Optimal. Leaf size=197 \[ \frac{4 \sqrt [4]{b} \sqrt{-\frac{d (a+b x)}{b c-a d}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right ),-1\right )}{d \sqrt{a+b x} \sqrt [4]{b c-a d}}+\frac{4 \sqrt{a+b x}}{\sqrt [4]{c+d x} (b c-a d)}-\frac{4 \sqrt [4]{b} \sqrt{-\frac{d (a+b x)}{b c-a d}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{d \sqrt{a+b x} \sqrt [4]{b c-a d}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.211094, antiderivative size = 197, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.421, Rules used = {51, 63, 307, 224, 221, 1200, 1199, 424} \[ \frac{4 \sqrt{a+b x}}{\sqrt [4]{c+d x} (b c-a d)}+\frac{4 \sqrt [4]{b} \sqrt{-\frac{d (a+b x)}{b c-a d}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{d \sqrt{a+b x} \sqrt [4]{b c-a d}}-\frac{4 \sqrt [4]{b} \sqrt{-\frac{d (a+b x)}{b c-a d}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{d \sqrt{a+b x} \sqrt [4]{b c-a d}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 51
Rule 63
Rule 307
Rule 224
Rule 221
Rule 1200
Rule 1199
Rule 424
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{a+b x} (c+d x)^{5/4}} \, dx &=\frac{4 \sqrt{a+b x}}{(b c-a d) \sqrt [4]{c+d x}}-\frac{b \int \frac{1}{\sqrt{a+b x} \sqrt [4]{c+d x}} \, dx}{b c-a d}\\ &=\frac{4 \sqrt{a+b x}}{(b c-a d) \sqrt [4]{c+d x}}-\frac{(4 b) \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{a-\frac{b c}{d}+\frac{b x^4}{d}}} \, dx,x,\sqrt [4]{c+d x}\right )}{d (b c-a d)}\\ &=\frac{4 \sqrt{a+b x}}{(b c-a d) \sqrt [4]{c+d x}}+\frac{\left (4 \sqrt{b}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a-\frac{b c}{d}+\frac{b x^4}{d}}} \, dx,x,\sqrt [4]{c+d x}\right )}{d \sqrt{b c-a d}}-\frac{\left (4 \sqrt{b}\right ) \operatorname{Subst}\left (\int \frac{1+\frac{\sqrt{b} x^2}{\sqrt{b c-a d}}}{\sqrt{a-\frac{b c}{d}+\frac{b x^4}{d}}} \, dx,x,\sqrt [4]{c+d x}\right )}{d \sqrt{b c-a d}}\\ &=\frac{4 \sqrt{a+b x}}{(b c-a d) \sqrt [4]{c+d x}}+\frac{\left (4 \sqrt{b} \sqrt{\frac{d (a+b x)}{-b c+a d}}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{b x^4}{\left (a-\frac{b c}{d}\right ) d}}} \, dx,x,\sqrt [4]{c+d x}\right )}{d \sqrt{b c-a d} \sqrt{a+b x}}-\frac{\left (4 \sqrt{b} \sqrt{\frac{d (a+b x)}{-b c+a d}}\right ) \operatorname{Subst}\left (\int \frac{1+\frac{\sqrt{b} x^2}{\sqrt{b c-a d}}}{\sqrt{1+\frac{b x^4}{\left (a-\frac{b c}{d}\right ) d}}} \, dx,x,\sqrt [4]{c+d x}\right )}{d \sqrt{b c-a d} \sqrt{a+b x}}\\ &=\frac{4 \sqrt{a+b x}}{(b c-a d) \sqrt [4]{c+d x}}+\frac{4 \sqrt [4]{b} \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 )}{d \sqrt [4]{b c-a d} \sqrt{a+b x}}-\frac{\left (4 \sqrt{b} \sqrt{\frac{d (a+b x)}{-b c+a d}}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1+\frac{\sqrt{b} x^2}{\sqrt{b c-a d}}}}{\sqrt{1-\frac{\sqrt{b} x^2}{\sqrt{b c-a d}}}} \, dx,x,\sqrt [4]{c+d x}\right )}{d \sqrt{b c-a d} \sqrt{a+b x}}\\ &=\frac{4 \sqrt{a+b x}}{(b c-a d) \sqrt [4]{c+d x}}-\frac{4 \sqrt [4]{b} \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 )}{d \sqrt [4]{b c-a d} \sqrt{a+b x}}+\frac{4 \sqrt [4]{b} \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 )}{d \sqrt [4]{b c-a d} \sqrt{a+b x}}\\ \end{align*}
Mathematica [C] time = 0.0348903, size = 71, normalized size = 0.36 \[ \frac{2 \sqrt{a+b x} \left (\frac{b (c+d x)}{b c-a d}\right )^{5/4} \, _2F_1\left (\frac{1}{2},\frac{5}{4};\frac{3}{2};\frac{d (a+b x)}{a d-b c}\right )}{b (c+d x)^{5/4}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.042, size = 0, normalized size = 0. \begin{align*} \int{{\frac{1}{\sqrt{bx+a}}} \left ( dx+c \right ) ^{-{\frac{5}{4}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{b x + a}{\left (d x + c\right )}^{\frac{5}{4}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{b x + a}{\left (d x + c\right )}^{\frac{3}{4}}}{b d^{2} x^{3} + a c^{2} +{\left (2 \, b c d + a d^{2}\right )} x^{2} +{\left (b c^{2} + 2 \, a c d\right )} x}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{a + b x} \left (c + d x\right )^{\frac{5}{4}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{b x + a}{\left (d x + c\right )}^{\frac{5}{4}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]