3.1511 \(\int \frac{1}{(a+b x)^{11/2} (c+d x)^{3/2}} \, dx\)

Optimal. Leaf size=206 \[ -\frac{512 d^5 \sqrt{a+b x}}{63 \sqrt{c+d x} (b c-a d)^6}-\frac{256 d^4}{63 \sqrt{a+b x} \sqrt{c+d x} (b c-a d)^5}+\frac{64 d^3}{63 (a+b x)^{3/2} \sqrt{c+d x} (b c-a d)^4}-\frac{32 d^2}{63 (a+b x)^{5/2} \sqrt{c+d x} (b c-a d)^3}+\frac{20 d}{63 (a+b x)^{7/2} \sqrt{c+d x} (b c-a d)^2}-\frac{2}{9 (a+b x)^{9/2} \sqrt{c+d x} (b c-a d)} \]

[Out]

-2/(9*(b*c - a*d)*(a + b*x)^(9/2)*Sqrt[c + d*x]) + (20*d)/(63*(b*c - a*d)^2*(a +
b*x)^(7/2)*Sqrt[c + d*x]) - (32*d^2)/(63*(b*c - a*d)^3*(a + b*x)^(5/2)*Sqrt[c +
d*x]) + (64*d^3)/(63*(b*c - a*d)^4*(a + b*x)^(3/2)*Sqrt[c + d*x]) - (256*d^4)/(
63*(b*c - a*d)^5*Sqrt[a + b*x]*Sqrt[c + d*x]) - (512*d^5*Sqrt[a + b*x])/(63*(b*c
- a*d)^6*Sqrt[c + d*x])

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Rubi [A]  time = 0.200007, antiderivative size = 206, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ -\frac{512 d^5 \sqrt{a+b x}}{63 \sqrt{c+d x} (b c-a d)^6}-\frac{256 d^4}{63 \sqrt{a+b x} \sqrt{c+d x} (b c-a d)^5}+\frac{64 d^3}{63 (a+b x)^{3/2} \sqrt{c+d x} (b c-a d)^4}-\frac{32 d^2}{63 (a+b x)^{5/2} \sqrt{c+d x} (b c-a d)^3}+\frac{20 d}{63 (a+b x)^{7/2} \sqrt{c+d x} (b c-a d)^2}-\frac{2}{9 (a+b x)^{9/2} \sqrt{c+d x} (b c-a d)} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b*x)^(11/2)*(c + d*x)^(3/2)),x]

[Out]

-2/(9*(b*c - a*d)*(a + b*x)^(9/2)*Sqrt[c + d*x]) + (20*d)/(63*(b*c - a*d)^2*(a +
b*x)^(7/2)*Sqrt[c + d*x]) - (32*d^2)/(63*(b*c - a*d)^3*(a + b*x)^(5/2)*Sqrt[c +
d*x]) + (64*d^3)/(63*(b*c - a*d)^4*(a + b*x)^(3/2)*Sqrt[c + d*x]) - (256*d^4)/(
63*(b*c - a*d)^5*Sqrt[a + b*x]*Sqrt[c + d*x]) - (512*d^5*Sqrt[a + b*x])/(63*(b*c
- a*d)^6*Sqrt[c + d*x])

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Rubi in Sympy [A]  time = 42.9238, size = 189, normalized size = 0.92 \[ - \frac{512 b d^{4} \sqrt{c + d x}}{63 \sqrt{a + b x} \left (a d - b c\right )^{6}} - \frac{256 b d^{3} \sqrt{c + d x}}{63 \left (a + b x\right )^{\frac{3}{2}} \left (a d - b c\right )^{5}} - \frac{64 d^{3}}{21 \left (a + b x\right )^{\frac{3}{2}} \sqrt{c + d x} \left (a d - b c\right )^{4}} + \frac{32 d^{2}}{63 \left (a + b x\right )^{\frac{5}{2}} \sqrt{c + d x} \left (a d - b c\right )^{3}} + \frac{20 d}{63 \left (a + b x\right )^{\frac{7}{2}} \sqrt{c + d x} \left (a d - b c\right )^{2}} + \frac{2}{9 \left (a + b x\right )^{\frac{9}{2}} \sqrt{c + d x} \left (a d - b c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(b*x+a)**(11/2)/(d*x+c)**(3/2),x)

[Out]

-512*b*d**4*sqrt(c + d*x)/(63*sqrt(a + b*x)*(a*d - b*c)**6) - 256*b*d**3*sqrt(c
+ d*x)/(63*(a + b*x)**(3/2)*(a*d - b*c)**5) - 64*d**3/(21*(a + b*x)**(3/2)*sqrt(
c + d*x)*(a*d - b*c)**4) + 32*d**2/(63*(a + b*x)**(5/2)*sqrt(c + d*x)*(a*d - b*c
)**3) + 20*d/(63*(a + b*x)**(7/2)*sqrt(c + d*x)*(a*d - b*c)**2) + 2/(9*(a + b*x)
**(9/2)*sqrt(c + d*x)*(a*d - b*c))

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Mathematica [A]  time = 0.368228, size = 143, normalized size = 0.69 \[ \frac{2 \sqrt{a+b x} \sqrt{c+d x} \left (\frac{65 b d^3 (b c-a d)}{(a+b x)^2}-\frac{33 b d^2 (b c-a d)^2}{(a+b x)^3}+\frac{17 b d (b c-a d)^3}{(a+b x)^4}-\frac{7 b (b c-a d)^4}{(a+b x)^5}-\frac{193 b d^4}{a+b x}-\frac{63 d^5}{c+d x}\right )}{63 (b c-a d)^6} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b*x)^(11/2)*(c + d*x)^(3/2)),x]

[Out]

(2*Sqrt[a + b*x]*Sqrt[c + d*x]*((-7*b*(b*c - a*d)^4)/(a + b*x)^5 + (17*b*d*(b*c
- a*d)^3)/(a + b*x)^4 - (33*b*d^2*(b*c - a*d)^2)/(a + b*x)^3 + (65*b*d^3*(b*c -
a*d))/(a + b*x)^2 - (193*b*d^4)/(a + b*x) - (63*d^5)/(c + d*x)))/(63*(b*c - a*d)
^6)

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Maple [B]  time = 0.017, size = 356, normalized size = 1.7 \[ -{\frac{512\,{b}^{5}{d}^{5}{x}^{5}+2304\,a{b}^{4}{d}^{5}{x}^{4}+256\,{b}^{5}c{d}^{4}{x}^{4}+4032\,{a}^{2}{b}^{3}{d}^{5}{x}^{3}+1152\,a{b}^{4}c{d}^{4}{x}^{3}-64\,{b}^{5}{c}^{2}{d}^{3}{x}^{3}+3360\,{a}^{3}{b}^{2}{d}^{5}{x}^{2}+2016\,{a}^{2}{b}^{3}c{d}^{4}{x}^{2}-288\,a{b}^{4}{c}^{2}{d}^{3}{x}^{2}+32\,{b}^{5}{c}^{3}{d}^{2}{x}^{2}+1260\,{a}^{4}b{d}^{5}x+1680\,{a}^{3}{b}^{2}c{d}^{4}x-504\,{a}^{2}{b}^{3}{c}^{2}{d}^{3}x+144\,a{b}^{4}{c}^{3}{d}^{2}x-20\,{b}^{5}{c}^{4}dx+126\,{a}^{5}{d}^{5}+630\,{a}^{4}bc{d}^{4}-420\,{a}^{3}{b}^{2}{c}^{2}{d}^{3}+252\,{a}^{2}{b}^{3}{c}^{3}{d}^{2}-90\,a{b}^{4}{c}^{4}d+14\,{b}^{5}{c}^{5}}{63\,{d}^{6}{a}^{6}-378\,b{d}^{5}c{a}^{5}+945\,{b}^{2}{d}^{4}{c}^{2}{a}^{4}-1260\,{b}^{3}{d}^{3}{c}^{3}{a}^{3}+945\,{b}^{4}{d}^{2}{c}^{4}{a}^{2}-378\,{b}^{5}d{c}^{5}a+63\,{b}^{6}{c}^{6}} \left ( bx+a \right ) ^{-{\frac{9}{2}}}{\frac{1}{\sqrt{dx+c}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b*x+a)^(11/2)/(d*x+c)^(3/2),x)

[Out]

-2/63*(256*b^5*d^5*x^5+1152*a*b^4*d^5*x^4+128*b^5*c*d^4*x^4+2016*a^2*b^3*d^5*x^3
+576*a*b^4*c*d^4*x^3-32*b^5*c^2*d^3*x^3+1680*a^3*b^2*d^5*x^2+1008*a^2*b^3*c*d^4*
x^2-144*a*b^4*c^2*d^3*x^2+16*b^5*c^3*d^2*x^2+630*a^4*b*d^5*x+840*a^3*b^2*c*d^4*x
-252*a^2*b^3*c^2*d^3*x+72*a*b^4*c^3*d^2*x-10*b^5*c^4*d*x+63*a^5*d^5+315*a^4*b*c*
d^4-210*a^3*b^2*c^2*d^3+126*a^2*b^3*c^3*d^2-45*a*b^4*c^4*d+7*b^5*c^5)/(b*x+a)^(9
/2)/(d*x+c)^(1/2)/(a^6*d^6-6*a^5*b*c*d^5+15*a^4*b^2*c^2*d^4-20*a^3*b^3*c^3*d^3+1
5*a^2*b^4*c^4*d^2-6*a*b^5*c^5*d+b^6*c^6)

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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(1/((b*x + a)^(11/2)*(d*x + c)^(3/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 6.56772, size = 1289, normalized size = 6.26 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^(11/2)*(d*x + c)^(3/2)),x, algorithm="fricas")

[Out]

-2/63*(256*b^5*d^5*x^5 + 7*b^5*c^5 - 45*a*b^4*c^4*d + 126*a^2*b^3*c^3*d^2 - 210*
a^3*b^2*c^2*d^3 + 315*a^4*b*c*d^4 + 63*a^5*d^5 + 128*(b^5*c*d^4 + 9*a*b^4*d^5)*x
^4 - 32*(b^5*c^2*d^3 - 18*a*b^4*c*d^4 - 63*a^2*b^3*d^5)*x^3 + 16*(b^5*c^3*d^2 -
9*a*b^4*c^2*d^3 + 63*a^2*b^3*c*d^4 + 105*a^3*b^2*d^5)*x^2 - 2*(5*b^5*c^4*d - 36*
a*b^4*c^3*d^2 + 126*a^2*b^3*c^2*d^3 - 420*a^3*b^2*c*d^4 - 315*a^4*b*d^5)*x)*sqrt
(b*x + a)*sqrt(d*x + c)/(a^5*b^6*c^7 - 6*a^6*b^5*c^6*d + 15*a^7*b^4*c^5*d^2 - 20
*a^8*b^3*c^4*d^3 + 15*a^9*b^2*c^3*d^4 - 6*a^10*b*c^2*d^5 + a^11*c*d^6 + (b^11*c^
6*d - 6*a*b^10*c^5*d^2 + 15*a^2*b^9*c^4*d^3 - 20*a^3*b^8*c^3*d^4 + 15*a^4*b^7*c^
2*d^5 - 6*a^5*b^6*c*d^6 + a^6*b^5*d^7)*x^6 + (b^11*c^7 - a*b^10*c^6*d - 15*a^2*b
^9*c^5*d^2 + 55*a^3*b^8*c^4*d^3 - 85*a^4*b^7*c^3*d^4 + 69*a^5*b^6*c^2*d^5 - 29*a
^6*b^5*c*d^6 + 5*a^7*b^4*d^7)*x^5 + 5*(a*b^10*c^7 - 4*a^2*b^9*c^6*d + 3*a^3*b^8*
c^5*d^2 + 10*a^4*b^7*c^4*d^3 - 25*a^5*b^6*c^3*d^4 + 24*a^6*b^5*c^2*d^5 - 11*a^7*
b^4*c*d^6 + 2*a^8*b^3*d^7)*x^4 + 10*(a^2*b^9*c^7 - 5*a^3*b^8*c^6*d + 9*a^4*b^7*c
^5*d^2 - 5*a^5*b^6*c^4*d^3 - 5*a^6*b^5*c^3*d^4 + 9*a^7*b^4*c^2*d^5 - 5*a^8*b^3*c
*d^6 + a^9*b^2*d^7)*x^3 + 5*(2*a^3*b^8*c^7 - 11*a^4*b^7*c^6*d + 24*a^5*b^6*c^5*d
^2 - 25*a^6*b^5*c^4*d^3 + 10*a^7*b^4*c^3*d^4 + 3*a^8*b^3*c^2*d^5 - 4*a^9*b^2*c*d
^6 + a^10*b*d^7)*x^2 + (5*a^4*b^7*c^7 - 29*a^5*b^6*c^6*d + 69*a^6*b^5*c^5*d^2 -
85*a^7*b^4*c^4*d^3 + 55*a^8*b^3*c^3*d^4 - 15*a^9*b^2*c^2*d^5 - a^10*b*c*d^6 + a^
11*d^7)*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(1/(b*x+a)**(11/2)/(d*x+c)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 1.22873, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^(11/2)*(d*x + c)^(3/2)),x, algorithm="giac")

[Out]

Done