3.1479 \(\int \frac{(c+d x)^{3/2}}{(a+b x)^{11/2}} \, dx\)

Optimal. Leaf size=101 \[ -\frac{16 d^2 (c+d x)^{5/2}}{315 (a+b x)^{5/2} (b c-a d)^3}+\frac{8 d (c+d x)^{5/2}}{63 (a+b x)^{7/2} (b c-a d)^2}-\frac{2 (c+d x)^{5/2}}{9 (a+b x)^{9/2} (b c-a d)} \]

[Out]

(-2*(c + d*x)^(5/2))/(9*(b*c - a*d)*(a + b*x)^(9/2)) + (8*d*(c + d*x)^(5/2))/(63
*(b*c - a*d)^2*(a + b*x)^(7/2)) - (16*d^2*(c + d*x)^(5/2))/(315*(b*c - a*d)^3*(a
+ b*x)^(5/2))

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Rubi [A]  time = 0.0811499, 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 \[ -\frac{16 d^2 (c+d x)^{5/2}}{315 (a+b x)^{5/2} (b c-a d)^3}+\frac{8 d (c+d x)^{5/2}}{63 (a+b x)^{7/2} (b c-a d)^2}-\frac{2 (c+d x)^{5/2}}{9 (a+b x)^{9/2} (b c-a d)} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x)^(3/2)/(a + b*x)^(11/2),x]

[Out]

(-2*(c + d*x)^(5/2))/(9*(b*c - a*d)*(a + b*x)^(9/2)) + (8*d*(c + d*x)^(5/2))/(63
*(b*c - a*d)^2*(a + b*x)^(7/2)) - (16*d^2*(c + d*x)^(5/2))/(315*(b*c - a*d)^3*(a
+ b*x)^(5/2))

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Rubi in Sympy [A]  time = 14.1675, size = 88, normalized size = 0.87 \[ \frac{16 d^{2} \left (c + d x\right )^{\frac{5}{2}}}{315 \left (a + b x\right )^{\frac{5}{2}} \left (a d - b c\right )^{3}} + \frac{8 d \left (c + d x\right )^{\frac{5}{2}}}{63 \left (a + b x\right )^{\frac{7}{2}} \left (a d - b c\right )^{2}} + \frac{2 \left (c + d x\right )^{\frac{5}{2}}}{9 \left (a + b x\right )^{\frac{9}{2}} \left (a d - b c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x+c)**(3/2)/(b*x+a)**(11/2),x)

[Out]

16*d**2*(c + d*x)**(5/2)/(315*(a + b*x)**(5/2)*(a*d - b*c)**3) + 8*d*(c + d*x)**
(5/2)/(63*(a + b*x)**(7/2)*(a*d - b*c)**2) + 2*(c + d*x)**(5/2)/(9*(a + b*x)**(9
/2)*(a*d - b*c))

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Mathematica [A]  time = 0.156644, size = 77, normalized size = 0.76 \[ \frac{2 (c+d x)^{5/2} \left (63 a^2 d^2+18 a b d (2 d x-5 c)+b^2 \left (35 c^2-20 c d x+8 d^2 x^2\right )\right )}{315 (a+b x)^{9/2} (a d-b c)^3} \]

Antiderivative was successfully verified.

[In]  Integrate[(c + d*x)^(3/2)/(a + b*x)^(11/2),x]

[Out]

(2*(c + d*x)^(5/2)*(63*a^2*d^2 + 18*a*b*d*(-5*c + 2*d*x) + b^2*(35*c^2 - 20*c*d*
x + 8*d^2*x^2)))/(315*(-(b*c) + a*d)^3*(a + b*x)^(9/2))

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Maple [A]  time = 0.01, size = 105, normalized size = 1. \[{\frac{16\,{b}^{2}{d}^{2}{x}^{2}+72\,ab{d}^{2}x-40\,{b}^{2}cdx+126\,{a}^{2}{d}^{2}-180\,abcd+70\,{b}^{2}{c}^{2}}{315\,{a}^{3}{d}^{3}-945\,{a}^{2}bc{d}^{2}+945\,a{b}^{2}{c}^{2}d-315\,{b}^{3}{c}^{3}} \left ( dx+c \right ) ^{{\frac{5}{2}}} \left ( bx+a \right ) ^{-{\frac{9}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x+c)^(3/2)/(b*x+a)^(11/2),x)

[Out]

2/315*(d*x+c)^(5/2)*(8*b^2*d^2*x^2+36*a*b*d^2*x-20*b^2*c*d*x+63*a^2*d^2-90*a*b*c
*d+35*b^2*c^2)/(b*x+a)^(9/2)/(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)^(3/2)/(b*x + a)^(11/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.26856, size = 575, normalized size = 5.69 \[ -\frac{2 \,{\left (8 \, b^{2} d^{4} x^{4} + 35 \, b^{2} c^{4} - 90 \, a b c^{3} d + 63 \, a^{2} c^{2} d^{2} - 4 \,{\left (b^{2} c d^{3} - 9 \, a b d^{4}\right )} x^{3} + 3 \,{\left (b^{2} c^{2} d^{2} - 6 \, a b c d^{3} + 21 \, a^{2} d^{4}\right )} x^{2} + 2 \,{\left (25 \, b^{2} c^{3} d - 72 \, a b c^{2} d^{2} + 63 \, a^{2} c d^{3}\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{315 \,{\left (a^{5} b^{3} c^{3} - 3 \, a^{6} b^{2} c^{2} d + 3 \, a^{7} b c d^{2} - a^{8} d^{3} +{\left (b^{8} c^{3} - 3 \, a b^{7} c^{2} d + 3 \, a^{2} b^{6} c d^{2} - a^{3} b^{5} d^{3}\right )} x^{5} + 5 \,{\left (a b^{7} c^{3} - 3 \, a^{2} b^{6} c^{2} d + 3 \, a^{3} b^{5} c d^{2} - a^{4} b^{4} d^{3}\right )} x^{4} + 10 \,{\left (a^{2} b^{6} c^{3} - 3 \, a^{3} b^{5} c^{2} d + 3 \, a^{4} b^{4} c d^{2} - a^{5} b^{3} d^{3}\right )} x^{3} + 10 \,{\left (a^{3} b^{5} c^{3} - 3 \, a^{4} b^{4} c^{2} d + 3 \, a^{5} b^{3} c d^{2} - a^{6} b^{2} d^{3}\right )} x^{2} + 5 \,{\left (a^{4} b^{4} c^{3} - 3 \, a^{5} b^{3} c^{2} d + 3 \, a^{6} b^{2} c d^{2} - a^{7} b d^{3}\right )} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^(3/2)/(b*x + a)^(11/2),x, algorithm="fricas")

[Out]

-2/315*(8*b^2*d^4*x^4 + 35*b^2*c^4 - 90*a*b*c^3*d + 63*a^2*c^2*d^2 - 4*(b^2*c*d^
3 - 9*a*b*d^4)*x^3 + 3*(b^2*c^2*d^2 - 6*a*b*c*d^3 + 21*a^2*d^4)*x^2 + 2*(25*b^2*
c^3*d - 72*a*b*c^2*d^2 + 63*a^2*c*d^3)*x)*sqrt(b*x + a)*sqrt(d*x + c)/(a^5*b^3*c
^3 - 3*a^6*b^2*c^2*d + 3*a^7*b*c*d^2 - a^8*d^3 + (b^8*c^3 - 3*a*b^7*c^2*d + 3*a^
2*b^6*c*d^2 - a^3*b^5*d^3)*x^5 + 5*(a*b^7*c^3 - 3*a^2*b^6*c^2*d + 3*a^3*b^5*c*d^
2 - a^4*b^4*d^3)*x^4 + 10*(a^2*b^6*c^3 - 3*a^3*b^5*c^2*d + 3*a^4*b^4*c*d^2 - a^5
*b^3*d^3)*x^3 + 10*(a^3*b^5*c^3 - 3*a^4*b^4*c^2*d + 3*a^5*b^3*c*d^2 - a^6*b^2*d^
3)*x^2 + 5*(a^4*b^4*c^3 - 3*a^5*b^3*c^2*d + 3*a^6*b^2*c*d^2 - a^7*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)**(3/2)/(b*x+a)**(11/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.425074, 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)^(11/2),x, algorithm="giac")

[Out]

Done