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

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

[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])

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Rubi [A]  time = 0.090962, antiderivative size = 216, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {78, 45, 37} \[ \frac{256 b^4 \sqrt{a+b x} (12 A b-13 a B)}{9009 a^6 x^{3/2}}-\frac{64 b^3 \sqrt{a+b x} (12 A b-13 a B)}{3003 a^5 x^{5/2}}+\frac{160 b^2 \sqrt{a+b x} (12 A b-13 a B)}{9009 a^4 x^{7/2}}-\frac{512 b^5 \sqrt{a+b x} (12 A b-13 a B)}{9009 a^7 \sqrt{x}}-\frac{20 b \sqrt{a+b x} (12 A b-13 a B)}{1287 a^3 x^{9/2}}+\frac{2 \sqrt{a+b x} (12 A b-13 a B)}{143 a^2 x^{11/2}}-\frac{2 A \sqrt{a+b x}}{13 a x^{13/2}} \]

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 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]))))

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 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]

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*}

Mathematica [A]  time = 0.0395407, size = 133, normalized size = 0.62 \[ -\frac{2 \sqrt{a+b x} \left (40 a^4 b^2 x^2 (21 A+26 B x)-96 a^3 b^3 x^3 (10 A+13 B x)+128 a^2 b^4 x^4 (9 A+13 B x)-14 a^5 b x (54 A+65 B x)+63 a^6 (11 A+13 B x)-256 a b^5 x^5 (6 A+13 B x)+3072 A b^6 x^6\right )}{9009 a^7 x^{13/2}} \]

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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Maple [A]  time = 0.005, size = 149, normalized size = 0.7 \begin{align*} -{\frac{6144\,A{b}^{6}{x}^{6}-6656\,Ba{b}^{5}{x}^{6}-3072\,Aa{b}^{5}{x}^{5}+3328\,B{a}^{2}{b}^{4}{x}^{5}+2304\,A{a}^{2}{b}^{4}{x}^{4}-2496\,B{a}^{3}{b}^{3}{x}^{4}-1920\,A{a}^{3}{b}^{3}{x}^{3}+2080\,B{a}^{4}{b}^{2}{x}^{3}+1680\,A{a}^{4}{b}^{2}{x}^{2}-1820\,B{a}^{5}b{x}^{2}-1512\,A{a}^{5}bx+1638\,B{a}^{6}x+1386\,A{a}^{6}}{9009\,{a}^{7}}\sqrt{bx+a}{x}^{-{\frac{13}{2}}}} \end{align*}

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 [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

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]

Exception raised: ValueError

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Fricas [A]  time = 2.58623, size = 362, normalized size = 1.68 \begin{align*} -\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{align*}

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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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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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Giac [A]  time = 1.87554, size = 343, normalized size = 1.59 \begin{align*} -\frac{{\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} b^{21}} - \frac{13 \,{\left (13 \, B a^{2} b^{12} - 12 \, A a b^{13}\right )}}{a^{7} b^{21}}\right )} + \frac{143 \,{\left (13 \, B a^{3} b^{12} - 12 \, A a^{2} b^{13}\right )}}{a^{7} b^{21}}\right )} - \frac{429 \,{\left (13 \, B a^{4} b^{12} - 12 \, A a^{3} b^{13}\right )}}{a^{7} b^{21}}\right )}{\left (b x + a\right )} + \frac{3003 \,{\left (13 \, B a^{5} b^{12} - 12 \, A a^{4} b^{13}\right )}}{a^{7} b^{21}}\right )}{\left (b x + a\right )} - \frac{3003 \,{\left (13 \, B a^{6} b^{12} - 12 \, A a^{5} b^{13}\right )}}{a^{7} b^{21}}\right )}{\left (b x + a\right )} + \frac{9009 \,{\left (B a^{7} b^{12} - A a^{6} b^{13}\right )}}{a^{7} b^{21}}\right )} \sqrt{b x + a} b}{6642155520 \,{\left ({\left (b x + a\right )} b - a b\right )}^{\frac{13}{2}}{\left | b \right |}} \end{align*}

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]

-1/6642155520*((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*b^21) - 13*(13*B*a^
2*b^12 - 12*A*a*b^13)/(a^7*b^21)) + 143*(13*B*a^3*b^12 - 12*A*a^2*b^13)/(a^7*b^21)) - 429*(13*B*a^4*b^12 - 12*
A*a^3*b^13)/(a^7*b^21))*(b*x + a) + 3003*(13*B*a^5*b^12 - 12*A*a^4*b^13)/(a^7*b^21))*(b*x + a) - 3003*(13*B*a^
6*b^12 - 12*A*a^5*b^13)/(a^7*b^21))*(b*x + a) + 9009*(B*a^7*b^12 - A*a^6*b^13)/(a^7*b^21))*sqrt(b*x + a)*b/(((
b*x + a)*b - a*b)^(13/2)*abs(b))