3.1501 \(\int \frac{1}{(a+b x)^{11/2} \sqrt{c+d x}} \, dx\)

Optimal. Leaf size=171 \[ -\frac{256 d^4 \sqrt{c+d x}}{315 \sqrt{a+b x} (b c-a d)^5}+\frac{128 d^3 \sqrt{c+d x}}{315 (a+b x)^{3/2} (b c-a d)^4}-\frac{32 d^2 \sqrt{c+d x}}{105 (a+b x)^{5/2} (b c-a d)^3}+\frac{16 d \sqrt{c+d x}}{63 (a+b x)^{7/2} (b c-a d)^2}-\frac{2 \sqrt{c+d x}}{9 (a+b x)^{9/2} (b c-a d)} \]

[Out]

(-2*Sqrt[c + d*x])/(9*(b*c - a*d)*(a + b*x)^(9/2)) + (16*d*Sqrt[c + d*x])/(63*(b
*c - a*d)^2*(a + b*x)^(7/2)) - (32*d^2*Sqrt[c + d*x])/(105*(b*c - a*d)^3*(a + b*
x)^(5/2)) + (128*d^3*Sqrt[c + d*x])/(315*(b*c - a*d)^4*(a + b*x)^(3/2)) - (256*d
^4*Sqrt[c + d*x])/(315*(b*c - a*d)^5*Sqrt[a + b*x])

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Rubi [A]  time = 0.15311, antiderivative size = 171, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ -\frac{256 d^4 \sqrt{c+d x}}{315 \sqrt{a+b x} (b c-a d)^5}+\frac{128 d^3 \sqrt{c+d x}}{315 (a+b x)^{3/2} (b c-a d)^4}-\frac{32 d^2 \sqrt{c+d x}}{105 (a+b x)^{5/2} (b c-a d)^3}+\frac{16 d \sqrt{c+d x}}{63 (a+b x)^{7/2} (b c-a d)^2}-\frac{2 \sqrt{c+d x}}{9 (a+b x)^{9/2} (b c-a d)} \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[c + d*x])/(9*(b*c - a*d)*(a + b*x)^(9/2)) + (16*d*Sqrt[c + d*x])/(63*(b
*c - a*d)^2*(a + b*x)^(7/2)) - (32*d^2*Sqrt[c + d*x])/(105*(b*c - a*d)^3*(a + b*
x)^(5/2)) + (128*d^3*Sqrt[c + d*x])/(315*(b*c - a*d)^4*(a + b*x)^(3/2)) - (256*d
^4*Sqrt[c + d*x])/(315*(b*c - a*d)^5*Sqrt[a + b*x])

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Rubi in Sympy [A]  time = 34.1677, size = 153, normalized size = 0.89 \[ \frac{256 d^{4} \sqrt{c + d x}}{315 \sqrt{a + b x} \left (a d - b c\right )^{5}} + \frac{128 d^{3} \sqrt{c + d x}}{315 \left (a + b x\right )^{\frac{3}{2}} \left (a d - b c\right )^{4}} + \frac{32 d^{2} \sqrt{c + d x}}{105 \left (a + b x\right )^{\frac{5}{2}} \left (a d - b c\right )^{3}} + \frac{16 d \sqrt{c + d x}}{63 \left (a + b x\right )^{\frac{7}{2}} \left (a d - b c\right )^{2}} + \frac{2 \sqrt{c + d x}}{9 \left (a + b x\right )^{\frac{9}{2}} \left (a d - b c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

256*d**4*sqrt(c + d*x)/(315*sqrt(a + b*x)*(a*d - b*c)**5) + 128*d**3*sqrt(c + d*
x)/(315*(a + b*x)**(3/2)*(a*d - b*c)**4) + 32*d**2*sqrt(c + d*x)/(105*(a + b*x)*
*(5/2)*(a*d - b*c)**3) + 16*d*sqrt(c + d*x)/(63*(a + b*x)**(7/2)*(a*d - b*c)**2)
+ 2*sqrt(c + d*x)/(9*(a + b*x)**(9/2)*(a*d - b*c))

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Mathematica [A]  time = 0.333191, size = 117, normalized size = 0.68 \[ \frac{2 \sqrt{c+d x} \left (64 d^3 (a+b x)^3 (b c-a d)-48 d^2 (a+b x)^2 (b c-a d)^2+40 d (a+b x) (b c-a d)^3-35 (b c-a d)^4-128 d^4 (a+b x)^4\right )}{315 (a+b x)^{9/2} (b c-a d)^5} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[c + d*x]*(-35*(b*c - a*d)^4 + 40*d*(b*c - a*d)^3*(a + b*x) - 48*d^2*(b*c
- a*d)^2*(a + b*x)^2 + 64*d^3*(b*c - a*d)*(a + b*x)^3 - 128*d^4*(a + b*x)^4))/(
315*(b*c - a*d)^5*(a + b*x)^(9/2))

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Maple [A]  time = 0.014, size = 256, normalized size = 1.5 \[{\frac{256\,{b}^{4}{d}^{4}{x}^{4}+1152\,a{b}^{3}{d}^{4}{x}^{3}-128\,{b}^{4}c{d}^{3}{x}^{3}+2016\,{a}^{2}{b}^{2}{d}^{4}{x}^{2}-576\,a{b}^{3}c{d}^{3}{x}^{2}+96\,{b}^{4}{c}^{2}{d}^{2}{x}^{2}+1680\,{a}^{3}b{d}^{4}x-1008\,{a}^{2}{b}^{2}c{d}^{3}x+432\,a{b}^{3}{c}^{2}{d}^{2}x-80\,{b}^{4}{c}^{3}dx+630\,{a}^{4}{d}^{4}-840\,{a}^{3}bc{d}^{3}+756\,{a}^{2}{b}^{2}{c}^{2}{d}^{2}-360\,a{b}^{3}{c}^{3}d+70\,{b}^{4}{c}^{4}}{315\,{a}^{5}{d}^{5}-1575\,{a}^{4}bc{d}^{4}+3150\,{a}^{3}{b}^{2}{c}^{2}{d}^{3}-3150\,{a}^{2}{b}^{3}{c}^{3}{d}^{2}+1575\,a{b}^{4}{c}^{4}d-315\,{b}^{5}{c}^{5}}\sqrt{dx+c} \left ( bx+a \right ) ^{-{\frac{9}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/315*(d*x+c)^(1/2)*(128*b^4*d^4*x^4+576*a*b^3*d^4*x^3-64*b^4*c*d^3*x^3+1008*a^2
*b^2*d^4*x^2-288*a*b^3*c*d^3*x^2+48*b^4*c^2*d^2*x^2+840*a^3*b*d^4*x-504*a^2*b^2*
c*d^3*x+216*a*b^3*c^2*d^2*x-40*b^4*c^3*d*x+315*a^4*d^4-420*a^3*b*c*d^3+378*a^2*b
^2*c^2*d^2-180*a*b^3*c^3*d+35*b^4*c^4)/(b*x+a)^(9/2)/(a^5*d^5-5*a^4*b*c*d^4+10*a
^3*b^2*c^2*d^3-10*a^2*b^3*c^3*d^2+5*a*b^4*c^4*d-b^5*c^5)

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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)*sqrt(d*x + c)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.16784, size = 861, normalized size = 5.04 \[ -\frac{2 \,{\left (128 \, b^{4} d^{4} x^{4} + 35 \, b^{4} c^{4} - 180 \, a b^{3} c^{3} d + 378 \, a^{2} b^{2} c^{2} d^{2} - 420 \, a^{3} b c d^{3} + 315 \, a^{4} d^{4} - 64 \,{\left (b^{4} c d^{3} - 9 \, a b^{3} d^{4}\right )} x^{3} + 48 \,{\left (b^{4} c^{2} d^{2} - 6 \, a b^{3} c d^{3} + 21 \, a^{2} b^{2} d^{4}\right )} x^{2} - 8 \,{\left (5 \, b^{4} c^{3} d - 27 \, a b^{3} c^{2} d^{2} + 63 \, a^{2} b^{2} c d^{3} - 105 \, a^{3} b d^{4}\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{315 \,{\left (a^{5} b^{5} c^{5} - 5 \, a^{6} b^{4} c^{4} d + 10 \, a^{7} b^{3} c^{3} d^{2} - 10 \, a^{8} b^{2} c^{2} d^{3} + 5 \, a^{9} b c d^{4} - a^{10} d^{5} +{\left (b^{10} c^{5} - 5 \, a b^{9} c^{4} d + 10 \, a^{2} b^{8} c^{3} d^{2} - 10 \, a^{3} b^{7} c^{2} d^{3} + 5 \, a^{4} b^{6} c d^{4} - a^{5} b^{5} d^{5}\right )} x^{5} + 5 \,{\left (a b^{9} c^{5} - 5 \, a^{2} b^{8} c^{4} d + 10 \, a^{3} b^{7} c^{3} d^{2} - 10 \, a^{4} b^{6} c^{2} d^{3} + 5 \, a^{5} b^{5} c d^{4} - a^{6} b^{4} d^{5}\right )} x^{4} + 10 \,{\left (a^{2} b^{8} c^{5} - 5 \, a^{3} b^{7} c^{4} d + 10 \, a^{4} b^{6} c^{3} d^{2} - 10 \, a^{5} b^{5} c^{2} d^{3} + 5 \, a^{6} b^{4} c d^{4} - a^{7} b^{3} d^{5}\right )} x^{3} + 10 \,{\left (a^{3} b^{7} c^{5} - 5 \, a^{4} b^{6} c^{4} d + 10 \, a^{5} b^{5} c^{3} d^{2} - 10 \, a^{6} b^{4} c^{2} d^{3} + 5 \, a^{7} b^{3} c d^{4} - a^{8} b^{2} d^{5}\right )} x^{2} + 5 \,{\left (a^{4} b^{6} c^{5} - 5 \, a^{5} b^{5} c^{4} d + 10 \, a^{6} b^{4} c^{3} d^{2} - 10 \, a^{7} b^{3} c^{2} d^{3} + 5 \, a^{8} b^{2} c d^{4} - a^{9} b d^{5}\right )} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/315*(128*b^4*d^4*x^4 + 35*b^4*c^4 - 180*a*b^3*c^3*d + 378*a^2*b^2*c^2*d^2 - 4
20*a^3*b*c*d^3 + 315*a^4*d^4 - 64*(b^4*c*d^3 - 9*a*b^3*d^4)*x^3 + 48*(b^4*c^2*d^
2 - 6*a*b^3*c*d^3 + 21*a^2*b^2*d^4)*x^2 - 8*(5*b^4*c^3*d - 27*a*b^3*c^2*d^2 + 63
*a^2*b^2*c*d^3 - 105*a^3*b*d^4)*x)*sqrt(b*x + a)*sqrt(d*x + c)/(a^5*b^5*c^5 - 5*
a^6*b^4*c^4*d + 10*a^7*b^3*c^3*d^2 - 10*a^8*b^2*c^2*d^3 + 5*a^9*b*c*d^4 - a^10*d
^5 + (b^10*c^5 - 5*a*b^9*c^4*d + 10*a^2*b^8*c^3*d^2 - 10*a^3*b^7*c^2*d^3 + 5*a^4
*b^6*c*d^4 - a^5*b^5*d^5)*x^5 + 5*(a*b^9*c^5 - 5*a^2*b^8*c^4*d + 10*a^3*b^7*c^3*
d^2 - 10*a^4*b^6*c^2*d^3 + 5*a^5*b^5*c*d^4 - a^6*b^4*d^5)*x^4 + 10*(a^2*b^8*c^5
- 5*a^3*b^7*c^4*d + 10*a^4*b^6*c^3*d^2 - 10*a^5*b^5*c^2*d^3 + 5*a^6*b^4*c*d^4 -
a^7*b^3*d^5)*x^3 + 10*(a^3*b^7*c^5 - 5*a^4*b^6*c^4*d + 10*a^5*b^5*c^3*d^2 - 10*a
^6*b^4*c^2*d^3 + 5*a^7*b^3*c*d^4 - a^8*b^2*d^5)*x^2 + 5*(a^4*b^6*c^5 - 5*a^5*b^5
*c^4*d + 10*a^6*b^4*c^3*d^2 - 10*a^7*b^3*c^2*d^3 + 5*a^8*b^2*c*d^4 - a^9*b*d^5)*
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)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.284503, size = 805, normalized size = 4.71 \[ -\frac{512 \,{\left (b^{8} c^{4} - 4 \, a b^{7} c^{3} d + 6 \, a^{2} b^{6} c^{2} d^{2} - 4 \, a^{3} b^{5} c d^{3} + a^{4} b^{4} d^{4} - 9 \,{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{2} b^{6} c^{3} + 27 \,{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{2} a b^{5} c^{2} d - 27 \,{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{2} a^{2} b^{4} c d^{2} + 9 \,{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{2} a^{3} b^{3} d^{3} + 36 \,{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{4} b^{4} c^{2} - 72 \,{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{4} a b^{3} c d + 36 \,{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{4} a^{2} b^{2} d^{2} - 84 \,{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{6} b^{2} c + 84 \,{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{6} a b d + 126 \,{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{8}\right )} \sqrt{b d} b^{5} d^{4}}{315 \,{\left (b^{2} c - a b d -{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{2}\right )}^{9}{\left | b \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-512/315*(b^8*c^4 - 4*a*b^7*c^3*d + 6*a^2*b^6*c^2*d^2 - 4*a^3*b^5*c*d^3 + a^4*b^
4*d^4 - 9*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*b^6*
c^3 + 27*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a*b^5
*c^2*d - 27*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^
2*b^4*c*d^2 + 9*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^
2*a^3*b^3*d^3 + 36*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d
))^4*b^4*c^2 - 72*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d)
)^4*a*b^3*c*d + 36*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d
))^4*a^2*b^2*d^2 - 84*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*
b*d))^6*b^2*c + 84*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d
))^6*a*b*d + 126*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))
^8)*sqrt(b*d)*b^5*d^4/((b^2*c - a*b*d - (sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c +
(b*x + a)*b*d - a*b*d))^2)^9*abs(b))