x} A b^{4}}{693 \, a^{5} x^{2}} - \frac {32 \, \sqrt {b x^{2} + a x} B b^{2}}{105 \, a^{3} x^{3}} + \frac {64 \, \sqrt {b x^{2} + a x} A b^{3}}{231 \, a^{4} x^{3}} + \frac {16 \, \sqrt {b x^{2} + a x} B b}{63 \, a^{2} x^{4}} - \frac {160 \, \sqrt {b x^{2} + a x} A b^{2}}{693 \, a^{3} x^{4}} - \frac {2 \, \sqrt {b x^{2} + a x} B}{9 \, a x^{5}} + \frac {20 \, \sqrt {b x^{2} + a x} A b}{99 \, a^{2} x^{5}} - \frac {2 \, \sqrt {b x^{2} + a x} A}{11 \, a x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/x^(13/2)/(b*x+a)^(1/2),x, algorithm="maxima")

[Out]

-256/315*sqrt(b*x^2 + a*x)*B*b^4/(a^5*x) + 512/693*sqrt(b*x^2 + a*x)*A*b^5/(a^6*x) + 128/315*sqrt(b*x^2 + a*x)
*B*b^3/(a^4*x^2) - 256/693*sqrt(b*x^2 + a*x)*A*b^4/(a^5*x^2) - 32/105*sqrt(b*x^2 + a*x)*B*b^2/(a^3*x^3) + 64/2
31*sqrt(b*x^2 + a*x)*A*b^3/(a^4*x^3) + 16/63*sqrt(b*x^2 + a*x)*B*b/(a^2*x^4) - 160/693*sqrt(b*x^2 + a*x)*A*b^2
/(a^3*x^4) - 2/9*sqrt(b*x^2 + a*x)*B/(a*x^5) + 20/99*sqrt(b*x^2 + a*x)*A*b/(a^2*x^5) - 2/11*sqrt(b*x^2 + a*x)*
A/(a*x^6)

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mupad [B]  time = 1.05, size = 117, normalized size = 0.64 \[ -\frac {\sqrt {a+b\,x}\,\left (\frac {2\,A}{11\,a}+\frac {x\,\left (770\,B\,a^5-700\,A\,a^4\,b\right )}{3465\,a^6}-\frac {32\,b^2\,x^3\,\left (10\,A\,b-11\,B\,a\right )}{1155\,a^4}+\frac {128\,b^3\,x^4\,\left (10\,A\,b-11\,B\,a\right )}{3465\,a^5}-\frac {256\,b^4\,x^5\,\left (10\,A\,b-11\,B\,a\right )}{3465\,a^6}+\frac {16\,b\,x^2\,\left (10\,A\,b-11\,B\,a\right )}{693\,a^3}\right )}{x^{11/2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*x)/(x^(13/2)*(a + b*x)^(1/2)),x)

[Out]

-((a + b*x)^(1/2)*((2*A)/(11*a) + (x*(770*B*a^5 - 700*A*a^4*b))/(3465*a^6) - (32*b^2*x^3*(10*A*b - 11*B*a))/(1
155*a^4) + (128*b^3*x^4*(10*A*b - 11*B*a))/(3465*a^5) - (256*b^4*x^5*(10*A*b - 11*B*a))/(3465*a^6) + (16*b*x^2
*(10*A*b - 11*B*a))/(693*a^3)))/x^(11/2)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/x**(13/2)/(b*x+a)**(1/2),x)

[Out]

Timed out

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