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

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

[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 = 0.267385, 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 \[ -\frac{512 b^5 \sqrt{a+b x} (12 A b-13 a B)}{9009 a^7 \sqrt{x}}+\frac{256 b^4 \sqrt{a+b x} (12 A b-13 a B)}{9009 a^6 x^{3/2}}-\frac{64 b^3 \sqrt{a+b x} (12 A b-13 a B)}{3003 a^5 x^{5/2}}+\frac{160 b^2 \sqrt{a+b x} (12 A b-13 a B)}{9009 a^4 x^{7/2}}-\frac{20 b \sqrt{a+b x} (12 A b-13 a B)}{1287 a^3 x^{9/2}}+\frac{2 \sqrt{a+b x} (12 A b-13 a B)}{143 a^2 x^{11/2}}-\frac{2 A \sqrt{a+b x}}{13 a x^{13/2}} \]

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])

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Rubi in Sympy [A]  time = 26.0808, size = 218, normalized size = 1.01 \[ - \frac{2 A \sqrt{a + b x}}{13 a x^{\frac{13}{2}}} + \frac{2 \sqrt{a + b x} \left (12 A b - 13 B a\right )}{143 a^{2} x^{\frac{11}{2}}} - \frac{20 b \sqrt{a + b x} \left (12 A b - 13 B a\right )}{1287 a^{3} x^{\frac{9}{2}}} + \frac{160 b^{2} \sqrt{a + b x} \left (12 A b - 13 B a\right )}{9009 a^{4} x^{\frac{7}{2}}} - \frac{64 b^{3} \sqrt{a + b x} \left (12 A b - 13 B a\right )}{3003 a^{5} x^{\frac{5}{2}}} + \frac{256 b^{4} \sqrt{a + b x} \left (12 A b - 13 B a\right )}{9009 a^{6} x^{\frac{3}{2}}} - \frac{512 b^{5} \sqrt{a + b x} \left (12 A b - 13 B a\right )}{9009 a^{7} \sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)/x**(15/2)/(b*x+a)**(1/2),x)

[Out]

-2*A*sqrt(a + b*x)/(13*a*x**(13/2)) + 2*sqrt(a + b*x)*(12*A*b - 13*B*a)/(143*a**
2*x**(11/2)) - 20*b*sqrt(a + b*x)*(12*A*b - 13*B*a)/(1287*a**3*x**(9/2)) + 160*b
**2*sqrt(a + b*x)*(12*A*b - 13*B*a)/(9009*a**4*x**(7/2)) - 64*b**3*sqrt(a + b*x)
*(12*A*b - 13*B*a)/(3003*a**5*x**(5/2)) + 256*b**4*sqrt(a + b*x)*(12*A*b - 13*B*
a)/(9009*a**6*x**(3/2)) - 512*b**5*sqrt(a + b*x)*(12*A*b - 13*B*a)/(9009*a**7*sq
rt(x))

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Mathematica [A]  time = 0.128731, size = 133, normalized size = 0.62 \[ -\frac{2 \sqrt{a+b x} \left (63 a^6 (11 A+13 B x)-14 a^5 b x (54 A+65 B x)+40 a^4 b^2 x^2 (21 A+26 B x)-96 a^3 b^3 x^3 (10 A+13 B x)+128 a^2 b^4 x^4 (9 A+13 B x)-256 a b^5 x^5 (6 A+13 B x)+3072 A b^6 x^6\right )}{9009 a^7 x^{13/2}} \]

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 = 0.01, size = 149, normalized size = 0.7 \[ -{\frac{6144\,A{b}^{6}{x}^{6}-6656\,Ba{b}^{5}{x}^{6}-3072\,Aa{b}^{5}{x}^{5}+3328\,B{a}^{2}{b}^{4}{x}^{5}+2304\,A{a}^{2}{b}^{4}{x}^{4}-2496\,B{a}^{3}{b}^{3}{x}^{4}-1920\,A{a}^{3}{b}^{3}{x}^{3}+2080\,B{a}^{4}{b}^{2}{x}^{3}+1680\,A{a}^{4}{b}^{2}{x}^{2}-1820\,B{a}^{5}b{x}^{2}-1512\,A{a}^{5}bx+1638\,B{a}^{6}x+1386\,A{a}^{6}}{9009\,{a}^{7}}\sqrt{bx+a}{x}^{-{\frac{13}{2}}}} \]

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]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/(sqrt(b*x + a)*x^(15/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.233347, size = 203, normalized size = 0.94 \[ -\frac{2 \,{\left (693 \, A a^{6} - 256 \,{\left (13 \, B a b^{5} - 12 \, A b^{6}\right )} x^{6} + 128 \,{\left (13 \, B a^{2} b^{4} - 12 \, A a b^{5}\right )} x^{5} - 96 \,{\left (13 \, B a^{3} b^{3} - 12 \, A a^{2} b^{4}\right )} x^{4} + 80 \,{\left (13 \, B a^{4} b^{2} - 12 \, A a^{3} b^{3}\right )} x^{3} - 70 \,{\left (13 \, B a^{5} b - 12 \, A a^{4} b^{2}\right )} x^{2} + 63 \,{\left (13 \, B a^{6} - 12 \, A a^{5} b\right )} x\right )} \sqrt{b x + a}}{9009 \, a^{7} x^{\frac{13}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/(sqrt(b*x + a)*x^(15/2)),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)*sq
rt(b*x + a)/(a^7*x^(13/2))

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

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/XCAS [A]  time = 0.236436, size = 343, normalized size = 1.59 \[ -\frac{{\left ({\left (2 \,{\left (8 \,{\left (2 \,{\left (b x + a\right )}{\left (4 \,{\left (b x + a\right )}{\left (\frac{2 \,{\left (13 \, B a b^{12} - 12 \, A b^{13}\right )}{\left (b x + a\right )}}{a^{7} b^{21}} - \frac{13 \,{\left (13 \, B a^{2} b^{12} - 12 \, A a b^{13}\right )}}{a^{7} b^{21}}\right )} + \frac{143 \,{\left (13 \, B a^{3} b^{12} - 12 \, A a^{2} b^{13}\right )}}{a^{7} b^{21}}\right )} - \frac{429 \,{\left (13 \, B a^{4} b^{12} - 12 \, A a^{3} b^{13}\right )}}{a^{7} b^{21}}\right )}{\left (b x + a\right )} + \frac{3003 \,{\left (13 \, B a^{5} b^{12} - 12 \, A a^{4} b^{13}\right )}}{a^{7} b^{21}}\right )}{\left (b x + a\right )} - \frac{3003 \,{\left (13 \, B a^{6} b^{12} - 12 \, A a^{5} b^{13}\right )}}{a^{7} b^{21}}\right )}{\left (b x + a\right )} + \frac{9009 \,{\left (B a^{7} b^{12} - A a^{6} b^{13}\right )}}{a^{7} b^{21}}\right )} \sqrt{b x + a} b}{6642155520 \,{\left ({\left (b x + a\right )} b - a b\right )}^{\frac{13}{2}}{\left | b \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/(sqrt(b*x + a)*x^(15/2)),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^2
1))*(b*x + a) + 3003*(13*B*a^5*b^12 - 12*A*a^4*b^13)/(a^7*b^21))*(b*x + a) - 300
3*(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))