3.1470 \(\int \frac{\sqrt{c+d x}}{(a+b x)^{13/2}} \, dx\)

Optimal. Leaf size=171 \[ -\frac{256 d^4 (c+d x)^{3/2}}{3465 (a+b x)^{3/2} (b c-a d)^5}+\frac{128 d^3 (c+d x)^{3/2}}{1155 (a+b x)^{5/2} (b c-a d)^4}-\frac{32 d^2 (c+d x)^{3/2}}{231 (a+b x)^{7/2} (b c-a d)^3}+\frac{16 d (c+d x)^{3/2}}{99 (a+b x)^{9/2} (b c-a d)^2}-\frac{2 (c+d x)^{3/2}}{11 (a+b x)^{11/2} (b c-a d)} \]

[Out]

(-2*(c + d*x)^(3/2))/(11*(b*c - a*d)*(a + b*x)^(11/2)) + (16*d*(c + d*x)^(3/2))/
(99*(b*c - a*d)^2*(a + b*x)^(9/2)) - (32*d^2*(c + d*x)^(3/2))/(231*(b*c - a*d)^3
*(a + b*x)^(7/2)) + (128*d^3*(c + d*x)^(3/2))/(1155*(b*c - a*d)^4*(a + b*x)^(5/2
)) - (256*d^4*(c + d*x)^(3/2))/(3465*(b*c - a*d)^5*(a + b*x)^(3/2))

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Rubi [A]  time = 0.158549, antiderivative size = 171, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ -\frac{256 d^4 (c+d x)^{3/2}}{3465 (a+b x)^{3/2} (b c-a d)^5}+\frac{128 d^3 (c+d x)^{3/2}}{1155 (a+b x)^{5/2} (b c-a d)^4}-\frac{32 d^2 (c+d x)^{3/2}}{231 (a+b x)^{7/2} (b c-a d)^3}+\frac{16 d (c+d x)^{3/2}}{99 (a+b x)^{9/2} (b c-a d)^2}-\frac{2 (c+d x)^{3/2}}{11 (a+b x)^{11/2} (b c-a d)} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[c + d*x]/(a + b*x)^(13/2),x]

[Out]

(-2*(c + d*x)^(3/2))/(11*(b*c - a*d)*(a + b*x)^(11/2)) + (16*d*(c + d*x)^(3/2))/
(99*(b*c - a*d)^2*(a + b*x)^(9/2)) - (32*d^2*(c + d*x)^(3/2))/(231*(b*c - a*d)^3
*(a + b*x)^(7/2)) + (128*d^3*(c + d*x)^(3/2))/(1155*(b*c - a*d)^4*(a + b*x)^(5/2
)) - (256*d^4*(c + d*x)^(3/2))/(3465*(b*c - a*d)^5*(a + b*x)^(3/2))

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Rubi in Sympy [A]  time = 30.5674, size = 153, normalized size = 0.89 \[ \frac{256 d^{4} \left (c + d x\right )^{\frac{3}{2}}}{3465 \left (a + b x\right )^{\frac{3}{2}} \left (a d - b c\right )^{5}} + \frac{128 d^{3} \left (c + d x\right )^{\frac{3}{2}}}{1155 \left (a + b x\right )^{\frac{5}{2}} \left (a d - b c\right )^{4}} + \frac{32 d^{2} \left (c + d x\right )^{\frac{3}{2}}}{231 \left (a + b x\right )^{\frac{7}{2}} \left (a d - b c\right )^{3}} + \frac{16 d \left (c + d x\right )^{\frac{3}{2}}}{99 \left (a + b x\right )^{\frac{9}{2}} \left (a d - b c\right )^{2}} + \frac{2 \left (c + d x\right )^{\frac{3}{2}}}{11 \left (a + b x\right )^{\frac{11}{2}} \left (a d - b c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x+c)**(1/2)/(b*x+a)**(13/2),x)

[Out]

256*d**4*(c + d*x)**(3/2)/(3465*(a + b*x)**(3/2)*(a*d - b*c)**5) + 128*d**3*(c +
d*x)**(3/2)/(1155*(a + b*x)**(5/2)*(a*d - b*c)**4) + 32*d**2*(c + d*x)**(3/2)/(
231*(a + b*x)**(7/2)*(a*d - b*c)**3) + 16*d*(c + d*x)**(3/2)/(99*(a + b*x)**(9/2
)*(a*d - b*c)**2) + 2*(c + d*x)**(3/2)/(11*(a + b*x)**(11/2)*(a*d - b*c))

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Mathematica [A]  time = 0.220419, size = 167, normalized size = 0.98 \[ \sqrt{a+b x} \sqrt{c+d x} \left (-\frac{256 d^5}{3465 b (a+b x) (b c-a d)^5}+\frac{128 d^4}{3465 b (a+b x)^2 (b c-a d)^4}-\frac{32 d^3}{1155 b (a+b x)^3 (b c-a d)^3}+\frac{16 d^2}{693 b (a+b x)^4 (b c-a d)^2}-\frac{2 d}{99 b (a+b x)^5 (b c-a d)}-\frac{2}{11 b (a+b x)^6}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[c + d*x]/(a + b*x)^(13/2),x]

[Out]

Sqrt[a + b*x]*Sqrt[c + d*x]*(-2/(11*b*(a + b*x)^6) - (2*d)/(99*b*(b*c - a*d)*(a
+ b*x)^5) + (16*d^2)/(693*b*(b*c - a*d)^2*(a + b*x)^4) - (32*d^3)/(1155*b*(b*c -
a*d)^3*(a + b*x)^3) + (128*d^4)/(3465*b*(b*c - a*d)^4*(a + b*x)^2) - (256*d^5)/
(3465*b*(b*c - a*d)^5*(a + b*x)))

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Maple [A]  time = 0.017, size = 256, normalized size = 1.5 \[{\frac{256\,{b}^{4}{d}^{4}{x}^{4}+1408\,a{b}^{3}{d}^{4}{x}^{3}-384\,{b}^{4}c{d}^{3}{x}^{3}+3168\,{a}^{2}{b}^{2}{d}^{4}{x}^{2}-2112\,a{b}^{3}c{d}^{3}{x}^{2}+480\,{b}^{4}{c}^{2}{d}^{2}{x}^{2}+3696\,{a}^{3}b{d}^{4}x-4752\,{a}^{2}{b}^{2}c{d}^{3}x+2640\,a{b}^{3}{c}^{2}{d}^{2}x-560\,{b}^{4}{c}^{3}dx+2310\,{a}^{4}{d}^{4}-5544\,{a}^{3}bc{d}^{3}+5940\,{a}^{2}{b}^{2}{c}^{2}{d}^{2}-3080\,a{b}^{3}{c}^{3}d+630\,{b}^{4}{c}^{4}}{3465\,{a}^{5}{d}^{5}-17325\,{a}^{4}bc{d}^{4}+34650\,{a}^{3}{b}^{2}{c}^{2}{d}^{3}-34650\,{a}^{2}{b}^{3}{c}^{3}{d}^{2}+17325\,a{b}^{4}{c}^{4}d-3465\,{b}^{5}{c}^{5}} \left ( dx+c \right ) ^{{\frac{3}{2}}} \left ( bx+a \right ) ^{-{\frac{11}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x+c)^(1/2)/(b*x+a)^(13/2),x)

[Out]

2/3465*(d*x+c)^(3/2)*(128*b^4*d^4*x^4+704*a*b^3*d^4*x^3-192*b^4*c*d^3*x^3+1584*a
^2*b^2*d^4*x^2-1056*a*b^3*c*d^3*x^2+240*b^4*c^2*d^2*x^2+1848*a^3*b*d^4*x-2376*a^
2*b^2*c*d^3*x+1320*a*b^3*c^2*d^2*x-280*b^4*c^3*d*x+1155*a^4*d^4-2772*a^3*b*c*d^3
+2970*a^2*b^2*c^2*d^2-1540*a*b^3*c^3*d+315*b^4*c^4)/(b*x+a)^(11/2)/(a^5*d^5-5*a^
4*b*c*d^4+10*a^3*b^2*c^2*d^3-10*a^2*b^3*c^3*d^2+5*a*b^4*c^4*d-b^5*c^5)

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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(sqrt(d*x + c)/(b*x + a)^(13/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 5.53639, size = 1054, normalized size = 6.16 \[ -\frac{2 \,{\left (128 \, b^{4} d^{5} x^{5} + 315 \, b^{4} c^{5} - 1540 \, a b^{3} c^{4} d + 2970 \, a^{2} b^{2} c^{3} d^{2} - 2772 \, a^{3} b c^{2} d^{3} + 1155 \, a^{4} c d^{4} - 64 \,{\left (b^{4} c d^{4} - 11 \, a b^{3} d^{5}\right )} x^{4} + 16 \,{\left (3 \, b^{4} c^{2} d^{3} - 22 \, a b^{3} c d^{4} + 99 \, a^{2} b^{2} d^{5}\right )} x^{3} - 8 \,{\left (5 \, b^{4} c^{3} d^{2} - 33 \, a b^{3} c^{2} d^{3} + 99 \, a^{2} b^{2} c d^{4} - 231 \, a^{3} b d^{5}\right )} x^{2} +{\left (35 \, b^{4} c^{4} d - 220 \, a b^{3} c^{3} d^{2} + 594 \, a^{2} b^{2} c^{2} d^{3} - 924 \, a^{3} b c d^{4} + 1155 \, a^{4} d^{5}\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{3465 \,{\left (a^{6} b^{5} c^{5} - 5 \, a^{7} b^{4} c^{4} d + 10 \, a^{8} b^{3} c^{3} d^{2} - 10 \, a^{9} b^{2} c^{2} d^{3} + 5 \, a^{10} b c d^{4} - a^{11} d^{5} +{\left (b^{11} c^{5} - 5 \, a b^{10} c^{4} d + 10 \, a^{2} b^{9} c^{3} d^{2} - 10 \, a^{3} b^{8} c^{2} d^{3} + 5 \, a^{4} b^{7} c d^{4} - a^{5} b^{6} d^{5}\right )} x^{6} + 6 \,{\left (a b^{10} c^{5} - 5 \, a^{2} b^{9} c^{4} d + 10 \, a^{3} b^{8} c^{3} d^{2} - 10 \, a^{4} b^{7} c^{2} d^{3} + 5 \, a^{5} b^{6} c d^{4} - a^{6} b^{5} d^{5}\right )} x^{5} + 15 \,{\left (a^{2} b^{9} c^{5} - 5 \, a^{3} b^{8} c^{4} d + 10 \, a^{4} b^{7} c^{3} d^{2} - 10 \, a^{5} b^{6} c^{2} d^{3} + 5 \, a^{6} b^{5} c d^{4} - a^{7} b^{4} d^{5}\right )} x^{4} + 20 \,{\left (a^{3} b^{8} c^{5} - 5 \, a^{4} b^{7} c^{4} d + 10 \, a^{5} b^{6} c^{3} d^{2} - 10 \, a^{6} b^{5} c^{2} d^{3} + 5 \, a^{7} b^{4} c d^{4} - a^{8} b^{3} d^{5}\right )} x^{3} + 15 \,{\left (a^{4} b^{7} c^{5} - 5 \, a^{5} b^{6} c^{4} d + 10 \, a^{6} b^{5} c^{3} d^{2} - 10 \, a^{7} b^{4} c^{2} d^{3} + 5 \, a^{8} b^{3} c d^{4} - a^{9} b^{2} d^{5}\right )} x^{2} + 6 \,{\left (a^{5} b^{6} c^{5} - 5 \, a^{6} b^{5} c^{4} d + 10 \, a^{7} b^{4} c^{3} d^{2} - 10 \, a^{8} b^{3} c^{2} d^{3} + 5 \, a^{9} b^{2} c d^{4} - a^{10} b d^{5}\right )} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(d*x + c)/(b*x + a)^(13/2),x, algorithm="fricas")

[Out]

-2/3465*(128*b^4*d^5*x^5 + 315*b^4*c^5 - 1540*a*b^3*c^4*d + 2970*a^2*b^2*c^3*d^2
- 2772*a^3*b*c^2*d^3 + 1155*a^4*c*d^4 - 64*(b^4*c*d^4 - 11*a*b^3*d^5)*x^4 + 16*
(3*b^4*c^2*d^3 - 22*a*b^3*c*d^4 + 99*a^2*b^2*d^5)*x^3 - 8*(5*b^4*c^3*d^2 - 33*a*
b^3*c^2*d^3 + 99*a^2*b^2*c*d^4 - 231*a^3*b*d^5)*x^2 + (35*b^4*c^4*d - 220*a*b^3*
c^3*d^2 + 594*a^2*b^2*c^2*d^3 - 924*a^3*b*c*d^4 + 1155*a^4*d^5)*x)*sqrt(b*x + a)
*sqrt(d*x + c)/(a^6*b^5*c^5 - 5*a^7*b^4*c^4*d + 10*a^8*b^3*c^3*d^2 - 10*a^9*b^2*
c^2*d^3 + 5*a^10*b*c*d^4 - a^11*d^5 + (b^11*c^5 - 5*a*b^10*c^4*d + 10*a^2*b^9*c^
3*d^2 - 10*a^3*b^8*c^2*d^3 + 5*a^4*b^7*c*d^4 - a^5*b^6*d^5)*x^6 + 6*(a*b^10*c^5
- 5*a^2*b^9*c^4*d + 10*a^3*b^8*c^3*d^2 - 10*a^4*b^7*c^2*d^3 + 5*a^5*b^6*c*d^4 -
a^6*b^5*d^5)*x^5 + 15*(a^2*b^9*c^5 - 5*a^3*b^8*c^4*d + 10*a^4*b^7*c^3*d^2 - 10*a
^5*b^6*c^2*d^3 + 5*a^6*b^5*c*d^4 - a^7*b^4*d^5)*x^4 + 20*(a^3*b^8*c^5 - 5*a^4*b^
7*c^4*d + 10*a^5*b^6*c^3*d^2 - 10*a^6*b^5*c^2*d^3 + 5*a^7*b^4*c*d^4 - a^8*b^3*d^
5)*x^3 + 15*(a^4*b^7*c^5 - 5*a^5*b^6*c^4*d + 10*a^6*b^5*c^3*d^2 - 10*a^7*b^4*c^2
*d^3 + 5*a^8*b^3*c*d^4 - a^9*b^2*d^5)*x^2 + 6*(a^5*b^6*c^5 - 5*a^6*b^5*c^4*d + 1
0*a^7*b^4*c^3*d^2 - 10*a^8*b^3*c^2*d^3 + 5*a^9*b^2*c*d^4 - a^10*b*d^5)*x)

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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((d*x+c)**(1/2)/(b*x+a)**(13/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.364064, size = 1, normalized size = 0.01 \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(d*x + c)/(b*x + a)^(13/2),x, algorithm="giac")

[Out]

Done