3.1672 \(\int \frac{(a+b x)^{7/2}}{(c+d x)^{9/4}} \, dx\)

Optimal. Leaf size=286 \[ -\frac{448 b^{5/4} (b c-a d)^{11/4} \sqrt{-\frac{d (a+b x)}{b c-a d}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{15 d^5 \sqrt{a+b x}}+\frac{448 b^{5/4} (b c-a d)^{11/4} \sqrt{-\frac{d (a+b x)}{b c-a d}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{15 d^5 \sqrt{a+b x}}-\frac{224 b^2 \sqrt{a+b x} (c+d x)^{3/4} (b c-a d)}{15 d^4}+\frac{112 b^2 (a+b x)^{3/2} (c+d x)^{3/4}}{9 d^3}-\frac{56 b (a+b x)^{5/2}}{5 d^2 \sqrt [4]{c+d x}}-\frac{4 (a+b x)^{7/2}}{5 d (c+d x)^{5/4}} \]

[Out]

(-4*(a + b*x)^(7/2))/(5*d*(c + d*x)^(5/4)) - (56*b*(a + b*x)^(5/2))/(5*d^2*(c +
d*x)^(1/4)) - (224*b^2*(b*c - a*d)*Sqrt[a + b*x]*(c + d*x)^(3/4))/(15*d^4) + (11
2*b^2*(a + b*x)^(3/2)*(c + d*x)^(3/4))/(9*d^3) + (448*b^(5/4)*(b*c - a*d)^(11/4)
*Sqrt[-((d*(a + b*x))/(b*c - a*d))]*EllipticE[ArcSin[(b^(1/4)*(c + d*x)^(1/4))/(
b*c - a*d)^(1/4)], -1])/(15*d^5*Sqrt[a + b*x]) - (448*b^(5/4)*(b*c - a*d)^(11/4)
*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])/(15*d^5*Sqrt[a + b*x])

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Rubi [A]  time = 0.849067, antiderivative size = 286, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 9, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.474 \[ -\frac{448 b^{5/4} (b c-a d)^{11/4} \sqrt{-\frac{d (a+b x)}{b c-a d}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{15 d^5 \sqrt{a+b x}}+\frac{448 b^{5/4} (b c-a d)^{11/4} \sqrt{-\frac{d (a+b x)}{b c-a d}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{15 d^5 \sqrt{a+b x}}-\frac{224 b^2 \sqrt{a+b x} (c+d x)^{3/4} (b c-a d)}{15 d^4}+\frac{112 b^2 (a+b x)^{3/2} (c+d x)^{3/4}}{9 d^3}-\frac{56 b (a+b x)^{5/2}}{5 d^2 \sqrt [4]{c+d x}}-\frac{4 (a+b x)^{7/2}}{5 d (c+d x)^{5/4}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^(7/2)/(c + d*x)^(9/4),x]

[Out]

(-4*(a + b*x)^(7/2))/(5*d*(c + d*x)^(5/4)) - (56*b*(a + b*x)^(5/2))/(5*d^2*(c +
d*x)^(1/4)) - (224*b^2*(b*c - a*d)*Sqrt[a + b*x]*(c + d*x)^(3/4))/(15*d^4) + (11
2*b^2*(a + b*x)^(3/2)*(c + d*x)^(3/4))/(9*d^3) + (448*b^(5/4)*(b*c - a*d)^(11/4)
*Sqrt[-((d*(a + b*x))/(b*c - a*d))]*EllipticE[ArcSin[(b^(1/4)*(c + d*x)^(1/4))/(
b*c - a*d)^(1/4)], -1])/(15*d^5*Sqrt[a + b*x]) - (448*b^(5/4)*(b*c - a*d)^(11/4)
*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])/(15*d^5*Sqrt[a + b*x])

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Rubi in Sympy [A]  time = 121.044, size = 478, normalized size = 1.67 \[ - \frac{448 b^{\frac{5}{4}} \sqrt{\frac{a d - b c + b \left (c + d x\right )}{\left (a d - b c\right ) \left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} + 1\right )^{2}}} \left (a d - b c\right )^{\frac{11}{4}} \left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} + 1\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt [4]{c + d x}}{\sqrt [4]{a d - b c}} \right )}\middle | \frac{1}{2}\right )}{15 d^{5} \sqrt{a - \frac{b c}{d} + \frac{b \left (c + d x\right )}{d}}} + \frac{224 b^{\frac{5}{4}} \sqrt{\frac{a d - b c + b \left (c + d x\right )}{\left (a d - b c\right ) \left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} + 1\right )^{2}}} \left (a d - b c\right )^{\frac{11}{4}} \left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} + 1\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt [4]{c + d x}}{\sqrt [4]{a d - b c}} \right )}\middle | \frac{1}{2}\right )}{15 d^{5} \sqrt{a - \frac{b c}{d} + \frac{b \left (c + d x\right )}{d}}} + \frac{448 b^{\frac{3}{2}} \sqrt [4]{c + d x} \left (a d - b c\right )^{\frac{3}{2}} \sqrt{a - \frac{b c}{d} + \frac{b \left (c + d x\right )}{d}}}{15 d^{4} \left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} + 1\right )} + \frac{112 b^{2} \left (a + b x\right )^{\frac{3}{2}} \left (c + d x\right )^{\frac{3}{4}}}{9 d^{3}} + \frac{224 b^{2} \sqrt{a + b x} \left (c + d x\right )^{\frac{3}{4}} \left (a d - b c\right )}{15 d^{4}} - \frac{56 b \left (a + b x\right )^{\frac{5}{2}}}{5 d^{2} \sqrt [4]{c + d x}} - \frac{4 \left (a + b x\right )^{\frac{7}{2}}}{5 d \left (c + d x\right )^{\frac{5}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**(7/2)/(d*x+c)**(9/4),x)

[Out]

-448*b**(5/4)*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)**(11/4)*(sqrt(b)*sqrt(c + d*x)/sqrt(a*d -
b*c) + 1)*elliptic_e(2*atan(b**(1/4)*(c + d*x)**(1/4)/(a*d - b*c)**(1/4)), 1/2)
/(15*d**5*sqrt(a - b*c/d + b*(c + d*x)/d)) + 224*b**(5/4)*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)**(11/4)*(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)/(15*d**5*sqrt(a - b*c/d + b*(c + d*
x)/d)) + 448*b**(3/2)*(c + d*x)**(1/4)*(a*d - b*c)**(3/2)*sqrt(a - b*c/d + b*(c
+ d*x)/d)/(15*d**4*(sqrt(b)*sqrt(c + d*x)/sqrt(a*d - b*c) + 1)) + 112*b**2*(a +
b*x)**(3/2)*(c + d*x)**(3/4)/(9*d**3) + 224*b**2*sqrt(a + b*x)*(c + d*x)**(3/4)*
(a*d - b*c)/(15*d**4) - 56*b*(a + b*x)**(5/2)/(5*d**2*(c + d*x)**(1/4)) - 4*(a +
b*x)**(7/2)/(5*d*(c + d*x)**(5/4))

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Mathematica [C]  time = 0.378175, size = 169, normalized size = 0.59 \[ \frac{4 (c+d x)^{3/4} \left (112 b^2 (b c-a d)^2 \sqrt{\frac{d (a+b x)}{a d-b c}} \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{7}{4};\frac{b (c+d x)}{b c-a d}\right )+\frac{d (a+b x) \left (-b^2 (c+d x)^2 (24 b c-29 a d)-153 b (c+d x) (b c-a d)^2+9 (b c-a d)^3+5 b^3 d x (c+d x)^2\right )}{(c+d x)^2}\right )}{45 d^5 \sqrt{a+b x}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^(7/2)/(c + d*x)^(9/4),x]

[Out]

(4*(c + d*x)^(3/4)*((d*(a + b*x)*(9*(b*c - a*d)^3 - 153*b*(b*c - a*d)^2*(c + d*x
) - b^2*(24*b*c - 29*a*d)*(c + d*x)^2 + 5*b^3*d*x*(c + d*x)^2))/(c + d*x)^2 + 11
2*b^2*(b*c - a*d)^2*Sqrt[(d*(a + b*x))/(-(b*c) + a*d)]*Hypergeometric2F1[1/2, 3/
4, 7/4, (b*(c + d*x))/(b*c - a*d)]))/(45*d^5*Sqrt[a + b*x])

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Maple [F]  time = 0.156, size = 0, normalized size = 0. \[ \int{1 \left ( bx+a \right ) ^{{\frac{7}{2}}} \left ( dx+c \right ) ^{-{\frac{9}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^(7/2)/(d*x+c)^(9/4),x)

[Out]

int((b*x+a)^(7/2)/(d*x+c)^(9/4),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x + a\right )}^{\frac{7}{2}}}{{\left (d x + c\right )}^{\frac{9}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(7/2)/(d*x + c)^(9/4),x, algorithm="maxima")

[Out]

integrate((b*x + a)^(7/2)/(d*x + c)^(9/4), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}\right )} \sqrt{b x + a}}{{\left (d^{2} x^{2} + 2 \, c d x + c^{2}\right )}{\left (d x + c\right )}^{\frac{1}{4}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(7/2)/(d*x + c)^(9/4),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^2*x^2 + 2*c
*d*x + c^2)*(d*x + c)^(1/4)), 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)**(9/4),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x + a\right )}^{\frac{7}{2}}}{{\left (d x + c\right )}^{\frac{9}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(7/2)/(d*x + c)^(9/4),x, algorithm="giac")

[Out]

integrate((b*x + a)^(7/2)/(d*x + c)^(9/4), x)