3.1513 \(\int \frac{(a+b x)^{7/2}}{(c+d x)^{5/2}} \, dx\)

Optimal. Leaf size=170 \[ \frac{35 b^{3/2} (b c-a d)^2 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{4 d^{9/2}}-\frac{35 b^2 \sqrt{a+b x} \sqrt{c+d x} (b c-a d)}{4 d^4}+\frac{35 b^2 (a+b x)^{3/2} \sqrt{c+d x}}{6 d^3}-\frac{14 b (a+b x)^{5/2}}{3 d^2 \sqrt{c+d x}}-\frac{2 (a+b x)^{7/2}}{3 d (c+d x)^{3/2}} \]

[Out]

(-2*(a + b*x)^(7/2))/(3*d*(c + d*x)^(3/2)) - (14*b*(a + b*x)^(5/2))/(3*d^2*Sqrt[
c + d*x]) - (35*b^2*(b*c - a*d)*Sqrt[a + b*x]*Sqrt[c + d*x])/(4*d^4) + (35*b^2*(
a + b*x)^(3/2)*Sqrt[c + d*x])/(6*d^3) + (35*b^(3/2)*(b*c - a*d)^2*ArcTanh[(Sqrt[
d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(4*d^(9/2))

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Rubi [A]  time = 0.21308, antiderivative size = 170, 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 \[ \frac{35 b^{3/2} (b c-a d)^2 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{4 d^{9/2}}-\frac{35 b^2 \sqrt{a+b x} \sqrt{c+d x} (b c-a d)}{4 d^4}+\frac{35 b^2 (a+b x)^{3/2} \sqrt{c+d x}}{6 d^3}-\frac{14 b (a+b x)^{5/2}}{3 d^2 \sqrt{c+d x}}-\frac{2 (a+b x)^{7/2}}{3 d (c+d x)^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^(7/2)/(c + d*x)^(5/2),x]

[Out]

(-2*(a + b*x)^(7/2))/(3*d*(c + d*x)^(3/2)) - (14*b*(a + b*x)^(5/2))/(3*d^2*Sqrt[
c + d*x]) - (35*b^2*(b*c - a*d)*Sqrt[a + b*x]*Sqrt[c + d*x])/(4*d^4) + (35*b^2*(
a + b*x)^(3/2)*Sqrt[c + d*x])/(6*d^3) + (35*b^(3/2)*(b*c - a*d)^2*ArcTanh[(Sqrt[
d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(4*d^(9/2))

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Rubi in Sympy [A]  time = 29.95, size = 158, normalized size = 0.93 \[ \frac{35 b^{\frac{3}{2}} \left (a d - b c\right )^{2} \operatorname{atanh}{\left (\frac{\sqrt{d} \sqrt{a + b x}}{\sqrt{b} \sqrt{c + d x}} \right )}}{4 d^{\frac{9}{2}}} + \frac{35 b^{2} \left (a + b x\right )^{\frac{3}{2}} \sqrt{c + d x}}{6 d^{3}} + \frac{35 b^{2} \sqrt{a + b x} \sqrt{c + d x} \left (a d - b c\right )}{4 d^{4}} - \frac{14 b \left (a + b x\right )^{\frac{5}{2}}}{3 d^{2} \sqrt{c + d x}} - \frac{2 \left (a + b x\right )^{\frac{7}{2}}}{3 d \left (c + d x\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**(7/2)/(d*x+c)**(5/2),x)

[Out]

35*b**(3/2)*(a*d - b*c)**2*atanh(sqrt(d)*sqrt(a + b*x)/(sqrt(b)*sqrt(c + d*x)))/
(4*d**(9/2)) + 35*b**2*(a + b*x)**(3/2)*sqrt(c + d*x)/(6*d**3) + 35*b**2*sqrt(a
+ b*x)*sqrt(c + d*x)*(a*d - b*c)/(4*d**4) - 14*b*(a + b*x)**(5/2)/(3*d**2*sqrt(c
+ d*x)) - 2*(a + b*x)**(7/2)/(3*d*(c + d*x)**(3/2))

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Mathematica [A]  time = 0.274719, size = 159, normalized size = 0.94 \[ \frac{35 b^{3/2} (b c-a d)^2 \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right )}{8 d^{9/2}}+\frac{\sqrt{a+b x} \left (-3 b^2 (c+d x)^2 (11 b c-13 a d)-80 b (c+d x) (b c-a d)^2+8 (b c-a d)^3+6 b^3 d x (c+d x)^2\right )}{12 d^4 (c+d x)^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^(7/2)/(c + d*x)^(5/2),x]

[Out]

(Sqrt[a + b*x]*(8*(b*c - a*d)^3 - 80*b*(b*c - a*d)^2*(c + d*x) - 3*b^2*(11*b*c -
13*a*d)*(c + d*x)^2 + 6*b^3*d*x*(c + d*x)^2))/(12*d^4*(c + d*x)^(3/2)) + (35*b^
(3/2)*(b*c - a*d)^2*Log[b*c + a*d + 2*b*d*x + 2*Sqrt[b]*Sqrt[d]*Sqrt[a + b*x]*Sq
rt[c + d*x]])/(8*d^(9/2))

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Maple [F]  time = 0.047, size = 0, normalized size = 0. \[ \int{1 \left ( bx+a \right ) ^{{\frac{7}{2}}} \left ( dx+c \right ) ^{-{\frac{5}{2}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^(7/2)/(d*x+c)^(5/2),x)

[Out]

int((b*x+a)^(7/2)/(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)^(7/2)/(d*x + c)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.695557, size = 1, normalized size = 0.01 \[ \left [\frac{105 \,{\left (b^{3} c^{4} - 2 \, a b^{2} c^{3} d + a^{2} b c^{2} d^{2} +{\left (b^{3} c^{2} d^{2} - 2 \, a b^{2} c d^{3} + a^{2} b d^{4}\right )} x^{2} + 2 \,{\left (b^{3} c^{3} d - 2 \, a b^{2} c^{2} d^{2} + a^{2} b c d^{3}\right )} x\right )} \sqrt{\frac{b}{d}} \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 4 \,{\left (2 \, b d^{2} x + b c d + a d^{2}\right )} \sqrt{b x + a} \sqrt{d x + c} \sqrt{\frac{b}{d}} + 8 \,{\left (b^{2} c d + a b d^{2}\right )} x\right ) + 4 \,{\left (6 \, b^{3} d^{3} x^{3} - 105 \, b^{3} c^{3} + 175 \, a b^{2} c^{2} d - 56 \, a^{2} b c d^{2} - 8 \, a^{3} d^{3} - 3 \,{\left (7 \, b^{3} c d^{2} - 13 \, a b^{2} d^{3}\right )} x^{2} - 2 \,{\left (70 \, b^{3} c^{2} d - 119 \, a b^{2} c d^{2} + 40 \, a^{2} b d^{3}\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{48 \,{\left (d^{6} x^{2} + 2 \, c d^{5} x + c^{2} d^{4}\right )}}, \frac{105 \,{\left (b^{3} c^{4} - 2 \, a b^{2} c^{3} d + a^{2} b c^{2} d^{2} +{\left (b^{3} c^{2} d^{2} - 2 \, a b^{2} c d^{3} + a^{2} b d^{4}\right )} x^{2} + 2 \,{\left (b^{3} c^{3} d - 2 \, a b^{2} c^{2} d^{2} + a^{2} b c d^{3}\right )} x\right )} \sqrt{-\frac{b}{d}} \arctan \left (\frac{2 \, b d x + b c + a d}{2 \, \sqrt{b x + a} \sqrt{d x + c} d \sqrt{-\frac{b}{d}}}\right ) + 2 \,{\left (6 \, b^{3} d^{3} x^{3} - 105 \, b^{3} c^{3} + 175 \, a b^{2} c^{2} d - 56 \, a^{2} b c d^{2} - 8 \, a^{3} d^{3} - 3 \,{\left (7 \, b^{3} c d^{2} - 13 \, a b^{2} d^{3}\right )} x^{2} - 2 \,{\left (70 \, b^{3} c^{2} d - 119 \, a b^{2} c d^{2} + 40 \, a^{2} b d^{3}\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{24 \,{\left (d^{6} x^{2} + 2 \, c d^{5} x + c^{2} d^{4}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(7/2)/(d*x + c)^(5/2),x, algorithm="fricas")

[Out]

[1/48*(105*(b^3*c^4 - 2*a*b^2*c^3*d + a^2*b*c^2*d^2 + (b^3*c^2*d^2 - 2*a*b^2*c*d
^3 + a^2*b*d^4)*x^2 + 2*(b^3*c^3*d - 2*a*b^2*c^2*d^2 + a^2*b*c*d^3)*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*(6*b^3*d
^3*x^3 - 105*b^3*c^3 + 175*a*b^2*c^2*d - 56*a^2*b*c*d^2 - 8*a^3*d^3 - 3*(7*b^3*c
*d^2 - 13*a*b^2*d^3)*x^2 - 2*(70*b^3*c^2*d - 119*a*b^2*c*d^2 + 40*a^2*b*d^3)*x)*
sqrt(b*x + a)*sqrt(d*x + c))/(d^6*x^2 + 2*c*d^5*x + c^2*d^4), 1/24*(105*(b^3*c^4
- 2*a*b^2*c^3*d + a^2*b*c^2*d^2 + (b^3*c^2*d^2 - 2*a*b^2*c*d^3 + a^2*b*d^4)*x^2
+ 2*(b^3*c^3*d - 2*a*b^2*c^2*d^2 + a^2*b*c*d^3)*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*(6*b^3*d^3*x^3 -
105*b^3*c^3 + 175*a*b^2*c^2*d - 56*a^2*b*c*d^2 - 8*a^3*d^3 - 3*(7*b^3*c*d^2 - 1
3*a*b^2*d^3)*x^2 - 2*(70*b^3*c^2*d - 119*a*b^2*c*d^2 + 40*a^2*b*d^3)*x)*sqrt(b*x
+ a)*sqrt(d*x + c))/(d^6*x^2 + 2*c*d^5*x + c^2*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)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.274017, size = 513, normalized size = 3.02 \[ \frac{{\left ({\left (3 \,{\left (b x + a\right )}{\left (\frac{2 \,{\left (b^{6} c d^{6} - a b^{5} d^{7}\right )}{\left (b x + a\right )}}{b^{2} c d^{7}{\left | b \right |} - a b d^{8}{\left | b \right |}} - \frac{7 \,{\left (b^{7} c^{2} d^{5} - 2 \, a b^{6} c d^{6} + a^{2} b^{5} d^{7}\right )}}{b^{2} c d^{7}{\left | b \right |} - a b d^{8}{\left | b \right |}}\right )} - \frac{140 \,{\left (b^{8} c^{3} d^{4} - 3 \, a b^{7} c^{2} d^{5} + 3 \, a^{2} b^{6} c d^{6} - a^{3} b^{5} d^{7}\right )}}{b^{2} c d^{7}{\left | b \right |} - a b d^{8}{\left | b \right |}}\right )}{\left (b x + a\right )} - \frac{105 \,{\left (b^{9} c^{4} d^{3} - 4 \, a b^{8} c^{3} d^{4} + 6 \, a^{2} b^{7} c^{2} d^{5} - 4 \, a^{3} b^{6} c d^{6} + a^{4} b^{5} d^{7}\right )}}{b^{2} c d^{7}{\left | b \right |} - a b d^{8}{\left | b \right |}}\right )} \sqrt{b x + a}}{12 \,{\left (b^{2} c +{\left (b x + a\right )} b d - a b d\right )}^{\frac{3}{2}}} - \frac{35 \,{\left (b^{5} c^{2} - 2 \, a b^{4} c d + a^{2} b^{3} d^{2}\right )}{\rm ln}\left ({\left | -\sqrt{b d} \sqrt{b x + a} + \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d} \right |}\right )}{4 \, \sqrt{b d} d^{4}{\left | b \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(7/2)/(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^2*c*d^7*abs(b) - a*b*
d^8*abs(b)) - 7*(b^7*c^2*d^5 - 2*a*b^6*c*d^6 + a^2*b^5*d^7)/(b^2*c*d^7*abs(b) -
a*b*d^8*abs(b))) - 140*(b^8*c^3*d^4 - 3*a*b^7*c^2*d^5 + 3*a^2*b^6*c*d^6 - a^3*b^
5*d^7)/(b^2*c*d^7*abs(b) - a*b*d^8*abs(b)))*(b*x + a) - 105*(b^9*c^4*d^3 - 4*a*b
^8*c^3*d^4 + 6*a^2*b^7*c^2*d^5 - 4*a^3*b^6*c*d^6 + a^4*b^5*d^7)/(b^2*c*d^7*abs(b
) - a*b*d^8*abs(b)))*sqrt(b*x + a)/(b^2*c + (b*x + a)*b*d - a*b*d)^(3/2) - 35/4*
(b^5*c^2 - 2*a*b^4*c*d + a^2*b^3*d^2)*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^4*abs(b))