3.1491 \(\int \frac{(c+d x)^{5/2}}{(a+b x)^{15/2}} \, dx\)

Optimal. Leaf size=136 \[ \frac{32 d^3 (c+d x)^{7/2}}{3003 (a+b x)^{7/2} (b c-a d)^4}-\frac{16 d^2 (c+d x)^{7/2}}{429 (a+b x)^{9/2} (b c-a d)^3}+\frac{12 d (c+d x)^{7/2}}{143 (a+b x)^{11/2} (b c-a d)^2}-\frac{2 (c+d x)^{7/2}}{13 (a+b x)^{13/2} (b c-a d)} \]

[Out]

(-2*(c + d*x)^(7/2))/(13*(b*c - a*d)*(a + b*x)^(13/2)) + (12*d*(c + d*x)^(7/2))/
(143*(b*c - a*d)^2*(a + b*x)^(11/2)) - (16*d^2*(c + d*x)^(7/2))/(429*(b*c - a*d)
^3*(a + b*x)^(9/2)) + (32*d^3*(c + d*x)^(7/2))/(3003*(b*c - a*d)^4*(a + b*x)^(7/
2))

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Rubi [A]  time = 0.111368, 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 \[ \frac{32 d^3 (c+d x)^{7/2}}{3003 (a+b x)^{7/2} (b c-a d)^4}-\frac{16 d^2 (c+d x)^{7/2}}{429 (a+b x)^{9/2} (b c-a d)^3}+\frac{12 d (c+d x)^{7/2}}{143 (a+b x)^{11/2} (b c-a d)^2}-\frac{2 (c+d x)^{7/2}}{13 (a+b x)^{13/2} (b c-a d)} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x)^(5/2)/(a + b*x)^(15/2),x]

[Out]

(-2*(c + d*x)^(7/2))/(13*(b*c - a*d)*(a + b*x)^(13/2)) + (12*d*(c + d*x)^(7/2))/
(143*(b*c - a*d)^2*(a + b*x)^(11/2)) - (16*d^2*(c + d*x)^(7/2))/(429*(b*c - a*d)
^3*(a + b*x)^(9/2)) + (32*d^3*(c + d*x)^(7/2))/(3003*(b*c - a*d)^4*(a + b*x)^(7/
2))

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Rubi in Sympy [A]  time = 22.3387, size = 121, normalized size = 0.89 \[ \frac{32 d^{3} \left (c + d x\right )^{\frac{7}{2}}}{3003 \left (a + b x\right )^{\frac{7}{2}} \left (a d - b c\right )^{4}} + \frac{16 d^{2} \left (c + d x\right )^{\frac{7}{2}}}{429 \left (a + b x\right )^{\frac{9}{2}} \left (a d - b c\right )^{3}} + \frac{12 d \left (c + d x\right )^{\frac{7}{2}}}{143 \left (a + b x\right )^{\frac{11}{2}} \left (a d - b c\right )^{2}} + \frac{2 \left (c + d x\right )^{\frac{7}{2}}}{13 \left (a + b x\right )^{\frac{13}{2}} \left (a d - b c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x+c)**(5/2)/(b*x+a)**(15/2),x)

[Out]

32*d**3*(c + d*x)**(7/2)/(3003*(a + b*x)**(7/2)*(a*d - b*c)**4) + 16*d**2*(c + d
*x)**(7/2)/(429*(a + b*x)**(9/2)*(a*d - b*c)**3) + 12*d*(c + d*x)**(7/2)/(143*(a
+ b*x)**(11/2)*(a*d - b*c)**2) + 2*(c + d*x)**(7/2)/(13*(a + b*x)**(13/2)*(a*d
- b*c))

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Mathematica [A]  time = 0.26358, size = 118, normalized size = 0.87 \[ \frac{2 (c+d x)^{7/2} \left (429 a^3 d^3+143 a^2 b d^2 (2 d x-7 c)+13 a b^2 d \left (63 c^2-28 c d x+8 d^2 x^2\right )+b^3 \left (-231 c^3+126 c^2 d x-56 c d^2 x^2+16 d^3 x^3\right )\right )}{3003 (a+b x)^{13/2} (b c-a d)^4} \]

Antiderivative was successfully verified.

[In]  Integrate[(c + d*x)^(5/2)/(a + b*x)^(15/2),x]

[Out]

(2*(c + d*x)^(7/2)*(429*a^3*d^3 + 143*a^2*b*d^2*(-7*c + 2*d*x) + 13*a*b^2*d*(63*
c^2 - 28*c*d*x + 8*d^2*x^2) + b^3*(-231*c^3 + 126*c^2*d*x - 56*c*d^2*x^2 + 16*d^
3*x^3)))/(3003*(b*c - a*d)^4*(a + b*x)^(13/2))

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Maple [A]  time = 0.015, size = 171, normalized size = 1.3 \[{\frac{32\,{b}^{3}{d}^{3}{x}^{3}+208\,a{b}^{2}{d}^{3}{x}^{2}-112\,{b}^{3}c{d}^{2}{x}^{2}+572\,{a}^{2}b{d}^{3}x-728\,a{b}^{2}c{d}^{2}x+252\,{b}^{3}{c}^{2}dx+858\,{a}^{3}{d}^{3}-2002\,{a}^{2}bc{d}^{2}+1638\,a{b}^{2}{c}^{2}d-462\,{b}^{3}{c}^{3}}{3003\,{d}^{4}{a}^{4}-12012\,b{d}^{3}c{a}^{3}+18018\,{b}^{2}{d}^{2}{c}^{2}{a}^{2}-12012\,{b}^{3}d{c}^{3}a+3003\,{b}^{4}{c}^{4}} \left ( dx+c \right ) ^{{\frac{7}{2}}} \left ( bx+a \right ) ^{-{\frac{13}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x+c)^(5/2)/(b*x+a)^(15/2),x)

[Out]

2/3003*(d*x+c)^(7/2)*(16*b^3*d^3*x^3+104*a*b^2*d^3*x^2-56*b^3*c*d^2*x^2+286*a^2*
b*d^3*x-364*a*b^2*c*d^2*x+126*b^3*c^2*d*x+429*a^3*d^3-1001*a^2*b*c*d^2+819*a*b^2
*c^2*d-231*b^3*c^3)/(b*x+a)^(13/2)/(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)^(5/2)/(b*x + a)^(15/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 13.7752, size = 1033, normalized size = 7.6 \[ \frac{2 \,{\left (16 \, b^{3} d^{6} x^{6} - 231 \, b^{3} c^{6} + 819 \, a b^{2} c^{5} d - 1001 \, a^{2} b c^{4} d^{2} + 429 \, a^{3} c^{3} d^{3} - 8 \,{\left (b^{3} c d^{5} - 13 \, a b^{2} d^{6}\right )} x^{5} + 2 \,{\left (3 \, b^{3} c^{2} d^{4} - 26 \, a b^{2} c d^{5} + 143 \, a^{2} b d^{6}\right )} x^{4} -{\left (5 \, b^{3} c^{3} d^{3} - 39 \, a b^{2} c^{2} d^{4} + 143 \, a^{2} b c d^{5} - 429 \, a^{3} d^{6}\right )} x^{3} -{\left (371 \, b^{3} c^{4} d^{2} - 1469 \, a b^{2} c^{3} d^{3} + 2145 \, a^{2} b c^{2} d^{4} - 1287 \, a^{3} c d^{5}\right )} x^{2} -{\left (567 \, b^{3} c^{5} d - 2093 \, a b^{2} c^{4} d^{2} + 2717 \, a^{2} b c^{3} d^{3} - 1287 \, a^{3} c^{2} d^{4}\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{3003 \,{\left (a^{7} b^{4} c^{4} - 4 \, a^{8} b^{3} c^{3} d + 6 \, a^{9} b^{2} c^{2} d^{2} - 4 \, a^{10} b c d^{3} + a^{11} d^{4} +{\left (b^{11} c^{4} - 4 \, a b^{10} c^{3} d + 6 \, a^{2} b^{9} c^{2} d^{2} - 4 \, a^{3} b^{8} c d^{3} + a^{4} b^{7} d^{4}\right )} x^{7} + 7 \,{\left (a b^{10} c^{4} - 4 \, a^{2} b^{9} c^{3} d + 6 \, a^{3} b^{8} c^{2} d^{2} - 4 \, a^{4} b^{7} c d^{3} + a^{5} b^{6} d^{4}\right )} x^{6} + 21 \,{\left (a^{2} b^{9} c^{4} - 4 \, a^{3} b^{8} c^{3} d + 6 \, a^{4} b^{7} c^{2} d^{2} - 4 \, a^{5} b^{6} c d^{3} + a^{6} b^{5} d^{4}\right )} x^{5} + 35 \,{\left (a^{3} b^{8} c^{4} - 4 \, a^{4} b^{7} c^{3} d + 6 \, a^{5} b^{6} c^{2} d^{2} - 4 \, a^{6} b^{5} c d^{3} + a^{7} b^{4} d^{4}\right )} x^{4} + 35 \,{\left (a^{4} b^{7} c^{4} - 4 \, a^{5} b^{6} c^{3} d + 6 \, a^{6} b^{5} c^{2} d^{2} - 4 \, a^{7} b^{4} c d^{3} + a^{8} b^{3} d^{4}\right )} x^{3} + 21 \,{\left (a^{5} b^{6} c^{4} - 4 \, a^{6} b^{5} c^{3} d + 6 \, a^{7} b^{4} c^{2} d^{2} - 4 \, a^{8} b^{3} c d^{3} + a^{9} b^{2} d^{4}\right )} x^{2} + 7 \,{\left (a^{6} b^{5} c^{4} - 4 \, a^{7} b^{4} c^{3} d + 6 \, a^{8} b^{3} c^{2} d^{2} - 4 \, a^{9} b^{2} c d^{3} + a^{10} b d^{4}\right )} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^(5/2)/(b*x + a)^(15/2),x, algorithm="fricas")

[Out]

2/3003*(16*b^3*d^6*x^6 - 231*b^3*c^6 + 819*a*b^2*c^5*d - 1001*a^2*b*c^4*d^2 + 42
9*a^3*c^3*d^3 - 8*(b^3*c*d^5 - 13*a*b^2*d^6)*x^5 + 2*(3*b^3*c^2*d^4 - 26*a*b^2*c
*d^5 + 143*a^2*b*d^6)*x^4 - (5*b^3*c^3*d^3 - 39*a*b^2*c^2*d^4 + 143*a^2*b*c*d^5
- 429*a^3*d^6)*x^3 - (371*b^3*c^4*d^2 - 1469*a*b^2*c^3*d^3 + 2145*a^2*b*c^2*d^4
- 1287*a^3*c*d^5)*x^2 - (567*b^3*c^5*d - 2093*a*b^2*c^4*d^2 + 2717*a^2*b*c^3*d^3
- 1287*a^3*c^2*d^4)*x)*sqrt(b*x + a)*sqrt(d*x + c)/(a^7*b^4*c^4 - 4*a^8*b^3*c^3
*d + 6*a^9*b^2*c^2*d^2 - 4*a^10*b*c*d^3 + a^11*d^4 + (b^11*c^4 - 4*a*b^10*c^3*d
+ 6*a^2*b^9*c^2*d^2 - 4*a^3*b^8*c*d^3 + a^4*b^7*d^4)*x^7 + 7*(a*b^10*c^4 - 4*a^2
*b^9*c^3*d + 6*a^3*b^8*c^2*d^2 - 4*a^4*b^7*c*d^3 + a^5*b^6*d^4)*x^6 + 21*(a^2*b^
9*c^4 - 4*a^3*b^8*c^3*d + 6*a^4*b^7*c^2*d^2 - 4*a^5*b^6*c*d^3 + a^6*b^5*d^4)*x^5
+ 35*(a^3*b^8*c^4 - 4*a^4*b^7*c^3*d + 6*a^5*b^6*c^2*d^2 - 4*a^6*b^5*c*d^3 + a^7
*b^4*d^4)*x^4 + 35*(a^4*b^7*c^4 - 4*a^5*b^6*c^3*d + 6*a^6*b^5*c^2*d^2 - 4*a^7*b^
4*c*d^3 + a^8*b^3*d^4)*x^3 + 21*(a^5*b^6*c^4 - 4*a^6*b^5*c^3*d + 6*a^7*b^4*c^2*d
^2 - 4*a^8*b^3*c*d^3 + a^9*b^2*d^4)*x^2 + 7*(a^6*b^5*c^4 - 4*a^7*b^4*c^3*d + 6*a
^8*b^3*c^2*d^2 - 4*a^9*b^2*c*d^3 + a^10*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)**(5/2)/(b*x+a)**(15/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.823631, 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)^(15/2),x, algorithm="giac")

[Out]

Done