3.1512 \(\int \frac{(a+b x)^{9/2}}{(c+d x)^{5/2}} \, dx\)

Optimal. Leaf size=204 \[ -\frac{105 b^{3/2} (b c-a d)^3 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{8 d^{11/2}}+\frac{105 b^2 \sqrt{a+b x} \sqrt{c+d x} (b c-a d)^2}{8 d^5}-\frac{35 b^2 (a+b x)^{3/2} \sqrt{c+d x} (b c-a d)}{4 d^4}+\frac{7 b^2 (a+b x)^{5/2} \sqrt{c+d x}}{d^3}-\frac{6 b (a+b x)^{7/2}}{d^2 \sqrt{c+d x}}-\frac{2 (a+b x)^{9/2}}{3 d (c+d x)^{3/2}} \]

[Out]

(-2*(a + b*x)^(9/2))/(3*d*(c + d*x)^(3/2)) - (6*b*(a + b*x)^(7/2))/(d^2*Sqrt[c +
d*x]) + (105*b^2*(b*c - a*d)^2*Sqrt[a + b*x]*Sqrt[c + d*x])/(8*d^5) - (35*b^2*(
b*c - a*d)*(a + b*x)^(3/2)*Sqrt[c + d*x])/(4*d^4) + (7*b^2*(a + b*x)^(5/2)*Sqrt[
c + d*x])/d^3 - (105*b^(3/2)*(b*c - a*d)^3*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt
[b]*Sqrt[c + d*x])])/(8*d^(11/2))

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Rubi [A]  time = 0.292248, antiderivative size = 204, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21 \[ -\frac{105 b^{3/2} (b c-a d)^3 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{8 d^{11/2}}+\frac{105 b^2 \sqrt{a+b x} \sqrt{c+d x} (b c-a d)^2}{8 d^5}-\frac{35 b^2 (a+b x)^{3/2} \sqrt{c+d x} (b c-a d)}{4 d^4}+\frac{7 b^2 (a+b x)^{5/2} \sqrt{c+d x}}{d^3}-\frac{6 b (a+b x)^{7/2}}{d^2 \sqrt{c+d x}}-\frac{2 (a+b x)^{9/2}}{3 d (c+d x)^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^(9/2)/(c + d*x)^(5/2),x]

[Out]

(-2*(a + b*x)^(9/2))/(3*d*(c + d*x)^(3/2)) - (6*b*(a + b*x)^(7/2))/(d^2*Sqrt[c +
d*x]) + (105*b^2*(b*c - a*d)^2*Sqrt[a + b*x]*Sqrt[c + d*x])/(8*d^5) - (35*b^2*(
b*c - a*d)*(a + b*x)^(3/2)*Sqrt[c + d*x])/(4*d^4) + (7*b^2*(a + b*x)^(5/2)*Sqrt[
c + d*x])/d^3 - (105*b^(3/2)*(b*c - a*d)^3*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt
[b]*Sqrt[c + d*x])])/(8*d^(11/2))

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Rubi in Sympy [A]  time = 42.4204, size = 190, normalized size = 0.93 \[ \frac{105 b^{\frac{3}{2}} \left (a d - b c\right )^{3} \operatorname{atanh}{\left (\frac{\sqrt{d} \sqrt{a + b x}}{\sqrt{b} \sqrt{c + d x}} \right )}}{8 d^{\frac{11}{2}}} + \frac{7 b^{2} \left (a + b x\right )^{\frac{5}{2}} \sqrt{c + d x}}{d^{3}} + \frac{35 b^{2} \left (a + b x\right )^{\frac{3}{2}} \sqrt{c + d x} \left (a d - b c\right )}{4 d^{4}} + \frac{105 b^{2} \sqrt{a + b x} \sqrt{c + d x} \left (a d - b c\right )^{2}}{8 d^{5}} - \frac{6 b \left (a + b x\right )^{\frac{7}{2}}}{d^{2} \sqrt{c + d x}} - \frac{2 \left (a + b x\right )^{\frac{9}{2}}}{3 d \left (c + d x\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**(9/2)/(d*x+c)**(5/2),x)

[Out]

105*b**(3/2)*(a*d - b*c)**3*atanh(sqrt(d)*sqrt(a + b*x)/(sqrt(b)*sqrt(c + d*x)))
/(8*d**(11/2)) + 7*b**2*(a + b*x)**(5/2)*sqrt(c + d*x)/d**3 + 35*b**2*(a + b*x)*
*(3/2)*sqrt(c + d*x)*(a*d - b*c)/(4*d**4) + 105*b**2*sqrt(a + b*x)*sqrt(c + d*x)
*(a*d - b*c)**2/(8*d**5) - 6*b*(a + b*x)**(7/2)/(d**2*sqrt(c + d*x)) - 2*(a + b*
x)**(9/2)/(3*d*(c + d*x)**(3/2))

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Mathematica [A]  time = 0.45911, size = 231, normalized size = 1.13 \[ \frac{\sqrt{a+b x} \left (-16 a^4 d^4-16 a^3 b d^3 (9 c+13 d x)+3 a^2 b^2 d^2 \left (231 c^2+318 c d x+55 d^2 x^2\right )-2 a b^3 d \left (420 c^3+567 c^2 d x+90 c d^2 x^2-25 d^3 x^3\right )+b^4 \left (315 c^4+420 c^3 d x+63 c^2 d^2 x^2-18 c d^3 x^3+8 d^4 x^4\right )\right )}{24 d^5 (c+d x)^{3/2}}-\frac{105 b^{3/2} (b c-a d)^3 \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right )}{16 d^{11/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^(9/2)/(c + d*x)^(5/2),x]

[Out]

(Sqrt[a + b*x]*(-16*a^4*d^4 - 16*a^3*b*d^3*(9*c + 13*d*x) + 3*a^2*b^2*d^2*(231*c
^2 + 318*c*d*x + 55*d^2*x^2) - 2*a*b^3*d*(420*c^3 + 567*c^2*d*x + 90*c*d^2*x^2 -
25*d^3*x^3) + b^4*(315*c^4 + 420*c^3*d*x + 63*c^2*d^2*x^2 - 18*c*d^3*x^3 + 8*d^
4*x^4)))/(24*d^5*(c + d*x)^(3/2)) - (105*b^(3/2)*(b*c - a*d)^3*Log[b*c + a*d + 2
*b*d*x + 2*Sqrt[b]*Sqrt[d]*Sqrt[a + b*x]*Sqrt[c + d*x]])/(16*d^(11/2))

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Maple [F]  time = 0.057, size = 0, normalized size = 0. \[ \int{1 \left ( bx+a \right ) ^{{\frac{9}{2}}} \left ( dx+c \right ) ^{-{\frac{5}{2}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^(9/2)/(d*x+c)^(5/2),x)

[Out]

int((b*x+a)^(9/2)/(d*x+c)^(5/2),x)

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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((b*x + a)^(9/2)/(d*x + c)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.26284, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(9/2)/(d*x + c)^(5/2),x, algorithm="fricas")

[Out]

[-1/96*(315*(b^4*c^5 - 3*a*b^3*c^4*d + 3*a^2*b^2*c^3*d^2 - a^3*b*c^2*d^3 + (b^4*
c^3*d^2 - 3*a*b^3*c^2*d^3 + 3*a^2*b^2*c*d^4 - a^3*b*d^5)*x^2 + 2*(b^4*c^4*d - 3*
a*b^3*c^3*d^2 + 3*a^2*b^2*c^2*d^3 - a^3*b*c*d^4)*x)*sqrt(b/d)*log(8*b^2*d^2*x^2
+ b^2*c^2 + 6*a*b*c*d + a^2*d^2 + 4*(2*b*d^2*x + b*c*d + a*d^2)*sqrt(b*x + a)*sq
rt(d*x + c)*sqrt(b/d) + 8*(b^2*c*d + a*b*d^2)*x) - 4*(8*b^4*d^4*x^4 + 315*b^4*c^
4 - 840*a*b^3*c^3*d + 693*a^2*b^2*c^2*d^2 - 144*a^3*b*c*d^3 - 16*a^4*d^4 - 2*(9*
b^4*c*d^3 - 25*a*b^3*d^4)*x^3 + 3*(21*b^4*c^2*d^2 - 60*a*b^3*c*d^3 + 55*a^2*b^2*
d^4)*x^2 + 2*(210*b^4*c^3*d - 567*a*b^3*c^2*d^2 + 477*a^2*b^2*c*d^3 - 104*a^3*b*
d^4)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(d^7*x^2 + 2*c*d^6*x + c^2*d^5), -1/48*(315
*(b^4*c^5 - 3*a*b^3*c^4*d + 3*a^2*b^2*c^3*d^2 - a^3*b*c^2*d^3 + (b^4*c^3*d^2 - 3
*a*b^3*c^2*d^3 + 3*a^2*b^2*c*d^4 - a^3*b*d^5)*x^2 + 2*(b^4*c^4*d - 3*a*b^3*c^3*d
^2 + 3*a^2*b^2*c^2*d^3 - a^3*b*c*d^4)*x)*sqrt(-b/d)*arctan(1/2*(2*b*d*x + b*c +
a*d)/(sqrt(b*x + a)*sqrt(d*x + c)*d*sqrt(-b/d))) - 2*(8*b^4*d^4*x^4 + 315*b^4*c^
4 - 840*a*b^3*c^3*d + 693*a^2*b^2*c^2*d^2 - 144*a^3*b*c*d^3 - 16*a^4*d^4 - 2*(9*
b^4*c*d^3 - 25*a*b^3*d^4)*x^3 + 3*(21*b^4*c^2*d^2 - 60*a*b^3*c*d^3 + 55*a^2*b^2*
d^4)*x^2 + 2*(210*b^4*c^3*d - 567*a*b^3*c^2*d^2 + 477*a^2*b^2*c*d^3 - 104*a^3*b*
d^4)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(d^7*x^2 + 2*c*d^6*x + c^2*d^5)]

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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((b*x+a)**(9/2)/(d*x+c)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.298346, size = 675, normalized size = 3.31 \[ \frac{{\left ({\left ({\left (2 \,{\left (b x + a\right )}{\left (\frac{4 \,{\left (b^{6} c d^{8} - a b^{5} d^{9}\right )}{\left (b x + a\right )}}{b^{2} c d^{9}{\left | b \right |} - a b d^{10}{\left | b \right |}} - \frac{9 \,{\left (b^{7} c^{2} d^{7} - 2 \, a b^{6} c d^{8} + a^{2} b^{5} d^{9}\right )}}{b^{2} c d^{9}{\left | b \right |} - a b d^{10}{\left | b \right |}}\right )} + \frac{63 \,{\left (b^{8} c^{3} d^{6} - 3 \, a b^{7} c^{2} d^{7} + 3 \, a^{2} b^{6} c d^{8} - a^{3} b^{5} d^{9}\right )}}{b^{2} c d^{9}{\left | b \right |} - a b d^{10}{\left | b \right |}}\right )}{\left (b x + a\right )} + \frac{420 \,{\left (b^{9} c^{4} d^{5} - 4 \, a b^{8} c^{3} d^{6} + 6 \, a^{2} b^{7} c^{2} d^{7} - 4 \, a^{3} b^{6} c d^{8} + a^{4} b^{5} d^{9}\right )}}{b^{2} c d^{9}{\left | b \right |} - a b d^{10}{\left | b \right |}}\right )}{\left (b x + a\right )} + \frac{315 \,{\left (b^{10} c^{5} d^{4} - 5 \, a b^{9} c^{4} d^{5} + 10 \, a^{2} b^{8} c^{3} d^{6} - 10 \, a^{3} b^{7} c^{2} d^{7} + 5 \, a^{4} b^{6} c d^{8} - a^{5} b^{5} d^{9}\right )}}{b^{2} c d^{9}{\left | b \right |} - a b d^{10}{\left | b \right |}}\right )} \sqrt{b x + a}}{24 \,{\left (b^{2} c +{\left (b x + a\right )} b d - a b d\right )}^{\frac{3}{2}}} + \frac{105 \,{\left (b^{6} c^{3} - 3 \, a b^{5} c^{2} d + 3 \, a^{2} b^{4} c d^{2} - a^{3} b^{3} d^{3}\right )}{\rm ln}\left ({\left | -\sqrt{b d} \sqrt{b x + a} + \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d} \right |}\right )}{8 \, \sqrt{b d} d^{5}{\left | b \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(9/2)/(d*x + c)^(5/2),x, algorithm="giac")

[Out]

1/24*(((2*(b*x + a)*(4*(b^6*c*d^8 - a*b^5*d^9)*(b*x + a)/(b^2*c*d^9*abs(b) - a*b
*d^10*abs(b)) - 9*(b^7*c^2*d^7 - 2*a*b^6*c*d^8 + a^2*b^5*d^9)/(b^2*c*d^9*abs(b)
- a*b*d^10*abs(b))) + 63*(b^8*c^3*d^6 - 3*a*b^7*c^2*d^7 + 3*a^2*b^6*c*d^8 - a^3*
b^5*d^9)/(b^2*c*d^9*abs(b) - a*b*d^10*abs(b)))*(b*x + a) + 420*(b^9*c^4*d^5 - 4*
a*b^8*c^3*d^6 + 6*a^2*b^7*c^2*d^7 - 4*a^3*b^6*c*d^8 + a^4*b^5*d^9)/(b^2*c*d^9*ab
s(b) - a*b*d^10*abs(b)))*(b*x + a) + 315*(b^10*c^5*d^4 - 5*a*b^9*c^4*d^5 + 10*a^
2*b^8*c^3*d^6 - 10*a^3*b^7*c^2*d^7 + 5*a^4*b^6*c*d^8 - a^5*b^5*d^9)/(b^2*c*d^9*a
bs(b) - a*b*d^10*abs(b)))*sqrt(b*x + a)/(b^2*c + (b*x + a)*b*d - a*b*d)^(3/2) +
105/8*(b^6*c^3 - 3*a*b^5*c^2*d + 3*a^2*b^4*c*d^2 - a^3*b^3*d^3)*ln(abs(-sqrt(b*d
)*sqrt(b*x + a) + sqrt(b^2*c + (b*x + a)*b*d - a*b*d)))/(sqrt(b*d)*d^5*abs(b))