3.1480 \(\int \frac{(c+d x)^{3/2}}{(a+b x)^{13/2}} \, dx\)

Optimal. Leaf size=136 \[ \frac{32 d^3 (c+d x)^{5/2}}{1155 (a+b x)^{5/2} (b c-a d)^4}-\frac{16 d^2 (c+d x)^{5/2}}{231 (a+b x)^{7/2} (b c-a d)^3}+\frac{4 d (c+d x)^{5/2}}{33 (a+b x)^{9/2} (b c-a d)^2}-\frac{2 (c+d x)^{5/2}}{11 (a+b x)^{11/2} (b c-a d)} \]

[Out]

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

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Rubi [A]  time = 0.112921, antiderivative size = 136, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ \frac{32 d^3 (c+d x)^{5/2}}{1155 (a+b x)^{5/2} (b c-a d)^4}-\frac{16 d^2 (c+d x)^{5/2}}{231 (a+b x)^{7/2} (b c-a d)^3}+\frac{4 d (c+d x)^{5/2}}{33 (a+b x)^{9/2} (b c-a d)^2}-\frac{2 (c+d x)^{5/2}}{11 (a+b x)^{11/2} (b c-a d)} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x)^(3/2)/(a + b*x)^(13/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 22.7216, size = 121, normalized size = 0.89 \[ \frac{32 d^{3} \left (c + d x\right )^{\frac{5}{2}}}{1155 \left (a + b x\right )^{\frac{5}{2}} \left (a d - b c\right )^{4}} + \frac{16 d^{2} \left (c + d x\right )^{\frac{5}{2}}}{231 \left (a + b x\right )^{\frac{7}{2}} \left (a d - b c\right )^{3}} + \frac{4 d \left (c + d x\right )^{\frac{5}{2}}}{33 \left (a + b x\right )^{\frac{9}{2}} \left (a d - b c\right )^{2}} + \frac{2 \left (c + d x\right )^{\frac{5}{2}}}{11 \left (a + b x\right )^{\frac{11}{2}} \left (a d - b c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x+c)**(3/2)/(b*x+a)**(13/2),x)

[Out]

32*d**3*(c + d*x)**(5/2)/(1155*(a + b*x)**(5/2)*(a*d - b*c)**4) + 16*d**2*(c + d
*x)**(5/2)/(231*(a + b*x)**(7/2)*(a*d - b*c)**3) + 4*d*(c + d*x)**(5/2)/(33*(a +
b*x)**(9/2)*(a*d - b*c)**2) + 2*(c + d*x)**(5/2)/(11*(a + b*x)**(11/2)*(a*d - b
*c))

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Mathematica [A]  time = 0.216005, size = 118, normalized size = 0.87 \[ \frac{2 (c+d x)^{5/2} \left (231 a^3 d^3+99 a^2 b d^2 (2 d x-5 c)+11 a b^2 d \left (35 c^2-20 c d x+8 d^2 x^2\right )+b^3 \left (-105 c^3+70 c^2 d x-40 c d^2 x^2+16 d^3 x^3\right )\right )}{1155 (a+b x)^{11/2} (b c-a d)^4} \]

Antiderivative was successfully verified.

[In]  Integrate[(c + d*x)^(3/2)/(a + b*x)^(13/2),x]

[Out]

(2*(c + d*x)^(5/2)*(231*a^3*d^3 + 99*a^2*b*d^2*(-5*c + 2*d*x) + 11*a*b^2*d*(35*c
^2 - 20*c*d*x + 8*d^2*x^2) + b^3*(-105*c^3 + 70*c^2*d*x - 40*c*d^2*x^2 + 16*d^3*
x^3)))/(1155*(b*c - a*d)^4*(a + b*x)^(11/2))

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Maple [A]  time = 0.013, size = 171, normalized size = 1.3 \[{\frac{32\,{b}^{3}{d}^{3}{x}^{3}+176\,a{b}^{2}{d}^{3}{x}^{2}-80\,{b}^{3}c{d}^{2}{x}^{2}+396\,{a}^{2}b{d}^{3}x-440\,a{b}^{2}c{d}^{2}x+140\,{b}^{3}{c}^{2}dx+462\,{a}^{3}{d}^{3}-990\,{a}^{2}bc{d}^{2}+770\,a{b}^{2}{c}^{2}d-210\,{b}^{3}{c}^{3}}{1155\,{d}^{4}{a}^{4}-4620\,b{d}^{3}c{a}^{3}+6930\,{b}^{2}{d}^{2}{c}^{2}{a}^{2}-4620\,{b}^{3}d{c}^{3}a+1155\,{b}^{4}{c}^{4}} \left ( dx+c \right ) ^{{\frac{5}{2}}} \left ( bx+a \right ) ^{-{\frac{11}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x+c)^(3/2)/(b*x+a)^(13/2),x)

[Out]

2/1155*(d*x+c)^(5/2)*(16*b^3*d^3*x^3+88*a*b^2*d^3*x^2-40*b^3*c*d^2*x^2+198*a^2*b
*d^3*x-220*a*b^2*c*d^2*x+70*b^3*c^2*d*x+231*a^3*d^3-495*a^2*b*c*d^2+385*a*b^2*c^
2*d-105*b^3*c^3)/(b*x+a)^(11/2)/(a^4*d^4-4*a^3*b*c*d^3+6*a^2*b^2*c^2*d^2-4*a*b^3
*c^3*d+b^4*c^4)

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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)^(3/2)/(b*x + a)^(13/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 5.71502, size = 876, normalized size = 6.44 \[ \frac{2 \,{\left (16 \, b^{3} d^{5} x^{5} - 105 \, b^{3} c^{5} + 385 \, a b^{2} c^{4} d - 495 \, a^{2} b c^{3} d^{2} + 231 \, a^{3} c^{2} d^{3} - 8 \,{\left (b^{3} c d^{4} - 11 \, a b^{2} d^{5}\right )} x^{4} + 2 \,{\left (3 \, b^{3} c^{2} d^{3} - 22 \, a b^{2} c d^{4} + 99 \, a^{2} b d^{5}\right )} x^{3} -{\left (5 \, b^{3} c^{3} d^{2} - 33 \, a b^{2} c^{2} d^{3} + 99 \, a^{2} b c d^{4} - 231 \, a^{3} d^{5}\right )} x^{2} - 2 \,{\left (70 \, b^{3} c^{4} d - 275 \, a b^{2} c^{3} d^{2} + 396 \, a^{2} b c^{2} d^{3} - 231 \, a^{3} c d^{4}\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{1155 \,{\left (a^{6} b^{4} c^{4} - 4 \, a^{7} b^{3} c^{3} d + 6 \, a^{8} b^{2} c^{2} d^{2} - 4 \, a^{9} b c d^{3} + a^{10} d^{4} +{\left (b^{10} c^{4} - 4 \, a b^{9} c^{3} d + 6 \, a^{2} b^{8} c^{2} d^{2} - 4 \, a^{3} b^{7} c d^{3} + a^{4} b^{6} d^{4}\right )} x^{6} + 6 \,{\left (a b^{9} c^{4} - 4 \, a^{2} b^{8} c^{3} d + 6 \, a^{3} b^{7} c^{2} d^{2} - 4 \, a^{4} b^{6} c d^{3} + a^{5} b^{5} d^{4}\right )} x^{5} + 15 \,{\left (a^{2} b^{8} c^{4} - 4 \, a^{3} b^{7} c^{3} d + 6 \, a^{4} b^{6} c^{2} d^{2} - 4 \, a^{5} b^{5} c d^{3} + a^{6} b^{4} d^{4}\right )} x^{4} + 20 \,{\left (a^{3} b^{7} c^{4} - 4 \, a^{4} b^{6} c^{3} d + 6 \, a^{5} b^{5} c^{2} d^{2} - 4 \, a^{6} b^{4} c d^{3} + a^{7} b^{3} d^{4}\right )} x^{3} + 15 \,{\left (a^{4} b^{6} c^{4} - 4 \, a^{5} b^{5} c^{3} d + 6 \, a^{6} b^{4} c^{2} d^{2} - 4 \, a^{7} b^{3} c d^{3} + a^{8} b^{2} d^{4}\right )} x^{2} + 6 \,{\left (a^{5} b^{5} c^{4} - 4 \, a^{6} b^{4} c^{3} d + 6 \, a^{7} b^{3} c^{2} d^{2} - 4 \, a^{8} b^{2} c d^{3} + a^{9} b d^{4}\right )} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^(3/2)/(b*x + a)^(13/2),x, algorithm="fricas")

[Out]

2/1155*(16*b^3*d^5*x^5 - 105*b^3*c^5 + 385*a*b^2*c^4*d - 495*a^2*b*c^3*d^2 + 231
*a^3*c^2*d^3 - 8*(b^3*c*d^4 - 11*a*b^2*d^5)*x^4 + 2*(3*b^3*c^2*d^3 - 22*a*b^2*c*
d^4 + 99*a^2*b*d^5)*x^3 - (5*b^3*c^3*d^2 - 33*a*b^2*c^2*d^3 + 99*a^2*b*c*d^4 - 2
31*a^3*d^5)*x^2 - 2*(70*b^3*c^4*d - 275*a*b^2*c^3*d^2 + 396*a^2*b*c^2*d^3 - 231*
a^3*c*d^4)*x)*sqrt(b*x + a)*sqrt(d*x + c)/(a^6*b^4*c^4 - 4*a^7*b^3*c^3*d + 6*a^8
*b^2*c^2*d^2 - 4*a^9*b*c*d^3 + a^10*d^4 + (b^10*c^4 - 4*a*b^9*c^3*d + 6*a^2*b^8*
c^2*d^2 - 4*a^3*b^7*c*d^3 + a^4*b^6*d^4)*x^6 + 6*(a*b^9*c^4 - 4*a^2*b^8*c^3*d +
6*a^3*b^7*c^2*d^2 - 4*a^4*b^6*c*d^3 + a^5*b^5*d^4)*x^5 + 15*(a^2*b^8*c^4 - 4*a^3
*b^7*c^3*d + 6*a^4*b^6*c^2*d^2 - 4*a^5*b^5*c*d^3 + a^6*b^4*d^4)*x^4 + 20*(a^3*b^
7*c^4 - 4*a^4*b^6*c^3*d + 6*a^5*b^5*c^2*d^2 - 4*a^6*b^4*c*d^3 + a^7*b^3*d^4)*x^3
+ 15*(a^4*b^6*c^4 - 4*a^5*b^5*c^3*d + 6*a^6*b^4*c^2*d^2 - 4*a^7*b^3*c*d^3 + a^8
*b^2*d^4)*x^2 + 6*(a^5*b^5*c^4 - 4*a^6*b^4*c^3*d + 6*a^7*b^3*c^2*d^2 - 4*a^8*b^2
*c*d^3 + a^9*b*d^4)*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)**(3/2)/(b*x+a)**(13/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.500726, size = 1, normalized size = 0.01 \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^(3/2)/(b*x + a)^(13/2),x, algorithm="giac")

[Out]

Done