3.1502 \(\int \frac{(a+b x)^{7/2}}{(c+d x)^{3/2}} \, dx\)

Optimal. Leaf size=174 \[ -\frac{35 \sqrt{b} (b c-a d)^3 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{8 d^{9/2}}+\frac{35 b \sqrt{a+b x} \sqrt{c+d x} (b c-a d)^2}{8 d^4}-\frac{35 b (a+b x)^{3/2} \sqrt{c+d x} (b c-a d)}{12 d^3}+\frac{7 b (a+b x)^{5/2} \sqrt{c+d x}}{3 d^2}-\frac{2 (a+b x)^{7/2}}{d \sqrt{c+d x}} \]

[Out]

(-2*(a + b*x)^(7/2))/(d*Sqrt[c + d*x]) + (35*b*(b*c - a*d)^2*Sqrt[a + b*x]*Sqrt[
c + d*x])/(8*d^4) - (35*b*(b*c - a*d)*(a + b*x)^(3/2)*Sqrt[c + d*x])/(12*d^3) +
(7*b*(a + b*x)^(5/2)*Sqrt[c + d*x])/(3*d^2) - (35*Sqrt[b]*(b*c - a*d)^3*ArcTanh[
(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(8*d^(9/2))

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Rubi [A]  time = 0.227519, antiderivative size = 174, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21 \[ -\frac{35 \sqrt{b} (b c-a d)^3 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{8 d^{9/2}}+\frac{35 b \sqrt{a+b x} \sqrt{c+d x} (b c-a d)^2}{8 d^4}-\frac{35 b (a+b x)^{3/2} \sqrt{c+d x} (b c-a d)}{12 d^3}+\frac{7 b (a+b x)^{5/2} \sqrt{c+d x}}{3 d^2}-\frac{2 (a+b x)^{7/2}}{d \sqrt{c+d x}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^(7/2)/(c + d*x)^(3/2),x]

[Out]

(-2*(a + b*x)^(7/2))/(d*Sqrt[c + d*x]) + (35*b*(b*c - a*d)^2*Sqrt[a + b*x]*Sqrt[
c + d*x])/(8*d^4) - (35*b*(b*c - a*d)*(a + b*x)^(3/2)*Sqrt[c + d*x])/(12*d^3) +
(7*b*(a + b*x)^(5/2)*Sqrt[c + d*x])/(3*d^2) - (35*Sqrt[b]*(b*c - a*d)^3*ArcTanh[
(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(8*d^(9/2))

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Rubi in Sympy [A]  time = 33.5876, size = 162, normalized size = 0.93 \[ \frac{35 \sqrt{b} \left (a d - b c\right )^{3} \operatorname{atanh}{\left (\frac{\sqrt{d} \sqrt{a + b x}}{\sqrt{b} \sqrt{c + d x}} \right )}}{8 d^{\frac{9}{2}}} + \frac{7 b \left (a + b x\right )^{\frac{5}{2}} \sqrt{c + d x}}{3 d^{2}} + \frac{35 b \left (a + b x\right )^{\frac{3}{2}} \sqrt{c + d x} \left (a d - b c\right )}{12 d^{3}} + \frac{35 b \sqrt{a + b x} \sqrt{c + d x} \left (a d - b c\right )^{2}}{8 d^{4}} - \frac{2 \left (a + b x\right )^{\frac{7}{2}}}{d \sqrt{c + d x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**(7/2)/(d*x+c)**(3/2),x)

[Out]

35*sqrt(b)*(a*d - b*c)**3*atanh(sqrt(d)*sqrt(a + b*x)/(sqrt(b)*sqrt(c + d*x)))/(
8*d**(9/2)) + 7*b*(a + b*x)**(5/2)*sqrt(c + d*x)/(3*d**2) + 35*b*(a + b*x)**(3/2
)*sqrt(c + d*x)*(a*d - b*c)/(12*d**3) + 35*b*sqrt(a + b*x)*sqrt(c + d*x)*(a*d -
b*c)**2/(8*d**4) - 2*(a + b*x)**(7/2)/(d*sqrt(c + d*x))

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Mathematica [A]  time = 0.337034, size = 165, normalized size = 0.95 \[ \frac{\sqrt{a+b x} \sqrt{c+d x} \left (b \left (87 a^2 d^2-136 a b c d+57 b^2 c^2\right )-2 b^2 d x (11 b c-19 a d)+\frac{48 (b c-a d)^3}{c+d x}+8 b^3 d^2 x^2\right )}{24 d^4}-\frac{35 \sqrt{b} (b c-a d)^3 \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right )}{16 d^{9/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^(7/2)/(c + d*x)^(3/2),x]

[Out]

(Sqrt[a + b*x]*Sqrt[c + d*x]*(b*(57*b^2*c^2 - 136*a*b*c*d + 87*a^2*d^2) - 2*b^2*
d*(11*b*c - 19*a*d)*x + 8*b^3*d^2*x^2 + (48*(b*c - a*d)^3)/(c + d*x)))/(24*d^4)
- (35*Sqrt[b]*(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^(9/2))

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Maple [F]  time = 0.048, size = 0, normalized size = 0. \[ \int{1 \left ( bx+a \right ) ^{{\frac{7}{2}}} \left ( dx+c \right ) ^{-{\frac{3}{2}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^(7/2)/(d*x+c)^(3/2),x)

[Out]

int((b*x+a)^(7/2)/(d*x+c)^(3/2),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)^(7/2)/(d*x + c)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.565602, size = 1, normalized size = 0.01 \[ \left [-\frac{105 \,{\left (b^{3} c^{4} - 3 \, a b^{2} c^{3} d + 3 \, a^{2} b c^{2} d^{2} - a^{3} c d^{3} +{\left (b^{3} c^{3} d - 3 \, a b^{2} c^{2} d^{2} + 3 \, a^{2} b c d^{3} - a^{3} d^{4}\right )} x\right )} \sqrt{\frac{b}{d}} \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 4 \,{\left (2 \, b d^{2} x + b c d + a d^{2}\right )} \sqrt{b x + a} \sqrt{d x + c} \sqrt{\frac{b}{d}} + 8 \,{\left (b^{2} c d + a b d^{2}\right )} x\right ) - 4 \,{\left (8 \, b^{3} d^{3} x^{3} + 105 \, b^{3} c^{3} - 280 \, a b^{2} c^{2} d + 231 \, a^{2} b c d^{2} - 48 \, a^{3} d^{3} - 2 \,{\left (7 \, b^{3} c d^{2} - 19 \, a b^{2} d^{3}\right )} x^{2} +{\left (35 \, b^{3} c^{2} d - 98 \, a b^{2} c d^{2} + 87 \, a^{2} b d^{3}\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{96 \,{\left (d^{5} x + c d^{4}\right )}}, -\frac{105 \,{\left (b^{3} c^{4} - 3 \, a b^{2} c^{3} d + 3 \, a^{2} b c^{2} d^{2} - a^{3} c d^{3} +{\left (b^{3} c^{3} d - 3 \, a b^{2} c^{2} d^{2} + 3 \, a^{2} b c d^{3} - a^{3} d^{4}\right )} x\right )} \sqrt{-\frac{b}{d}} \arctan \left (\frac{2 \, b d x + b c + a d}{2 \, \sqrt{b x + a} \sqrt{d x + c} d \sqrt{-\frac{b}{d}}}\right ) - 2 \,{\left (8 \, b^{3} d^{3} x^{3} + 105 \, b^{3} c^{3} - 280 \, a b^{2} c^{2} d + 231 \, a^{2} b c d^{2} - 48 \, a^{3} d^{3} - 2 \,{\left (7 \, b^{3} c d^{2} - 19 \, a b^{2} d^{3}\right )} x^{2} +{\left (35 \, b^{3} c^{2} d - 98 \, a b^{2} c d^{2} + 87 \, a^{2} b d^{3}\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{48 \,{\left (d^{5} x + c d^{4}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(7/2)/(d*x + c)^(3/2),x, algorithm="fricas")

[Out]

[-1/96*(105*(b^3*c^4 - 3*a*b^2*c^3*d + 3*a^2*b*c^2*d^2 - a^3*c*d^3 + (b^3*c^3*d
- 3*a*b^2*c^2*d^2 + 3*a^2*b*c*d^3 - a^3*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)*sqrt(d
*x + c)*sqrt(b/d) + 8*(b^2*c*d + a*b*d^2)*x) - 4*(8*b^3*d^3*x^3 + 105*b^3*c^3 -
280*a*b^2*c^2*d + 231*a^2*b*c*d^2 - 48*a^3*d^3 - 2*(7*b^3*c*d^2 - 19*a*b^2*d^3)*
x^2 + (35*b^3*c^2*d - 98*a*b^2*c*d^2 + 87*a^2*b*d^3)*x)*sqrt(b*x + a)*sqrt(d*x +
c))/(d^5*x + c*d^4), -1/48*(105*(b^3*c^4 - 3*a*b^2*c^3*d + 3*a^2*b*c^2*d^2 - a^
3*c*d^3 + (b^3*c^3*d - 3*a*b^2*c^2*d^2 + 3*a^2*b*c*d^3 - a^3*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^3*d^3*x^3 + 105*b^3*c^3 - 280*a*b^2*c^2*d + 231*a^2*b*c*d^2 - 48*a^3*d^3 -
2*(7*b^3*c*d^2 - 19*a*b^2*d^3)*x^2 + (35*b^3*c^2*d - 98*a*b^2*c*d^2 + 87*a^2*b*
d^3)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(d^5*x + c*d^4)]

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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)**(7/2)/(d*x+c)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.261277, size = 423, normalized size = 2.43 \[ \frac{{\left ({\left (2 \,{\left (\frac{4 \,{\left (b x + a\right )} b^{2} d^{6}}{b^{10} c d^{8} - a b^{9} d^{9}} - \frac{7 \,{\left (b^{3} c d^{5} - a b^{2} d^{6}\right )}}{b^{10} c d^{8} - a b^{9} d^{9}}\right )}{\left (b x + a\right )} + \frac{35 \,{\left (b^{4} c^{2} d^{4} - 2 \, a b^{3} c d^{5} + a^{2} b^{2} d^{6}\right )}}{b^{10} c d^{8} - a b^{9} d^{9}}\right )}{\left (b x + a\right )} + \frac{105 \,{\left (b^{5} c^{3} d^{3} - 3 \, a b^{4} c^{2} d^{4} + 3 \, a^{2} b^{3} c d^{5} - a^{3} b^{2} d^{6}\right )}}{b^{10} c d^{8} - a b^{9} d^{9}}\right )} \sqrt{b x + a}}{184320 \, \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}} + \frac{7 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )}{\rm ln}\left ({\left | -\sqrt{b d} \sqrt{b x + a} + \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d} \right |}\right )}{12288 \, \sqrt{b d} b^{7} d^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(7/2)/(d*x + c)^(3/2),x, algorithm="giac")

[Out]

1/184320*((2*(4*(b*x + a)*b^2*d^6/(b^10*c*d^8 - a*b^9*d^9) - 7*(b^3*c*d^5 - a*b^
2*d^6)/(b^10*c*d^8 - a*b^9*d^9))*(b*x + a) + 35*(b^4*c^2*d^4 - 2*a*b^3*c*d^5 + a
^2*b^2*d^6)/(b^10*c*d^8 - a*b^9*d^9))*(b*x + a) + 105*(b^5*c^3*d^3 - 3*a*b^4*c^2
*d^4 + 3*a^2*b^3*c*d^5 - a^3*b^2*d^6)/(b^10*c*d^8 - a*b^9*d^9))*sqrt(b*x + a)/sq
rt(b^2*c + (b*x + a)*b*d - a*b*d) + 7/12288*(b^2*c^2 - 2*a*b*c*d + a^2*d^2)*ln(a
bs(-sqrt(b*d)*sqrt(b*x + a) + sqrt(b^2*c + (b*x + a)*b*d - a*b*d)))/(sqrt(b*d)*b
^7*d^5)