3.1490 \(\int \frac{(c+d x)^{5/2}}{(a+b x)^{13/2}} \, dx\)

Optimal. Leaf size=101 \[ -\frac{16 d^2 (c+d x)^{7/2}}{693 (a+b x)^{7/2} (b c-a d)^3}+\frac{8 d (c+d x)^{7/2}}{99 (a+b x)^{9/2} (b c-a d)^2}-\frac{2 (c+d x)^{7/2}}{11 (a+b x)^{11/2} (b c-a d)} \]

[Out]

(-2*(c + d*x)^(7/2))/(11*(b*c - a*d)*(a + b*x)^(11/2)) + (8*d*(c + d*x)^(7/2))/(
99*(b*c - a*d)^2*(a + b*x)^(9/2)) - (16*d^2*(c + d*x)^(7/2))/(693*(b*c - a*d)^3*
(a + b*x)^(7/2))

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Rubi [A]  time = 0.0795382, antiderivative size = 101, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ -\frac{16 d^2 (c+d x)^{7/2}}{693 (a+b x)^{7/2} (b c-a d)^3}+\frac{8 d (c+d x)^{7/2}}{99 (a+b x)^{9/2} (b c-a d)^2}-\frac{2 (c+d x)^{7/2}}{11 (a+b x)^{11/2} (b c-a d)} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x)^(5/2)/(a + b*x)^(13/2),x]

[Out]

(-2*(c + d*x)^(7/2))/(11*(b*c - a*d)*(a + b*x)^(11/2)) + (8*d*(c + d*x)^(7/2))/(
99*(b*c - a*d)^2*(a + b*x)^(9/2)) - (16*d^2*(c + d*x)^(7/2))/(693*(b*c - a*d)^3*
(a + b*x)^(7/2))

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Rubi in Sympy [A]  time = 14.6113, size = 88, normalized size = 0.87 \[ \frac{16 d^{2} \left (c + d x\right )^{\frac{7}{2}}}{693 \left (a + b x\right )^{\frac{7}{2}} \left (a d - b c\right )^{3}} + \frac{8 d \left (c + d x\right )^{\frac{7}{2}}}{99 \left (a + b x\right )^{\frac{9}{2}} \left (a d - b c\right )^{2}} + \frac{2 \left (c + d x\right )^{\frac{7}{2}}}{11 \left (a + b x\right )^{\frac{11}{2}} \left (a d - b c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x+c)**(5/2)/(b*x+a)**(13/2),x)

[Out]

16*d**2*(c + d*x)**(7/2)/(693*(a + b*x)**(7/2)*(a*d - b*c)**3) + 8*d*(c + d*x)**
(7/2)/(99*(a + b*x)**(9/2)*(a*d - b*c)**2) + 2*(c + d*x)**(7/2)/(11*(a + b*x)**(
11/2)*(a*d - b*c))

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Mathematica [A]  time = 0.217843, size = 77, normalized size = 0.76 \[ \frac{2 (c+d x)^{7/2} \left (99 a^2 d^2+22 a b d (2 d x-7 c)+b^2 \left (63 c^2-28 c d x+8 d^2 x^2\right )\right )}{693 (a+b x)^{11/2} (a d-b c)^3} \]

Antiderivative was successfully verified.

[In]  Integrate[(c + d*x)^(5/2)/(a + b*x)^(13/2),x]

[Out]

(2*(c + d*x)^(7/2)*(99*a^2*d^2 + 22*a*b*d*(-7*c + 2*d*x) + b^2*(63*c^2 - 28*c*d*
x + 8*d^2*x^2)))/(693*(-(b*c) + a*d)^3*(a + b*x)^(11/2))

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Maple [A]  time = 0.01, size = 105, normalized size = 1. \[{\frac{16\,{b}^{2}{d}^{2}{x}^{2}+88\,ab{d}^{2}x-56\,{b}^{2}cdx+198\,{a}^{2}{d}^{2}-308\,abcd+126\,{b}^{2}{c}^{2}}{693\,{a}^{3}{d}^{3}-2079\,{a}^{2}bc{d}^{2}+2079\,a{b}^{2}{c}^{2}d-693\,{b}^{3}{c}^{3}} \left ( dx+c \right ) ^{{\frac{7}{2}}} \left ( bx+a \right ) ^{-{\frac{11}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x+c)^(5/2)/(b*x+a)^(13/2),x)

[Out]

2/693*(d*x+c)^(7/2)*(8*b^2*d^2*x^2+44*a*b*d^2*x-28*b^2*c*d*x+99*a^2*d^2-154*a*b*
c*d+63*b^2*c^2)/(b*x+a)^(11/2)/(a^3*d^3-3*a^2*b*c*d^2+3*a*b^2*c^2*d-b^3*c^3)

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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((d*x + c)^(5/2)/(b*x + a)^(13/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 5.69704, size = 693, normalized size = 6.86 \[ -\frac{2 \,{\left (8 \, b^{2} d^{5} x^{5} + 63 \, b^{2} c^{5} - 154 \, a b c^{4} d + 99 \, a^{2} c^{3} d^{2} - 4 \,{\left (b^{2} c d^{4} - 11 \, a b d^{5}\right )} x^{4} +{\left (3 \, b^{2} c^{2} d^{3} - 22 \, a b c d^{4} + 99 \, a^{2} d^{5}\right )} x^{3} +{\left (113 \, b^{2} c^{3} d^{2} - 330 \, a b c^{2} d^{3} + 297 \, a^{2} c d^{4}\right )} x^{2} +{\left (161 \, b^{2} c^{4} d - 418 \, a b c^{3} d^{2} + 297 \, a^{2} c^{2} d^{3}\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{693 \,{\left (a^{6} b^{3} c^{3} - 3 \, a^{7} b^{2} c^{2} d + 3 \, a^{8} b c d^{2} - a^{9} d^{3} +{\left (b^{9} c^{3} - 3 \, a b^{8} c^{2} d + 3 \, a^{2} b^{7} c d^{2} - a^{3} b^{6} d^{3}\right )} x^{6} + 6 \,{\left (a b^{8} c^{3} - 3 \, a^{2} b^{7} c^{2} d + 3 \, a^{3} b^{6} c d^{2} - a^{4} b^{5} d^{3}\right )} x^{5} + 15 \,{\left (a^{2} b^{7} c^{3} - 3 \, a^{3} b^{6} c^{2} d + 3 \, a^{4} b^{5} c d^{2} - a^{5} b^{4} d^{3}\right )} x^{4} + 20 \,{\left (a^{3} b^{6} c^{3} - 3 \, a^{4} b^{5} c^{2} d + 3 \, a^{5} b^{4} c d^{2} - a^{6} b^{3} d^{3}\right )} x^{3} + 15 \,{\left (a^{4} b^{5} c^{3} - 3 \, a^{5} b^{4} c^{2} d + 3 \, a^{6} b^{3} c d^{2} - a^{7} b^{2} d^{3}\right )} x^{2} + 6 \,{\left (a^{5} b^{4} c^{3} - 3 \, a^{6} b^{3} c^{2} d + 3 \, a^{7} b^{2} c d^{2} - a^{8} b d^{3}\right )} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^(5/2)/(b*x + a)^(13/2),x, algorithm="fricas")

[Out]

-2/693*(8*b^2*d^5*x^5 + 63*b^2*c^5 - 154*a*b*c^4*d + 99*a^2*c^3*d^2 - 4*(b^2*c*d
^4 - 11*a*b*d^5)*x^4 + (3*b^2*c^2*d^3 - 22*a*b*c*d^4 + 99*a^2*d^5)*x^3 + (113*b^
2*c^3*d^2 - 330*a*b*c^2*d^3 + 297*a^2*c*d^4)*x^2 + (161*b^2*c^4*d - 418*a*b*c^3*
d^2 + 297*a^2*c^2*d^3)*x)*sqrt(b*x + a)*sqrt(d*x + c)/(a^6*b^3*c^3 - 3*a^7*b^2*c
^2*d + 3*a^8*b*c*d^2 - a^9*d^3 + (b^9*c^3 - 3*a*b^8*c^2*d + 3*a^2*b^7*c*d^2 - a^
3*b^6*d^3)*x^6 + 6*(a*b^8*c^3 - 3*a^2*b^7*c^2*d + 3*a^3*b^6*c*d^2 - a^4*b^5*d^3)
*x^5 + 15*(a^2*b^7*c^3 - 3*a^3*b^6*c^2*d + 3*a^4*b^5*c*d^2 - a^5*b^4*d^3)*x^4 +
20*(a^3*b^6*c^3 - 3*a^4*b^5*c^2*d + 3*a^5*b^4*c*d^2 - a^6*b^3*d^3)*x^3 + 15*(a^4
*b^5*c^3 - 3*a^5*b^4*c^2*d + 3*a^6*b^3*c*d^2 - a^7*b^2*d^3)*x^2 + 6*(a^5*b^4*c^3
- 3*a^6*b^3*c^2*d + 3*a^7*b^2*c*d^2 - a^8*b*d^3)*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)**(5/2)/(b*x+a)**(13/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.665762, size = 1, normalized size = 0.01 \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^(5/2)/(b*x + a)^(13/2),x, algorithm="giac")

[Out]

Done