3.629 \(\int \frac{x^2 (a+b x)^{3/2}}{(c+d x)^{5/2}} \, dx\)

Optimal. Leaf size=267 \[ \frac{\left (3 a^2 d^2-30 a b c d+35 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{4 \sqrt{b} d^{9/2}}-\frac{\sqrt{a+b x} \sqrt{c+d x} \left (3 a^2 d^2-30 a b c d+35 b^2 c^2\right )}{4 d^4 (b c-a d)}+\frac{(a+b x)^{3/2} \sqrt{c+d x} \left (3 a^2 d^2-30 a b c d+35 b^2 c^2\right )}{6 d^3 (b c-a d)^2}+\frac{2 c^2 (a+b x)^{5/2}}{3 d^2 (c+d x)^{3/2} (b c-a d)}-\frac{4 c (a+b x)^{5/2} (4 b c-3 a d)}{3 d^2 \sqrt{c+d x} (b c-a d)^2} \]

[Out]

(2*c^2*(a + b*x)^(5/2))/(3*d^2*(b*c - a*d)*(c + d*x)^(3/2)) - (4*c*(4*b*c - 3*a*
d)*(a + b*x)^(5/2))/(3*d^2*(b*c - a*d)^2*Sqrt[c + d*x]) - ((35*b^2*c^2 - 30*a*b*
c*d + 3*a^2*d^2)*Sqrt[a + b*x]*Sqrt[c + d*x])/(4*d^4*(b*c - a*d)) + ((35*b^2*c^2
- 30*a*b*c*d + 3*a^2*d^2)*(a + b*x)^(3/2)*Sqrt[c + d*x])/(6*d^3*(b*c - a*d)^2)
+ ((35*b^2*c^2 - 30*a*b*c*d + 3*a^2*d^2)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b
]*Sqrt[c + d*x])])/(4*Sqrt[b]*d^(9/2))

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Rubi [A]  time = 0.656536, antiderivative size = 267, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ \frac{\left (3 a^2 d^2-30 a b c d+35 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{4 \sqrt{b} d^{9/2}}-\frac{\sqrt{a+b x} \sqrt{c+d x} \left (3 a^2 d^2-30 a b c d+35 b^2 c^2\right )}{4 d^4 (b c-a d)}+\frac{(a+b x)^{3/2} \sqrt{c+d x} \left (3 a^2 d^2-30 a b c d+35 b^2 c^2\right )}{6 d^3 (b c-a d)^2}+\frac{2 c^2 (a+b x)^{5/2}}{3 d^2 (c+d x)^{3/2} (b c-a d)}-\frac{4 c (a+b x)^{5/2} (4 b c-3 a d)}{3 d^2 \sqrt{c+d x} (b c-a d)^2} \]

Antiderivative was successfully verified.

[In]  Int[(x^2*(a + b*x)^(3/2))/(c + d*x)^(5/2),x]

[Out]

(2*c^2*(a + b*x)^(5/2))/(3*d^2*(b*c - a*d)*(c + d*x)^(3/2)) - (4*c*(4*b*c - 3*a*
d)*(a + b*x)^(5/2))/(3*d^2*(b*c - a*d)^2*Sqrt[c + d*x]) - ((35*b^2*c^2 - 30*a*b*
c*d + 3*a^2*d^2)*Sqrt[a + b*x]*Sqrt[c + d*x])/(4*d^4*(b*c - a*d)) + ((35*b^2*c^2
- 30*a*b*c*d + 3*a^2*d^2)*(a + b*x)^(3/2)*Sqrt[c + d*x])/(6*d^3*(b*c - a*d)^2)
+ ((35*b^2*c^2 - 30*a*b*c*d + 3*a^2*d^2)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b
]*Sqrt[c + d*x])])/(4*Sqrt[b]*d^(9/2))

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Rubi in Sympy [A]  time = 50.8423, size = 253, normalized size = 0.95 \[ - \frac{2 c^{2} \left (a + b x\right )^{\frac{5}{2}}}{3 d^{2} \left (c + d x\right )^{\frac{3}{2}} \left (a d - b c\right )} + \frac{4 c \left (a + b x\right )^{\frac{5}{2}} \left (3 a d - 4 b c\right )}{3 d^{2} \sqrt{c + d x} \left (a d - b c\right )^{2}} + \frac{\left (a + b x\right )^{\frac{3}{2}} \sqrt{c + d x} \left (3 a^{2} d^{2} - 30 a b c d + 35 b^{2} c^{2}\right )}{6 d^{3} \left (a d - b c\right )^{2}} + \frac{\sqrt{a + b x} \sqrt{c + d x} \left (3 a^{2} d^{2} - 30 a b c d + 35 b^{2} c^{2}\right )}{4 d^{4} \left (a d - b c\right )} + \frac{\left (3 a^{2} d^{2} - 30 a b c d + 35 b^{2} c^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{d} \sqrt{a + b x}}{\sqrt{b} \sqrt{c + d x}} \right )}}{4 \sqrt{b} d^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**2*(b*x+a)**(3/2)/(d*x+c)**(5/2),x)

[Out]

-2*c**2*(a + b*x)**(5/2)/(3*d**2*(c + d*x)**(3/2)*(a*d - b*c)) + 4*c*(a + b*x)**
(5/2)*(3*a*d - 4*b*c)/(3*d**2*sqrt(c + d*x)*(a*d - b*c)**2) + (a + b*x)**(3/2)*s
qrt(c + d*x)*(3*a**2*d**2 - 30*a*b*c*d + 35*b**2*c**2)/(6*d**3*(a*d - b*c)**2) +
sqrt(a + b*x)*sqrt(c + d*x)*(3*a**2*d**2 - 30*a*b*c*d + 35*b**2*c**2)/(4*d**4*(
a*d - b*c)) + (3*a**2*d**2 - 30*a*b*c*d + 35*b**2*c**2)*atanh(sqrt(d)*sqrt(a + b
*x)/(sqrt(b)*sqrt(c + d*x)))/(4*sqrt(b)*d**(9/2))

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Mathematica [A]  time = 0.212597, size = 162, normalized size = 0.61 \[ \frac{\left (3 a^2 d^2-30 a b c d+35 b^2 c^2\right ) \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right )}{8 \sqrt{b} d^{9/2}}+\frac{\sqrt{a+b x} \left (a d \left (55 c^2+78 c d x+15 d^2 x^2\right )-b \left (105 c^3+140 c^2 d x+21 c d^2 x^2-6 d^3 x^3\right )\right )}{12 d^4 (c+d x)^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^2*(a + b*x)^(3/2))/(c + d*x)^(5/2),x]

[Out]

(Sqrt[a + b*x]*(a*d*(55*c^2 + 78*c*d*x + 15*d^2*x^2) - b*(105*c^3 + 140*c^2*d*x
+ 21*c*d^2*x^2 - 6*d^3*x^3)))/(12*d^4*(c + d*x)^(3/2)) + ((35*b^2*c^2 - 30*a*b*c
*d + 3*a^2*d^2)*Log[b*c + a*d + 2*b*d*x + 2*Sqrt[b]*Sqrt[d]*Sqrt[a + b*x]*Sqrt[c
+ d*x]])/(8*Sqrt[b]*d^(9/2))

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Maple [B]  time = 0.037, size = 676, normalized size = 2.5 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2*(b*x+a)^(3/2)/(d*x+c)^(5/2),x)

[Out]

1/24*(b*x+a)^(1/2)*(9*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+
b*c)/(b*d)^(1/2))*x^2*a^2*d^4-90*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)
^(1/2)+a*d+b*c)/(b*d)^(1/2))*x^2*a*b*c*d^3+105*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c
))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*x^2*b^2*c^2*d^2+12*x^3*b*d^3*((b*x+a)
*(d*x+c))^(1/2)*(b*d)^(1/2)+18*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(
1/2)+a*d+b*c)/(b*d)^(1/2))*x*a^2*c*d^3-180*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(
1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*x*a*b*c^2*d^2+210*ln(1/2*(2*b*d*x+2*((b*x
+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*x*b^2*c^3*d+30*x^2*a*d^3*((
b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-42*x^2*b*c*d^2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(
1/2)+9*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2
))*a^2*c^2*d^2-90*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)
/(b*d)^(1/2))*a*b*c^3*d+105*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2
)+a*d+b*c)/(b*d)^(1/2))*b^2*c^4+156*x*a*c*d^2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2
)-280*x*b*c^2*d*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+110*a*c^2*d*((b*x+a)*(d*x+c)
)^(1/2)*(b*d)^(1/2)-210*b*c^3*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2))/(b*d)^(1/2)/(
(b*x+a)*(d*x+c))^(1/2)/d^4/(d*x+c)^(3/2)

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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)^(3/2)*x^2/(d*x + c)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.710259, size = 1, normalized size = 0. \[ \left [\frac{4 \,{\left (6 \, b d^{3} x^{3} - 105 \, b c^{3} + 55 \, a c^{2} d - 3 \,{\left (7 \, b c d^{2} - 5 \, a d^{3}\right )} x^{2} - 2 \,{\left (70 \, b c^{2} d - 39 \, a c d^{2}\right )} x\right )} \sqrt{b d} \sqrt{b x + a} \sqrt{d x + c} + 3 \,{\left (35 \, b^{2} c^{4} - 30 \, a b c^{3} d + 3 \, a^{2} c^{2} d^{2} +{\left (35 \, b^{2} c^{2} d^{2} - 30 \, a b c d^{3} + 3 \, a^{2} d^{4}\right )} x^{2} + 2 \,{\left (35 \, b^{2} c^{3} d - 30 \, a b c^{2} d^{2} + 3 \, a^{2} c d^{3}\right )} x\right )} \log \left (4 \,{\left (2 \, b^{2} d^{2} x + b^{2} c d + a b d^{2}\right )} \sqrt{b x + a} \sqrt{d x + c} +{\left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 8 \,{\left (b^{2} c d + a b d^{2}\right )} x\right )} \sqrt{b d}\right )}{48 \,{\left (d^{6} x^{2} + 2 \, c d^{5} x + c^{2} d^{4}\right )} \sqrt{b d}}, \frac{2 \,{\left (6 \, b d^{3} x^{3} - 105 \, b c^{3} + 55 \, a c^{2} d - 3 \,{\left (7 \, b c d^{2} - 5 \, a d^{3}\right )} x^{2} - 2 \,{\left (70 \, b c^{2} d - 39 \, a c d^{2}\right )} x\right )} \sqrt{-b d} \sqrt{b x + a} \sqrt{d x + c} + 3 \,{\left (35 \, b^{2} c^{4} - 30 \, a b c^{3} d + 3 \, a^{2} c^{2} d^{2} +{\left (35 \, b^{2} c^{2} d^{2} - 30 \, a b c d^{3} + 3 \, a^{2} d^{4}\right )} x^{2} + 2 \,{\left (35 \, b^{2} c^{3} d - 30 \, a b c^{2} d^{2} + 3 \, a^{2} c d^{3}\right )} x\right )} \arctan \left (\frac{{\left (2 \, b d x + b c + a d\right )} \sqrt{-b d}}{2 \, \sqrt{b x + a} \sqrt{d x + c} b d}\right )}{24 \,{\left (d^{6} x^{2} + 2 \, c d^{5} x + c^{2} d^{4}\right )} \sqrt{-b d}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(3/2)*x^2/(d*x + c)^(5/2),x, algorithm="fricas")

[Out]

[1/48*(4*(6*b*d^3*x^3 - 105*b*c^3 + 55*a*c^2*d - 3*(7*b*c*d^2 - 5*a*d^3)*x^2 - 2
*(70*b*c^2*d - 39*a*c*d^2)*x)*sqrt(b*d)*sqrt(b*x + a)*sqrt(d*x + c) + 3*(35*b^2*
c^4 - 30*a*b*c^3*d + 3*a^2*c^2*d^2 + (35*b^2*c^2*d^2 - 30*a*b*c*d^3 + 3*a^2*d^4)
*x^2 + 2*(35*b^2*c^3*d - 30*a*b*c^2*d^2 + 3*a^2*c*d^3)*x)*log(4*(2*b^2*d^2*x + b
^2*c*d + a*b*d^2)*sqrt(b*x + a)*sqrt(d*x + c) + (8*b^2*d^2*x^2 + b^2*c^2 + 6*a*b
*c*d + a^2*d^2 + 8*(b^2*c*d + a*b*d^2)*x)*sqrt(b*d)))/((d^6*x^2 + 2*c*d^5*x + c^
2*d^4)*sqrt(b*d)), 1/24*(2*(6*b*d^3*x^3 - 105*b*c^3 + 55*a*c^2*d - 3*(7*b*c*d^2
- 5*a*d^3)*x^2 - 2*(70*b*c^2*d - 39*a*c*d^2)*x)*sqrt(-b*d)*sqrt(b*x + a)*sqrt(d*
x + c) + 3*(35*b^2*c^4 - 30*a*b*c^3*d + 3*a^2*c^2*d^2 + (35*b^2*c^2*d^2 - 30*a*b
*c*d^3 + 3*a^2*d^4)*x^2 + 2*(35*b^2*c^3*d - 30*a*b*c^2*d^2 + 3*a^2*c*d^3)*x)*arc
tan(1/2*(2*b*d*x + b*c + a*d)*sqrt(-b*d)/(sqrt(b*x + a)*sqrt(d*x + c)*b*d)))/((d
^6*x^2 + 2*c*d^5*x + c^2*d^4)*sqrt(-b*d))]

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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(x**2*(b*x+a)**(3/2)/(d*x+c)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.274877, size = 531, normalized size = 1.99 \[ \frac{{\left ({\left (3 \,{\left (b x + a\right )}{\left (\frac{2 \,{\left (b^{6} c d^{6} - a b^{5} d^{7}\right )}{\left (b x + a\right )}}{b^{4} c d^{7}{\left | b \right |} - a b^{3} d^{8}{\left | b \right |}} - \frac{7 \, b^{7} c^{2} d^{5} - 6 \, a b^{6} c d^{6} - a^{2} b^{5} d^{7}}{b^{4} c d^{7}{\left | b \right |} - a b^{3} d^{8}{\left | b \right |}}\right )} - \frac{4 \,{\left (35 \, b^{8} c^{3} d^{4} - 65 \, a b^{7} c^{2} d^{5} + 33 \, a^{2} b^{6} c d^{6} - 3 \, a^{3} b^{5} d^{7}\right )}}{b^{4} c d^{7}{\left | b \right |} - a b^{3} d^{8}{\left | b \right |}}\right )}{\left (b x + a\right )} - \frac{3 \,{\left (35 \, b^{9} c^{4} d^{3} - 100 \, a b^{8} c^{3} d^{4} + 98 \, a^{2} b^{7} c^{2} d^{5} - 36 \, a^{3} b^{6} c d^{6} + 3 \, a^{4} b^{5} d^{7}\right )}}{b^{4} c d^{7}{\left | b \right |} - a b^{3} d^{8}{\left | b \right |}}\right )} \sqrt{b x + a}}{12 \,{\left (b^{2} c +{\left (b x + a\right )} b d - a b d\right )}^{\frac{3}{2}}} - \frac{{\left (35 \, b^{3} c^{2} - 30 \, a b^{2} c d + 3 \, a^{2} b d^{2}\right )}{\rm ln}\left ({\left | -\sqrt{b d} \sqrt{b x + a} + \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d} \right |}\right )}{4 \, \sqrt{b d} d^{4}{\left | b \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(3/2)*x^2/(d*x + c)^(5/2),x, algorithm="giac")

[Out]

1/12*((3*(b*x + a)*(2*(b^6*c*d^6 - a*b^5*d^7)*(b*x + a)/(b^4*c*d^7*abs(b) - a*b^
3*d^8*abs(b)) - (7*b^7*c^2*d^5 - 6*a*b^6*c*d^6 - a^2*b^5*d^7)/(b^4*c*d^7*abs(b)
- a*b^3*d^8*abs(b))) - 4*(35*b^8*c^3*d^4 - 65*a*b^7*c^2*d^5 + 33*a^2*b^6*c*d^6 -
3*a^3*b^5*d^7)/(b^4*c*d^7*abs(b) - a*b^3*d^8*abs(b)))*(b*x + a) - 3*(35*b^9*c^4
*d^3 - 100*a*b^8*c^3*d^4 + 98*a^2*b^7*c^2*d^5 - 36*a^3*b^6*c*d^6 + 3*a^4*b^5*d^7
)/(b^4*c*d^7*abs(b) - a*b^3*d^8*abs(b)))*sqrt(b*x + a)/(b^2*c + (b*x + a)*b*d -
a*b*d)^(3/2) - 1/4*(35*b^3*c^2 - 30*a*b^2*c*d + 3*a^2*b*d^2)*ln(abs(-sqrt(b*d)*s
qrt(b*x + a) + sqrt(b^2*c + (b*x + a)*b*d - a*b*d)))/(sqrt(b*d)*d^4*abs(b))