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

Optimal. Leaf size=319 \[ -\frac{5 (b c-a d) \left (a^2 d^2-14 a b c d+21 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{8 \sqrt{b} d^{11/2}}+\frac{5 \sqrt{a+b x} \sqrt{c+d x} \left (a^2 d^2-14 a b c d+21 b^2 c^2\right )}{8 d^5}-\frac{5 (a+b x)^{3/2} \sqrt{c+d x} \left (a^2 d^2-14 a b c d+21 b^2 c^2\right )}{12 d^4 (b c-a d)}+\frac{(a+b x)^{5/2} \sqrt{c+d x} \left (a^2 d^2-14 a b c d+21 b^2 c^2\right )}{3 d^3 (b c-a d)^2}+\frac{2 c^2 (a+b x)^{7/2}}{3 d^2 (c+d x)^{3/2} (b c-a d)}-\frac{4 c (a+b x)^{7/2} (5 b c-3 a d)}{3 d^2 \sqrt{c+d x} (b c-a d)^2} \]

[Out]

(2*c^2*(a + b*x)^(7/2))/(3*d^2*(b*c - a*d)*(c + d*x)^(3/2)) - (4*c*(5*b*c - 3*a*
d)*(a + b*x)^(7/2))/(3*d^2*(b*c - a*d)^2*Sqrt[c + d*x]) + (5*(21*b^2*c^2 - 14*a*
b*c*d + a^2*d^2)*Sqrt[a + b*x]*Sqrt[c + d*x])/(8*d^5) - (5*(21*b^2*c^2 - 14*a*b*
c*d + a^2*d^2)*(a + b*x)^(3/2)*Sqrt[c + d*x])/(12*d^4*(b*c - a*d)) + ((21*b^2*c^
2 - 14*a*b*c*d + a^2*d^2)*(a + b*x)^(5/2)*Sqrt[c + d*x])/(3*d^3*(b*c - a*d)^2) -
(5*(b*c - a*d)*(21*b^2*c^2 - 14*a*b*c*d + a^2*d^2)*ArcTanh[(Sqrt[d]*Sqrt[a + b*
x])/(Sqrt[b]*Sqrt[c + d*x])])/(8*Sqrt[b]*d^(11/2))

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Rubi [A]  time = 0.81547, antiderivative size = 319, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ -\frac{5 (b c-a d) \left (a^2 d^2-14 a b c d+21 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{8 \sqrt{b} d^{11/2}}+\frac{5 \sqrt{a+b x} \sqrt{c+d x} \left (a^2 d^2-14 a b c d+21 b^2 c^2\right )}{8 d^5}-\frac{5 (a+b x)^{3/2} \sqrt{c+d x} \left (a^2 d^2-14 a b c d+21 b^2 c^2\right )}{12 d^4 (b c-a d)}+\frac{(a+b x)^{5/2} \sqrt{c+d x} \left (a^2 d^2-14 a b c d+21 b^2 c^2\right )}{3 d^3 (b c-a d)^2}+\frac{2 c^2 (a+b x)^{7/2}}{3 d^2 (c+d x)^{3/2} (b c-a d)}-\frac{4 c (a+b x)^{7/2} (5 b c-3 a d)}{3 d^2 \sqrt{c+d x} (b c-a d)^2} \]

Antiderivative was successfully verified.

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

[Out]

(2*c^2*(a + b*x)^(7/2))/(3*d^2*(b*c - a*d)*(c + d*x)^(3/2)) - (4*c*(5*b*c - 3*a*
d)*(a + b*x)^(7/2))/(3*d^2*(b*c - a*d)^2*Sqrt[c + d*x]) + (5*(21*b^2*c^2 - 14*a*
b*c*d + a^2*d^2)*Sqrt[a + b*x]*Sqrt[c + d*x])/(8*d^5) - (5*(21*b^2*c^2 - 14*a*b*
c*d + a^2*d^2)*(a + b*x)^(3/2)*Sqrt[c + d*x])/(12*d^4*(b*c - a*d)) + ((21*b^2*c^
2 - 14*a*b*c*d + a^2*d^2)*(a + b*x)^(5/2)*Sqrt[c + d*x])/(3*d^3*(b*c - a*d)^2) -
(5*(b*c - a*d)*(21*b^2*c^2 - 14*a*b*c*d + a^2*d^2)*ArcTanh[(Sqrt[d]*Sqrt[a + b*
x])/(Sqrt[b]*Sqrt[c + d*x])])/(8*Sqrt[b]*d^(11/2))

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Rubi in Sympy [A]  time = 64.4578, size = 306, normalized size = 0.96 \[ - \frac{2 c^{2} \left (a + b x\right )^{\frac{7}{2}}}{3 d^{2} \left (c + d x\right )^{\frac{3}{2}} \left (a d - b c\right )} + \frac{4 c \left (a + b x\right )^{\frac{7}{2}} \left (3 a d - 5 b c\right )}{3 d^{2} \sqrt{c + d x} \left (a d - b c\right )^{2}} + \frac{\left (a + b x\right )^{\frac{5}{2}} \sqrt{c + d x} \left (a^{2} d^{2} - 14 a b c d + 21 b^{2} c^{2}\right )}{3 d^{3} \left (a d - b c\right )^{2}} + \frac{5 \left (a + b x\right )^{\frac{3}{2}} \sqrt{c + d x} \left (a^{2} d^{2} - 14 a b c d + 21 b^{2} c^{2}\right )}{12 d^{4} \left (a d - b c\right )} + \frac{5 \sqrt{a + b x} \sqrt{c + d x} \left (a^{2} d^{2} - 14 a b c d + 21 b^{2} c^{2}\right )}{8 d^{5}} + \frac{5 \left (a d - b c\right ) \left (a^{2} d^{2} - 14 a b c d + 21 b^{2} c^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{d} \sqrt{a + b x}}{\sqrt{b} \sqrt{c + d x}} \right )}}{8 \sqrt{b} d^{\frac{11}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*c**2*(a + b*x)**(7/2)/(3*d**2*(c + d*x)**(3/2)*(a*d - b*c)) + 4*c*(a + b*x)**
(7/2)*(3*a*d - 5*b*c)/(3*d**2*sqrt(c + d*x)*(a*d - b*c)**2) + (a + b*x)**(5/2)*s
qrt(c + d*x)*(a**2*d**2 - 14*a*b*c*d + 21*b**2*c**2)/(3*d**3*(a*d - b*c)**2) + 5
*(a + b*x)**(3/2)*sqrt(c + d*x)*(a**2*d**2 - 14*a*b*c*d + 21*b**2*c**2)/(12*d**4
*(a*d - b*c)) + 5*sqrt(a + b*x)*sqrt(c + d*x)*(a**2*d**2 - 14*a*b*c*d + 21*b**2*
c**2)/(8*d**5) + 5*(a*d - b*c)*(a**2*d**2 - 14*a*b*c*d + 21*b**2*c**2)*atanh(sqr
t(d)*sqrt(a + b*x)/(sqrt(b)*sqrt(c + d*x)))/(8*sqrt(b)*d**(11/2))

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Mathematica [A]  time = 0.325133, size = 220, normalized size = 0.69 \[ \frac{5 (a d-b c) \left (a^2 d^2-14 a b c d+21 b^2 c^2\right ) \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right )}{16 \sqrt{b} d^{11/2}}+\frac{\sqrt{a+b x} \left (a^2 d^2 \left (113 c^2+162 c d x+33 d^2 x^2\right )-2 a b d \left (210 c^3+287 c^2 d x+48 c d^2 x^2-13 d^3 x^3\right )+b^2 \left (315 c^4+420 c^3 d x+63 c^2 d^2 x^2-18 c d^3 x^3+8 d^4 x^4\right )\right )}{24 d^5 (c+d x)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[a + b*x]*(a^2*d^2*(113*c^2 + 162*c*d*x + 33*d^2*x^2) - 2*a*b*d*(210*c^3 +
287*c^2*d*x + 48*c*d^2*x^2 - 13*d^3*x^3) + b^2*(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)) + (5*(-(b*c) + a*d
)*(21*b^2*c^2 - 14*a*b*c*d + 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]])/(16*Sqrt[b]*d^(11/2))

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Maple [B]  time = 0.043, size = 1002, normalized size = 3.1 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/48*(b*x+a)^(1/2)*(16*x^4*b^2*d^4*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+15*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^3*d^
5-225*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*b*c*d^4+525*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^2*c^2*d^3-315*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^3*c^3*d^2+52*x^3*a*b*d^4*((b*x+a)*(d*x
+c))^(1/2)*(b*d)^(1/2)-36*x^3*b^2*c*d^3*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+30*l
n(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^3
*c*d^4-450*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*b*c^2*d^3+1050*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^2*c^3*d^2-630*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^3*c^4*d+66*x^2*a^2*d^4*((b*x+a)*(d*x
+c))^(1/2)*(b*d)^(1/2)-192*x^2*a*b*c*d^3*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+126
*x^2*b^2*c^2*d^2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+15*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^3*c^2*d^3-225*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*b*c^3*d^2+52
5*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^2*c^4*d-315*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^3*c^5+324*x*a^2*c*d^3*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-1148*x*a*b
*c^2*d^2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+840*x*b^2*c^3*d*((b*x+a)*(d*x+c))^(
1/2)*(b*d)^(1/2)+226*a^2*c^2*d^2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-840*a*b*c^3
*d*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+630*b^2*c^4*((b*x+a)*(d*x+c))^(1/2)*(b*d)
^(1/2))/((b*x+a)*(d*x+c))^(1/2)/(b*d)^(1/2)/(d*x+c)^(3/2)/d^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((b*x + a)^(5/2)*x^2/(d*x + c)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.34826, 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)^(5/2)*x^2/(d*x + c)^(5/2),x, algorithm="fricas")

[Out]

[1/96*(4*(8*b^2*d^4*x^4 + 315*b^2*c^4 - 420*a*b*c^3*d + 113*a^2*c^2*d^2 - 2*(9*b
^2*c*d^3 - 13*a*b*d^4)*x^3 + 3*(21*b^2*c^2*d^2 - 32*a*b*c*d^3 + 11*a^2*d^4)*x^2
+ 2*(210*b^2*c^3*d - 287*a*b*c^2*d^2 + 81*a^2*c*d^3)*x)*sqrt(b*d)*sqrt(b*x + a)*
sqrt(d*x + c) - 15*(21*b^3*c^5 - 35*a*b^2*c^4*d + 15*a^2*b*c^3*d^2 - a^3*c^2*d^3
+ (21*b^3*c^3*d^2 - 35*a*b^2*c^2*d^3 + 15*a^2*b*c*d^4 - a^3*d^5)*x^2 + 2*(21*b^
3*c^4*d - 35*a*b^2*c^3*d^2 + 15*a^2*b*c^2*d^3 - a^3*c*d^4)*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^7*x^2 + 2*c*d^6*x
+ c^2*d^5)*sqrt(b*d)), 1/48*(2*(8*b^2*d^4*x^4 + 315*b^2*c^4 - 420*a*b*c^3*d + 11
3*a^2*c^2*d^2 - 2*(9*b^2*c*d^3 - 13*a*b*d^4)*x^3 + 3*(21*b^2*c^2*d^2 - 32*a*b*c*
d^3 + 11*a^2*d^4)*x^2 + 2*(210*b^2*c^3*d - 287*a*b*c^2*d^2 + 81*a^2*c*d^3)*x)*sq
rt(-b*d)*sqrt(b*x + a)*sqrt(d*x + c) - 15*(21*b^3*c^5 - 35*a*b^2*c^4*d + 15*a^2*
b*c^3*d^2 - a^3*c^2*d^3 + (21*b^3*c^3*d^2 - 35*a*b^2*c^2*d^3 + 15*a^2*b*c*d^4 -
a^3*d^5)*x^2 + 2*(21*b^3*c^4*d - 35*a*b^2*c^3*d^2 + 15*a^2*b*c^2*d^3 - a^3*c*d^4
)*x)*arctan(1/2*(2*b*d*x + b*c + a*d)*sqrt(-b*d)/(sqrt(b*x + a)*sqrt(d*x + c)*b*
d)))/((d^7*x^2 + 2*c*d^6*x + c^2*d^5)*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)**(5/2)/(d*x+c)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.274546, size = 694, normalized size = 2.18 \[ \frac{{\left ({\left ({\left (2 \,{\left (b x + a\right )}{\left (\frac{4 \,{\left (b^{6} c d^{8} - a b^{5} d^{9}\right )}{\left (b x + a\right )}}{b^{4} c d^{9}{\left | b \right |} - a b^{3} d^{10}{\left | b \right |}} - \frac{3 \,{\left (3 \, b^{7} c^{2} d^{7} - 2 \, a b^{6} c d^{8} - a^{2} b^{5} d^{9}\right )}}{b^{4} c d^{9}{\left | b \right |} - a b^{3} d^{10}{\left | b \right |}}\right )} + \frac{3 \,{\left (21 \, b^{8} c^{3} d^{6} - 35 \, a b^{7} c^{2} d^{7} + 15 \, a^{2} b^{6} c d^{8} - a^{3} b^{5} d^{9}\right )}}{b^{4} c d^{9}{\left | b \right |} - a b^{3} d^{10}{\left | b \right |}}\right )}{\left (b x + a\right )} + \frac{20 \,{\left (21 \, b^{9} c^{4} d^{5} - 56 \, a b^{8} c^{3} d^{6} + 50 \, a^{2} b^{7} c^{2} d^{7} - 16 \, a^{3} b^{6} c d^{8} + a^{4} b^{5} d^{9}\right )}}{b^{4} c d^{9}{\left | b \right |} - a b^{3} d^{10}{\left | b \right |}}\right )}{\left (b x + a\right )} + \frac{15 \,{\left (21 \, b^{10} c^{5} d^{4} - 77 \, a b^{9} c^{4} d^{5} + 106 \, a^{2} b^{8} c^{3} d^{6} - 66 \, a^{3} b^{7} c^{2} d^{7} + 17 \, a^{4} b^{6} c d^{8} - a^{5} b^{5} d^{9}\right )}}{b^{4} c d^{9}{\left | b \right |} - a b^{3} d^{10}{\left | b \right |}}\right )} \sqrt{b x + a}}{24 \,{\left (b^{2} c +{\left (b x + a\right )} b d - a b d\right )}^{\frac{3}{2}}} + \frac{5 \,{\left (21 \, b^{4} c^{3} - 35 \, a b^{3} c^{2} d + 15 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )}{\rm ln}\left ({\left | -\sqrt{b d} \sqrt{b x + a} + \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d} \right |}\right )}{8 \, \sqrt{b d} d^{5}{\left | b \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(5/2)*x^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^4*c*d^9*abs(b) - a*b
^3*d^10*abs(b)) - 3*(3*b^7*c^2*d^7 - 2*a*b^6*c*d^8 - a^2*b^5*d^9)/(b^4*c*d^9*abs
(b) - a*b^3*d^10*abs(b))) + 3*(21*b^8*c^3*d^6 - 35*a*b^7*c^2*d^7 + 15*a^2*b^6*c*
d^8 - a^3*b^5*d^9)/(b^4*c*d^9*abs(b) - a*b^3*d^10*abs(b)))*(b*x + a) + 20*(21*b^
9*c^4*d^5 - 56*a*b^8*c^3*d^6 + 50*a^2*b^7*c^2*d^7 - 16*a^3*b^6*c*d^8 + a^4*b^5*d
^9)/(b^4*c*d^9*abs(b) - a*b^3*d^10*abs(b)))*(b*x + a) + 15*(21*b^10*c^5*d^4 - 77
*a*b^9*c^4*d^5 + 106*a^2*b^8*c^3*d^6 - 66*a^3*b^7*c^2*d^7 + 17*a^4*b^6*c*d^8 - a
^5*b^5*d^9)/(b^4*c*d^9*abs(b) - a*b^3*d^10*abs(b)))*sqrt(b*x + a)/(b^2*c + (b*x
+ a)*b*d - a*b*d)^(3/2) + 5/8*(21*b^4*c^3 - 35*a*b^3*c^2*d + 15*a^2*b^2*c*d^2 -
a^3*b*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))