3.1666 \(\int \frac{(a+b x)^{7/2}}{(c+d x)^{7/4}} \, dx\)

Optimal. Leaf size=207 \[ -\frac{320 b^{3/4} (b c-a d)^{13/4} \sqrt{-\frac{d (a+b x)}{b c-a d}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{33 d^5 \sqrt{a+b x}}+\frac{160 b \sqrt{a+b x} \sqrt [4]{c+d x} (b c-a d)^2}{33 d^4}-\frac{80 b (a+b x)^{3/2} \sqrt [4]{c+d x} (b c-a d)}{33 d^3}+\frac{56 b (a+b x)^{5/2} \sqrt [4]{c+d x}}{33 d^2}-\frac{4 (a+b x)^{7/2}}{3 d (c+d x)^{3/4}} \]

[Out]

(-4*(a + b*x)^(7/2))/(3*d*(c + d*x)^(3/4)) + (160*b*(b*c - a*d)^2*Sqrt[a + b*x]*
(c + d*x)^(1/4))/(33*d^4) - (80*b*(b*c - a*d)*(a + b*x)^(3/2)*(c + d*x)^(1/4))/(
33*d^3) + (56*b*(a + b*x)^(5/2)*(c + d*x)^(1/4))/(33*d^2) - (320*b^(3/4)*(b*c -
a*d)^(13/4)*Sqrt[-((d*(a + b*x))/(b*c - a*d))]*EllipticF[ArcSin[(b^(1/4)*(c + d*
x)^(1/4))/(b*c - a*d)^(1/4)], -1])/(33*d^5*Sqrt[a + b*x])

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Rubi [A]  time = 0.342726, antiderivative size = 207, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263 \[ -\frac{320 b^{3/4} (b c-a d)^{13/4} \sqrt{-\frac{d (a+b x)}{b c-a d}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{33 d^5 \sqrt{a+b x}}+\frac{160 b \sqrt{a+b x} \sqrt [4]{c+d x} (b c-a d)^2}{33 d^4}-\frac{80 b (a+b x)^{3/2} \sqrt [4]{c+d x} (b c-a d)}{33 d^3}+\frac{56 b (a+b x)^{5/2} \sqrt [4]{c+d x}}{33 d^2}-\frac{4 (a+b x)^{7/2}}{3 d (c+d x)^{3/4}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^(7/2)/(c + d*x)^(7/4),x]

[Out]

(-4*(a + b*x)^(7/2))/(3*d*(c + d*x)^(3/4)) + (160*b*(b*c - a*d)^2*Sqrt[a + b*x]*
(c + d*x)^(1/4))/(33*d^4) - (80*b*(b*c - a*d)*(a + b*x)^(3/2)*(c + d*x)^(1/4))/(
33*d^3) + (56*b*(a + b*x)^(5/2)*(c + d*x)^(1/4))/(33*d^2) - (320*b^(3/4)*(b*c -
a*d)^(13/4)*Sqrt[-((d*(a + b*x))/(b*c - a*d))]*EllipticF[ArcSin[(b^(1/4)*(c + d*
x)^(1/4))/(b*c - a*d)^(1/4)], -1])/(33*d^5*Sqrt[a + b*x])

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Rubi in Sympy [A]  time = 51.0033, size = 260, normalized size = 1.26 \[ \frac{160 b^{\frac{3}{4}} \sqrt{\frac{a d - b c + b \left (c + d x\right )}{\left (a d - b c\right ) \left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} + 1\right )^{2}}} \left (a d - b c\right )^{\frac{13}{4}} \left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} + 1\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt [4]{c + d x}}{\sqrt [4]{a d - b c}} \right )}\middle | \frac{1}{2}\right )}{33 d^{5} \sqrt{a - \frac{b c}{d} + \frac{b \left (c + d x\right )}{d}}} + \frac{56 b \left (a + b x\right )^{\frac{5}{2}} \sqrt [4]{c + d x}}{33 d^{2}} + \frac{80 b \left (a + b x\right )^{\frac{3}{2}} \sqrt [4]{c + d x} \left (a d - b c\right )}{33 d^{3}} + \frac{160 b \sqrt{a + b x} \sqrt [4]{c + d x} \left (a d - b c\right )^{2}}{33 d^{4}} - \frac{4 \left (a + b x\right )^{\frac{7}{2}}}{3 d \left (c + d x\right )^{\frac{3}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**(7/2)/(d*x+c)**(7/4),x)

[Out]

160*b**(3/4)*sqrt((a*d - b*c + b*(c + d*x))/((a*d - b*c)*(sqrt(b)*sqrt(c + d*x)/
sqrt(a*d - b*c) + 1)**2))*(a*d - b*c)**(13/4)*(sqrt(b)*sqrt(c + d*x)/sqrt(a*d -
b*c) + 1)*elliptic_f(2*atan(b**(1/4)*(c + d*x)**(1/4)/(a*d - b*c)**(1/4)), 1/2)/
(33*d**5*sqrt(a - b*c/d + b*(c + d*x)/d)) + 56*b*(a + b*x)**(5/2)*(c + d*x)**(1/
4)/(33*d**2) + 80*b*(a + b*x)**(3/2)*(c + d*x)**(1/4)*(a*d - b*c)/(33*d**3) + 16
0*b*sqrt(a + b*x)*(c + d*x)**(1/4)*(a*d - b*c)**2/(33*d**4) - 4*(a + b*x)**(7/2)
/(3*d*(c + d*x)**(3/4))

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Mathematica [C]  time = 0.340966, size = 181, normalized size = 0.87 \[ \frac{4 \sqrt [4]{c+d x} \left (\frac{d (a+b x) \left (b (c+d x) \left (41 a^2 d^2-67 a b c d+29 b^2 c^2\right )-3 b^2 d x (c+d x) (3 b c-5 a d)+11 (b c-a d)^3+3 b^3 d^2 x^2 (c+d x)\right )}{c+d x}-80 b (b c-a d)^3 \sqrt{\frac{d (a+b x)}{a d-b c}} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{5}{4};\frac{b (c+d x)}{b c-a d}\right )\right )}{33 d^5 \sqrt{a+b x}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^(7/2)/(c + d*x)^(7/4),x]

[Out]

(4*(c + d*x)^(1/4)*((d*(a + b*x)*(11*(b*c - a*d)^3 + b*(29*b^2*c^2 - 67*a*b*c*d
+ 41*a^2*d^2)*(c + d*x) - 3*b^2*d*(3*b*c - 5*a*d)*x*(c + d*x) + 3*b^3*d^2*x^2*(c
+ d*x)))/(c + d*x) - 80*b*(b*c - a*d)^3*Sqrt[(d*(a + b*x))/(-(b*c) + a*d)]*Hype
rgeometric2F1[1/4, 1/2, 5/4, (b*(c + d*x))/(b*c - a*d)]))/(33*d^5*Sqrt[a + b*x])

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Maple [F]  time = 0.093, size = 0, normalized size = 0. \[ \int{1 \left ( bx+a \right ) ^{{\frac{7}{2}}} \left ( dx+c \right ) ^{-{\frac{7}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^(7/2)/(d*x+c)^(7/4),x)

[Out]

int((b*x+a)^(7/2)/(d*x+c)^(7/4),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x + a\right )}^{\frac{7}{2}}}{{\left (d x + c\right )}^{\frac{7}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(7/2)/(d*x + c)^(7/4),x, algorithm="maxima")

[Out]

integrate((b*x + a)^(7/2)/(d*x + c)^(7/4), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}\right )} \sqrt{b x + a}}{{\left (d x + c\right )}^{\frac{7}{4}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(7/2)/(d*x + c)^(7/4),x, algorithm="fricas")

[Out]

integral((b^3*x^3 + 3*a*b^2*x^2 + 3*a^2*b*x + a^3)*sqrt(b*x + a)/(d*x + c)^(7/4)
, x)

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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)**(7/4),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x + a\right )}^{\frac{7}{2}}}{{\left (d x + c\right )}^{\frac{7}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(7/2)/(d*x + c)^(7/4),x, algorithm="giac")

[Out]

integrate((b*x + a)^(7/2)/(d*x + c)^(7/4), x)